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Bifurcation study on fractional non-smooth oscillator containing clearance constraints:

Bifurcation study on fractional non-smooth oscillator containing clearance constraints: Bifurcation characteristics of a fractional non-smooth oscillator containing clearance constraints under sinusoidal exci- tation are investigated. First, the bifurcation response equation of the fractional non-smooth system is obtained via the K–B method. Second, the stability of the bifurcation response equation is analyzed, and parametric conditions for stability are acquired. The bifurcation characteristics of the fractional non-smooth system are then studied using sin- gularity theory, and the transition set and bifurcation diagram under six different constrained parameters are acquired. Finally, the analysis of the influence of fractional terms on the dynamic characteristics of the system is emphasized through numerical simulation. Local bifurcation diagrams of the system under different fractional coefficients and orders verify that the system will present various motions, such as periodic motion, multiple periodic motion, and chaos, with the change in fractional coefficient and order. This manifestation indicates that fractional parameters have a direct effect on the motion form of this non-smooth system. Thus, these results provide a theoretical reference for investigating and repressing oscillation problems of similar systems. Keywords Bifurcation, fractional non-smooth oscillator, clearance constraint, singularity theory Introduction It has been more than 300 years since the concept of fractional calculus was first proposed. In recent years, the fractional derivatives have been widely concerned. He defined a new fractal derive and two examples are used to illustrate the use of fractal derivatives to establish governing equations and how to solve these equations. He introduced the fractal-Cantorian spacetime and fractional calculus, then He studied further fractal calculus and its geometrical explanation. What’s more Wang and Wang considered the fractional heat transfer equations and obtained the approximate analytical solutions. Fan et al. applied this theory to wool fiber, indicating that wool fiber has an excellent warm retention property. Wang et al. used Adomian’s decomposition method and varia- tional iteration method to study the fractional KdV–Burgers–Kuramo equation. Wang et al. established a fractal modification of the telegraph equation with fractal derivatives and the approximate analytical solution is obtained by two-scale transform method and He’s homotopy perturbation method. He analyzed a generalized KdV– Burgers equation with fractal derivatives and provides conservation laws in an energy form and possible solution structures. He and Ain clarified the properties of fractal calculus and revealed the relationship between the fractal calculus and traditional calculus using the two-scale transform. State Key Laboratory of Structural Mechanical Behavior and System Safety for Transportation Engineering Co-Constructed by Province and Ministry of Shijiazhuang Tiedao University, Shijiazhuang, China Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang, China Corresponding author: Yongjun Shen, Shijiazhuang Tiedao University, No. 17, Beierhuan Dong Road, Shijiazhuang 050043, Hebei Province, China. Email: shenyongjun@126.com Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https:// creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 2 1118 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Analytical method is an important means to study nonlinear systems, many methods such as the Hamiltonian approach method, He’s frequency formulation method, K–B method, multi-scale method, harmonic balance method, variational iteration method, and homotopy analysis method are adopted. Forsat studied the nonlinear control equation based on beam theory using Hamiltonian method, and the nonlinear vibration frequencies are obtained. He’s frequency method comes from ancient Chinese mathematics, by using He’s amplitude–frequency formula. Zhang obtained the analytical solution of nonlinear oscillator with discontinuity by using He’s ampli- tude–frequency formula, which has a high accuracy. Some methods are also extended to solve the fractional-order systems. Niu et al. studied the approximate analytical solution of a fractional-order single degree-of-freedom oscillator with clearance by using the KBM asymptotic method. Xu et al. proposed a new method to solve the second-order approximate solution of fractional Duffing oscillator by combining L–P method and multi-scale method. Xiao et al. established the approximate expression of the analytical solution of fractional Vander Pol oscillator by harmonic balance method. At the same time, in 1998, the variational iteration method was an effective tool for factional calculus. In 2007, Momani and Odibat adopted the homotopy perturbation method for fractional differential equations with great success. Most engineering materials are not ideal solids or liquids but viscous–elastic. Therefore, the model, which is established by only considering elastic properties or damping characteristics, fails to reflect the material essence completely. Nearly, all physical systems can be expressed by a fractional model, and related studies also show that the model established using fractional calculus is accurate and can effectively reflect constitutive relations. 16–19 20 Therefore, fractional calculus has been extensively applied in different engineering fields. Sun et al. intro- duced fractional derivative to establish a model for oil and gas suspension. They proved that the fractional model could more accurately describe the characteristics of the oil and gas suspension than the integer-order model through numerical and experimental verifications. Lewandowski and Lasecka-Plura described a viscoelastic damper through a fractional model, conducted a theoretical analysis of design sensitivity of this damper, and verified their results through illustration. Li et al. introduced an FKV-based constitutive fractional vibration attenuation model for viscoelastic suspension and found that this viscoelastic suspension response was of global correlation and memorability. Wu and Sangguan proved that the model containing fractional derivative could effectively predict the dynamic characteristics of rubber vibration isolators. They also described that the corre- lation between dynamic characteristics and frequency of the rubber vibration isolator through the fractional derivative was reasonable and accurate. Non-smooth motion is extensively used in many oscillatory systems. However, dynamic behaviors and mech- anisms of some important non-smooth systems remain unclear. Particularly, dynamic behaviors, such as bifur- cation and chaos, which aggravate machines, have imposed potential safety hazards on machine operation. Therefore, conducting an in-depth analysis of non-smooth systems is necessary. Scholars have already investi- gated non-smooth systems. Zhong and Chen used singularity theory to study the bifurcation of resonant solutions of clearance-containing suspension system models. Wei et al. analyzed the transformation of single- degree-of-freedom piecewise smooth systems from n 1 periodic motion into chaotic behaviors through period- doubling bifurcation and then investigated the control of system chaotic motions. Zhang et al. established a generalized piecewise linear model of Chua’s circuit and studied codimension-1 unconventional fold bifurcation generated during the passage of the system trajectory through the interface. They also analyzed the resulting bursting oscillation phenomenon. Zhang et al. used Melnikov theory for the oscillation model of clearance- containing gear systems to investigate global bifurcation conditions for the heteroclinic orbit and the stability of its periodic motion. Li and Zhao obtained Melnikov function for the subharmonic orbit of piecewise smooth systems to investigate the existence of subharmonic orbits. Huang and Xu acquired frequency response and stability conditions for a controlled piecewise smooth system with negative stiffness and explored symmetrical breaking bifurcation and chaotic motion in this system. Hou et al. established a roller model for rolling mills containing nonlinear constraints and analyzed bifurcation characteristics of the system under autonomous and non-autonomous conditions via stability systems of singular points and singularity theory. Studies regarding fractional non-smooth systems containing clearance constraints are currently limited. Dynamic characteristics are complex due to the strong nonlinearity and singularity of such systems, and most 31,32 investigations have adopted numerical methods. Numerical and analytical methods are the main research methods for fractional systems, wherein the former generally only provide solutions under specific parameters, while the latter can acquire definite relational expressions between system characteristics and parameters to realize quantitative analysis. Hence, analytical and numerical methods are combined in this study to investigate frac- tional non-smooth oscillators containing clearance constraints. This paper is organized as follows: first is the Introduction part; the next section introduces the physical model of the system and the motion equations; then the Wang et al. Wang et al. 1119 3 next section obtains the bifurcation response equation of the system and its stability through the analytical method and examines the bifurcations of the system under different parameters using singularity theory; following section presents the analysis of chaotic motions of the system under fractional coefficient and order via the numerical method; and the last section draws the main conclusions. System model and motion equation The fractional non-smooth system model containing clearance constraints is shown in Figure 1. This model comprises a mass, a damper, a linear spring, a nonlinear spring, and the nonlinear constraints with clearance symmetrically, where m, c, k , k , and k are the system mass, linear damping coefficient, linear stiffness coeffi- 1 2 3 cient, nonlinear stiffness coefficient, and nonlinear constraint stiffness coefficient, respectively. ðÞ ðÞ KD ½xt �B cos xt is the p-order derivative of x(t) to t with the fractional coefficient K. The non-smooth system is subject to a force excitation. The nonlinearity k and fractional derivative may be caused by the physical properties of the material; the nonlinear spring k may be caused by the restraint. When the displacement of m is less than clearance D, then only spring k , k and damping c are involved, if the displacement of m exceeds clearance D, the nonlinear spring k is 1 2 3 engaged. The differential equation of the system motion is presented as follows 3 p € _ ðÞ mx þ k x þ k x þ cx þ fx ðÞþKD ½xðtÞ� ¼ B cos xt (1) 1 2 0 where > ðÞ k x þ D x < � D fx ðÞ ¼ 0 � D � x � D ðÞ k x � D x > D The following parameters are inputted into equation (1) rffiffiffiffiffi k k k c c 1 2 3 1 2 x ¼ ; ek ¼ ; ek ¼ ; 2el ¼ ; 2el ¼ ; 0 1 2 1 2 m m m m m c B K 3 0 2el ¼ ; eB ¼ ; eK ¼ m m m Equation (1) can be transformed into 2 3 p x € þ x x þ ek x þ 2elx_ þ egx þ eK D ½xðtÞ� ¼ eBcosðÞ xt (2) ðÞ 1 1 > ðÞ k x þ D x < � D gx ¼ (3) ðÞ 0 � D � x � D ðÞ k x � D x > D Solution of the system bifurcation response Solution of the bifurcation response equation through the K–B method Based on the K–B method, which means the excitation frequency is close to the natural one, that is, x � x,a detuning parameter r is introduced to illustrate the proximate degree 2 2 x ¼ x þ ed 0 4 1120 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 1. Fractional non-smooth system model containing clearance constraints. Then, equation (2) is transformed into the following formula �� 2 3 p € ðÞ _ x þ x x ¼ e fcos xt þ dx � k x � 2lx � gx ðÞ � K D ½x� (4) 1 1 Equation (4) is assumed to satisfy the following conditions x ¼ at ðÞcos/ (5) x_ ¼�at ðÞxsin/ where / ¼ xt þ hðÞ t . at ðÞ and hðÞ t are slowly varying functions of time t. Equation (5) is substituted into equation (4) to obtain the following �� 3 3 p _ ðÞ a ¼� fcos xt þ racos/ þ 2laxsin/ � k a cos / � f ðÞ acos/ � eK D ½� acos/ sin/ 1 3 1 �� (6) 3 3 p > _ : ah ¼� fcosðÞ xt þ racos/ þ 2laxsin/ � k a cos / � f ðÞ acos/ � eK D ½� acos/ cos/ 1 3 1 Integrals are taken for a_ and ah, and averaging values are taken as follows 2p > � � 3 3 ðÞ > a_ ¼� fcos xt þ racos/ þ 2laxsin/ � k a cos / � f ðÞ acos/ sin/ d/ 1 3 2px > T � lim K D ½� acos/ sin/ d/; T!1 Tx Z (7) 2p � � > e > 3 3 ah ¼� fcosðÞ xt þ racos/ þ 2laxsin/ � k a cos / � f ðÞ acos/ cos/ d/ > 1 3 > 2px > e > � lim K D ½� acos/ cos/ d/ T!1 Tx The fractional differential is defined as Caputo, and the integral of the fractional part can be obtained through 33,34 residue theorem and contour integral shown as below p�1 e aeKx sinðpp=2Þ � lim K D ½� acos/ sin/ d/ ¼� ; T!1 Tx 2 Z (8) T p�1 e aeKx cosðpp=2Þ � lim K D ½� acos/ cos/ d/ ¼ T!1 Tx 2 0 Wang et al. Wang et al. 1121 5 The solved integrals are substituted into the original parameters to obtain the following B sinh a a_ ¼� � c þ c ðÞ 2mx 2m �� (9) ax B cosh a 3a ah ¼� � þ k þ k þ k þ k 1 2 2 2mx 2mx 4 where k is the piecewise equivalent stiffness �� a k 1 k ¼ 18/ þ 12/ cos2/ � 14sin2/ � sin4/ ; 0 0 0 0 0 4p 2 / ¼ arccos c is fractional equivalent damping p�1 c ¼ Kx sinðpp=2Þ and k is fractional equivalent stiffness k ¼ Kx cosðpp=2Þ 3a 0 Let cc ¼ c þ c, kk ¼ k þ ðÞ k þ k þ k then equation (9) can be rewritten as 1 1 2 3 B sinh a a_ ¼� � cc 2mx 2m (10) ax B cosh a ah ¼� � þ kk 2 2mx 2mx _ � Let a_ ¼ ah ¼ 0. After eliminating h (phase of the steady-state response) from equation (10), the amplitude– frequency equation of the bifurcation response can be obtained as follows hi 2 2 2 2 2 ðÞ a x cc þ mx � kk � B ¼ 0 (11) where a � is amplitude of the steady-state response. Stability analysis of system response equation The system is stable when a � D. Thus, only the situation under a > D is considered. a ¼ a þ Da and h ¼ h þ Dh are substituted into equation (1) to investigate the stability of the response equa- tion, and then � � dDa � cc ac � cc ðÞ a � B cosh 2 0 ¼� þ Da � � Dh dt 2m 2m 2mx �� (12) � � � dDh Fcosh k kk ðÞ a � Fsinh ¼ þ Da � þ Dh dt 2a � mx 2mx 2ma �x 2 3 cc ac � cc xa � a � � � � k � kk 6 7 6 2m 2m 2 2mx 7 det ¼ 0 (13) 4 5 x kk k kk cc � þ þ � � k 2a � 2mxa � 2mx 2m 6 1122 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Wang et al. 7 The following characteristic equations are obtained according to equation (13) �� 0 2 0 0 ac  cc þ 2cc cc þ ccac  cc 2kk þ ak  kk 2 2 2 k þ k þ � 2m 4m 4m (14) 2 2 2 2 0 2 0 ðÞ ðÞ cc x þ kk � mx þ ac  ccx cc þ ak  kk � mx kk 2 2 þ ¼ 0 2 2 4m x where 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi c x D 3 4 5 2 3 3 cc ¼ ðDa þ 2D Þ 1 � � 3a / 2 2 a p a sffiffiffiffiffiffiffiffiffiffiffiffiffiffi �� 2 4 2 D D D (15) Dk 33 � 28 þ 8 þ 42ak / 1 � 3 3 2 4 2 a a a kk ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4p 1 � Defining Figure 2. Transition set and bifurcation diagram under constrained parameters B and B . 1 4 ac  cc þ 2cc P ¼ 2m Let a ¼ y � . Then, equation (16) is transformed into the following form 6b and 6 4 3 2 y þ B y þ B y þ B y þ B y þ l ¼ 0 (17) 4 3 2 1 2 0 0 cc þ ccac  cc 2kk þ ak  kk 2 2 Q ¼ � where 4m 4m 2 2 2 2 0 2 0 ðÞ ðÞ �� cc x þ kk � mx þ ac  ccx cc þ ak  kk � mx kk 2 2 4 2 2 3 þ b b b � 6b b b þ 27b b b � 108b b 1 5 5 4 5 6 3 5 6 2 6 2 2 4m x B ¼ þ b 324b 6 6 �� 3 2 b b 5b � 24b b b þ 72b b 2 5 5 4 5 6 3 6 the stability conditions for the bifurcation response equation are as follows: B ¼ � b 144b 6 6 P > 0 and Q > 0 b 5b � 18b b b 3 5 4 5 6 B ¼ þ b 27b 6 6 Bifurcation characteristic analysis b 5b 4 5 B ¼ � b 12b 6 6 The bifurcation response equation of the system is analyzed using singularity theory. The bifurcation response �� 3 2 2 4 3 5 b b 1296b b � 216b b b � 7776b b þ 36b b b � 5b 0 5 5 6 3 5 6 1 6 4 5 6 5 equation is transformed into the following l ¼ þ b 46656b 6 6 6 5 4 3 2 b a þ b a þ b a þ b a þ b a þ b a þ b ¼ 0 (16) 6 5 4 3 2 1 0 Singularity theory indicates that equation (17) is the universal unfolding of form y þ l ¼ 0. The durability of this universal unfolding is thus analyzed. This analysis is conducted by constraining the parameters due to where numerous constraints. �� 6 3 D k D k pp 3 3 2 2 2 p�1 b ¼ � B b ¼ 2k p � D k � 2mpx þ 2pKx cos ; (1) Constrained parameters B and B 1 4 0 0 1 1 3 2 2 p p 2 3 4 256 Bifurcation point set, B þ ðÞ B ¼ 0 3 2 4 2 1 63D k 4096B 4 3 2 2 2 2 2 2 2 2p�2 2 Hysteretic point set, ¼ B [ B ¼ 0 ðÞ 3 2 ðÞ b ¼ k þ x m x þ c � 2k m þ K x 1 � x ðÞ 1 þ cospp � 729 2 1 1 1 2 3 4 256 2 4p �� Double-limit point set, B þ ðÞ B ¼ 0 3 2 �� 1 pp D k pp 2 4 2 2 2 p�1 2 pþ1 The transition set and bifurcation diagram under constrained parameters B and B are shown in Figure 2. 1 4 þ D k k þ 6D k þ D k mx þ Kx cos k � m x � þ c Kx sin 1 3 3 3 1 1 1 p 2 p 2 �� (2) Constrained parameters B and B 2 4 �� Dk pp 2 2 p Hysteretic point set is ðÞ ðÞ b ¼ D 16k þ 3pk � 3pk þ 4Kp 3p � 8 k � mx x cos ; 3 3 2 3 1 2p 2 �� () �� �� 64 3 3 3 pp 2 2 p�1 3 ðÞ ðÞ ðÞ b ¼ D k k � k þ 65k þ 9k p þ k þ k k � mx þ K k þ k x sin ; 5B B 4 3 3 2 3 3 2 3 1 2 3 1 3 p 4 2 2 2 ¼� [ fg B ¼ 0 9 25 3 9 ðÞ b ¼ Dk ðÞ k þ k 3p � 8 ; b ¼ ðÞ k þ k 5 3 2 3 6 2 3 2 16 Wang et al. Wang et al. 1123 7 Figure 2. Transition set and bifurcation diagram under constrained parameters B and B . 1 4 Let a ¼ y � . Then, equation (16) is transformed into the following form 6b 6 4 3 2 y þ B y þ B y þ B y þ B y þ l ¼ 0 (17) 4 3 2 1 where �� 4 2 2 3 b b b � 6b b b þ 27b b b � 108b b 1 5 5 4 5 6 3 5 6 2 6 B ¼ þ b 324b 6 6 �� 3 2 b b 5b � 24b b b þ 72b b 2 5 5 4 5 6 3 6 B ¼ � b 144b 6 6 b 5b � 18b b b 3 5 4 5 6 B ¼ þ b 27b 6 6 b 5b 4 5 B ¼ � b 12b 6 6 �� 3 2 2 4 3 5 b b 1296b b � 216b b b � 7776b b þ 36b b b � 5b 0 5 5 6 3 5 6 1 6 4 5 6 5 l ¼ þ b 46656b 6 6 Singularity theory indicates that equation (17) is the universal unfolding of form y þ l ¼ 0. The durability of this universal unfolding is thus analyzed. This analysis is conducted by constraining the parameters due to numerous constraints. (1) Constrained parameters B and B 1 4 4 256 Bifurcation point set, B þ ðÞ B ¼ 0 3 2 4096B 4 Hysteretic point set, ¼ B [ B ¼ 0 3 2 4 256 Double-limit point set, B þ ðÞ B ¼ 0 3 2 The transition set and bifurcation diagram under constrained parameters B and B are shown in Figure 2. 1 4 (2) Constrained parameters B and B 2 4 Hysteretic point set is () �� 5B B 1 3 ¼� [ fg B ¼ 0 9 25 8 1124 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Wang et al. 9 Figure 5. Transition set and bifurcation diagram under constrained parameters B and B . 2 3 Figure 3. Transition set and bifurcation diagram under constrained parameters B and B . 2 4 Figure 6. Transition set and bifurcation diagram under constrained parameters B and B . 3 4 Figure 4. Transition set and bifurcation diagram under constrained parameters B and B . 1 2 (3) Constrained parameters B and B and double-limit point set is 1 2 64 3 3 Hysteretic point set, B þ ¼ 0; B < 0 4 4 729 2 5 4 Double-limit point set, U 3 5 B ¼� B 1 3 The transition set and bifurcation diagram under constrained parameters B and B are shown in Figure 4. 1 2 27 25 (4) Constrained parameters B and B 2 3 2 5 Hysteretic point set, 3125B þ 512B ¼ 0 1 4 3125 2 5 Double-limit point set, B þ 27B ¼ 0 1 4 The transition set and bifurcation diagram under constrained parameters B and B are shown in 4 2 4 The transition set and bifurcation diagram under constrained parameters B and B are shown in Figure 5. 2 3 Figure 3. Wang et al. Wang et al. 1125 9 Figure 5. Transition set and bifurcation diagram under constrained parameters B and B . 2 3 Figure 6. Transition set and bifurcation diagram under constrained parameters B and B . 3 4 (3) Constrained parameters B and B 1 2 64 3 3 Hysteretic point set, B þ ¼ 0; B < 0 4 4 729 2 Double-limit point set, U The transition set and bifurcation diagram under constrained parameters B and B are shown in Figure 4. 1 2 (4) Constrained parameters B and B 2 3 2 5 Hysteretic point set, 3125B þ 512B ¼ 0 1 4 3125 2 5 Double-limit point set, B þ 27B ¼ 0 1 4 The transition set and bifurcation diagram under constrained parameters B and B are shown in Figure 5. 2 3 10 1126 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Wang et al. 11 Figure 7. Transition set and bifurcation diagram under constrained parameters B and B . 1 3 Figure 9. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under K¼�0.7 (single-periodic motion). Figure 8. Local bifurcation diagram of the system under varying fractional coefficient K. (5) Constrained parameters B and B 3 4 625 4 5 Hysteretic point set, B þ B ¼ 0; B < 0 1 2 2 625 4 1 5 Double-limit point set, B þ B ¼ 0 1 2 256 5 The transition set and bifurcation diagram under constrained parameters B and B are shown in Figure 6. 3 4 (6) Constrained parameters B and B 1 3 fg Hysteretic point set, 3B � B ¼ 0; B < 0 [ B ¼ 0 2 4 4 2 1 2 Double-limit point set, B ¼ B 2 4 Figure 10. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under K¼�0.74 (double- The transition set and bifurcation diagram under constrained parameters B and B are shown in Figure 7. 1 3 periodic motion). Wang et al. Wang et al. 1127 11 Figure 9. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under K¼�0.7 (single-periodic motion). Figure 10. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under K¼�0.74 (double- periodic motion). 12 1128 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Wang et al. 13 Figure 12. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under K¼�1 (chaotic motion). Figure 11. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under K¼�0.752 (four- periodic motion). Numerical analysis System bifurcation and path to chaos under the influence of fractional parameter K The following system parameters are selected: m¼ 1, c¼ 0.3, F¼ 0.5, k ¼ 0.5, k ¼ 0.5, k ¼ 0.1, d¼ 0.2, and 1 2 3 p¼ 0.1. Under different values of fractional parameter K, the local bifurcation diagram of the system is shown in Figure 8. Figure 8 shows that the path for the system to chaos under varying fractional coefficient K is as follows: chaotic motion ! alternation of multiperiodic and chaotic motions ! chaotic motion ! from period-doubling motion to single-periodic state. Figures 9–12 demonstrate that the system presents different motion states under different values of fractional Figure 13. Local bifurcation diagram under varying fractional order p. coefficient K. Under K¼�0.7, system phase diagram, displacement time history diagram, and Poincare section are shown in Figure 9. This figure reveals that the phase trajectory of the system is an ellipse and the Poincare section is a dot. Meanwhile, the displacement time history diagram shows that the system is under stable single- periodic motion. Under K¼�0.74, the phase trajectory of the system is turned into two ellipses: the Poincare System bifurcation and path to chaos under the influence of fractional order p section is also turned into two dots, and the time history diagram also displays regular two-periodic motions. Therefore, the system is under a double-periodic motion state. Figure 11 shows that the phase trajectory of the The following system parameters are selected: m¼ 1, c¼ 0.3, F¼ 0.3, k ¼ 0.5, k ¼ 0.5, k ¼ 0.1, d¼ 0.2, and 1 2 3 system under K¼�0.752 becomes four ellipses. The system is also under a four-periodic motion state based on K¼�0.6. The local bifurcation diagram, which varies with fractional order p, is shown in Figure 13. the Poincare section and the time history diagram. When K¼�1, the phase trajectories are intertwined without Paroxysmal bifurcation and chaos are generated in the system with the change in fractional order p. As p overlapping, the Poincare section is disorderly and unsystematic, and the displacement time history diagram also turns from 0 to 0.0045, the system enters a periodic motion state from the chaotic motion state; when p becomes displays a disorderly state. These findings verify the chaotic motion state of the system. 0.007, the system again enters a chaotic motion state from periodic motion. Under p¼ 0.01, the system is Wang et al. Wang et al. 1129 13 Figure 12. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under K¼�1 (chaotic motion). Figure 13. Local bifurcation diagram under varying fractional order p. System bifurcation and path to chaos under the influence of fractional order p The following system parameters are selected: m¼ 1, c¼ 0.3, F¼ 0.3, k ¼ 0.5, k ¼ 0.5, k ¼ 0.1, d¼ 0.2, and 1 2 3 K¼�0.6. The local bifurcation diagram, which varies with fractional order p, is shown in Figure 13. Paroxysmal bifurcation and chaos are generated in the system with the change in fractional order p. As p turns from 0 to 0.0045, the system enters a periodic motion state from the chaotic motion state; when p becomes 0.007, the system again enters a chaotic motion state from periodic motion. Under p¼ 0.01, the system is 14 1130 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Wang et al. 15 Figure 14. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under p¼�0.004 (chaotic Figure 16. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under p¼�0.012 (five- motion). periodic motion). transformed from a chaotic motion state into a periodic motion state once again. With the change in fractional order p, this fractional non-smooth system presents alternate motion states between periodic and chaotic motions. The system demonstrates different motion states under different values of fractional order p. Figure 14 shows the phase diagram, displacement time history diagram, and Poincare section of the system under p¼ 0.004. The phase trajectory of the system comprises numerous ellipses, which are disorderly, irregular, and are not over- lapping. Thus, the system is under a chaotic motion state. The phase diagram, displacement time history diagram, and Poincare section under p¼ 0.0045 are presented in Figure 15. The amplified phase diagram shows that the phase trajectory comprises four ellipses, and the Poincare section comprises four independent dots. Meanwhile, the system is under a four-periodic motion state according to the displacement time history diagram. When p¼ 0.012 (Figure 16), the amplified phase diagram indicates the transformation of the phase trajectory into five ellipses and the Poincare section into five independent dots. Thus, the system is under a five-periodic motion state from the displacement time history diagram. Conclusions The fractional non-smooth oscillator containing clearance constraints was taken as the study object, and the nonlinear kinetic equation of the system was established. The dynamic behaviors of the system, such as bifurca- tion and chaos, were also investigated through analytical and numerical methods. The main conclusions are presented as follows. (1) The system bifurcation equation was analyzed via singularity theory, and the transition set and bifurcation diagram of the system under six different constraint parameters are obtained. On each bifurcation diagram, Figure 15. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under p¼�0.0045 (four- the transition set divides the system into multiple sub-regions, and the bifurcation curves of different sub- periodic motion). regions represent different bifurcation characteristics of the system under different parameter conditions. Wang et al. Wang et al. 1131 15 Figure 16. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under p¼�0.012 (five- periodic motion). transformed from a chaotic motion state into a periodic motion state once again. With the change in fractional order p, this fractional non-smooth system presents alternate motion states between periodic and chaotic motions. The system demonstrates different motion states under different values of fractional order p. Figure 14 shows the phase diagram, displacement time history diagram, and Poincare section of the system under p¼ 0.004. The phase trajectory of the system comprises numerous ellipses, which are disorderly, irregular, and are not over- lapping. Thus, the system is under a chaotic motion state. The phase diagram, displacement time history diagram, and Poincare section under p¼ 0.0045 are presented in Figure 15. The amplified phase diagram shows that the phase trajectory comprises four ellipses, and the Poincare section comprises four independent dots. Meanwhile, the system is under a four-periodic motion state according to the displacement time history diagram. When p¼ 0.012 (Figure 16), the amplified phase diagram indicates the transformation of the phase trajectory into five ellipses and the Poincare section into five independent dots. Thus, the system is under a five-periodic motion state from the displacement time history diagram. Conclusions The fractional non-smooth oscillator containing clearance constraints was taken as the study object, and the nonlinear kinetic equation of the system was established. The dynamic behaviors of the system, such as bifurca- tion and chaos, were also investigated through analytical and numerical methods. The main conclusions are presented as follows. (1) The system bifurcation equation was analyzed via singularity theory, and the transition set and bifurcation diagram of the system under six different constraint parameters are obtained. On each bifurcation diagram, the transition set divides the system into multiple sub-regions, and the bifurcation curves of different sub- regions represent different bifurcation characteristics of the system under different parameter conditions. 16 1132 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) The bifurcation of the system in each region is persistent. On the dividing line, the bifurcation of the system is not persistent, and the bifurcation graphs determined by parameters in each subregion are topologically equivalent. Therefore, dynamic behaviors of the system can be changed by varying its parameters to provide a theoretical reference for the reasonable selection of system parameters. (2) The numerical results show that besides the route to chaos through period-doubling with the change of fractional order p, with the change of fractional-order coefficient K, the system still exists Paroxysmal bifur- cation, in which the periodic motion and chaotic motion of the system alternately appear. At the same time, the above bifurcation and chaos phenomena are further confirmed by the displacement time history diagram, phase portrait, and Poincare section. Hence, the dynamic behaviors of the fractional non-smooth system containing clearance constraints can be changed by varying fractional coefficient and order, thus a foundation for repressing oscillations of similar systems has been established. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (nos 11802183, 11772206, 11872256 and U1934201). ORCID iD Yongjun Shen https://orcid.org/0000-0002-8768-1958 References 1. He JH. A new fractal derivation. Therm Sci 2011; 15: 145–147. 2. He JH. A tutorial review on fractal spacetime and fractional calculus. Int J Theor Phys 2014; 53: 3698–3718. 3. He JH. Fractal calculus and its geometrical explanation. 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Investigating nonlinear vibrations of higher-order hyper-elastic beams using the Hamiltonian method. Acta Mech 2020; 231: 125–138. 11. Zhang HL. Application of He’s amplitude–frequency formulation to a nonlinear oscillator with discontinuity. Comput Math Appl 2009; 58: 2197–2198. 12. Niu JC, Zhao ZS, Xing HJ, et al. Forced vibration of a fractional-order single degree-of-freedom oscillator with clearance. J Vib Shock 2020; 39: 251–256. 13. Xu Y, Li YG, Liu D, et al. Responses of Duffing oscillator with fractional damping and random phase. Nonlinear Dyn 2013; 74: 745–753. 14. Xiao M, Zheng WX and Cao J. Approximate expressions of a fractional order Van der Pol oscillator by the residue harmonic balance method. Math Comput Simul 2013; 89: 1–12. 15. Momani S and Odibat Z. Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys Lett 2007; 365: 345–350. 16. Gabano JD and Poinot T. Fractional modelling and identification of thermal systems. 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Abstract

Bifurcation characteristics of a fractional non-smooth oscillator containing clearance constraints under sinusoidal exci- tation are investigated. First, the bifurcation response equation of the fractional non-smooth system is obtained via the K–B method. Second, the stability of the bifurcation response equation is analyzed, and parametric conditions for stability are acquired. The bifurcation characteristics of the fractional non-smooth system are then studied using sin- gularity theory, and the transition set and bifurcation diagram under six different constrained parameters are acquired. Finally, the analysis of the influence of fractional terms on the dynamic characteristics of the system is emphasized through numerical simulation. Local bifurcation diagrams of the system under different fractional coefficients and orders verify that the system will present various motions, such as periodic motion, multiple periodic motion, and chaos, with the change in fractional coefficient and order. This manifestation indicates that fractional parameters have a direct effect on the motion form of this non-smooth system. Thus, these results provide a theoretical reference for investigating and repressing oscillation problems of similar systems. Keywords Bifurcation, fractional non-smooth oscillator, clearance constraint, singularity theory Introduction It has been more than 300 years since the concept of fractional calculus was first proposed. In recent years, the fractional derivatives have been widely concerned. He defined a new fractal derive and two examples are used to illustrate the use of fractal derivatives to establish governing equations and how to solve these equations. He introduced the fractal-Cantorian spacetime and fractional calculus, then He studied further fractal calculus and its geometrical explanation. What’s more Wang and Wang considered the fractional heat transfer equations and obtained the approximate analytical solutions. Fan et al. applied this theory to wool fiber, indicating that wool fiber has an excellent warm retention property. Wang et al. used Adomian’s decomposition method and varia- tional iteration method to study the fractional KdV–Burgers–Kuramo equation. Wang et al. established a fractal modification of the telegraph equation with fractal derivatives and the approximate analytical solution is obtained by two-scale transform method and He’s homotopy perturbation method. He analyzed a generalized KdV– Burgers equation with fractal derivatives and provides conservation laws in an energy form and possible solution structures. He and Ain clarified the properties of fractal calculus and revealed the relationship between the fractal calculus and traditional calculus using the two-scale transform. State Key Laboratory of Structural Mechanical Behavior and System Safety for Transportation Engineering Co-Constructed by Province and Ministry of Shijiazhuang Tiedao University, Shijiazhuang, China Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang, China Corresponding author: Yongjun Shen, Shijiazhuang Tiedao University, No. 17, Beierhuan Dong Road, Shijiazhuang 050043, Hebei Province, China. Email: shenyongjun@126.com Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https:// creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 2 1118 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Analytical method is an important means to study nonlinear systems, many methods such as the Hamiltonian approach method, He’s frequency formulation method, K–B method, multi-scale method, harmonic balance method, variational iteration method, and homotopy analysis method are adopted. Forsat studied the nonlinear control equation based on beam theory using Hamiltonian method, and the nonlinear vibration frequencies are obtained. He’s frequency method comes from ancient Chinese mathematics, by using He’s amplitude–frequency formula. Zhang obtained the analytical solution of nonlinear oscillator with discontinuity by using He’s ampli- tude–frequency formula, which has a high accuracy. Some methods are also extended to solve the fractional-order systems. Niu et al. studied the approximate analytical solution of a fractional-order single degree-of-freedom oscillator with clearance by using the KBM asymptotic method. Xu et al. proposed a new method to solve the second-order approximate solution of fractional Duffing oscillator by combining L–P method and multi-scale method. Xiao et al. established the approximate expression of the analytical solution of fractional Vander Pol oscillator by harmonic balance method. At the same time, in 1998, the variational iteration method was an effective tool for factional calculus. In 2007, Momani and Odibat adopted the homotopy perturbation method for fractional differential equations with great success. Most engineering materials are not ideal solids or liquids but viscous–elastic. Therefore, the model, which is established by only considering elastic properties or damping characteristics, fails to reflect the material essence completely. Nearly, all physical systems can be expressed by a fractional model, and related studies also show that the model established using fractional calculus is accurate and can effectively reflect constitutive relations. 16–19 20 Therefore, fractional calculus has been extensively applied in different engineering fields. Sun et al. intro- duced fractional derivative to establish a model for oil and gas suspension. They proved that the fractional model could more accurately describe the characteristics of the oil and gas suspension than the integer-order model through numerical and experimental verifications. Lewandowski and Lasecka-Plura described a viscoelastic damper through a fractional model, conducted a theoretical analysis of design sensitivity of this damper, and verified their results through illustration. Li et al. introduced an FKV-based constitutive fractional vibration attenuation model for viscoelastic suspension and found that this viscoelastic suspension response was of global correlation and memorability. Wu and Sangguan proved that the model containing fractional derivative could effectively predict the dynamic characteristics of rubber vibration isolators. They also described that the corre- lation between dynamic characteristics and frequency of the rubber vibration isolator through the fractional derivative was reasonable and accurate. Non-smooth motion is extensively used in many oscillatory systems. However, dynamic behaviors and mech- anisms of some important non-smooth systems remain unclear. Particularly, dynamic behaviors, such as bifur- cation and chaos, which aggravate machines, have imposed potential safety hazards on machine operation. Therefore, conducting an in-depth analysis of non-smooth systems is necessary. Scholars have already investi- gated non-smooth systems. Zhong and Chen used singularity theory to study the bifurcation of resonant solutions of clearance-containing suspension system models. Wei et al. analyzed the transformation of single- degree-of-freedom piecewise smooth systems from n 1 periodic motion into chaotic behaviors through period- doubling bifurcation and then investigated the control of system chaotic motions. Zhang et al. established a generalized piecewise linear model of Chua’s circuit and studied codimension-1 unconventional fold bifurcation generated during the passage of the system trajectory through the interface. They also analyzed the resulting bursting oscillation phenomenon. Zhang et al. used Melnikov theory for the oscillation model of clearance- containing gear systems to investigate global bifurcation conditions for the heteroclinic orbit and the stability of its periodic motion. Li and Zhao obtained Melnikov function for the subharmonic orbit of piecewise smooth systems to investigate the existence of subharmonic orbits. Huang and Xu acquired frequency response and stability conditions for a controlled piecewise smooth system with negative stiffness and explored symmetrical breaking bifurcation and chaotic motion in this system. Hou et al. established a roller model for rolling mills containing nonlinear constraints and analyzed bifurcation characteristics of the system under autonomous and non-autonomous conditions via stability systems of singular points and singularity theory. Studies regarding fractional non-smooth systems containing clearance constraints are currently limited. Dynamic characteristics are complex due to the strong nonlinearity and singularity of such systems, and most 31,32 investigations have adopted numerical methods. Numerical and analytical methods are the main research methods for fractional systems, wherein the former generally only provide solutions under specific parameters, while the latter can acquire definite relational expressions between system characteristics and parameters to realize quantitative analysis. Hence, analytical and numerical methods are combined in this study to investigate frac- tional non-smooth oscillators containing clearance constraints. This paper is organized as follows: first is the Introduction part; the next section introduces the physical model of the system and the motion equations; then the Wang et al. Wang et al. 1119 3 next section obtains the bifurcation response equation of the system and its stability through the analytical method and examines the bifurcations of the system under different parameters using singularity theory; following section presents the analysis of chaotic motions of the system under fractional coefficient and order via the numerical method; and the last section draws the main conclusions. System model and motion equation The fractional non-smooth system model containing clearance constraints is shown in Figure 1. This model comprises a mass, a damper, a linear spring, a nonlinear spring, and the nonlinear constraints with clearance symmetrically, where m, c, k , k , and k are the system mass, linear damping coefficient, linear stiffness coeffi- 1 2 3 cient, nonlinear stiffness coefficient, and nonlinear constraint stiffness coefficient, respectively. ðÞ ðÞ KD ½xt �B cos xt is the p-order derivative of x(t) to t with the fractional coefficient K. The non-smooth system is subject to a force excitation. The nonlinearity k and fractional derivative may be caused by the physical properties of the material; the nonlinear spring k may be caused by the restraint. When the displacement of m is less than clearance D, then only spring k , k and damping c are involved, if the displacement of m exceeds clearance D, the nonlinear spring k is 1 2 3 engaged. The differential equation of the system motion is presented as follows 3 p € _ ðÞ mx þ k x þ k x þ cx þ fx ðÞþKD ½xðtÞ� ¼ B cos xt (1) 1 2 0 where > ðÞ k x þ D x < � D fx ðÞ ¼ 0 � D � x � D ðÞ k x � D x > D The following parameters are inputted into equation (1) rffiffiffiffiffi k k k c c 1 2 3 1 2 x ¼ ; ek ¼ ; ek ¼ ; 2el ¼ ; 2el ¼ ; 0 1 2 1 2 m m m m m c B K 3 0 2el ¼ ; eB ¼ ; eK ¼ m m m Equation (1) can be transformed into 2 3 p x € þ x x þ ek x þ 2elx_ þ egx þ eK D ½xðtÞ� ¼ eBcosðÞ xt (2) ðÞ 1 1 > ðÞ k x þ D x < � D gx ¼ (3) ðÞ 0 � D � x � D ðÞ k x � D x > D Solution of the system bifurcation response Solution of the bifurcation response equation through the K–B method Based on the K–B method, which means the excitation frequency is close to the natural one, that is, x � x,a detuning parameter r is introduced to illustrate the proximate degree 2 2 x ¼ x þ ed 0 4 1120 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 1. Fractional non-smooth system model containing clearance constraints. Then, equation (2) is transformed into the following formula �� 2 3 p € ðÞ _ x þ x x ¼ e fcos xt þ dx � k x � 2lx � gx ðÞ � K D ½x� (4) 1 1 Equation (4) is assumed to satisfy the following conditions x ¼ at ðÞcos/ (5) x_ ¼�at ðÞxsin/ where / ¼ xt þ hðÞ t . at ðÞ and hðÞ t are slowly varying functions of time t. Equation (5) is substituted into equation (4) to obtain the following �� 3 3 p _ ðÞ a ¼� fcos xt þ racos/ þ 2laxsin/ � k a cos / � f ðÞ acos/ � eK D ½� acos/ sin/ 1 3 1 �� (6) 3 3 p > _ : ah ¼� fcosðÞ xt þ racos/ þ 2laxsin/ � k a cos / � f ðÞ acos/ � eK D ½� acos/ cos/ 1 3 1 Integrals are taken for a_ and ah, and averaging values are taken as follows 2p > � � 3 3 ðÞ > a_ ¼� fcos xt þ racos/ þ 2laxsin/ � k a cos / � f ðÞ acos/ sin/ d/ 1 3 2px > T � lim K D ½� acos/ sin/ d/; T!1 Tx Z (7) 2p � � > e > 3 3 ah ¼� fcosðÞ xt þ racos/ þ 2laxsin/ � k a cos / � f ðÞ acos/ cos/ d/ > 1 3 > 2px > e > � lim K D ½� acos/ cos/ d/ T!1 Tx The fractional differential is defined as Caputo, and the integral of the fractional part can be obtained through 33,34 residue theorem and contour integral shown as below p�1 e aeKx sinðpp=2Þ � lim K D ½� acos/ sin/ d/ ¼� ; T!1 Tx 2 Z (8) T p�1 e aeKx cosðpp=2Þ � lim K D ½� acos/ cos/ d/ ¼ T!1 Tx 2 0 Wang et al. Wang et al. 1121 5 The solved integrals are substituted into the original parameters to obtain the following B sinh a a_ ¼� � c þ c ðÞ 2mx 2m �� (9) ax B cosh a 3a ah ¼� � þ k þ k þ k þ k 1 2 2 2mx 2mx 4 where k is the piecewise equivalent stiffness �� a k 1 k ¼ 18/ þ 12/ cos2/ � 14sin2/ � sin4/ ; 0 0 0 0 0 4p 2 / ¼ arccos c is fractional equivalent damping p�1 c ¼ Kx sinðpp=2Þ and k is fractional equivalent stiffness k ¼ Kx cosðpp=2Þ 3a 0 Let cc ¼ c þ c, kk ¼ k þ ðÞ k þ k þ k then equation (9) can be rewritten as 1 1 2 3 B sinh a a_ ¼� � cc 2mx 2m (10) ax B cosh a ah ¼� � þ kk 2 2mx 2mx _ � Let a_ ¼ ah ¼ 0. After eliminating h (phase of the steady-state response) from equation (10), the amplitude– frequency equation of the bifurcation response can be obtained as follows hi 2 2 2 2 2 ðÞ a x cc þ mx � kk � B ¼ 0 (11) where a � is amplitude of the steady-state response. Stability analysis of system response equation The system is stable when a � D. Thus, only the situation under a > D is considered. a ¼ a þ Da and h ¼ h þ Dh are substituted into equation (1) to investigate the stability of the response equa- tion, and then � � dDa � cc ac � cc ðÞ a � B cosh 2 0 ¼� þ Da � � Dh dt 2m 2m 2mx �� (12) � � � dDh Fcosh k kk ðÞ a � Fsinh ¼ þ Da � þ Dh dt 2a � mx 2mx 2ma �x 2 3 cc ac � cc xa � a � � � � k � kk 6 7 6 2m 2m 2 2mx 7 det ¼ 0 (13) 4 5 x kk k kk cc � þ þ � � k 2a � 2mxa � 2mx 2m 6 1122 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Wang et al. 7 The following characteristic equations are obtained according to equation (13) �� 0 2 0 0 ac  cc þ 2cc cc þ ccac  cc 2kk þ ak  kk 2 2 2 k þ k þ � 2m 4m 4m (14) 2 2 2 2 0 2 0 ðÞ ðÞ cc x þ kk � mx þ ac  ccx cc þ ak  kk � mx kk 2 2 þ ¼ 0 2 2 4m x where 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi c x D 3 4 5 2 3 3 cc ¼ ðDa þ 2D Þ 1 � � 3a / 2 2 a p a sffiffiffiffiffiffiffiffiffiffiffiffiffiffi �� 2 4 2 D D D (15) Dk 33 � 28 þ 8 þ 42ak / 1 � 3 3 2 4 2 a a a kk ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4p 1 � Defining Figure 2. Transition set and bifurcation diagram under constrained parameters B and B . 1 4 ac  cc þ 2cc P ¼ 2m Let a ¼ y � . Then, equation (16) is transformed into the following form 6b and 6 4 3 2 y þ B y þ B y þ B y þ B y þ l ¼ 0 (17) 4 3 2 1 2 0 0 cc þ ccac  cc 2kk þ ak  kk 2 2 Q ¼ � where 4m 4m 2 2 2 2 0 2 0 ðÞ ðÞ �� cc x þ kk � mx þ ac  ccx cc þ ak  kk � mx kk 2 2 4 2 2 3 þ b b b � 6b b b þ 27b b b � 108b b 1 5 5 4 5 6 3 5 6 2 6 2 2 4m x B ¼ þ b 324b 6 6 �� 3 2 b b 5b � 24b b b þ 72b b 2 5 5 4 5 6 3 6 the stability conditions for the bifurcation response equation are as follows: B ¼ � b 144b 6 6 P > 0 and Q > 0 b 5b � 18b b b 3 5 4 5 6 B ¼ þ b 27b 6 6 Bifurcation characteristic analysis b 5b 4 5 B ¼ � b 12b 6 6 The bifurcation response equation of the system is analyzed using singularity theory. The bifurcation response �� 3 2 2 4 3 5 b b 1296b b � 216b b b � 7776b b þ 36b b b � 5b 0 5 5 6 3 5 6 1 6 4 5 6 5 equation is transformed into the following l ¼ þ b 46656b 6 6 6 5 4 3 2 b a þ b a þ b a þ b a þ b a þ b a þ b ¼ 0 (16) 6 5 4 3 2 1 0 Singularity theory indicates that equation (17) is the universal unfolding of form y þ l ¼ 0. The durability of this universal unfolding is thus analyzed. This analysis is conducted by constraining the parameters due to where numerous constraints. �� 6 3 D k D k pp 3 3 2 2 2 p�1 b ¼ � B b ¼ 2k p � D k � 2mpx þ 2pKx cos ; (1) Constrained parameters B and B 1 4 0 0 1 1 3 2 2 p p 2 3 4 256 Bifurcation point set, B þ ðÞ B ¼ 0 3 2 4 2 1 63D k 4096B 4 3 2 2 2 2 2 2 2 2p�2 2 Hysteretic point set, ¼ B [ B ¼ 0 ðÞ 3 2 ðÞ b ¼ k þ x m x þ c � 2k m þ K x 1 � x ðÞ 1 þ cospp � 729 2 1 1 1 2 3 4 256 2 4p �� Double-limit point set, B þ ðÞ B ¼ 0 3 2 �� 1 pp D k pp 2 4 2 2 2 p�1 2 pþ1 The transition set and bifurcation diagram under constrained parameters B and B are shown in Figure 2. 1 4 þ D k k þ 6D k þ D k mx þ Kx cos k � m x � þ c Kx sin 1 3 3 3 1 1 1 p 2 p 2 �� (2) Constrained parameters B and B 2 4 �� Dk pp 2 2 p Hysteretic point set is ðÞ ðÞ b ¼ D 16k þ 3pk � 3pk þ 4Kp 3p � 8 k � mx x cos ; 3 3 2 3 1 2p 2 �� () �� �� 64 3 3 3 pp 2 2 p�1 3 ðÞ ðÞ ðÞ b ¼ D k k � k þ 65k þ 9k p þ k þ k k � mx þ K k þ k x sin ; 5B B 4 3 3 2 3 3 2 3 1 2 3 1 3 p 4 2 2 2 ¼� [ fg B ¼ 0 9 25 3 9 ðÞ b ¼ Dk ðÞ k þ k 3p � 8 ; b ¼ ðÞ k þ k 5 3 2 3 6 2 3 2 16 Wang et al. Wang et al. 1123 7 Figure 2. Transition set and bifurcation diagram under constrained parameters B and B . 1 4 Let a ¼ y � . Then, equation (16) is transformed into the following form 6b 6 4 3 2 y þ B y þ B y þ B y þ B y þ l ¼ 0 (17) 4 3 2 1 where �� 4 2 2 3 b b b � 6b b b þ 27b b b � 108b b 1 5 5 4 5 6 3 5 6 2 6 B ¼ þ b 324b 6 6 �� 3 2 b b 5b � 24b b b þ 72b b 2 5 5 4 5 6 3 6 B ¼ � b 144b 6 6 b 5b � 18b b b 3 5 4 5 6 B ¼ þ b 27b 6 6 b 5b 4 5 B ¼ � b 12b 6 6 �� 3 2 2 4 3 5 b b 1296b b � 216b b b � 7776b b þ 36b b b � 5b 0 5 5 6 3 5 6 1 6 4 5 6 5 l ¼ þ b 46656b 6 6 Singularity theory indicates that equation (17) is the universal unfolding of form y þ l ¼ 0. The durability of this universal unfolding is thus analyzed. This analysis is conducted by constraining the parameters due to numerous constraints. (1) Constrained parameters B and B 1 4 4 256 Bifurcation point set, B þ ðÞ B ¼ 0 3 2 4096B 4 Hysteretic point set, ¼ B [ B ¼ 0 3 2 4 256 Double-limit point set, B þ ðÞ B ¼ 0 3 2 The transition set and bifurcation diagram under constrained parameters B and B are shown in Figure 2. 1 4 (2) Constrained parameters B and B 2 4 Hysteretic point set is () �� 5B B 1 3 ¼� [ fg B ¼ 0 9 25 8 1124 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Wang et al. 9 Figure 5. Transition set and bifurcation diagram under constrained parameters B and B . 2 3 Figure 3. Transition set and bifurcation diagram under constrained parameters B and B . 2 4 Figure 6. Transition set and bifurcation diagram under constrained parameters B and B . 3 4 Figure 4. Transition set and bifurcation diagram under constrained parameters B and B . 1 2 (3) Constrained parameters B and B and double-limit point set is 1 2 64 3 3 Hysteretic point set, B þ ¼ 0; B < 0 4 4 729 2 5 4 Double-limit point set, U 3 5 B ¼� B 1 3 The transition set and bifurcation diagram under constrained parameters B and B are shown in Figure 4. 1 2 27 25 (4) Constrained parameters B and B 2 3 2 5 Hysteretic point set, 3125B þ 512B ¼ 0 1 4 3125 2 5 Double-limit point set, B þ 27B ¼ 0 1 4 The transition set and bifurcation diagram under constrained parameters B and B are shown in 4 2 4 The transition set and bifurcation diagram under constrained parameters B and B are shown in Figure 5. 2 3 Figure 3. Wang et al. Wang et al. 1125 9 Figure 5. Transition set and bifurcation diagram under constrained parameters B and B . 2 3 Figure 6. Transition set and bifurcation diagram under constrained parameters B and B . 3 4 (3) Constrained parameters B and B 1 2 64 3 3 Hysteretic point set, B þ ¼ 0; B < 0 4 4 729 2 Double-limit point set, U The transition set and bifurcation diagram under constrained parameters B and B are shown in Figure 4. 1 2 (4) Constrained parameters B and B 2 3 2 5 Hysteretic point set, 3125B þ 512B ¼ 0 1 4 3125 2 5 Double-limit point set, B þ 27B ¼ 0 1 4 The transition set and bifurcation diagram under constrained parameters B and B are shown in Figure 5. 2 3 10 1126 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Wang et al. 11 Figure 7. Transition set and bifurcation diagram under constrained parameters B and B . 1 3 Figure 9. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under K¼�0.7 (single-periodic motion). Figure 8. Local bifurcation diagram of the system under varying fractional coefficient K. (5) Constrained parameters B and B 3 4 625 4 5 Hysteretic point set, B þ B ¼ 0; B < 0 1 2 2 625 4 1 5 Double-limit point set, B þ B ¼ 0 1 2 256 5 The transition set and bifurcation diagram under constrained parameters B and B are shown in Figure 6. 3 4 (6) Constrained parameters B and B 1 3 fg Hysteretic point set, 3B � B ¼ 0; B < 0 [ B ¼ 0 2 4 4 2 1 2 Double-limit point set, B ¼ B 2 4 Figure 10. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under K¼�0.74 (double- The transition set and bifurcation diagram under constrained parameters B and B are shown in Figure 7. 1 3 periodic motion). Wang et al. Wang et al. 1127 11 Figure 9. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under K¼�0.7 (single-periodic motion). Figure 10. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under K¼�0.74 (double- periodic motion). 12 1128 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Wang et al. 13 Figure 12. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under K¼�1 (chaotic motion). Figure 11. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under K¼�0.752 (four- periodic motion). Numerical analysis System bifurcation and path to chaos under the influence of fractional parameter K The following system parameters are selected: m¼ 1, c¼ 0.3, F¼ 0.5, k ¼ 0.5, k ¼ 0.5, k ¼ 0.1, d¼ 0.2, and 1 2 3 p¼ 0.1. Under different values of fractional parameter K, the local bifurcation diagram of the system is shown in Figure 8. Figure 8 shows that the path for the system to chaos under varying fractional coefficient K is as follows: chaotic motion ! alternation of multiperiodic and chaotic motions ! chaotic motion ! from period-doubling motion to single-periodic state. Figures 9–12 demonstrate that the system presents different motion states under different values of fractional Figure 13. Local bifurcation diagram under varying fractional order p. coefficient K. Under K¼�0.7, system phase diagram, displacement time history diagram, and Poincare section are shown in Figure 9. This figure reveals that the phase trajectory of the system is an ellipse and the Poincare section is a dot. Meanwhile, the displacement time history diagram shows that the system is under stable single- periodic motion. Under K¼�0.74, the phase trajectory of the system is turned into two ellipses: the Poincare System bifurcation and path to chaos under the influence of fractional order p section is also turned into two dots, and the time history diagram also displays regular two-periodic motions. Therefore, the system is under a double-periodic motion state. Figure 11 shows that the phase trajectory of the The following system parameters are selected: m¼ 1, c¼ 0.3, F¼ 0.3, k ¼ 0.5, k ¼ 0.5, k ¼ 0.1, d¼ 0.2, and 1 2 3 system under K¼�0.752 becomes four ellipses. The system is also under a four-periodic motion state based on K¼�0.6. The local bifurcation diagram, which varies with fractional order p, is shown in Figure 13. the Poincare section and the time history diagram. When K¼�1, the phase trajectories are intertwined without Paroxysmal bifurcation and chaos are generated in the system with the change in fractional order p. As p overlapping, the Poincare section is disorderly and unsystematic, and the displacement time history diagram also turns from 0 to 0.0045, the system enters a periodic motion state from the chaotic motion state; when p becomes displays a disorderly state. These findings verify the chaotic motion state of the system. 0.007, the system again enters a chaotic motion state from periodic motion. Under p¼ 0.01, the system is Wang et al. Wang et al. 1129 13 Figure 12. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under K¼�1 (chaotic motion). Figure 13. Local bifurcation diagram under varying fractional order p. System bifurcation and path to chaos under the influence of fractional order p The following system parameters are selected: m¼ 1, c¼ 0.3, F¼ 0.3, k ¼ 0.5, k ¼ 0.5, k ¼ 0.1, d¼ 0.2, and 1 2 3 K¼�0.6. The local bifurcation diagram, which varies with fractional order p, is shown in Figure 13. Paroxysmal bifurcation and chaos are generated in the system with the change in fractional order p. As p turns from 0 to 0.0045, the system enters a periodic motion state from the chaotic motion state; when p becomes 0.007, the system again enters a chaotic motion state from periodic motion. Under p¼ 0.01, the system is 14 1130 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) Wang et al. 15 Figure 14. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under p¼�0.004 (chaotic Figure 16. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under p¼�0.012 (five- motion). periodic motion). transformed from a chaotic motion state into a periodic motion state once again. With the change in fractional order p, this fractional non-smooth system presents alternate motion states between periodic and chaotic motions. The system demonstrates different motion states under different values of fractional order p. Figure 14 shows the phase diagram, displacement time history diagram, and Poincare section of the system under p¼ 0.004. The phase trajectory of the system comprises numerous ellipses, which are disorderly, irregular, and are not over- lapping. Thus, the system is under a chaotic motion state. The phase diagram, displacement time history diagram, and Poincare section under p¼ 0.0045 are presented in Figure 15. The amplified phase diagram shows that the phase trajectory comprises four ellipses, and the Poincare section comprises four independent dots. Meanwhile, the system is under a four-periodic motion state according to the displacement time history diagram. When p¼ 0.012 (Figure 16), the amplified phase diagram indicates the transformation of the phase trajectory into five ellipses and the Poincare section into five independent dots. Thus, the system is under a five-periodic motion state from the displacement time history diagram. Conclusions The fractional non-smooth oscillator containing clearance constraints was taken as the study object, and the nonlinear kinetic equation of the system was established. The dynamic behaviors of the system, such as bifurca- tion and chaos, were also investigated through analytical and numerical methods. The main conclusions are presented as follows. (1) The system bifurcation equation was analyzed via singularity theory, and the transition set and bifurcation diagram of the system under six different constraint parameters are obtained. On each bifurcation diagram, Figure 15. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under p¼�0.0045 (four- the transition set divides the system into multiple sub-regions, and the bifurcation curves of different sub- periodic motion). regions represent different bifurcation characteristics of the system under different parameter conditions. Wang et al. Wang et al. 1131 15 Figure 16. System (a) phase diagram, (b) displacement time history diagram, and (c) Poincare section under p¼�0.012 (five- periodic motion). transformed from a chaotic motion state into a periodic motion state once again. With the change in fractional order p, this fractional non-smooth system presents alternate motion states between periodic and chaotic motions. The system demonstrates different motion states under different values of fractional order p. Figure 14 shows the phase diagram, displacement time history diagram, and Poincare section of the system under p¼ 0.004. The phase trajectory of the system comprises numerous ellipses, which are disorderly, irregular, and are not over- lapping. Thus, the system is under a chaotic motion state. The phase diagram, displacement time history diagram, and Poincare section under p¼ 0.0045 are presented in Figure 15. The amplified phase diagram shows that the phase trajectory comprises four ellipses, and the Poincare section comprises four independent dots. Meanwhile, the system is under a four-periodic motion state according to the displacement time history diagram. When p¼ 0.012 (Figure 16), the amplified phase diagram indicates the transformation of the phase trajectory into five ellipses and the Poincare section into five independent dots. Thus, the system is under a five-periodic motion state from the displacement time history diagram. Conclusions The fractional non-smooth oscillator containing clearance constraints was taken as the study object, and the nonlinear kinetic equation of the system was established. The dynamic behaviors of the system, such as bifurca- tion and chaos, were also investigated through analytical and numerical methods. The main conclusions are presented as follows. (1) The system bifurcation equation was analyzed via singularity theory, and the transition set and bifurcation diagram of the system under six different constraint parameters are obtained. On each bifurcation diagram, the transition set divides the system into multiple sub-regions, and the bifurcation curves of different sub- regions represent different bifurcation characteristics of the system under different parameter conditions. 16 1132 Journal of Low Frequency Noise, Vibration and Active Control 40(3) Journal of Low Frequency Noise, Vibration and Active Control 0(0) The bifurcation of the system in each region is persistent. On the dividing line, the bifurcation of the system is not persistent, and the bifurcation graphs determined by parameters in each subregion are topologically equivalent. Therefore, dynamic behaviors of the system can be changed by varying its parameters to provide a theoretical reference for the reasonable selection of system parameters. (2) The numerical results show that besides the route to chaos through period-doubling with the change of fractional order p, with the change of fractional-order coefficient K, the system still exists Paroxysmal bifur- cation, in which the periodic motion and chaotic motion of the system alternately appear. At the same time, the above bifurcation and chaos phenomena are further confirmed by the displacement time history diagram, phase portrait, and Poincare section. Hence, the dynamic behaviors of the fractional non-smooth system containing clearance constraints can be changed by varying fractional coefficient and order, thus a foundation for repressing oscillations of similar systems has been established. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (nos 11802183, 11772206, 11872256 and U1934201). ORCID iD Yongjun Shen https://orcid.org/0000-0002-8768-1958 References 1. He JH. A new fractal derivation. Therm Sci 2011; 15: 145–147. 2. He JH. A tutorial review on fractal spacetime and fractional calculus. Int J Theor Phys 2014; 53: 3698–3718. 3. He JH. Fractal calculus and its geometrical explanation. 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Journal

"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE

Published: Sep 30, 2020

Keywords: Bifurcation; fractional non-smooth oscillator; clearance constraint; singularity theory

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