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Energy flow characteristics of friction-induced nonlinear vibrations in a water-lubricated bearing-shaft coupled system

Energy flow characteristics of friction-induced nonlinear vibrations in a water-lubricated... Based on the energy flow theory of nonlinear dynamical system, the stabilities, bifurcations, possible periodical/chaotic motions of nonlinear water-lubricated bearing-shaft coupled systems are investigated in this paper. It is revealed that the energy flow characteristics around the equlibrium point of system behaving in the three types with different friction-para- mters. (a) Energy flow matrix has two negative and one positive energy flow factors, constructing an attractive local zero- energy flow surface, in which free vibrations by initial disturbances show damped modulated oscillations with the system tending its equlibrium state, while forced vibrations by external forces show stable oscillations. (b) Energy flow matrix has one negative and two positive energy flow factors, spaning a divergence local zero-energy flow surface, so that the both free and forced vibrations are divergence oscillations with the system being unstable. (c) Energy flow matrix has a zero-energy flow factor and two opposite factors, which constructes a local zero-energy flow surface dividing the local phase space into stable, unstable and central subspace, and the simulation shows friction self-induced unstable vibrations for both free and forced cases. For a set of friction parameters, the system behaves a periodical oscillation, where the bearing motion tends zero and the shaft motion reaches a stable limit circle in phase space with the instant energy flow tending a constant and the time averaged one tending zero. Numerical simulations have not found any possible chaotic motions of the system. It is discovered that the damping matrices of cases (a), (b) and (c) respectively have positive, negative and zero diagonal ele- ments, resulting in the different dynamic behavour of system, which gives a giderline to design the water-lubricated bearing unit with expected performance by adjusting the friction parameters for applications. Keywords Nonlinear friction-induced vibrations · Nonlinear energy flows · Nonlinear water-lubricated bearing-shaft systems · Bifucation friction parameters · Energy flow matrices · Periodical oscilation 1 Introduction Water-lubricated bearings have been more and more used in the marine and pump industries to eliminate the pollution of metal bearings lubricated by oil and grease as well as Executive Editor: Li-Feng Wang to increase efficiency and reliability of marine propulsion systems. The first effective water-lubricated bearing made * Jing Tang Xing jtxing@soton.ac.uk of lignum vitae was invented by John Penn in the 1840s while man-made plastic bearings and natural rubber bear- Key Laboratory of Metallurgical Equipment and Control ings appeared in the 1920s, which had gradually predomi- Technology, School of Machinery & Automation, Wuhan nated in pumps, naval and many commercial ships by the University of Science and Technology, Wuhan 430081, China 1940s. The need for improved water-lubricated bearings was greatly recognised in 1942 when a number of U.S. ships Hubei Key Laboratory of Hydroelectric Machinery Design & Maintenance, Three Gorges University, Yichang 443002, suffered extensive combat damage at the Battle of Midway, China after which, USA navy department began to carry out an Maritime Engineering Group, CMEE, School of Engineering, extensive research on water-lubricated bearings and obtained Faculty of Engineering & Physical Sciences, University a series of achievements and the only military specification of Southampton, Southampton SO17 1BJ, UK Vol.:(0123456789) 1 3 680 L. Qin et al. of the water lubricated rubber bearings was created by MIL- performance of this type of bearing systems, it is essential DTL-17901C (SH) in 2005 [1]. Since the friction perfor- to investigate its integrated nonlinear characteristics. In mance plays a key role in bearing vibration or noise, this the area on friction induced vibrations, Ibrahim [20, 21] specification especially describes the requirements for the contributed two important review papers, of which the first test rig and scope of friction coefficients of bearing synthetic one concerns the mechanics of contact and friction and rubber facings. the second one discusses the dynamics and modeling of Stern-tube bearing noise is associated with both the quan- friction induced vibrations. Both of papers provide the tity of friction and the slope of friction–velocity curve [2], comprehensive account of the main theorems and mecha- which involves bearing materials. To reduce bearing noises, nisms developed in the historical literatures concerning many historical experimental and analytical investigations friction-induced noise and vibrations with a quite wide were reported [3–5]. For examples, Krauter [3, 4] focused references listed. Furthermore, Ding and Zhai [22] pre- on a three-degree-of-freedom (3-DOF) system to incorporate sented a review paper on the advances of friction dynamics with the essential elements of the friction and vibration phe- in mechanics system, in which the common used friction nomena in their experimental model, in which a linear analy- models, friction related self-excited vibrations and their sis was considered to determine the instability conditions controls were presented. The interested readers may refer of the system that may lead to frictional vibrations. Sinou these three review papers for more information and refer- et al. [6] proposed a numerical model incorporating realistic ences in this area. laws of local friction in base of experimental results to char- More directly linking the topic of this paper, Simpson acterize the dynamics of a lubricated system and to study and Ibrahim [2] investigated the dynamic behaviour of its complex global responses triggered by the local inter- nonlinear friction-induced vibration in water-lubricated facial behaviour. Graf and Ostermeyer [7] have shown how bearings based on the traditional method dealing with the stability of an oscillator sliding on a belt will change, nonlinear dynamical systems. Their paper developed an if a dynamic friction law with inner variable is considered analytical nonlinear, two-degree-of-freedom model, which instead of a velocity-dependent coefficient of friction, which emulates the stern-tube bearing. Typical friction-speed demonstrates the unstable vibration can even be found in curves were adopted based on the experimental results of the case of a positive velocity-dependency of friction coef- Krauter [23]. The stability conditions of equilibrium were ficient. Baramsky et al. [8 ] contributed the analytical and predicated. In the unstable cases, the nonlinear response experimental results for the occurrence of friction-induced behavior was examined by using numerical integration of vibrations during tightening of bolted joints of the most the coupled equations of motion. The dependence of the used machine elements. Niknam and Farhang [9] employed relative sliding speed on time and its effect on the friction a two-degree-of-freedom single mass-on-belt model to study force was included in the numerical simulation. A con- friction-induced instability due to mode-coupling, in which trol parameter that combines the influence of the normal numerical analyses are used to tackle the effect of three load and the slope of the friction-speed curve was used to parameters related to belt velocity, friction coefficient, and construct a bifurcation diagram which separated different normal load on the mass response. Ghorbel et al. [10] pro- response characteristics, including squeal, limit cycle, and posed a minimal 2-DOF disk brake model to investigate the stable regions. effects of different parameters on mode-coupling instabil - To reveal nonlinear behavior of nonlinear dynamical sys- ity, which considers self-excited vibration, gyroscopic effect, tems, such as as discussed in the books by Guckenheimer friction-induced damping, and brake pad geometry. and Holmes [24], Thompson and Stewart [25], Chen et al. In designs of this type of berings, two research directions [26] and Liu and Chen [27], Xing [28] developed a genral- have been developing: one aims to obtain good friction char- ised energy flow theory to investigate nonlinear dynamical acteristics using suitable geometric parameters with differ - systems governed by ordinary differential equation in phase ent facing layers and lubrications [11–14], and another is to space. Important nonlinear phenomena such as, stabilities, create new bearing materials [15–18]. Recently, Smith [19] periodical orbits, bifurcations and chaos can be revealed presented a paper on the design and lubrication of water- from the energy flow behavors of the systems [29]. The aim lubricated, rubber, cutlass bearings systems, which gave a and motivation of this paper are to investigate the 2-DOF methodology to predict the minimum film thickness between nonlinear model of the stern-tube bearing proposed in Ref. the journal and heavily loaded stave, including a 3D finite [2] to reveal its energy flow characteristics governing its element (FE) model to predict generated pressures in cutlass nonlinear dynamical behavours, which is also as an example bearings, and comparing with experimental data. to further develop energy flow approach based on two sca- From the point view of sciences, the water-lubricated lar varaibles, genralised potentional and kinetic energies, to bearings system is a typical nonlinear system concerning tackle complex nonlinear dynamical systems, especially in friction-induced vibration noises, so that for good noise multi-dimensional phase spaces. 1 3 Energy flow characteristics of friction‑induced nonlinear vibrations… 681 the disk radius and 𝜃 is the torsional velocity of the disk rela- tive to the bearing. Considering only oscillatory motions of Bearing housing both the shaft and bearing, and representing the motion by Rubber stave the linear displacement, we can denote the relative sliding Gap filled with water Rotor shaft speed of the friction pair in the form 𝜈 = R𝜃 = ẋ −ẋ = z − z , z = ẋ , z = ẋ . (2) 2 1 2 1 1 1 2 2 Substituting Eq. (2) into Eq. (1) and then taking the deriv- ative of the resultant equation with respect to the relative sliding speed, we obtain the visous damping coefficient as dF Fig. 1 Typical grooved stern tube bearing structure −av b = =−Na  −  e , (3) 0 1 dv −av which, when the function e is expanded into a Taylor series, becomes 2 2 3 b =− Na  −  1 − av + a v + o(v ) 0 1 1 2 3 =− Na  −  1 − a z − z + a z − z + o z − z . 0 1 2 1 2 1 2 1 (4) 1.1.2 Dynamic equations of the friction pair system Fig. 2 Analytical 2-DOF model of stern tube bearing Using the model shown in Fig. 2 and the second Newton’s law, we obtain the following two equations of motion of the system 1.1 Dynamic modelling and equations m x ̈ + c x ̇ + k x = b(x ̇ − x ̇ )+ F , 1 1 1 1 1 1 2 1 1 of water‑lubricated bearings (5) m x ̈ + c x ̇ + k x =−b(x ̇ − x ̇ )+ F , 2 2 2 2 2 2 2 1 2 The marine propeller shafts are supported in stern tube bear- which can be rewritten in a matrix form ings, which are lubricated by water and are almost all grooved, of which the scketched structure is shown in Fig. 1. The 2-DOF ̈ ̇ ̂ MX + CX + KX = F, (6) analytical model used in Ref. [2] is shown in Fig. 2, where the where the mass, damping and stiffness matrices as well as mass, stiffness, and damping coefficient of the bearing (sub- the displacement and force vectors are resepctively defined script 1) and the flexible shaft (subscript 2) are denoted by as follows m , m , k , k , c and c , respectively. Here, the forces F and 1 2 1 2 1 2 1 F denote possible prescribed external forces applied to the m 0 c + b −b 1 1 bearing and the shaft, respectively, while the force F denotes M = , C = , 0 m −bc + b 2 2 the friction force acted to the bearing by the shaft, which is k 0 x F assumed in the viscous type propotional to the relative sliding 1 1 1 K = , X = , F = . speed of the friction pair with a viscous damping coefficient 0 k x F 2 2 2 (7) b , so that F = b(x ̇ − x ̇ ). f 2 1 Introducing the nondimensional parameters 1.1.1 Viscous damping coefficient c 𝜔 k m i 2 i 1 𝜁 = , 𝛾 = , 𝜔 = , 𝜂 = , i 21 i 12 𝜔 m m Historically, the friction force was formulated by the friction 2 k m 1 i 2 i i model developed by Smith et al. [5] and used by Simpson and x F i i y = , f = , i = j = 1, 2, Ibrahim [2] in their 2-DOF analytical model, that is i i R 𝛽 N −aR𝜃 Na F = N[𝜇 +(𝜇 −𝜇 )e ], (1) 𝜏 = 𝜔 t, 𝜇 = 𝜇 − 𝜇 , 𝛼 = , f 1 0 1 1 01 0 1 m 𝜔 1 1 where N denotes the normal bearing load; a is a constant m 𝜔 R 𝛽 = , 𝜓 = 𝛼𝛽 = aR𝜔 , with units of inverse velocity;  and  are the static and 1 0 1 N (8) dynamic coefficients of friction, respectively; R represents 1 3 682 L. Qin et al. where  denote dimensionless damping coefficients,  ratio ̃ is called the generalised force of system. Here, J denotes the i 21 T T T of circular frequencies,  natural frequencies of the bear- i coefficient matrix of the state variable vector [y z ] , which ing and the shaft subsystems,  mass ratio, y dimension- ̃ involves the nonlinear damping parameter b . The solution of 12 i less displacements,  dimensionless time,  dimensionless Eq. (11) with a prescribed initial condition −1 friction coefficient,  dimensionless normal bearing load T T and  the dimensionless value of constant  . Using these y(0)= y ̂, z(0)= z ̂, r = y ̂ z ̂ , (13) dimensionless parameters, the dynamic equation of system gives an orbit starting from the position r in the phase space can be written as 0 as shown in Fig. 3. my ̈ + cy ̇ + ky = f , (9) The parameters of system, such as k , c and c , can be 1 1 2 functions of normal load and temperature. Krauter [4] devel- where oped empirical relationships for these parameters 10 ◦ 2𝜁 0 10 1 k = 116.52N + 10535.4 at 35 C, 2 1 m = , k = 𝛾 , c = c ̂ + c ̃, c ̂ = 2𝛾 𝜁 , 21 2 01∕𝜂 0 𝜂 m  ln 2 (14) 𝜂 12 1 2 c =−0.618T + 8.474, c = . 1 2 πN y f 1 1 ̃ ̃ ̂ −𝜓 Iz c ̃ =bI, y = , f = , z = y ̇ , , b =−𝛼𝜇 e 2 2 Typical values of these parameters were selected for con- 1 −1 ̂ stant values of normal load and temperature, i.e. N = 120 N −𝜓 Iz ̂ ̃ I = −11 , I = , 𝜕 b∕𝜕 z = 𝛼𝜇 𝜓 e 1 01 −11 (27 lb) and T =35 ℃, by Simpson and Ibrahim [2] to investi- ̇ −𝜓 Iz gate the nonlinear dynamical behaviour of system. We also use ̃ ̃ ̃ ̂ =− 𝜓 b =−𝜕 b∕𝜕 z , b = 𝛼𝜇 𝜓 e Iż . 2 01 these chosen parameters to reveal the energy flow character - (10a) istics of system, and to rewrite Eqs. (11) and (12) in the form For the approximation as shown in Eq. (4) adopted by Simpson and Ibrahim [2], the parameter b and its derivatives ẏ y ̃ ̃ ̃ ̃ ̃ = J + f = F, J = J + J , 0 b are given by the following approximated formulations ż z 1 0 I 00 2 T ̃ ̂ ̃ ̃ ̃ b ≈−𝛼𝜇 (1 − 𝜓 Iz + 𝜓 z Iz), J = , J = , 01 0 b 2  0 b (15) ̇ T ̃ ̂ ̃ ̃ ̂ b ≈ 𝛼𝜇 𝜓 (Iż − 𝜓 ż Iz), 𝜕 b∕𝜕 z =−𝛼𝜇 𝜓 (1 − 𝜓 Iz) −10 −2𝜁 0 01 1 01 = ,  = , 2 2 2 ̃ ̃ 0 −𝛾 0 −2𝛾 𝜁 =−𝜕 b∕𝜕 z , 𝜕 b∕𝜕 z =−𝛼𝜇 𝜓 21 2 2 01 2 2 2 2 ̃ ̃ −11 =−𝜕 b∕𝜕 z , 𝜕 b∕𝜕 z 𝜕 z = 𝛼𝜇 𝜓 , ̃ ̃ 1 2 01 2 (10b) b = b,  = , 𝜂 −𝜂 12 12 implying that the parameter b is approximated by a quadratic from which in association with Eq. (10) we obtain function of the relative velocity (z − z ) , its first derivatives 2 1 are linear, and the second derivatives are constants. y ̈ y ̇ y ̇ ̇ ̃ ̃ ̃ = J + J + f , (16a) z ̈ z ̇ z 1.2 Dynamic equations and equilibrium points 1.2.1 Equations in phase space ̇ ̇ ̃ ̃ J = J = , (16b) 0 b Using the state variables y and z , we can rewrite Eq. (9) in the form of the phase space, i.e. y ̇ = F = z, (11) −1 ̃ ̃ ̂ ż = F = m [−(c ̂ + bI)z − ky + f], of which, the vector F y ̃ ̃ F = = J + f , F z 0 I 0 ̃ ̃ J = , f = , −1 −1 ̃ −1 ̃ ̂ −m k −m (c ̂ + bI) m f (12) Fig. 3 Orbit, position and tangent vectors at a point on the orbit of a dynamical system in phase space 1 3 Energy flow characteristics of friction‑induced nonlinear vibrations… 683 ̇ 2 2 T T ̃ ̃ (16c) E = r ∕2, r = y y + z z, (22) b = b. while the generalised kinetical energy equals the half square of the tangent vector, the velocity, at a point, i.e. 1.2.2 Equilibrium point and Jacobian matrix 2 2 T T K = r ̇  ∕2, r ̇  = y ̇ y ̇ + z ̇ z ̇ . (23) The point at which F = 0 is called the equilibrium point of system. Obviously, if there is no external force ( f = 0 ), t he Therefore, these two non-negative real scalars can be used origin (y = 0 = z) of phase space is an equilibrium point of to describe the postion and velocity at a point on the solution system. orbit of system in phase space, and to reveal related nonlin- By using Eqs. (15) and (16), the Jacobian matrix of the ear dynamical behavours of system. system can be derived as 2.2 Energy flow equation and its time average J = F∇ = Jr + f = J + J, T T y z (17) Pre-multiplicating Eqs. (13) and (15) by a row vector J J b b T T T T J = r r = (Iz) . T T r = y z and its initial value r ̂ (0) , respectively, we y z 0 b obtain the energy flow equation Here the gradient operator ∇ is defined by T T T T ̇ � � � E = P, P = r F = r Jr + p ̂, p ̂ = r f = z f , T T T T ∕y    T T T ̈ ̇ ̂ T T T E = ṙ F + r F = 2K + r r ̈, E(0)= E = r ̂ r ̂∕2 =(y ̂ y ̂ + z ̂ � z)∕2, ∇= = , ∇ = = , r = y z . T T y z r ∕z T T ̇ ̇ � (18) E = Edt ∕T = (r Jr + p ̂)dt ∕T, 0 0 Substituting Eq. (15) into Eq. (17), we finally obtain the (24a) Jacobian matrix of which, P denotes the power done by the generalised force ̃ ̂ ̃ of system, p ̂ is the power of the external forces and E rep- J = J +[1 + 𝜓 (Iz)]J . (19) 0 b resents the time change rate of generalised potential energy, ̃ ̃ Here the matrices J and are defined by Eq. (15). called as the energy flow of system. Physically, the energy 0 b flow equation in Eq. (24a) implies that the energy flow of system equals the power done by the generalised force. 2 Energy flow formulations of system The energy flow equation in Eq. (24a) can be rewritten in the form 2.1 Generalised potential energy and kinetic T T T ̇ ̃ ̃ ̃ ̃ E = r (E + U)r + p ̂ = r Er + p ̂, r Ur = 0, energy T T ̃ ̃ ̃ ̃ ̃ ̃ E =(J + J )∕2, U =(J − J )∕2, (24b) We define two scalar variables: the generalised potential energy and generalised kinetical energy of system in the ̇ ̃ E = (r Er + p ̂)dt ∕T. following forms. Generalised potential energy and its time average ̃ ̃ Here E is a real symmetrical matrix, while is a real anti- T T symmetrical matrix. 1 1 1 1 T T T T E = r r = (y y + z z), ⟨E⟩ = Edt = r rdt. 2 2 T 2T 0 0 2.3 Kinetic energy flow equation and its time (20) average Generalised kinetical energy and its time average T T Pre-multiplying Eq. (16a) by r ̇ and using Eqs. (14) and 1 1 1 1 T T T T K = r ̇ r ̇ = (y ̇ y ̇ + z ̇ z ̇ ), ⟨K⟩ = Kdt = r ̇ r ̇ dt. (16c), we obtain the kinetic energy flow and its time average 2 2 T 2T 0 0 (21) ̇ ̇ ̇ ̇ T T T T T T T ̇ ̃ ̃ ̃ ̃ ̃ ̃ K = r ̇ r ̈ = r ̇ Jṙ + r ̇ Jr + r ̇ f = r ̇ Eṙ + z ̇ bz + r ̇ f , Physically, these two scalars may not practical potential T T and kinetical energies, therefore we use the word general- ̇ ̇ T T T T ̃ ̃ ̃ K = r ̇ r ̈dt ∕T = (r ̇ Eṙ + z ̇ bz + r ̇ f)dt ∕T. ised to distinguish them with practical physical quantities. 0 0 Geometrically, as shown in Fig. 3, the generalised potential (25) energy equals the half square of the distance of a point to the origin of phase space, i.e. 1 3 684 L. Qin et al. ̇ ̇ 2.4 Zero‑energy flow surfaces and equilibrium ΔE = E(r + )=(r + ) F(r + )=(r + 𝜀 )F (r + 𝜀 ), i i i j j points =(r + 𝜀 ) F + F 𝜀 + 0.5F 𝜀 𝜀 + O(𝜀 ) i i i i,j j i,jk j k = v𝜀 F + r F 𝜀 + 𝜀 F 𝜀 + 0.5r F 𝜀 𝜀 Generally, the energy flow of system is a function of time i i i i,j j i i,j j i i,jk j k T T T T T and the position of a point in phase space, which generates =  p +  (E + U) +  E  =  p +  (E + E ), 1 1 a scalar field called as the energy flow field of the nonlin- (27a) ear dynamical system. Equation where the energy flow gradient vector p , the energy flow matrices E and E , and spin matric U are given by ̇ 1 E = 0, (26) T T T p = F + J r, E =(J + J )∕2, U =(J − J )∕2, defines a generalised surface or subspace in phase space, � � which is called as a zero-energy flow surface on which the E = r ∇∇ F =  +  + B, B = , 1 i i 4×4 4×4 energy flow vanishes. If an orbit of the nonlinear dynamical 2 2 2 2 ̃ ̃ 𝜕 𝜕 𝜕 b 𝜕 b ⎡ ⎤ ⎡ ⎤ system is on a zero-energy flow surface, the distance of a 𝜕 z 𝜕 z 𝜕 z 𝜕 z 𝜕 z 𝜕 z 𝜕 z 𝜕 z ⎢ 1 1 1 2 ⎥ ̃ ̂ ⎢ 1 1 1 2 ⎥ ̂ b =(z − 𝜂 z ) (bIz)= (z − 𝜂 z ) (Iz) 2 2 2 2 1 12 2 1 12 2 ̃ ̃ 𝜕 𝜕 𝜕 b 𝜕 b ⎢ ⎥ ⎢ ⎥ point on the orbit is not changed with time. 𝜕 z 𝜕 z 𝜕 z 𝜕 z 𝜕 z 𝜕 z 𝜕 z 𝜕 z ⎣ ⎦ ⎣ ⎦ 2 1 2 2 2 1 2 2 � � � � Based on the theory of extreme values of a function and 1 −1 1 −1 2 2 T ̃ ̂ ̃ =(z − 𝜂 z )𝜓 b (Iz)= 𝜓 bz 𝜂 z . 1 12 2 the geometrical meaning of generalised potential energyE , −1 1 −1 1 we can conclude that the characteristics of orbit at a point (27b) on the zero-energy flow surface as follows. Here, we have used Eqs. (10a) and (15) and noticed that the components F and F of the vector F are lin- 1 2 • E > 0 , the generalised potential energy takes a local ear functions of y and z , so that their second derivatives minimum-extreme value at this point compared with the vanish. points in the local domain around it. Therefore, with time For an equilibrium point satisfying F(r)= 0 , Eq. (27a) going, the value of E increases and the orbit backwards becomes the origin of phase space. • E = 0 , it cannot determine that generalised potential T ̇ ̇ ΔE = E(r, )= J r +  (E + E ), (28a) energy takes a local minimum- or maximum-extreme value at this point, and higher time derivatives are needed which is further reduced to to give a solution. ̇ ̇ ΔE = E()=  E, (28b) • E < 0 , the generalised potential energy takes a local maximum-extreme value at this point compared with the for an equilibrium point r = 0, and so that E  = 0. At the points in the local domain around it. Therefore, with time equilibrium point r = 0 the second time derivative of the going, the value of E decreases and the orbit towards the generalised energy in Eq. (24a) vanishes, so that we need to origin of phase space. consider the variation of E around the point r = 0 to investi- gate the orbit behaviour. From Eq. (28a) and the geometrical If no external force p ̂ = 0, there are the following three meaning in Eq. (22), we consider the variation of energy cases satisfying E = 0 in Eq. (26). flow E to determine the characteristics of orbit as follows. Case 1:r = 0 , that represents the origin of phase space, If r +  < r, then E(r + ) < E(r), so that ΔE > 0 at which the generalised potential energy is defined as implies the flow towards the zero-energy flow surface, zero; while ΔE < 0 indicates the flow backwards the zero- • Case 2:F = 0 , implying an equilibrium point of system, energy flow surface; so that equilibrium points are on the zero-energy flow If r +  > r, then E(r + ) > E(r), so that ΔE < 0 surface; implies the flow towards the sero-energy flow surface, Case 3: P = 0, r ≠ 0 ≠ F, correseponding a generalised while ΔE > 0 indicates the flow backwards the zero- zero-energy flow surface. energy flow surface; If the flows from both sides of the zero-energy flow sur - Assume that r denotes a point on an orbit on a zero- face tend to it, this surface is an attracting surface. energy flow surface, P(t, r)= r F = r F = 0 , and  is an i i The local ability of equilibrium point r = 0 can be deter- small orbit variation around r , generally, the variation of mined by the behaviour of energy flow matrix E as fol- energy flow caused by the orbit variation can be approxi- lows: mated to the quantities of  in the form 1 3 Energy flow characteristics of friction‑induced nonlinear vibrations… 685 𝜆 as the energy flow characteristic factors and we can definitely - negative, asymptotic stable, conclude for a point of the orbit of nonlinear system. E ∶ semi - definitely - negative, stable, definitly or semi - definitly - positive, unstable. • The positive, zero and negative value of the factors 𝜆 (29) respectively implies the energy flow increase, unchanged and decrease caused by the disturbance in the I-th Figure  4 shows a case where the orbit intersects at a principal direction, from which the behaviour of energy point r on the energy flow surface. Since ΔE > 0 above flow increments at this point can be identified to judge the surface and ΔE < 0 under the surface, so that the flow the local dynamical behaviour, such as for local stability along the orbit backwards this point and this point is an of orbits around an equilibrium point r = 0, we have unstable point. negative , stable subspace, ∶ zero, central flow subspace, (33) 2.5 Energy flow matrix and energy flow positive, unstable subspace. characteristic factor • If its energy flow characteristic factors are not all semi- Both of matrices E and E are real symmetrical matri- negative or not all semi-positive, there will exist a small ces, called as the energy flow matrices, the former is for subdomain around this point in the phase space deter- the total energy flow in Eq. ( 24b), while the later for the mined by E = 0 , from which a zero-energy flow surface incremental energy flow relative to the zero-energy flow can be obtained. surface in Eq. (27b). The real symmetrical matrix has its real eigenvalues  and corresponding eigenvector  sat- I I 2.6 Spin matrices and periodical orbit isfying the orthogonal relationships, such as for matrix E in Eq. (24b), we have The matrices U in Eq. (24b) and U in Eq. (27b) are two T T real skew-symmetrical matrices, called as spin matrices. ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ = I,  E = diag(𝜆 ),  = 𝝃 𝝃 𝝃 𝝃 . I 1 2 3 4 For periodical orbits, it is neccesary that there exsist a time (30) period T and its corrsponding closed orbit Γ such that the These eigenvectors span an energy flow space in which following integrations along the closed orbit hold, i.e. the vector r can be represented by � � 1 1 ̂ ̂ T+t T+t ̇ ̇ ̂ ̃ E = Edt = 0, ⟨E⟩ = Edt = E, r = 𝝋 ̃ , (31) ̂t ̂t T T Γ Γ which, when substituted into Eq. (24b), gives (34a) � � 1 1 ̂ ̂ 4 T+t T+t ̇ ̇ ̂ K = Kdt = 0, ⟨K⟩ = Kdt = K, ̂ ̂ t t T 2 T ̇ ̃ ̃ T T E =  ̃ diag(𝜆 ) ̃ = 𝜆 𝜑 ̃ ,  ̃ = 𝜑 ̃ 𝜑 ̃ 𝜑 ̃ 𝜑 ̃ . Γ Γ I I 1 2 3 4 I=1 ̂ ̂ where E and K are two positive constants. The stability of (32) this periodical orbit can be determined by the values of the Therefore, 𝜆 represents the energy flow variation energy flow ΔE around each point of the orbit as discussed caused by a unit disturbance 𝜑 ̃ = 1 in the I-th principal for the zero-energy flow surface. direction 𝜑 ̃ of the energy flow matrix. Therefore, we call The time averaged kinetic energy can be re-written as 1 1 1 ̂ T ̂ T T+t T+t ⟨K⟩ = r ̇ r ̇ dt = r ̇ dr = . (34b) ̂t ̂t 2T 2T 2T Γ Γ Physically, this integration denotes the circulation inte- gral of velocity field r ̇ along the closed orbit and involves the skew-symmetrical spin matrix. To clarify this, we consider a 3-D vector field y ̇ = f , t he curlf , or ∇× f at a point O is explained in Fig. 5. Here  is a unit vector, the projection of the curlf onto  is defined as the limit of a closed line integral along the curve C in a plane orthogonal to  , that is Fig. 4 Zero-energy flow surface and an unstable point r determined by ΔE 1 3 ̃𝜑 686 L. Qin et al. � � � � T+t ̇ ̇ E = lim Edt = 0, T→∞ � � T+̂t ⟨E⟩ = lim Edt = E, T→∞ � � � � T+̂t ̇ ̇ K = lim Kdt = 0, T→∞ � � T+t ⟨K⟩ = lim Kdt = K. ̂t T→∞ (35) Fig. 5 Circulation integration along a path C, of which the positive This implies that the time averaged mechanical energy direction obeys the right-hand rule, to define the cur1f ⟨E + K⟩ tends a constant when the average time tends to infinite. Also, for a chaotic motion, flows are restricted in a finite volume, so that the space averaged rate of volume (∇× f) = (curlf) = lim f ⋅ dy 𝜈 𝜈 strain of phase space must not be positive, i.e. A→0 1 2 = lim ẏ ⋅ ẏ dt = lim Kdt , 1 1 A→0  A→0  = dV = 𝜆 dV ≤ 0. A A (36) V I C C � � V V V V I=1 (34c) where A is the area enclosed by curve C. The curlf can be denoted in a tensor form [30, 31] 3 Investigations of nonlinear (∇× f) = e f , i ijk k,j (34d) water‑lubricated bearing system where e is the permutation tensor. The vector curlf is a ijk ̂ ̂ 3.1 Zero‑energy flow surface (f = 0 = f ) dual vector of a skew-symmetrical matrix U , spin matrix, 1 2 satisfying the following relationship Using Eq.  (24b), we obtain the energy flow surface −1 −1 ̇ ̃ E = r Er = 0 of the system, i.e. (U) = U = e (∇× f) = e e f ij ij ijk k ijk krs s,r 2 2 ⎡ ⎤ 00 0 0 (34e) ⎢ ⎥ T 00 0 (1 − 𝛾 )∕2 f − f ⎢ ⎥ i,j j,i T −1 J − J 21 r r ⎢ ⎥ = (  −   )f = = . ̃ ̃ ir js is jr s,r 00 −2𝜁 − b (1 + 𝜂 )b∕2 ⎢ 1 12 ⎥ 2 2 2 ij ⎢ ⎥ ̃ ̃ 0 (1 − 𝛾 )∕2 (1 + 𝜂 )b∕2 −2𝛾 𝜁 − 𝜂 b 12 21 2 12 ⎣ ⎦ 2 2 2 Therefore, a positive time-averaged kinetic energy implies ̃ ̃ ̃ = −(2𝜁 + b)z −(2𝛾 𝜁 + 𝜂 b)z +(1 + 𝜂 )bz z +(1 − 𝛾 )z y = 0, 1 21 2 12 12 1 2 2 2 1 2 21 there must be a non-zero spin matrix for periodical orbits. (37a) where the nonlinear friction parameter 2.7 Bifurcation and chaos −𝜓 (z −z ) ̃ 2 1 b =−𝛼𝜇 e , (37b) The energy flow of a nonlinear dynamical system is affected or its approximation to a second order of sliding speed by the bifurcation parameters to reveal the birfurcation char- acteristics of orbits. For example, different parameters result 2 T ̃ ̂ ̃ b ≈−𝛼𝜇 (1 − 𝜓 Iz + 𝜓 z Iz). (37c) in different equilibrium points, zero-energy flow surfaces 2 and energy flow characteristic factors etc. The zero-energy flow surface in Eq. (37a) is independ- Xing [28, 29] has discovered that a chaotic motion of ent on the variable y , therefore, the axis o − y is a zero- 1 1 nonlinear dynamical system can be considered as a periodi- energy line of the system. On this line defined by the posi- cal motion with an infinte long time period and the following T tion vector r = y 0 00 , the energy flow E = 0. For the integrations hold prescribed parameters in Eq. (41), Eq. (37) governs the 1 3 ̇𝜐 ̇𝜐 Energy flow characteristics of friction‑induced nonlinear vibrations… 687 1 3 Table 1 Energy flow characteristic factors at the equilibrium point of the system (  = 0.008325) ξ 0.1116 0.111245 0.1111 0.1110 α λ λ λ λ λ λ λ λ λ λ λ λ 2 3 4 2 3 4 2 3 4 2 3 4 1.5 − 0.507725 − 0.055150 0.506133 − 0.507713 − 0.054460 0.506141 − 0.507708 − 0.054178 0.506144 − 0.507705 − 0.053983 0.506146 2.0 − 0.512170 − 0.001144 0.512114 − 0.512154 − 0.000467 0.512131 − 0.512147 − 0.000191 0.512137 − 0.512142 0 0.512142 2.2 − 0.514088 0.019815 0.515290 − 0.514069 0.0204843 0.515312 − 0.514062 0.020758 0.515320 − 0.514057 0.020947 0.515327 2.5 − 0.517085 0.050374 0.521052 − 0.517064 0.0510321 0.521083 − 0.517055 0.051301 0.521096 − 0.517049 0.051486 0.521105 2.6 − 0.516113 0.060295 0.523 267 − 0.518091 0.0609483 (0.523302 − 0.518082 0.061215 0.523 317 − 0.518076 0.061399 0.523327 2.7 − 0.519153 0.070069 0.525642 − 0.519131 0.0707175 0.525682 − 0.519122 0.070982 0.525698 − 0.519115 0.071165 0.525709 3.0 − 0.522349 0.090427 0.533805 − 0.522324 0.0990577 0.533059 − 0.522314 0.099315 0.533 881 − 0.522307 0.099493 0.533897 3.5 − 0.527885 0.141957 0.551352 − 0.527856 0.1425522 0.551439 − 0.527845 0.142795 0.551475 − 0.527837 0.142963 0.551499 4.0 − 0.533642 0.179975 0.574634 − 0.533611 0.1805229 0.574764 − 0.533598 0.180747 0.574817 − 0.533589 0.180901 0.574854 4.5 − 0.539584 0.211801 0.604291 − 0.539550 0.2122948 0.604474 0.539536 0.212496 0.604540 − 0.539527 0.212635 0.604600 Table 2 Energy flow characteristic factors at the equilibrium point of the system (  = 0.00) ξ 0.1116 0.111245 0.1111 0.1110 α λ λ λ λ λ λ λ λ λ λ λ λ 2 3 4 2 3 4 2 3 4 2 3 4 2.0 − 0.512177 − 0.001144 0.512107 − 0.512160 − 0.000467 0.512124 − 0.512154 − 0.000191 0.512131 − 0.512149 0 4.512135 2.2 − 0.514045 0.019915 0.515283 − 0.514076 0.020484 0.515345 − 0.514468 0.020758 0.515314 − 0.514063 0.020947 0.515320 688 L. Qin et al. Table 3 Energy flow characteristic factors at the equilibrium point of the system (  = 0.009) ξ 2.0 2.12 2.5 α λ λ λ λ λ λ λ λ λ 2 3 4 2 3 4 2 3 4 0.119720 − 0.517573 − 0.010078 0.511718 − 0.513740 − 0.003913 0.513519 − 0.517589 0.035312 0.520352 forms of zero-energy surfaces of system with the different E − 𝜆I = 0. (40c) values of  and  listed in Tables 1, 2 and 3, which will be given in Sect. 3.2. The solutions of Eqs. (40b) and (40c) are three eigen- values and eigenvectors 𝜆 ,  (I = 2, 3, 4), I I ̂ ̂ 3.1.1 Linearazation at the fixed point (f = 0 = f ) 1 2 𝜆 ,  = ⃗ , I = 2, 3, 4. (40d) I I Vanishing the generalised kinetical energy in Eq. (21) and using Eq. (15), we have 00 1 0 y 3.2 Bifurcation of zero‑energy flow surfaces ⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ 00 0 1 y around the fixed point = 0, ⎢ ⎥⎢ ⎥ (38) ̃ ̃ −10 −2𝜁 − b b z 1 1 ⎢ ⎥⎢ ⎥ ̃ ̃ ⎣ ⎦⎣ ⎦ 0 −𝛾 𝜂 b −2𝛾 𝜁 − 𝜂 b z The dimensionless parameters studied in Ref. [2] are listed 12 21 2 12 2 as follows which gives an equilibrium point r = 0 . From Eqs. (10), =0.111;  = 0.1116, 0.111245, 0.1111;  = 0.008325;  = 0.01; ̃ ̃ ̃ 01 1 2 21 (19), and (27b), at this point b =−𝛼𝜇 , J = J + J , 01 0 b E = 0 , so that the energy flow matrix and the spin matrix in Eq. (27b) become −4 =7.5 × 10 ;  = 3040.2;  = 2.0, 2.5, 3.0, 3.5, 4.0; � � (41) E = , based on which the obtained energy flow characteristic fac- 0 E tors around the fixed point are given in Tables  1, 2 and 3, 00 (1 −  )∕2 ⎡ ⎤ which shows the bifurcation of zero-energy flow surface ⎢ ⎥ E = 0  − 2 − (1 +  )∕2 , 01 1 01 12 affected by the bifurcation parameters: the dimensionless ⎢ ⎥ (1 −  )∕2 − (1 +  )∕2   − 2 ⎣ ⎦ 01 12 12 01 21 2 friction coefficient  and the dimensionless damping coef- (39a) ficients  and  . 1 2 It is found that the all energy flow characteristic factors 00 1 0 ⎡ ⎤ � � (I = 2, 3, 4) , for the investigated parameters, include 2 I ⎢ 00 0 1 +  ∕2 ⎥ � 21 � U = . negative, positive and zero real numbers, so that the small ⎢ ⎥ −10 0 − 1 −  ∕2 01 12 � � � � ⎢ ⎥ local domain around the equilibrium point can be divided into ⎣ ⎦ 0 − 1 +  ∕2  1 −  ∕20 01 12 a stable (𝜆 < 0 ), an unstable (𝜆 > 0 ) and a central flow sub- I I (39b) space ( = 0 ), to reveal the local dynamic behaviour around the equilibrium point. In the subspace span by the corresponding energy flow 3.1.2 Energy flow characteristic factors and vectors characteristic vectors 𝜉 around the equilibrium point, the energy flow at point  =    can be formulated by 2 3 4 For the approximated energy flow matrix E , using Eqs. Eq. (28), i.e. (30)-(32), we can solve its energy flow characteristic fac- 2 2 2 tors and vectors. Obviously, we have a zero characteristic E = 𝜆 𝜙 + 𝜆 𝜙 + 𝜆 𝜙 , (42a) 2 3 4 2 3 4 factor and its vector in the form and the zero-energy flow surface is governed by = 0,  = 10 00 . (40a) 1 1 2 2 2 E = 𝜆 𝜙 + 𝜆 𝜙 + 𝜆 𝜙 = 0. (42b) 2 3 4 2 3 4 The rest three energy flow characteristic factors and vectors Based on the results listed in Tables 1, 2 and 3, in a lin- are obtained by solving the eigenvalue problem earised approximation of system around the equilibrium point, there are following three local structures of zero- (E − 𝜆I )⃗r = 0, ⃗r = y z z , (40b) 2 1 2 energy flow surfaces with its local dynamic behaviour. of which the characteristic equation is 1 3 Energy flow characteristics of friction‑induced nonlinear vibrations… 689 3.2.1 Case (a): two negative and one positive characteristic Eq. (42b) becomes factor (˛ = 2.0 ,  = 0.1116,  = 0.009) 1 2 2 2 2 0.512177 + 0.001144 = 0.512107 , (43a) 2 3 4 In this case, such as for parameters  = 2.0 ,  = 0.1116 which is rewritten in the form and  = 0.009 , the equation of zero-energy flow surface in 2 2 2 2 2 𝜑 ̃ 𝜑 ̃ 𝜑 ̃ 𝜑 ̃ 𝜑 ̃ 2 3 4 2 3 + = , + = 𝜑 ̃ , 2 2 2 2 2 a a a R R 2 3 4 2 3 1 1 1 1 1 1 a = = , a = = , a = , � √ � √ √ √ 2 3 4 (43b) � � 0.512177 � � 0.001144 0.512107 𝜆 𝜆 �2� �3� √ √ 𝜆 𝜆 a a 4 3 4 R = = , R = = , � � 2 3 a a 4 4 � � � � 𝜆 𝜆 �2� �3� Fig. 6 Zero-energy flow surface around equilibrium point for case (a) ( = 2.0,  = 0.1116,  = 0.009) based on linearized Eq.  (43a): 1 2 a sketched one, b calculated one based on Eq. (43a), and c one based on nonlinear Eq. (37) 1 3 ̃𝜑 ̃𝜑 ̃𝜑 690 L. Qin et al. of which the zero-energy flow surface and energy flow vari- 0.0012 0.2220 c = , (43c) ations in the small domain around the equilibrium point is 0.2220 0.0180 sketched in Fig. 6a, and the computer simulating surface is with the positive diagonal damping coefficients in the sys- drawn in Fig. 6b, while the zero-energy surface based on Eq. (37) for nonlinear case is given in Fig. 6c. tem, so that the motions caused by initial conditions will be ⃗ ⃗ ⃗ damped. In Fig. 6a, the three characteristic vectors 𝜉 , 𝜉 and 𝜉 2 3 4 respectively with the three factors span a subspace in phase By using the MATLAB program in Ref. [28] and the nonlinear friction parameter in Eq.  (37b), the results space o − y z z . This local linearized zero-energy flow 2 1 2 ⃗ ⃗ surface is symmetrical to the coordinate plane o − 𝜉 𝜉 and of numerical simulations with the initial conditions 2 3 −6 y = 10 11 11 for this case are given in Figs. 7, 8, 9. axis-symmetrical to the axis o − 𝜉 , so that a plane perpen- dicular to axis o − 𝜉 intersects the zero-energy flow surface Figure 7 gives the time histories of displacement and veloc- 𝜑 ̃ R > 𝜑 ̃ R ity of the bearing and the shaft, which shows the damped in an elliptical curve of semi-axis , as shown 4 3  4 2 ⃗ ⃗ in Fig. 6 for the ones at 𝜑 ̃ = 1 . The plane o − 𝜉 𝜉 inter- oscillations. Figure 8 presents the phase diagram y(1)–y(3) 4 3 4 of bearing unit and the one y(2)–y(4) of shaft unit, as well sects the zero-energy flow surface in the straight lines of 𝜑 ̃ =±𝜑 ̃ R . Based on Eq. (42a), at a disturbed point outside as the time histories of generalized potential energy and the 3 3 distance of phase point to the origin, which indicates the the zero-energy flow surface on the plane 𝜑 ̃ =const., the energy flow E < 0 , so that it intends to reduce the distance orbit points in both phase diagrams move from outside to the origin, and the generalized potential energy and the distance to the origin of space, implying the flow towards the zero- energy surface. For a disturbed point inside the zero-energy of orbit point to the origin tend to zero with time increasing. Figure 9 shows the instant energy flow behaving a damped flow surface on the plane 𝜑 ̃ =const., the energy flow E > 0 and it intends to increase the distance to the origin of space oscillation and the time history of phase space volume strain decreasing with time. implying the flow also towards the zero-energy surface. Therefore, the zero-energy surface is an attracting surface. 3.2.2 Case (b): one negative and two positive characteristic In this case, from Eq. (10a), the damping matrix of system at the fixed point takes factors (˛ = 2.2 ,  = 0.111245 ,  = 0.008325) 1 2 Case 2 corresponds a zero-energy flow surface in Eq. (42b) governed by equation 2 2 2 0.514069 = 0.0204843 + 0.515312 , (44a) 2 3 4 Fig. 7 Time histories of displacement and velocity of bearing and shaft ( = 2.0,  = 0.1116,  = 0.009) 1 2 1 3 ̃𝜑 ̃𝜑 ̃𝜑 Energy flow characteristics of friction‑induced nonlinear vibrations… 691 Fig. 8 Phase diagram y(1)–y(3) of bearing and the one y(2)–y(4) of shaft, and time histories of generalized potential energy and distance of phase point to origin ( = 2.0,  = 0.1116,  = 0.009) 1 2 -11 -3 Instant Total Energy Flow Phase space volume strain 10 10 -2 -2 -4 -4 -6 -6 00.5 1 1.5 2 2.53 3.5 4 4.5 5 00.5 1 1.5 2 2.53 3.5 4 4.5 5 -4 -4 t t 10 Fig. 9 Time history of instant energy flow, and phase space volume strain ( = 2.0,  = 0.1116,  = 0.009) 1 2 which is rewritten in the form 2 2 2 2 2 𝜑 ̃ 𝜑 ̃ 𝜑 ̃ 𝜑 ̃ 𝜑 ̃ 2 3 4 3 4 = + , 𝜑 ̃ = + , 2 2 2 2 2 2 2 a a a a ∕a a ∕a 2 3 4 3 2 4 2 1 1 1 1 1 1 a = = , a = = , a = = , � √ √ √ √ √ 2 3 4 (44b) 𝜆 𝜆 � � 0.514069 0.020483 0.515312 3 4 � � � � � � � � 𝜆 𝜆 2 2 a � � a � � R = = , R = = . √ √ 4 3 a a 2 𝜆 2 𝜆 4 3 1 3 dEe/dt Vs 692 L. Qin et al. The sketched zero-energy flow surface and energy flow variations around the equilibrium point is shown in Fig.  10a, while the computer drawn surface based on Eq. (44a) is given in Fig.  10b. The one for the non- linear equation is like to the one shown in Fig.  6c, so neglected. The local linearised surface is symmetrical ⃗ ⃗ about the coordinate plane o − 𝜉 𝜉 and axis-symmetrical 3 4 to the axis o − 𝜉 , so that a plane perpendicular to axis o − 𝜉 intersects the surface in an elliptical curve of semi- 𝜑 ̃ R > 𝜑 ̃ R axis , as shown in Fig. 10a for the ones at 2 3  2 4 ⃗ ⃗ 𝜑 ̃ = 1 . The plane o − 𝜉 𝜉 intersects the surface in the 2 2 3 two straight lines of =±𝜑 ̃ R . At a disturbed point out- 3 2 3 side the zero-energy flow surface on the plane perpendicu- ⃗ ̇ lar to the axis o − 𝜉 , the energy flow E > 0 by Eq. (42a), so that this disturbance intends to increase the distance to the origin of space, implying the flow backwards the zero-energy surface. For a disturbed point inside the zero- energy flow surface, the energy flow E < 0 and the distur- bance intends to reduce the distance to the origin of space implying the flow also backwards the zero-energy surface. Therefore, this zero-energy surface is a divergence surface. In fact, for this case, the damping matrix of the system in Eq. (10a) becomes −0.0217 0.2442 c = , (44c) 0.2442 −0.0222 that shows the negative diagonal damping coefficients for both bearing and shaft, so that the motions caused by initial conditions will be divergence. The results of numerical simulations with the initial Fig. 10 Zero-energy flow surface around equilibrium point for case −6 conditions y = 10 11 11 for this case are given in (b) ( = 2.2  = 0.111245,  = 0.008325) based on Eq.  (44a): 1 2 a sketched one and b calculated one Figs. 11–13. Figure 11 gives the time histories of displace- ment and velocity of the bearing and shaft, which shows the Fig. 11 Time histories of displacement and velocity of bearing and shaft ( = 2.2  = 0.111245,  = 0.008325) 1 2 1 3 ̃𝜑 Energy flow characteristics of friction‑induced nonlinear vibrations… 693 Fig. 12 Phase diagram y(1)–y(3) of bearing and the one y(2)–y(4) of shaft, and time histories of generalized potential energy and distance of phase point to origin ( = 2.2  = 0.111245,  = 0.008325) 1 2 ab Fig. 13 Time histories of a instant energy flow and b phase space volume strain ( = 2.2  = 0.111245,  = 0.008325) 1 2 Fig. 14 Zero energy flow surface Eq.  (45a) around equilibrium point for case 3: a sketched one, b calculated one ( = 2.0  = 0.111,  = 0.008325) 1 2 1 3 694 L. Qin et al. divergence oscillations. Figure 12 presents the phase dia- discussed below. In this case, the central energy flow inves- gram y(1)–y(3) of bearing and the one y(2)–y(4) of shaft, tigation by considering higher-order terms is needed [28]. as well as the time histories of generalized potential energy The zero energy flow serface around the origin becomes and the distance of phase point to the origin, which indicates 2 2 0.512142𝜑 ̃ = 0.512142 , 𝜑 ̃ =±𝜑 ̃ , (45a) the orbit points in both phase diagrams move far away to 2 4 2 4 the origin, and the generalized potential energy as well as corresponding two orthogonal planes perpendicular to the distance of orbit point to the origin increase. Figure 13 ⃗ ⃗ the coordinate place o − 𝜉 𝜉 with the intersect lines and 2 4 shows the instant energy flow and the time history of phase =−𝜑 ̃ , respectively, as shown in Fig. 14. In the domain 2 4 space volume strain, of which both positively increase very ⃗ ̇ containing axis o − 𝜉 , the energy flow E < 0 , so that the fast at about non-dimension time 350. orbit disturbance along o − 𝜉 axis will be towards the zero-energy flow surface, while in the domain containing 3.2.3 Case (c): one zero and two opposite factors ⃗ ̇ ⃗ o − 𝜉 axis, E > 0 , the disturbance along o − 𝜉 axis will be 4 4 (˛ = 2.0,  = 0.111,  = 0.008325) 1 2 backwards the zero-energy surface. Therefore, the dynamic behaviour of the system cannot be predicted only based on In this case, since there exsists a zero-energy flow charac- linear approximation. The theoretical analysis relies on cen- teristic factor, the dynamic behavour of the system may not tral energy flow theorem, which is more complex, so that be determined only based on linearised approximation as Fig. 15 Time histories of displacement and velocity of bearing and shaft ( = 2.0  = 0.111,  = 0.008325) 1 2 -9 10 Generalised Energy Potential 0 12 3 4 5 6 7 89 10 -4 10 Distance to Origin 0.5 0 12 3 4 5 6 7 89 10 t 10 Fig. 16 Phase diagram y(1)–y(3) of bearing and the one y(2)–y(4) of shaft, and time histories of generalized potential energy and distance of phase point to origin ( = 2.0  = 0.111,  = 0.008325) 1 2 1 3 DD GEP ̃𝜑 ̃𝜑 Energy flow characteristics of friction‑induced nonlinear vibrations… 695 ab Fig. 17 Time histories of a instant energy flow and b phase space volume strain ( = 2.0  = 0.111,  = 0.008325) 1 2 Fig. 18 Enlarged time histories of displacement and velocity in modulated styles for bearing and shaft in initial time-period before breaking time shown in Fig. 15 ( = 2.0  = 0.111,  = 0.008325) 1 2 here we will reveal the solution by numerically simulat- displacement and velocity of bearing increase, while the ing the case with above parameters. Also, from linearised ones of shaft show oscillations, and both are in modu- approximation, the summation of diagonal elements of the lated style as indicated by enlarged picture in Fig. 18. The energy flow matrix E vanish, so that its time change rate of generalised potential energy and the orbit point distance phase volume strain approximately vanishes, i.e. to the origin of phase space are modulated oscillations with no obviously-change, and the phase diagram of shaft tends a periodical orbit as shown in Fig. 16. The energy ≈ 𝜆 = 0, (45b) flow shown in Fig.  17a has small values with no obvi- I=1 ously change. When the simulation time reaches about implying the volume of phase diagram should not change, 9.4  ×  10 s, the displacement and velocity of bearing which might not be true for the nonlinear equation with no are extreme large so that the simulation is stopped. Fig- linearazation. ure  17b shows the time history of phase space volume Now, we show the numerical simulation results to explain strain, indicating it increases with time and takes extreme above discussion. The damping matrix at fixed point takes large value at after the stop time. To explain this phenom- value enon, we check the friction damping matrix in Eq. (10a), −𝜓 Iz which is propotional the factor b =−𝛼𝜇 e . Refer r ing 0 0.2220 c = , to the Fig.  15, since the amplitude of bearing velocity (45c) 0.2220 0 z = y(3) increases while the amplitude of shaft velocity z = y(4) is nearly unchanged, so that the power of fric- of which the diagonal damping coefficients of system vanish. 2 −Iz −(z −z ) 2 1 tion function e = e will takes increased positive Figure  15 shows the time histories of displacement value, because of this, nonlinear friction force extremely and velocity for both bearing and shaft units, from which increased, which causes a self-excited oscillation of the it has been found that before the time 9.4  ×  10 s, the nonlinear bearing-shaft friction system. 1 3 ̇𝜐 696 L. Qin et al. Fig. 19 Time histories of displacement and velocity of bearing and shaft ( = 2.0  = 0.111,  = 0.008325) 1 2 Fig. 20 Phase diagrams y(1)–y(3) of bearing and y(2)–y(4) of shaft, and time histories of generalized potential energy and distance of phase point to origin ( = 2.0  = 0.111,  = 0.008325) 1 2 -11 Instant Total Energy Flow -3 Phase Space Volume Strain 10 10 ab 5 4 0 0 -2 -5 -4 00.2 0.40.6 0.8 11.2 1.4 1.61.8 2 00.2 0.40.6 0.8 11.2 1.4 1.61.8 2 t 10 t Fig. 21 Time histories of a instant energy flow and b phase space volume strain ( = 2.0  = 0.111,  = 0.008325) 1 2 1 3 dEe/dt Vs Energy flow characteristics of friction‑induced nonlinear vibrations… 697 Fig. 22 Time histories of time-averaged energy flow ( = 2.0  = 0.111,  = 0.008325) 1 2 From the discussion on the zero-energy flow surfaces of may conclude the stablity of system about the equlilibrium three cases and the corresponding numerical simulations point y = 0 as follows: based on the nonlinear friction function in Eq. (37b), but not the approximated one in Eq. (37c) used in Ref. [2], we Fig. 23 Dynamic response time histories of displacements and velocities of bearing and shaft of the system subject an external force: I) on bear- T T ̂ ̂ ̂ ̂ ing f = 00 f 0 and II) on shaftf = 000 f 1 3 698 L. Qin et al. Fig. 24 Phase diagrams y(1)–y(3) of bearing and y(2)–y(4) of shaft, and time histories of generalized potential energy and distance of phase T T ̂ ̂ ̂ ̂ point to origin: I) force on bearing f = 00 f 0 and II) force on shaft f = 000 f a b I) II) ̂ ̂ Fig. 25 Time histories of a instant energy flow and b phase space volume strain in forced vibration: I) force on bearing f = 00 f 0 and II) ̂ ̂ force on shaft f = 000 f 1 3 Energy flow characteristics of friction‑induced nonlinear vibrations… 699 Fig. 26 Curves of time-averaged energy flow vs average time for forced vibration: I) force on bearing f = and II) force on shaft 00 f 0 f = 000 f • Case (a) when the energy flow matrix at the equilib- conclusion is correct. For example, the three cases rium point has two negative and one positive char- w i t h  = 2.0,  = 0.1116, 0.1111, 0.111245, stud - acteristic factor, the local zero-energy flow surface ied based on the friction force approximation in in the small domain around the equilibrium point Eq. (37c) in Ref. [2] were reported the stable case for behaves an attractive surface, and the system is sta-  = 0.1116 but unstable cases for the last two values of ble to initial disturbance. We also simulated some  = 0.1111, 0.111245 . We re-check these three cases cases reported in Ref. [2], which further confirm this by using the friction force Eq. (37b), and the simulation Fig. 27 Dynamic response time histories of displacements and velocities of bearing and shaft in the system subject an external force: I) on bear- T T ̂ ̂ ̂ ̂ ing f = 000 f and II) on shaft f = 000 f 1 3 700 L. Qin et al. Fig. 28 Time histories of generalized potential energy, distance of phase point to origin and phase space volume strain in forced vibration: I) T T ̂ ̂ ̂ ̂ force on bearing f = 000 f and II) force on shaft f = 000 f results confirm that for each case, the energy flow matrix simulation result shows a divergence self-excited oscil- has two negative and one positive characteristic factor lation, which might be miss-judged as a stable oscillation given in Tables 1, 2 and 3, respectively, and the equilib- if the simulation time is not enough large. For example, rium point is stable. Fig. 18, cut out from Fig. 15 ( = 2.0,  = 0.111 ), shows Case (b) when the energy flow matrix at the equilibrium the enlarged time histories of displacement and velocity point has one negative and two positive characteristic of both bearing and shaft in the initial time before the factors, the local zero-energy flow surface in the small simulation breaking, which seems a stable picture but domain around the equilibrium point behaves a diver- unstable indicated by Fig. 15. gence surface, and the system is unstable. Case (c) when the energy flow matrix at the equilibrium 3.3 Periodical oscillation (f = 0) point has one zero factor and two same absolute value factors, one negative and another positive, the local zero- From Sect. 2.6, we know that for a possible periodical orbit, energy flow surface in the small domain around the equi- a necessary condition is that the spin matrix U of the system librium point behaves divergence in a sub-domain and must not vanish. From Eqs. (15), (19) and (27b), we obtain attractive in another sub-domain as shown in Fig. 14. The the spin matrix at the equilibrium point dynamical behaviour about this case requires to undergo a higher order approximation analysis. The numerical 1 3 Energy flow characteristics of friction‑induced nonlinear vibrations… 701 Fig. 29 Dynamic response time histories of displacements and velocities of bearing and shaft in the system subject an external force: I) on bear- T T ̂ ̂ ̂ ̂ ing f = and II) on shaft f = 000 f 000 f of phase point to the origin behave modulated oscillation 𝟎𝐈 − 𝐔 = ≠ 𝟎 , (46) without obviously amplitude variations, as shown in Fig. 20. −𝐈 𝟎 The instant energy flow curve behaviours a stable oscilla- implying possible periodical orbits of the sys - tion in Fig.  21a, and its time averaged one tends to zero tem. We simulate a case with parameters with average time increasing as shown in Fig. 22. The phase ( = 2.0  = 0.111,  = 0.008325) , of which there are two space volume train descreases with time shown in Fig. 21b. 1 2 negative and one positive energy flow characteristic factors According to the energy flow theory discussed in Sect.  2.6, and a damping matrix at equilibrium point this is a stable periodical motion of the system excited by initial conditions. 0.0012 0.2220 We have simulated the cases with param - c = . (47) 0.2220 0 e ter s ( = 2.0  = 0.1111,  = 0.008325) and 1 2 ( = 2.12  = 0.119725,  = 0.0009) [2], of which the 1 2 Figure 19 shows that the displacement and velocity of energy o fl w matrix has two negative and one positive energy bearing decrease with time, while the shaft ones are in a flow characteristic factors and both systems also behave the periodical motion. The phase diagram of bearing tends similar stable periodical oscillations as the one presented to the origin while the one of shaft shows a closed orbit, herein. and the generalised potential energy as well as the distance 1 3 702 L. Qin et al. Fig. 30 Enlarged response time histories, before simulation breaking, of displacements and velocities of the system subject an external force: I) T T ̂ ̂ ̂ ̂ on bearing f = 000 f and II) on shaft f = 000 f ̃ the phase diagrams tend periodical orbits in Fig. 24, as well 3.4 Forced oscillations (f ≠ 0) as the instant energy flows and the phase space volume −6 strains in Fig.  25 also tend stable curves. Figure  26 con- We respectively add an excitation force of amplitude 10 , firms that the time-averaged energy flows tend zero with −6 frequency 0.5 and phase angle 0, i.e. f = 10 cos(0.5𝜏 ), on average time increasing. These typical results demonstrate the bearing mass and the shaft mass to investigate the forced that the forced vibrations are periodical motions of the sys- dynamic response of the system. The simulated results are tem although the different curve styles due to different force given as follows. added potions. Since the positive damping of the system, the free vibration components due to initial conditions are 3.4.1 Case (a): parameters gradually damped, and a stable forced vibration is obtained. ˛ = 2.0  = 0.1116,  = 0.009 1 2 However, since the system is nonlinear, the frequency of the forced vibration does not equal the force frequency, so that Figures 23–26 give the simulation results, in which, with the response curves behave modulated pictures compositing time increasing, the time histories of displacements and of two frequencies response. velocities shown in Fig.  23 tend stable forced modulated oscillations, the generalised potential energies and the dis- tances of phase points to the origin tend stable pictures and 1 3 Energy flow characteristics of friction‑induced nonlinear vibrations… 703 3.4.2 Case (b): parameters discovered: (a) two negative and one positive factors, con- ˛ = 2.2,  = 0.111245,  = 0.008325 structing an attractive local zero-energy flow surface, in 1 2 which free vibrations show damped modulated oscillations In this case with main negative damping coefficients, allowing the system returning to its equlibrium state, while the system behaves divergence oscillations, of which the forced vibrations show stable oscillations; (b) one negative dynamic response time histories of displacements and and two positive factors, spanning a divergence local zero- velocities in Fig.  27, the generalised potential energies energy flow surface, so that the both free / forced vibrations and the distances of phase point to origin as well as the are in divergence resulting an unstable system; (c) one zero phase space volume strains in Fig. 28 increase very fast and two opposite factors, constructing a local zero-energy with time going. flow surface dividing the phase space into stable, unstable and central subspace, and the simulation shows friction self- 3.4.3 Case (c): parameters induced unstable vibrations in both free and forced cases. ˛ = 2.0,  = 0.111,  = 0.008325 For a set of friction parameters, the system behaves a peri- 1 2 odical oscillation, in which the bearing motion tends zero The forced vibration in this case is like the free vibration and the shaft motion tends a stable limit circle in phase space discussed for case 3 in Sect. 3.3, as shown in Fig. 29, the with the instant energy flow tends a constant and its time dynamic responses of displacements and velocities of both averaged one tends zero. Numerical simulations have not bearing and shaft increase until about time 3.3 × 10 when found any possible chaotic motions of the system. It is dis- the responses sharply increased and simulations stopped. covered that the damping matrices of cases (a), (b) and (c) To show the response curve details, Fig. 30 provides local respectively have positive, negative and zero diagonal ele- enlarged curves before breaking time in Fig.  29, from ments, dominating the dynamic behavour in different cases which it is observed the dynamic responses behave the of the system, which provides an approach to design the modulated vibrations before the simulation breaking. The water-lubricated bearing unit with expecting performance in curves of generalised potential energy, the phase diagram, marine engineering applications by choosing suitable fric- etc. also show the characteristics, so they are neglected tion parameters to generate the expected damping matrix. herein. Acknowledgements We gratefully acknowledge NSFC (51509194) and CSC for providing finacial support eanabling Li Qin and Hongling Qin 3.5 Chaotic motions to visit the University of Southampton to engage the related research. With different friction parameters, we have tested several Open Access This article is licensed under a Creative Commons Attri- cases for free vibrations induced by given initial distur- bution 4.0 International License, which permits use, sharing, adapta- tion, distribution and reproduction in any medium or format, as long bances and for forced vibrations excited by external forces as you give appropriate credit to the original author(s) and the source, acted on the bearing and shaft masses in order to see if cha- provide a link to the Creative Commons licence, and indicate if changes otic motions may be found. The simulations results have not were made. The images or other third party material in this article are shown any chaotic motions of this system. included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will 4 Conclusion need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativ ecommons .or g/licenses/b y/4.0/. Following the detailed describption of this investigation given above, we may conclude the main contributions as follows. The paper is a first one to study a nonlinear water- References lubricated bearing-shaft coupled system using the energy flow approach, which practices and further confirmed an 1. USA Department of Defense: Bearing Components, Bonded Syn- efective energy-flow means to tackle varous complex non - thetic Rubber, Water Lubricated. MIL-DTL-17901C (SH) (2005) 2. Simpson, T.A., Ibrahim, R.A.: Nonlinear friction-induced vibra- linear systems and to investigate their nonlinear character- tion in water-lubricated bearings. J. Vib. Control 2, 87–113 (1996) istics: stability, periodical motion, birfurcation and possible 3. Krauter, A.I.: Squeal of water-lubricated elastomeric bearings— chaos.The results obtained for the investigated system have an exploratory experimentation. Technical Report No. 77-TR-25. revealed that the dynamic behaviour about its equilibrium Shaker Research Corp., Ballston Lake, New York (1977) 4. Krauter, A.I.: Brower, squeal of water-lubricated elastomeric bear- point is fully determined by the energy flow matrix and its ings: A quantitative laboratory investigation. Technical Report characteristic factors. 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Peng, J., Wang, J., Yang, M.: Research on modifying mechanic 28. Xing, J.T.: Energy Flow Theory of Nonlinear Dynamical Systems performance of water lubricated plastic alloy bearings material. with Applications. Springer, Heidelberg (2015) Lubr. Eng. 6, 80–82 (2004) 29. Xing, J.T.: Generalised energy conservation law for chaotic phe- 14. Orndorff, R.L.: Water-lubricated rubber bearings, history and new nomena. Acta. Mech. Sin. 35(6), 1257–1268 (2019) developments. Naval Engineers Journal 10, 39–52 (1985) 30. Fung, Y.C.: A First Course in Cintinuum Mechanics. Prentice- 15. Qin, H., Zhou, X., Yan, Z.: Effect of thickness and hardness and Hall, New Jerscy, Englewood Cliffs (1977) their interaction of rubber layer of stern bearing on the friction 31. Xing, J.T.: Fluid-Solid Interaction Dynamics, Theory, Variational performance. Acta Armamentarii 34(3), 318–323 (2013) Principles, Numerical Methods, and Applications. Beijing & 16. 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Energy flow characteristics of friction-induced nonlinear vibrations in a water-lubricated bearing-shaft coupled system

Qin, Li; Qin, Hongling; Xing, Jing Tang
"Acta Mechanica Sinica" , Volume 37 (4) – Mar 4, 2021
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Abstract

Based on the energy flow theory of nonlinear dynamical system, the stabilities, bifurcations, possible periodical/chaotic motions of nonlinear water-lubricated bearing-shaft coupled systems are investigated in this paper. It is revealed that the energy flow characteristics around the equlibrium point of system behaving in the three types with different friction-para- mters. (a) Energy flow matrix has two negative and one positive energy flow factors, constructing an attractive local zero- energy flow surface, in which free vibrations by initial disturbances show damped modulated oscillations with the system tending its equlibrium state, while forced vibrations by external forces show stable oscillations. (b) Energy flow matrix has one negative and two positive energy flow factors, spaning a divergence local zero-energy flow surface, so that the both free and forced vibrations are divergence oscillations with the system being unstable. (c) Energy flow matrix has a zero-energy flow factor and two opposite factors, which constructes a local zero-energy flow surface dividing the local phase space into stable, unstable and central subspace, and the simulation shows friction self-induced unstable vibrations for both free and forced cases. For a set of friction parameters, the system behaves a periodical oscillation, where the bearing motion tends zero and the shaft motion reaches a stable limit circle in phase space with the instant energy flow tending a constant and the time averaged one tending zero. Numerical simulations have not found any possible chaotic motions of the system. It is discovered that the damping matrices of cases (a), (b) and (c) respectively have positive, negative and zero diagonal ele- ments, resulting in the different dynamic behavour of system, which gives a giderline to design the water-lubricated bearing unit with expected performance by adjusting the friction parameters for applications. Keywords Nonlinear friction-induced vibrations · Nonlinear energy flows · Nonlinear water-lubricated bearing-shaft systems · Bifucation friction parameters · Energy flow matrices · Periodical oscilation 1 Introduction Water-lubricated bearings have been more and more used in the marine and pump industries to eliminate the pollution of metal bearings lubricated by oil and grease as well as Executive Editor: Li-Feng Wang to increase efficiency and reliability of marine propulsion systems. The first effective water-lubricated bearing made * Jing Tang Xing jtxing@soton.ac.uk of lignum vitae was invented by John Penn in the 1840s while man-made plastic bearings and natural rubber bear- Key Laboratory of Metallurgical Equipment and Control ings appeared in the 1920s, which had gradually predomi- Technology, School of Machinery & Automation, Wuhan nated in pumps, naval and many commercial ships by the University of Science and Technology, Wuhan 430081, China 1940s. The need for improved water-lubricated bearings was greatly recognised in 1942 when a number of U.S. ships Hubei Key Laboratory of Hydroelectric Machinery Design & Maintenance, Three Gorges University, Yichang 443002, suffered extensive combat damage at the Battle of Midway, China after which, USA navy department began to carry out an Maritime Engineering Group, CMEE, School of Engineering, extensive research on water-lubricated bearings and obtained Faculty of Engineering & Physical Sciences, University a series of achievements and the only military specification of Southampton, Southampton SO17 1BJ, UK Vol.:(0123456789) 1 3 680 L. Qin et al. of the water lubricated rubber bearings was created by MIL- performance of this type of bearing systems, it is essential DTL-17901C (SH) in 2005 [1]. Since the friction perfor- to investigate its integrated nonlinear characteristics. In mance plays a key role in bearing vibration or noise, this the area on friction induced vibrations, Ibrahim [20, 21] specification especially describes the requirements for the contributed two important review papers, of which the first test rig and scope of friction coefficients of bearing synthetic one concerns the mechanics of contact and friction and rubber facings. the second one discusses the dynamics and modeling of Stern-tube bearing noise is associated with both the quan- friction induced vibrations. Both of papers provide the tity of friction and the slope of friction–velocity curve [2], comprehensive account of the main theorems and mecha- which involves bearing materials. To reduce bearing noises, nisms developed in the historical literatures concerning many historical experimental and analytical investigations friction-induced noise and vibrations with a quite wide were reported [3–5]. For examples, Krauter [3, 4] focused references listed. Furthermore, Ding and Zhai [22] pre- on a three-degree-of-freedom (3-DOF) system to incorporate sented a review paper on the advances of friction dynamics with the essential elements of the friction and vibration phe- in mechanics system, in which the common used friction nomena in their experimental model, in which a linear analy- models, friction related self-excited vibrations and their sis was considered to determine the instability conditions controls were presented. The interested readers may refer of the system that may lead to frictional vibrations. Sinou these three review papers for more information and refer- et al. [6] proposed a numerical model incorporating realistic ences in this area. laws of local friction in base of experimental results to char- More directly linking the topic of this paper, Simpson acterize the dynamics of a lubricated system and to study and Ibrahim [2] investigated the dynamic behaviour of its complex global responses triggered by the local inter- nonlinear friction-induced vibration in water-lubricated facial behaviour. Graf and Ostermeyer [7] have shown how bearings based on the traditional method dealing with the stability of an oscillator sliding on a belt will change, nonlinear dynamical systems. Their paper developed an if a dynamic friction law with inner variable is considered analytical nonlinear, two-degree-of-freedom model, which instead of a velocity-dependent coefficient of friction, which emulates the stern-tube bearing. Typical friction-speed demonstrates the unstable vibration can even be found in curves were adopted based on the experimental results of the case of a positive velocity-dependency of friction coef- Krauter [23]. The stability conditions of equilibrium were ficient. Baramsky et al. [8 ] contributed the analytical and predicated. In the unstable cases, the nonlinear response experimental results for the occurrence of friction-induced behavior was examined by using numerical integration of vibrations during tightening of bolted joints of the most the coupled equations of motion. The dependence of the used machine elements. Niknam and Farhang [9] employed relative sliding speed on time and its effect on the friction a two-degree-of-freedom single mass-on-belt model to study force was included in the numerical simulation. A con- friction-induced instability due to mode-coupling, in which trol parameter that combines the influence of the normal numerical analyses are used to tackle the effect of three load and the slope of the friction-speed curve was used to parameters related to belt velocity, friction coefficient, and construct a bifurcation diagram which separated different normal load on the mass response. Ghorbel et al. [10] pro- response characteristics, including squeal, limit cycle, and posed a minimal 2-DOF disk brake model to investigate the stable regions. effects of different parameters on mode-coupling instabil - To reveal nonlinear behavior of nonlinear dynamical sys- ity, which considers self-excited vibration, gyroscopic effect, tems, such as as discussed in the books by Guckenheimer friction-induced damping, and brake pad geometry. and Holmes [24], Thompson and Stewart [25], Chen et al. In designs of this type of berings, two research directions [26] and Liu and Chen [27], Xing [28] developed a genral- have been developing: one aims to obtain good friction char- ised energy flow theory to investigate nonlinear dynamical acteristics using suitable geometric parameters with differ - systems governed by ordinary differential equation in phase ent facing layers and lubrications [11–14], and another is to space. Important nonlinear phenomena such as, stabilities, create new bearing materials [15–18]. Recently, Smith [19] periodical orbits, bifurcations and chaos can be revealed presented a paper on the design and lubrication of water- from the energy flow behavors of the systems [29]. The aim lubricated, rubber, cutlass bearings systems, which gave a and motivation of this paper are to investigate the 2-DOF methodology to predict the minimum film thickness between nonlinear model of the stern-tube bearing proposed in Ref. the journal and heavily loaded stave, including a 3D finite [2] to reveal its energy flow characteristics governing its element (FE) model to predict generated pressures in cutlass nonlinear dynamical behavours, which is also as an example bearings, and comparing with experimental data. to further develop energy flow approach based on two sca- From the point view of sciences, the water-lubricated lar varaibles, genralised potentional and kinetic energies, to bearings system is a typical nonlinear system concerning tackle complex nonlinear dynamical systems, especially in friction-induced vibration noises, so that for good noise multi-dimensional phase spaces. 1 3 Energy flow characteristics of friction‑induced nonlinear vibrations… 681 the disk radius and 𝜃 is the torsional velocity of the disk rela- tive to the bearing. Considering only oscillatory motions of Bearing housing both the shaft and bearing, and representing the motion by Rubber stave the linear displacement, we can denote the relative sliding Gap filled with water Rotor shaft speed of the friction pair in the form 𝜈 = R𝜃 = ẋ −ẋ = z − z , z = ẋ , z = ẋ . (2) 2 1 2 1 1 1 2 2 Substituting Eq. (2) into Eq. (1) and then taking the deriv- ative of the resultant equation with respect to the relative sliding speed, we obtain the visous damping coefficient as dF Fig. 1 Typical grooved stern tube bearing structure −av b = =−Na  −  e , (3) 0 1 dv −av which, when the function e is expanded into a Taylor series, becomes 2 2 3 b =− Na  −  1 − av + a v + o(v ) 0 1 1 2 3 =− Na  −  1 − a z − z + a z − z + o z − z . 0 1 2 1 2 1 2 1 (4) 1.1.2 Dynamic equations of the friction pair system Fig. 2 Analytical 2-DOF model of stern tube bearing Using the model shown in Fig. 2 and the second Newton’s law, we obtain the following two equations of motion of the system 1.1 Dynamic modelling and equations m x ̈ + c x ̇ + k x = b(x ̇ − x ̇ )+ F , 1 1 1 1 1 1 2 1 1 of water‑lubricated bearings (5) m x ̈ + c x ̇ + k x =−b(x ̇ − x ̇ )+ F , 2 2 2 2 2 2 2 1 2 The marine propeller shafts are supported in stern tube bear- which can be rewritten in a matrix form ings, which are lubricated by water and are almost all grooved, of which the scketched structure is shown in Fig. 1. The 2-DOF ̈ ̇ ̂ MX + CX + KX = F, (6) analytical model used in Ref. [2] is shown in Fig. 2, where the where the mass, damping and stiffness matrices as well as mass, stiffness, and damping coefficient of the bearing (sub- the displacement and force vectors are resepctively defined script 1) and the flexible shaft (subscript 2) are denoted by as follows m , m , k , k , c and c , respectively. Here, the forces F and 1 2 1 2 1 2 1 F denote possible prescribed external forces applied to the m 0 c + b −b 1 1 bearing and the shaft, respectively, while the force F denotes M = , C = , 0 m −bc + b 2 2 the friction force acted to the bearing by the shaft, which is k 0 x F assumed in the viscous type propotional to the relative sliding 1 1 1 K = , X = , F = . speed of the friction pair with a viscous damping coefficient 0 k x F 2 2 2 (7) b , so that F = b(x ̇ − x ̇ ). f 2 1 Introducing the nondimensional parameters 1.1.1 Viscous damping coefficient c 𝜔 k m i 2 i 1 𝜁 = , 𝛾 = , 𝜔 = , 𝜂 = , i 21 i 12 𝜔 m m Historically, the friction force was formulated by the friction 2 k m 1 i 2 i i model developed by Smith et al. [5] and used by Simpson and x F i i y = , f = , i = j = 1, 2, Ibrahim [2] in their 2-DOF analytical model, that is i i R 𝛽 N −aR𝜃 Na F = N[𝜇 +(𝜇 −𝜇 )e ], (1) 𝜏 = 𝜔 t, 𝜇 = 𝜇 − 𝜇 , 𝛼 = , f 1 0 1 1 01 0 1 m 𝜔 1 1 where N denotes the normal bearing load; a is a constant m 𝜔 R 𝛽 = , 𝜓 = 𝛼𝛽 = aR𝜔 , with units of inverse velocity;  and  are the static and 1 0 1 N (8) dynamic coefficients of friction, respectively; R represents 1 3 682 L. Qin et al. where  denote dimensionless damping coefficients,  ratio ̃ is called the generalised force of system. Here, J denotes the i 21 T T T of circular frequencies,  natural frequencies of the bear- i coefficient matrix of the state variable vector [y z ] , which ing and the shaft subsystems,  mass ratio, y dimension- ̃ involves the nonlinear damping parameter b . The solution of 12 i less displacements,  dimensionless time,  dimensionless Eq. (11) with a prescribed initial condition −1 friction coefficient,  dimensionless normal bearing load T T and  the dimensionless value of constant  . Using these y(0)= y ̂, z(0)= z ̂, r = y ̂ z ̂ , (13) dimensionless parameters, the dynamic equation of system gives an orbit starting from the position r in the phase space can be written as 0 as shown in Fig. 3. my ̈ + cy ̇ + ky = f , (9) The parameters of system, such as k , c and c , can be 1 1 2 functions of normal load and temperature. Krauter [4] devel- where oped empirical relationships for these parameters 10 ◦ 2𝜁 0 10 1 k = 116.52N + 10535.4 at 35 C, 2 1 m = , k = 𝛾 , c = c ̂ + c ̃, c ̂ = 2𝛾 𝜁 , 21 2 01∕𝜂 0 𝜂 m  ln 2 (14) 𝜂 12 1 2 c =−0.618T + 8.474, c = . 1 2 πN y f 1 1 ̃ ̃ ̂ −𝜓 Iz c ̃ =bI, y = , f = , z = y ̇ , , b =−𝛼𝜇 e 2 2 Typical values of these parameters were selected for con- 1 −1 ̂ stant values of normal load and temperature, i.e. N = 120 N −𝜓 Iz ̂ ̃ I = −11 , I = , 𝜕 b∕𝜕 z = 𝛼𝜇 𝜓 e 1 01 −11 (27 lb) and T =35 ℃, by Simpson and Ibrahim [2] to investi- ̇ −𝜓 Iz gate the nonlinear dynamical behaviour of system. We also use ̃ ̃ ̃ ̂ =− 𝜓 b =−𝜕 b∕𝜕 z , b = 𝛼𝜇 𝜓 e Iż . 2 01 these chosen parameters to reveal the energy flow character - (10a) istics of system, and to rewrite Eqs. (11) and (12) in the form For the approximation as shown in Eq. (4) adopted by Simpson and Ibrahim [2], the parameter b and its derivatives ẏ y ̃ ̃ ̃ ̃ ̃ = J + f = F, J = J + J , 0 b are given by the following approximated formulations ż z 1 0 I 00 2 T ̃ ̂ ̃ ̃ ̃ b ≈−𝛼𝜇 (1 − 𝜓 Iz + 𝜓 z Iz), J = , J = , 01 0 b 2  0 b (15) ̇ T ̃ ̂ ̃ ̃ ̂ b ≈ 𝛼𝜇 𝜓 (Iż − 𝜓 ż Iz), 𝜕 b∕𝜕 z =−𝛼𝜇 𝜓 (1 − 𝜓 Iz) −10 −2𝜁 0 01 1 01 = ,  = , 2 2 2 ̃ ̃ 0 −𝛾 0 −2𝛾 𝜁 =−𝜕 b∕𝜕 z , 𝜕 b∕𝜕 z =−𝛼𝜇 𝜓 21 2 2 01 2 2 2 2 ̃ ̃ −11 =−𝜕 b∕𝜕 z , 𝜕 b∕𝜕 z 𝜕 z = 𝛼𝜇 𝜓 , ̃ ̃ 1 2 01 2 (10b) b = b,  = , 𝜂 −𝜂 12 12 implying that the parameter b is approximated by a quadratic from which in association with Eq. (10) we obtain function of the relative velocity (z − z ) , its first derivatives 2 1 are linear, and the second derivatives are constants. y ̈ y ̇ y ̇ ̇ ̃ ̃ ̃ = J + J + f , (16a) z ̈ z ̇ z 1.2 Dynamic equations and equilibrium points 1.2.1 Equations in phase space ̇ ̇ ̃ ̃ J = J = , (16b) 0 b Using the state variables y and z , we can rewrite Eq. (9) in the form of the phase space, i.e. y ̇ = F = z, (11) −1 ̃ ̃ ̂ ż = F = m [−(c ̂ + bI)z − ky + f], of which, the vector F y ̃ ̃ F = = J + f , F z 0 I 0 ̃ ̃ J = , f = , −1 −1 ̃ −1 ̃ ̂ −m k −m (c ̂ + bI) m f (12) Fig. 3 Orbit, position and tangent vectors at a point on the orbit of a dynamical system in phase space 1 3 Energy flow characteristics of friction‑induced nonlinear vibrations… 683 ̇ 2 2 T T ̃ ̃ (16c) E = r ∕2, r = y y + z z, (22) b = b. while the generalised kinetical energy equals the half square of the tangent vector, the velocity, at a point, i.e. 1.2.2 Equilibrium point and Jacobian matrix 2 2 T T K = r ̇  ∕2, r ̇  = y ̇ y ̇ + z ̇ z ̇ . (23) The point at which F = 0 is called the equilibrium point of system. Obviously, if there is no external force ( f = 0 ), t he Therefore, these two non-negative real scalars can be used origin (y = 0 = z) of phase space is an equilibrium point of to describe the postion and velocity at a point on the solution system. orbit of system in phase space, and to reveal related nonlin- By using Eqs. (15) and (16), the Jacobian matrix of the ear dynamical behavours of system. system can be derived as 2.2 Energy flow equation and its time average J = F∇ = Jr + f = J + J, T T y z (17) Pre-multiplicating Eqs. (13) and (15) by a row vector J J b b T T T T J = r r = (Iz) . T T r = y z and its initial value r ̂ (0) , respectively, we y z 0 b obtain the energy flow equation Here the gradient operator ∇ is defined by T T T T ̇ � � � E = P, P = r F = r Jr + p ̂, p ̂ = r f = z f , T T T T ∕y    T T T ̈ ̇ ̂ T T T E = ṙ F + r F = 2K + r r ̈, E(0)= E = r ̂ r ̂∕2 =(y ̂ y ̂ + z ̂ � z)∕2, ∇= = , ∇ = = , r = y z . T T y z r ∕z T T ̇ ̇ � (18) E = Edt ∕T = (r Jr + p ̂)dt ∕T, 0 0 Substituting Eq. (15) into Eq. (17), we finally obtain the (24a) Jacobian matrix of which, P denotes the power done by the generalised force ̃ ̂ ̃ of system, p ̂ is the power of the external forces and E rep- J = J +[1 + 𝜓 (Iz)]J . (19) 0 b resents the time change rate of generalised potential energy, ̃ ̃ Here the matrices J and are defined by Eq. (15). called as the energy flow of system. Physically, the energy 0 b flow equation in Eq. (24a) implies that the energy flow of system equals the power done by the generalised force. 2 Energy flow formulations of system The energy flow equation in Eq. (24a) can be rewritten in the form 2.1 Generalised potential energy and kinetic T T T ̇ ̃ ̃ ̃ ̃ E = r (E + U)r + p ̂ = r Er + p ̂, r Ur = 0, energy T T ̃ ̃ ̃ ̃ ̃ ̃ E =(J + J )∕2, U =(J − J )∕2, (24b) We define two scalar variables: the generalised potential energy and generalised kinetical energy of system in the ̇ ̃ E = (r Er + p ̂)dt ∕T. following forms. Generalised potential energy and its time average ̃ ̃ Here E is a real symmetrical matrix, while is a real anti- T T symmetrical matrix. 1 1 1 1 T T T T E = r r = (y y + z z), ⟨E⟩ = Edt = r rdt. 2 2 T 2T 0 0 2.3 Kinetic energy flow equation and its time (20) average Generalised kinetical energy and its time average T T Pre-multiplying Eq. (16a) by r ̇ and using Eqs. (14) and 1 1 1 1 T T T T K = r ̇ r ̇ = (y ̇ y ̇ + z ̇ z ̇ ), ⟨K⟩ = Kdt = r ̇ r ̇ dt. (16c), we obtain the kinetic energy flow and its time average 2 2 T 2T 0 0 (21) ̇ ̇ ̇ ̇ T T T T T T T ̇ ̃ ̃ ̃ ̃ ̃ ̃ K = r ̇ r ̈ = r ̇ Jṙ + r ̇ Jr + r ̇ f = r ̇ Eṙ + z ̇ bz + r ̇ f , Physically, these two scalars may not practical potential T T and kinetical energies, therefore we use the word general- ̇ ̇ T T T T ̃ ̃ ̃ K = r ̇ r ̈dt ∕T = (r ̇ Eṙ + z ̇ bz + r ̇ f)dt ∕T. ised to distinguish them with practical physical quantities. 0 0 Geometrically, as shown in Fig. 3, the generalised potential (25) energy equals the half square of the distance of a point to the origin of phase space, i.e. 1 3 684 L. Qin et al. ̇ ̇ 2.4 Zero‑energy flow surfaces and equilibrium ΔE = E(r + )=(r + ) F(r + )=(r + 𝜀 )F (r + 𝜀 ), i i i j j points =(r + 𝜀 ) F + F 𝜀 + 0.5F 𝜀 𝜀 + O(𝜀 ) i i i i,j j i,jk j k = v𝜀 F + r F 𝜀 + 𝜀 F 𝜀 + 0.5r F 𝜀 𝜀 Generally, the energy flow of system is a function of time i i i i,j j i i,j j i i,jk j k T T T T T and the position of a point in phase space, which generates =  p +  (E + U) +  E  =  p +  (E + E ), 1 1 a scalar field called as the energy flow field of the nonlin- (27a) ear dynamical system. Equation where the energy flow gradient vector p , the energy flow matrices E and E , and spin matric U are given by ̇ 1 E = 0, (26) T T T p = F + J r, E =(J + J )∕2, U =(J − J )∕2, defines a generalised surface or subspace in phase space, � � which is called as a zero-energy flow surface on which the E = r ∇∇ F =  +  + B, B = , 1 i i 4×4 4×4 energy flow vanishes. If an orbit of the nonlinear dynamical 2 2 2 2 ̃ ̃ 𝜕 𝜕 𝜕 b 𝜕 b ⎡ ⎤ ⎡ ⎤ system is on a zero-energy flow surface, the distance of a 𝜕 z 𝜕 z 𝜕 z 𝜕 z 𝜕 z 𝜕 z 𝜕 z 𝜕 z ⎢ 1 1 1 2 ⎥ ̃ ̂ ⎢ 1 1 1 2 ⎥ ̂ b =(z − 𝜂 z ) (bIz)= (z − 𝜂 z ) (Iz) 2 2 2 2 1 12 2 1 12 2 ̃ ̃ 𝜕 𝜕 𝜕 b 𝜕 b ⎢ ⎥ ⎢ ⎥ point on the orbit is not changed with time. 𝜕 z 𝜕 z 𝜕 z 𝜕 z 𝜕 z 𝜕 z 𝜕 z 𝜕 z ⎣ ⎦ ⎣ ⎦ 2 1 2 2 2 1 2 2 � � � � Based on the theory of extreme values of a function and 1 −1 1 −1 2 2 T ̃ ̂ ̃ =(z − 𝜂 z )𝜓 b (Iz)= 𝜓 bz 𝜂 z . 1 12 2 the geometrical meaning of generalised potential energyE , −1 1 −1 1 we can conclude that the characteristics of orbit at a point (27b) on the zero-energy flow surface as follows. Here, we have used Eqs. (10a) and (15) and noticed that the components F and F of the vector F are lin- 1 2 • E > 0 , the generalised potential energy takes a local ear functions of y and z , so that their second derivatives minimum-extreme value at this point compared with the vanish. points in the local domain around it. Therefore, with time For an equilibrium point satisfying F(r)= 0 , Eq. (27a) going, the value of E increases and the orbit backwards becomes the origin of phase space. • E = 0 , it cannot determine that generalised potential T ̇ ̇ ΔE = E(r, )= J r +  (E + E ), (28a) energy takes a local minimum- or maximum-extreme value at this point, and higher time derivatives are needed which is further reduced to to give a solution. ̇ ̇ ΔE = E()=  E, (28b) • E < 0 , the generalised potential energy takes a local maximum-extreme value at this point compared with the for an equilibrium point r = 0, and so that E  = 0. At the points in the local domain around it. Therefore, with time equilibrium point r = 0 the second time derivative of the going, the value of E decreases and the orbit towards the generalised energy in Eq. (24a) vanishes, so that we need to origin of phase space. consider the variation of E around the point r = 0 to investi- gate the orbit behaviour. From Eq. (28a) and the geometrical If no external force p ̂ = 0, there are the following three meaning in Eq. (22), we consider the variation of energy cases satisfying E = 0 in Eq. (26). flow E to determine the characteristics of orbit as follows. Case 1:r = 0 , that represents the origin of phase space, If r +  < r, then E(r + ) < E(r), so that ΔE > 0 at which the generalised potential energy is defined as implies the flow towards the zero-energy flow surface, zero; while ΔE < 0 indicates the flow backwards the zero- • Case 2:F = 0 , implying an equilibrium point of system, energy flow surface; so that equilibrium points are on the zero-energy flow If r +  > r, then E(r + ) > E(r), so that ΔE < 0 surface; implies the flow towards the sero-energy flow surface, Case 3: P = 0, r ≠ 0 ≠ F, correseponding a generalised while ΔE > 0 indicates the flow backwards the zero- zero-energy flow surface. energy flow surface; If the flows from both sides of the zero-energy flow sur - Assume that r denotes a point on an orbit on a zero- face tend to it, this surface is an attracting surface. energy flow surface, P(t, r)= r F = r F = 0 , and  is an i i The local ability of equilibrium point r = 0 can be deter- small orbit variation around r , generally, the variation of mined by the behaviour of energy flow matrix E as fol- energy flow caused by the orbit variation can be approxi- lows: mated to the quantities of  in the form 1 3 Energy flow characteristics of friction‑induced nonlinear vibrations… 685 𝜆 as the energy flow characteristic factors and we can definitely - negative, asymptotic stable, conclude for a point of the orbit of nonlinear system. E ∶ semi - definitely - negative, stable, definitly or semi - definitly - positive, unstable. • The positive, zero and negative value of the factors 𝜆 (29) respectively implies the energy flow increase, unchanged and decrease caused by the disturbance in the I-th Figure  4 shows a case where the orbit intersects at a principal direction, from which the behaviour of energy point r on the energy flow surface. Since ΔE > 0 above flow increments at this point can be identified to judge the surface and ΔE < 0 under the surface, so that the flow the local dynamical behaviour, such as for local stability along the orbit backwards this point and this point is an of orbits around an equilibrium point r = 0, we have unstable point. negative , stable subspace, ∶ zero, central flow subspace, (33) 2.5 Energy flow matrix and energy flow positive, unstable subspace. characteristic factor • If its energy flow characteristic factors are not all semi- Both of matrices E and E are real symmetrical matri- negative or not all semi-positive, there will exist a small ces, called as the energy flow matrices, the former is for subdomain around this point in the phase space deter- the total energy flow in Eq. ( 24b), while the later for the mined by E = 0 , from which a zero-energy flow surface incremental energy flow relative to the zero-energy flow can be obtained. surface in Eq. (27b). The real symmetrical matrix has its real eigenvalues  and corresponding eigenvector  sat- I I 2.6 Spin matrices and periodical orbit isfying the orthogonal relationships, such as for matrix E in Eq. (24b), we have The matrices U in Eq. (24b) and U in Eq. (27b) are two T T real skew-symmetrical matrices, called as spin matrices. ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ = I,  E = diag(𝜆 ),  = 𝝃 𝝃 𝝃 𝝃 . I 1 2 3 4 For periodical orbits, it is neccesary that there exsist a time (30) period T and its corrsponding closed orbit Γ such that the These eigenvectors span an energy flow space in which following integrations along the closed orbit hold, i.e. the vector r can be represented by � � 1 1 ̂ ̂ T+t T+t ̇ ̇ ̂ ̃ E = Edt = 0, ⟨E⟩ = Edt = E, r = 𝝋 ̃ , (31) ̂t ̂t T T Γ Γ which, when substituted into Eq. (24b), gives (34a) � � 1 1 ̂ ̂ 4 T+t T+t ̇ ̇ ̂ K = Kdt = 0, ⟨K⟩ = Kdt = K, ̂ ̂ t t T 2 T ̇ ̃ ̃ T T E =  ̃ diag(𝜆 ) ̃ = 𝜆 𝜑 ̃ ,  ̃ = 𝜑 ̃ 𝜑 ̃ 𝜑 ̃ 𝜑 ̃ . Γ Γ I I 1 2 3 4 I=1 ̂ ̂ where E and K are two positive constants. The stability of (32) this periodical orbit can be determined by the values of the Therefore, 𝜆 represents the energy flow variation energy flow ΔE around each point of the orbit as discussed caused by a unit disturbance 𝜑 ̃ = 1 in the I-th principal for the zero-energy flow surface. direction 𝜑 ̃ of the energy flow matrix. Therefore, we call The time averaged kinetic energy can be re-written as 1 1 1 ̂ T ̂ T T+t T+t ⟨K⟩ = r ̇ r ̇ dt = r ̇ dr = . (34b) ̂t ̂t 2T 2T 2T Γ Γ Physically, this integration denotes the circulation inte- gral of velocity field r ̇ along the closed orbit and involves the skew-symmetrical spin matrix. To clarify this, we consider a 3-D vector field y ̇ = f , t he curlf , or ∇× f at a point O is explained in Fig. 5. Here  is a unit vector, the projection of the curlf onto  is defined as the limit of a closed line integral along the curve C in a plane orthogonal to  , that is Fig. 4 Zero-energy flow surface and an unstable point r determined by ΔE 1 3 ̃𝜑 686 L. Qin et al. � � � � T+t ̇ ̇ E = lim Edt = 0, T→∞ � � T+̂t ⟨E⟩ = lim Edt = E, T→∞ � � � � T+̂t ̇ ̇ K = lim Kdt = 0, T→∞ � � T+t ⟨K⟩ = lim Kdt = K. ̂t T→∞ (35) Fig. 5 Circulation integration along a path C, of which the positive This implies that the time averaged mechanical energy direction obeys the right-hand rule, to define the cur1f ⟨E + K⟩ tends a constant when the average time tends to infinite. Also, for a chaotic motion, flows are restricted in a finite volume, so that the space averaged rate of volume (∇× f) = (curlf) = lim f ⋅ dy 𝜈 𝜈 strain of phase space must not be positive, i.e. A→0 1 2 = lim ẏ ⋅ ẏ dt = lim Kdt , 1 1 A→0  A→0  = dV = 𝜆 dV ≤ 0. A A (36) V I C C � � V V V V I=1 (34c) where A is the area enclosed by curve C. The curlf can be denoted in a tensor form [30, 31] 3 Investigations of nonlinear (∇× f) = e f , i ijk k,j (34d) water‑lubricated bearing system where e is the permutation tensor. The vector curlf is a ijk ̂ ̂ 3.1 Zero‑energy flow surface (f = 0 = f ) dual vector of a skew-symmetrical matrix U , spin matrix, 1 2 satisfying the following relationship Using Eq.  (24b), we obtain the energy flow surface −1 −1 ̇ ̃ E = r Er = 0 of the system, i.e. (U) = U = e (∇× f) = e e f ij ij ijk k ijk krs s,r 2 2 ⎡ ⎤ 00 0 0 (34e) ⎢ ⎥ T 00 0 (1 − 𝛾 )∕2 f − f ⎢ ⎥ i,j j,i T −1 J − J 21 r r ⎢ ⎥ = (  −   )f = = . ̃ ̃ ir js is jr s,r 00 −2𝜁 − b (1 + 𝜂 )b∕2 ⎢ 1 12 ⎥ 2 2 2 ij ⎢ ⎥ ̃ ̃ 0 (1 − 𝛾 )∕2 (1 + 𝜂 )b∕2 −2𝛾 𝜁 − 𝜂 b 12 21 2 12 ⎣ ⎦ 2 2 2 Therefore, a positive time-averaged kinetic energy implies ̃ ̃ ̃ = −(2𝜁 + b)z −(2𝛾 𝜁 + 𝜂 b)z +(1 + 𝜂 )bz z +(1 − 𝛾 )z y = 0, 1 21 2 12 12 1 2 2 2 1 2 21 there must be a non-zero spin matrix for periodical orbits. (37a) where the nonlinear friction parameter 2.7 Bifurcation and chaos −𝜓 (z −z ) ̃ 2 1 b =−𝛼𝜇 e , (37b) The energy flow of a nonlinear dynamical system is affected or its approximation to a second order of sliding speed by the bifurcation parameters to reveal the birfurcation char- acteristics of orbits. For example, different parameters result 2 T ̃ ̂ ̃ b ≈−𝛼𝜇 (1 − 𝜓 Iz + 𝜓 z Iz). (37c) in different equilibrium points, zero-energy flow surfaces 2 and energy flow characteristic factors etc. The zero-energy flow surface in Eq. (37a) is independ- Xing [28, 29] has discovered that a chaotic motion of ent on the variable y , therefore, the axis o − y is a zero- 1 1 nonlinear dynamical system can be considered as a periodi- energy line of the system. On this line defined by the posi- cal motion with an infinte long time period and the following T tion vector r = y 0 00 , the energy flow E = 0. For the integrations hold prescribed parameters in Eq. (41), Eq. (37) governs the 1 3 ̇𝜐 ̇𝜐 Energy flow characteristics of friction‑induced nonlinear vibrations… 687 1 3 Table 1 Energy flow characteristic factors at the equilibrium point of the system (  = 0.008325) ξ 0.1116 0.111245 0.1111 0.1110 α λ λ λ λ λ λ λ λ λ λ λ λ 2 3 4 2 3 4 2 3 4 2 3 4 1.5 − 0.507725 − 0.055150 0.506133 − 0.507713 − 0.054460 0.506141 − 0.507708 − 0.054178 0.506144 − 0.507705 − 0.053983 0.506146 2.0 − 0.512170 − 0.001144 0.512114 − 0.512154 − 0.000467 0.512131 − 0.512147 − 0.000191 0.512137 − 0.512142 0 0.512142 2.2 − 0.514088 0.019815 0.515290 − 0.514069 0.0204843 0.515312 − 0.514062 0.020758 0.515320 − 0.514057 0.020947 0.515327 2.5 − 0.517085 0.050374 0.521052 − 0.517064 0.0510321 0.521083 − 0.517055 0.051301 0.521096 − 0.517049 0.051486 0.521105 2.6 − 0.516113 0.060295 0.523 267 − 0.518091 0.0609483 (0.523302 − 0.518082 0.061215 0.523 317 − 0.518076 0.061399 0.523327 2.7 − 0.519153 0.070069 0.525642 − 0.519131 0.0707175 0.525682 − 0.519122 0.070982 0.525698 − 0.519115 0.071165 0.525709 3.0 − 0.522349 0.090427 0.533805 − 0.522324 0.0990577 0.533059 − 0.522314 0.099315 0.533 881 − 0.522307 0.099493 0.533897 3.5 − 0.527885 0.141957 0.551352 − 0.527856 0.1425522 0.551439 − 0.527845 0.142795 0.551475 − 0.527837 0.142963 0.551499 4.0 − 0.533642 0.179975 0.574634 − 0.533611 0.1805229 0.574764 − 0.533598 0.180747 0.574817 − 0.533589 0.180901 0.574854 4.5 − 0.539584 0.211801 0.604291 − 0.539550 0.2122948 0.604474 0.539536 0.212496 0.604540 − 0.539527 0.212635 0.604600 Table 2 Energy flow characteristic factors at the equilibrium point of the system (  = 0.00) ξ 0.1116 0.111245 0.1111 0.1110 α λ λ λ λ λ λ λ λ λ λ λ λ 2 3 4 2 3 4 2 3 4 2 3 4 2.0 − 0.512177 − 0.001144 0.512107 − 0.512160 − 0.000467 0.512124 − 0.512154 − 0.000191 0.512131 − 0.512149 0 4.512135 2.2 − 0.514045 0.019915 0.515283 − 0.514076 0.020484 0.515345 − 0.514468 0.020758 0.515314 − 0.514063 0.020947 0.515320 688 L. Qin et al. Table 3 Energy flow characteristic factors at the equilibrium point of the system (  = 0.009) ξ 2.0 2.12 2.5 α λ λ λ λ λ λ λ λ λ 2 3 4 2 3 4 2 3 4 0.119720 − 0.517573 − 0.010078 0.511718 − 0.513740 − 0.003913 0.513519 − 0.517589 0.035312 0.520352 forms of zero-energy surfaces of system with the different E − 𝜆I = 0. (40c) values of  and  listed in Tables 1, 2 and 3, which will be given in Sect. 3.2. The solutions of Eqs. (40b) and (40c) are three eigen- values and eigenvectors 𝜆 ,  (I = 2, 3, 4), I I ̂ ̂ 3.1.1 Linearazation at the fixed point (f = 0 = f ) 1 2 𝜆 ,  = ⃗ , I = 2, 3, 4. (40d) I I Vanishing the generalised kinetical energy in Eq. (21) and using Eq. (15), we have 00 1 0 y 3.2 Bifurcation of zero‑energy flow surfaces ⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ 00 0 1 y around the fixed point = 0, ⎢ ⎥⎢ ⎥ (38) ̃ ̃ −10 −2𝜁 − b b z 1 1 ⎢ ⎥⎢ ⎥ ̃ ̃ ⎣ ⎦⎣ ⎦ 0 −𝛾 𝜂 b −2𝛾 𝜁 − 𝜂 b z The dimensionless parameters studied in Ref. [2] are listed 12 21 2 12 2 as follows which gives an equilibrium point r = 0 . From Eqs. (10), =0.111;  = 0.1116, 0.111245, 0.1111;  = 0.008325;  = 0.01; ̃ ̃ ̃ 01 1 2 21 (19), and (27b), at this point b =−𝛼𝜇 , J = J + J , 01 0 b E = 0 , so that the energy flow matrix and the spin matrix in Eq. (27b) become −4 =7.5 × 10 ;  = 3040.2;  = 2.0, 2.5, 3.0, 3.5, 4.0; � � (41) E = , based on which the obtained energy flow characteristic fac- 0 E tors around the fixed point are given in Tables  1, 2 and 3, 00 (1 −  )∕2 ⎡ ⎤ which shows the bifurcation of zero-energy flow surface ⎢ ⎥ E = 0  − 2 − (1 +  )∕2 , 01 1 01 12 affected by the bifurcation parameters: the dimensionless ⎢ ⎥ (1 −  )∕2 − (1 +  )∕2   − 2 ⎣ ⎦ 01 12 12 01 21 2 friction coefficient  and the dimensionless damping coef- (39a) ficients  and  . 1 2 It is found that the all energy flow characteristic factors 00 1 0 ⎡ ⎤ � � (I = 2, 3, 4) , for the investigated parameters, include 2 I ⎢ 00 0 1 +  ∕2 ⎥ � 21 � U = . negative, positive and zero real numbers, so that the small ⎢ ⎥ −10 0 − 1 −  ∕2 01 12 � � � � ⎢ ⎥ local domain around the equilibrium point can be divided into ⎣ ⎦ 0 − 1 +  ∕2  1 −  ∕20 01 12 a stable (𝜆 < 0 ), an unstable (𝜆 > 0 ) and a central flow sub- I I (39b) space ( = 0 ), to reveal the local dynamic behaviour around the equilibrium point. In the subspace span by the corresponding energy flow 3.1.2 Energy flow characteristic factors and vectors characteristic vectors 𝜉 around the equilibrium point, the energy flow at point  =    can be formulated by 2 3 4 For the approximated energy flow matrix E , using Eqs. Eq. (28), i.e. (30)-(32), we can solve its energy flow characteristic fac- 2 2 2 tors and vectors. Obviously, we have a zero characteristic E = 𝜆 𝜙 + 𝜆 𝜙 + 𝜆 𝜙 , (42a) 2 3 4 2 3 4 factor and its vector in the form and the zero-energy flow surface is governed by = 0,  = 10 00 . (40a) 1 1 2 2 2 E = 𝜆 𝜙 + 𝜆 𝜙 + 𝜆 𝜙 = 0. (42b) 2 3 4 2 3 4 The rest three energy flow characteristic factors and vectors Based on the results listed in Tables 1, 2 and 3, in a lin- are obtained by solving the eigenvalue problem earised approximation of system around the equilibrium point, there are following three local structures of zero- (E − 𝜆I )⃗r = 0, ⃗r = y z z , (40b) 2 1 2 energy flow surfaces with its local dynamic behaviour. of which the characteristic equation is 1 3 Energy flow characteristics of friction‑induced nonlinear vibrations… 689 3.2.1 Case (a): two negative and one positive characteristic Eq. (42b) becomes factor (˛ = 2.0 ,  = 0.1116,  = 0.009) 1 2 2 2 2 0.512177 + 0.001144 = 0.512107 , (43a) 2 3 4 In this case, such as for parameters  = 2.0 ,  = 0.1116 which is rewritten in the form and  = 0.009 , the equation of zero-energy flow surface in 2 2 2 2 2 𝜑 ̃ 𝜑 ̃ 𝜑 ̃ 𝜑 ̃ 𝜑 ̃ 2 3 4 2 3 + = , + = 𝜑 ̃ , 2 2 2 2 2 a a a R R 2 3 4 2 3 1 1 1 1 1 1 a = = , a = = , a = , � √ � √ √ √ 2 3 4 (43b) � � 0.512177 � � 0.001144 0.512107 𝜆 𝜆 �2� �3� √ √ 𝜆 𝜆 a a 4 3 4 R = = , R = = , � � 2 3 a a 4 4 � � � � 𝜆 𝜆 �2� �3� Fig. 6 Zero-energy flow surface around equilibrium point for case (a) ( = 2.0,  = 0.1116,  = 0.009) based on linearized Eq.  (43a): 1 2 a sketched one, b calculated one based on Eq. (43a), and c one based on nonlinear Eq. (37) 1 3 ̃𝜑 ̃𝜑 ̃𝜑 690 L. Qin et al. of which the zero-energy flow surface and energy flow vari- 0.0012 0.2220 c = , (43c) ations in the small domain around the equilibrium point is 0.2220 0.0180 sketched in Fig. 6a, and the computer simulating surface is with the positive diagonal damping coefficients in the sys- drawn in Fig. 6b, while the zero-energy surface based on Eq. (37) for nonlinear case is given in Fig. 6c. tem, so that the motions caused by initial conditions will be ⃗ ⃗ ⃗ damped. In Fig. 6a, the three characteristic vectors 𝜉 , 𝜉 and 𝜉 2 3 4 respectively with the three factors span a subspace in phase By using the MATLAB program in Ref. [28] and the nonlinear friction parameter in Eq.  (37b), the results space o − y z z . This local linearized zero-energy flow 2 1 2 ⃗ ⃗ surface is symmetrical to the coordinate plane o − 𝜉 𝜉 and of numerical simulations with the initial conditions 2 3 −6 y = 10 11 11 for this case are given in Figs. 7, 8, 9. axis-symmetrical to the axis o − 𝜉 , so that a plane perpen- dicular to axis o − 𝜉 intersects the zero-energy flow surface Figure 7 gives the time histories of displacement and veloc- 𝜑 ̃ R > 𝜑 ̃ R ity of the bearing and the shaft, which shows the damped in an elliptical curve of semi-axis , as shown 4 3  4 2 ⃗ ⃗ in Fig. 6 for the ones at 𝜑 ̃ = 1 . The plane o − 𝜉 𝜉 inter- oscillations. Figure 8 presents the phase diagram y(1)–y(3) 4 3 4 of bearing unit and the one y(2)–y(4) of shaft unit, as well sects the zero-energy flow surface in the straight lines of 𝜑 ̃ =±𝜑 ̃ R . Based on Eq. (42a), at a disturbed point outside as the time histories of generalized potential energy and the 3 3 distance of phase point to the origin, which indicates the the zero-energy flow surface on the plane 𝜑 ̃ =const., the energy flow E < 0 , so that it intends to reduce the distance orbit points in both phase diagrams move from outside to the origin, and the generalized potential energy and the distance to the origin of space, implying the flow towards the zero- energy surface. For a disturbed point inside the zero-energy of orbit point to the origin tend to zero with time increasing. Figure 9 shows the instant energy flow behaving a damped flow surface on the plane 𝜑 ̃ =const., the energy flow E > 0 and it intends to increase the distance to the origin of space oscillation and the time history of phase space volume strain decreasing with time. implying the flow also towards the zero-energy surface. Therefore, the zero-energy surface is an attracting surface. 3.2.2 Case (b): one negative and two positive characteristic In this case, from Eq. (10a), the damping matrix of system at the fixed point takes factors (˛ = 2.2 ,  = 0.111245 ,  = 0.008325) 1 2 Case 2 corresponds a zero-energy flow surface in Eq. (42b) governed by equation 2 2 2 0.514069 = 0.0204843 + 0.515312 , (44a) 2 3 4 Fig. 7 Time histories of displacement and velocity of bearing and shaft ( = 2.0,  = 0.1116,  = 0.009) 1 2 1 3 ̃𝜑 ̃𝜑 ̃𝜑 Energy flow characteristics of friction‑induced nonlinear vibrations… 691 Fig. 8 Phase diagram y(1)–y(3) of bearing and the one y(2)–y(4) of shaft, and time histories of generalized potential energy and distance of phase point to origin ( = 2.0,  = 0.1116,  = 0.009) 1 2 -11 -3 Instant Total Energy Flow Phase space volume strain 10 10 -2 -2 -4 -4 -6 -6 00.5 1 1.5 2 2.53 3.5 4 4.5 5 00.5 1 1.5 2 2.53 3.5 4 4.5 5 -4 -4 t t 10 Fig. 9 Time history of instant energy flow, and phase space volume strain ( = 2.0,  = 0.1116,  = 0.009) 1 2 which is rewritten in the form 2 2 2 2 2 𝜑 ̃ 𝜑 ̃ 𝜑 ̃ 𝜑 ̃ 𝜑 ̃ 2 3 4 3 4 = + , 𝜑 ̃ = + , 2 2 2 2 2 2 2 a a a a ∕a a ∕a 2 3 4 3 2 4 2 1 1 1 1 1 1 a = = , a = = , a = = , � √ √ √ √ √ 2 3 4 (44b) 𝜆 𝜆 � � 0.514069 0.020483 0.515312 3 4 � � � � � � � � 𝜆 𝜆 2 2 a � � a � � R = = , R = = . √ √ 4 3 a a 2 𝜆 2 𝜆 4 3 1 3 dEe/dt Vs 692 L. Qin et al. The sketched zero-energy flow surface and energy flow variations around the equilibrium point is shown in Fig.  10a, while the computer drawn surface based on Eq. (44a) is given in Fig.  10b. The one for the non- linear equation is like to the one shown in Fig.  6c, so neglected. The local linearised surface is symmetrical ⃗ ⃗ about the coordinate plane o − 𝜉 𝜉 and axis-symmetrical 3 4 to the axis o − 𝜉 , so that a plane perpendicular to axis o − 𝜉 intersects the surface in an elliptical curve of semi- 𝜑 ̃ R > 𝜑 ̃ R axis , as shown in Fig. 10a for the ones at 2 3  2 4 ⃗ ⃗ 𝜑 ̃ = 1 . The plane o − 𝜉 𝜉 intersects the surface in the 2 2 3 two straight lines of =±𝜑 ̃ R . At a disturbed point out- 3 2 3 side the zero-energy flow surface on the plane perpendicu- ⃗ ̇ lar to the axis o − 𝜉 , the energy flow E > 0 by Eq. (42a), so that this disturbance intends to increase the distance to the origin of space, implying the flow backwards the zero-energy surface. For a disturbed point inside the zero- energy flow surface, the energy flow E < 0 and the distur- bance intends to reduce the distance to the origin of space implying the flow also backwards the zero-energy surface. Therefore, this zero-energy surface is a divergence surface. In fact, for this case, the damping matrix of the system in Eq. (10a) becomes −0.0217 0.2442 c = , (44c) 0.2442 −0.0222 that shows the negative diagonal damping coefficients for both bearing and shaft, so that the motions caused by initial conditions will be divergence. The results of numerical simulations with the initial Fig. 10 Zero-energy flow surface around equilibrium point for case −6 conditions y = 10 11 11 for this case are given in (b) ( = 2.2  = 0.111245,  = 0.008325) based on Eq.  (44a): 1 2 a sketched one and b calculated one Figs. 11–13. Figure 11 gives the time histories of displace- ment and velocity of the bearing and shaft, which shows the Fig. 11 Time histories of displacement and velocity of bearing and shaft ( = 2.2  = 0.111245,  = 0.008325) 1 2 1 3 ̃𝜑 Energy flow characteristics of friction‑induced nonlinear vibrations… 693 Fig. 12 Phase diagram y(1)–y(3) of bearing and the one y(2)–y(4) of shaft, and time histories of generalized potential energy and distance of phase point to origin ( = 2.2  = 0.111245,  = 0.008325) 1 2 ab Fig. 13 Time histories of a instant energy flow and b phase space volume strain ( = 2.2  = 0.111245,  = 0.008325) 1 2 Fig. 14 Zero energy flow surface Eq.  (45a) around equilibrium point for case 3: a sketched one, b calculated one ( = 2.0  = 0.111,  = 0.008325) 1 2 1 3 694 L. Qin et al. divergence oscillations. Figure 12 presents the phase dia- discussed below. In this case, the central energy flow inves- gram y(1)–y(3) of bearing and the one y(2)–y(4) of shaft, tigation by considering higher-order terms is needed [28]. as well as the time histories of generalized potential energy The zero energy flow serface around the origin becomes and the distance of phase point to the origin, which indicates 2 2 0.512142𝜑 ̃ = 0.512142 , 𝜑 ̃ =±𝜑 ̃ , (45a) the orbit points in both phase diagrams move far away to 2 4 2 4 the origin, and the generalized potential energy as well as corresponding two orthogonal planes perpendicular to the distance of orbit point to the origin increase. Figure 13 ⃗ ⃗ the coordinate place o − 𝜉 𝜉 with the intersect lines and 2 4 shows the instant energy flow and the time history of phase =−𝜑 ̃ , respectively, as shown in Fig. 14. In the domain 2 4 space volume strain, of which both positively increase very ⃗ ̇ containing axis o − 𝜉 , the energy flow E < 0 , so that the fast at about non-dimension time 350. orbit disturbance along o − 𝜉 axis will be towards the zero-energy flow surface, while in the domain containing 3.2.3 Case (c): one zero and two opposite factors ⃗ ̇ ⃗ o − 𝜉 axis, E > 0 , the disturbance along o − 𝜉 axis will be 4 4 (˛ = 2.0,  = 0.111,  = 0.008325) 1 2 backwards the zero-energy surface. Therefore, the dynamic behaviour of the system cannot be predicted only based on In this case, since there exsists a zero-energy flow charac- linear approximation. The theoretical analysis relies on cen- teristic factor, the dynamic behavour of the system may not tral energy flow theorem, which is more complex, so that be determined only based on linearised approximation as Fig. 15 Time histories of displacement and velocity of bearing and shaft ( = 2.0  = 0.111,  = 0.008325) 1 2 -9 10 Generalised Energy Potential 0 12 3 4 5 6 7 89 10 -4 10 Distance to Origin 0.5 0 12 3 4 5 6 7 89 10 t 10 Fig. 16 Phase diagram y(1)–y(3) of bearing and the one y(2)–y(4) of shaft, and time histories of generalized potential energy and distance of phase point to origin ( = 2.0  = 0.111,  = 0.008325) 1 2 1 3 DD GEP ̃𝜑 ̃𝜑 Energy flow characteristics of friction‑induced nonlinear vibrations… 695 ab Fig. 17 Time histories of a instant energy flow and b phase space volume strain ( = 2.0  = 0.111,  = 0.008325) 1 2 Fig. 18 Enlarged time histories of displacement and velocity in modulated styles for bearing and shaft in initial time-period before breaking time shown in Fig. 15 ( = 2.0  = 0.111,  = 0.008325) 1 2 here we will reveal the solution by numerically simulat- displacement and velocity of bearing increase, while the ing the case with above parameters. Also, from linearised ones of shaft show oscillations, and both are in modu- approximation, the summation of diagonal elements of the lated style as indicated by enlarged picture in Fig. 18. The energy flow matrix E vanish, so that its time change rate of generalised potential energy and the orbit point distance phase volume strain approximately vanishes, i.e. to the origin of phase space are modulated oscillations with no obviously-change, and the phase diagram of shaft tends a periodical orbit as shown in Fig. 16. The energy ≈ 𝜆 = 0, (45b) flow shown in Fig.  17a has small values with no obvi- I=1 ously change. When the simulation time reaches about implying the volume of phase diagram should not change, 9.4  ×  10 s, the displacement and velocity of bearing which might not be true for the nonlinear equation with no are extreme large so that the simulation is stopped. Fig- linearazation. ure  17b shows the time history of phase space volume Now, we show the numerical simulation results to explain strain, indicating it increases with time and takes extreme above discussion. The damping matrix at fixed point takes large value at after the stop time. To explain this phenom- value enon, we check the friction damping matrix in Eq. (10a), −𝜓 Iz which is propotional the factor b =−𝛼𝜇 e . Refer r ing 0 0.2220 c = , to the Fig.  15, since the amplitude of bearing velocity (45c) 0.2220 0 z = y(3) increases while the amplitude of shaft velocity z = y(4) is nearly unchanged, so that the power of fric- of which the diagonal damping coefficients of system vanish. 2 −Iz −(z −z ) 2 1 tion function e = e will takes increased positive Figure  15 shows the time histories of displacement value, because of this, nonlinear friction force extremely and velocity for both bearing and shaft units, from which increased, which causes a self-excited oscillation of the it has been found that before the time 9.4  ×  10 s, the nonlinear bearing-shaft friction system. 1 3 ̇𝜐 696 L. Qin et al. Fig. 19 Time histories of displacement and velocity of bearing and shaft ( = 2.0  = 0.111,  = 0.008325) 1 2 Fig. 20 Phase diagrams y(1)–y(3) of bearing and y(2)–y(4) of shaft, and time histories of generalized potential energy and distance of phase point to origin ( = 2.0  = 0.111,  = 0.008325) 1 2 -11 Instant Total Energy Flow -3 Phase Space Volume Strain 10 10 ab 5 4 0 0 -2 -5 -4 00.2 0.40.6 0.8 11.2 1.4 1.61.8 2 00.2 0.40.6 0.8 11.2 1.4 1.61.8 2 t 10 t Fig. 21 Time histories of a instant energy flow and b phase space volume strain ( = 2.0  = 0.111,  = 0.008325) 1 2 1 3 dEe/dt Vs Energy flow characteristics of friction‑induced nonlinear vibrations… 697 Fig. 22 Time histories of time-averaged energy flow ( = 2.0  = 0.111,  = 0.008325) 1 2 From the discussion on the zero-energy flow surfaces of may conclude the stablity of system about the equlilibrium three cases and the corresponding numerical simulations point y = 0 as follows: based on the nonlinear friction function in Eq. (37b), but not the approximated one in Eq. (37c) used in Ref. [2], we Fig. 23 Dynamic response time histories of displacements and velocities of bearing and shaft of the system subject an external force: I) on bear- T T ̂ ̂ ̂ ̂ ing f = 00 f 0 and II) on shaftf = 000 f 1 3 698 L. Qin et al. Fig. 24 Phase diagrams y(1)–y(3) of bearing and y(2)–y(4) of shaft, and time histories of generalized potential energy and distance of phase T T ̂ ̂ ̂ ̂ point to origin: I) force on bearing f = 00 f 0 and II) force on shaft f = 000 f a b I) II) ̂ ̂ Fig. 25 Time histories of a instant energy flow and b phase space volume strain in forced vibration: I) force on bearing f = 00 f 0 and II) ̂ ̂ force on shaft f = 000 f 1 3 Energy flow characteristics of friction‑induced nonlinear vibrations… 699 Fig. 26 Curves of time-averaged energy flow vs average time for forced vibration: I) force on bearing f = and II) force on shaft 00 f 0 f = 000 f • Case (a) when the energy flow matrix at the equilib- conclusion is correct. For example, the three cases rium point has two negative and one positive char- w i t h  = 2.0,  = 0.1116, 0.1111, 0.111245, stud - acteristic factor, the local zero-energy flow surface ied based on the friction force approximation in in the small domain around the equilibrium point Eq. (37c) in Ref. [2] were reported the stable case for behaves an attractive surface, and the system is sta-  = 0.1116 but unstable cases for the last two values of ble to initial disturbance. We also simulated some  = 0.1111, 0.111245 . We re-check these three cases cases reported in Ref. [2], which further confirm this by using the friction force Eq. (37b), and the simulation Fig. 27 Dynamic response time histories of displacements and velocities of bearing and shaft in the system subject an external force: I) on bear- T T ̂ ̂ ̂ ̂ ing f = 000 f and II) on shaft f = 000 f 1 3 700 L. Qin et al. Fig. 28 Time histories of generalized potential energy, distance of phase point to origin and phase space volume strain in forced vibration: I) T T ̂ ̂ ̂ ̂ force on bearing f = 000 f and II) force on shaft f = 000 f results confirm that for each case, the energy flow matrix simulation result shows a divergence self-excited oscil- has two negative and one positive characteristic factor lation, which might be miss-judged as a stable oscillation given in Tables 1, 2 and 3, respectively, and the equilib- if the simulation time is not enough large. For example, rium point is stable. Fig. 18, cut out from Fig. 15 ( = 2.0,  = 0.111 ), shows Case (b) when the energy flow matrix at the equilibrium the enlarged time histories of displacement and velocity point has one negative and two positive characteristic of both bearing and shaft in the initial time before the factors, the local zero-energy flow surface in the small simulation breaking, which seems a stable picture but domain around the equilibrium point behaves a diver- unstable indicated by Fig. 15. gence surface, and the system is unstable. Case (c) when the energy flow matrix at the equilibrium 3.3 Periodical oscillation (f = 0) point has one zero factor and two same absolute value factors, one negative and another positive, the local zero- From Sect. 2.6, we know that for a possible periodical orbit, energy flow surface in the small domain around the equi- a necessary condition is that the spin matrix U of the system librium point behaves divergence in a sub-domain and must not vanish. From Eqs. (15), (19) and (27b), we obtain attractive in another sub-domain as shown in Fig. 14. The the spin matrix at the equilibrium point dynamical behaviour about this case requires to undergo a higher order approximation analysis. The numerical 1 3 Energy flow characteristics of friction‑induced nonlinear vibrations… 701 Fig. 29 Dynamic response time histories of displacements and velocities of bearing and shaft in the system subject an external force: I) on bear- T T ̂ ̂ ̂ ̂ ing f = and II) on shaft f = 000 f 000 f of phase point to the origin behave modulated oscillation 𝟎𝐈 − 𝐔 = ≠ 𝟎 , (46) without obviously amplitude variations, as shown in Fig. 20. −𝐈 𝟎 The instant energy flow curve behaviours a stable oscilla- implying possible periodical orbits of the sys - tion in Fig.  21a, and its time averaged one tends to zero tem. We simulate a case with parameters with average time increasing as shown in Fig. 22. The phase ( = 2.0  = 0.111,  = 0.008325) , of which there are two space volume train descreases with time shown in Fig. 21b. 1 2 negative and one positive energy flow characteristic factors According to the energy flow theory discussed in Sect.  2.6, and a damping matrix at equilibrium point this is a stable periodical motion of the system excited by initial conditions. 0.0012 0.2220 We have simulated the cases with param - c = . (47) 0.2220 0 e ter s ( = 2.0  = 0.1111,  = 0.008325) and 1 2 ( = 2.12  = 0.119725,  = 0.0009) [2], of which the 1 2 Figure 19 shows that the displacement and velocity of energy o fl w matrix has two negative and one positive energy bearing decrease with time, while the shaft ones are in a flow characteristic factors and both systems also behave the periodical motion. The phase diagram of bearing tends similar stable periodical oscillations as the one presented to the origin while the one of shaft shows a closed orbit, herein. and the generalised potential energy as well as the distance 1 3 702 L. Qin et al. Fig. 30 Enlarged response time histories, before simulation breaking, of displacements and velocities of the system subject an external force: I) T T ̂ ̂ ̂ ̂ on bearing f = 000 f and II) on shaft f = 000 f ̃ the phase diagrams tend periodical orbits in Fig. 24, as well 3.4 Forced oscillations (f ≠ 0) as the instant energy flows and the phase space volume −6 strains in Fig.  25 also tend stable curves. Figure  26 con- We respectively add an excitation force of amplitude 10 , firms that the time-averaged energy flows tend zero with −6 frequency 0.5 and phase angle 0, i.e. f = 10 cos(0.5𝜏 ), on average time increasing. These typical results demonstrate the bearing mass and the shaft mass to investigate the forced that the forced vibrations are periodical motions of the sys- dynamic response of the system. The simulated results are tem although the different curve styles due to different force given as follows. added potions. Since the positive damping of the system, the free vibration components due to initial conditions are 3.4.1 Case (a): parameters gradually damped, and a stable forced vibration is obtained. ˛ = 2.0  = 0.1116,  = 0.009 1 2 However, since the system is nonlinear, the frequency of the forced vibration does not equal the force frequency, so that Figures 23–26 give the simulation results, in which, with the response curves behave modulated pictures compositing time increasing, the time histories of displacements and of two frequencies response. velocities shown in Fig.  23 tend stable forced modulated oscillations, the generalised potential energies and the dis- tances of phase points to the origin tend stable pictures and 1 3 Energy flow characteristics of friction‑induced nonlinear vibrations… 703 3.4.2 Case (b): parameters discovered: (a) two negative and one positive factors, con- ˛ = 2.2,  = 0.111245,  = 0.008325 structing an attractive local zero-energy flow surface, in 1 2 which free vibrations show damped modulated oscillations In this case with main negative damping coefficients, allowing the system returning to its equlibrium state, while the system behaves divergence oscillations, of which the forced vibrations show stable oscillations; (b) one negative dynamic response time histories of displacements and and two positive factors, spanning a divergence local zero- velocities in Fig.  27, the generalised potential energies energy flow surface, so that the both free / forced vibrations and the distances of phase point to origin as well as the are in divergence resulting an unstable system; (c) one zero phase space volume strains in Fig. 28 increase very fast and two opposite factors, constructing a local zero-energy with time going. flow surface dividing the phase space into stable, unstable and central subspace, and the simulation shows friction self- 3.4.3 Case (c): parameters induced unstable vibrations in both free and forced cases. ˛ = 2.0,  = 0.111,  = 0.008325 For a set of friction parameters, the system behaves a peri- 1 2 odical oscillation, in which the bearing motion tends zero The forced vibration in this case is like the free vibration and the shaft motion tends a stable limit circle in phase space discussed for case 3 in Sect. 3.3, as shown in Fig. 29, the with the instant energy flow tends a constant and its time dynamic responses of displacements and velocities of both averaged one tends zero. Numerical simulations have not bearing and shaft increase until about time 3.3 × 10 when found any possible chaotic motions of the system. It is dis- the responses sharply increased and simulations stopped. covered that the damping matrices of cases (a), (b) and (c) To show the response curve details, Fig. 30 provides local respectively have positive, negative and zero diagonal ele- enlarged curves before breaking time in Fig.  29, from ments, dominating the dynamic behavour in different cases which it is observed the dynamic responses behave the of the system, which provides an approach to design the modulated vibrations before the simulation breaking. The water-lubricated bearing unit with expecting performance in curves of generalised potential energy, the phase diagram, marine engineering applications by choosing suitable fric- etc. also show the characteristics, so they are neglected tion parameters to generate the expected damping matrix. herein. Acknowledgements We gratefully acknowledge NSFC (51509194) and CSC for providing finacial support eanabling Li Qin and Hongling Qin 3.5 Chaotic motions to visit the University of Southampton to engage the related research. With different friction parameters, we have tested several Open Access This article is licensed under a Creative Commons Attri- cases for free vibrations induced by given initial distur- bution 4.0 International License, which permits use, sharing, adapta- tion, distribution and reproduction in any medium or format, as long bances and for forced vibrations excited by external forces as you give appropriate credit to the original author(s) and the source, acted on the bearing and shaft masses in order to see if cha- provide a link to the Creative Commons licence, and indicate if changes otic motions may be found. The simulations results have not were made. The images or other third party material in this article are shown any chaotic motions of this system. included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will 4 Conclusion need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativ ecommons .or g/licenses/b y/4.0/. Following the detailed describption of this investigation given above, we may conclude the main contributions as follows. The paper is a first one to study a nonlinear water- References lubricated bearing-shaft coupled system using the energy flow approach, which practices and further confirmed an 1. 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Journal

"Acta Mechanica Sinica" – Springer Journals

Published: Mar 4, 2021

Keywords: Nonlinear friction-induced vibrations; Nonlinear energy flows; Nonlinear water-lubricated bearing-shaft systems; Bifucation friction parameters; Energy flow matrices; Periodical oscilation

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