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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2012, Article ID 268964, 10 pages doi:10.1155/2012/268964 Research Article Effects of Structural Parameters on the Dynamics of a Beam Structure with a Beam-Type Vibration Absorber 1 2 1 3 Mothanna Y. Abd, Azma Putra, Nawal A. A. Jalil, and Jamaludin Noorzaei Department of Mechanical and Manufacturing Engineering, Faculty of Engineering, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia Department of Structure and Materials, Faculty of Mechanical Engineering, Universiti Teknikal Malaysia Melaka, Hang Tuah Jaya, 76100 Durian Tunggal, Melaka, Malaysia Department of Civil Engineering, Faculty of Engineering, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia Correspondence should be addressed to Nawal A. A. Jalil, nawal@eng.upm.edu.my Received 30 April 2012; Revised 21 September 2012; Accepted 23 September 2012 Academic Editor: Abul Azad Copyright © 2012 Mothanna Y. Abd et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A beam-type absorber has been known as one of the dynamic vibration absorbers used to suppress excessive vibration of an engineering structure. This paper studies an absorbing beam which is attached through a visco-elastic layer on a primary beam structure. Solutions of the dynamic response are presented at the midspan of the primary and absorbing beams in simply supported edges subjected to a stationary harmonic load. The eﬀect of structural parameters, namely, rigidity ratio, mass ratio, and damping of the layer and the structure as well as the layer stiﬀness on the response is investigated to reduce the vibration amplitude at the fundamental frequency of the original single primary beam. It is found that this can considerably reduce the amplitude at the corresponding troublesome frequency, but compromised situation should be noted by controlling the structural parameters. The model is also validated with measured data with reasonable agreement. 1. Introduction of double-cantilever visco-elastic beam connected by spring and viscous damper. The auxiliary beam is attached to the A beam-type absorber is one of the techniques to reduce center of the main beam excited at its end by a sinusoidal undesirable vibration of many vibrating systems, such as force. It is found that the amplitude at resonances of the main a synchronous machine, mounting structure for a sensitive beam is sensitive to the stiﬀness and mass of the absorbing instrument, and other continuous structure in engineering. beam. The damping ratio was formulated as a function The absorber system usually consists of a beam attached of mass and layer stiﬀness of the absorber. Vu et al. [2] to the host structure using an elastic element. The natural studied the distributed vibration absorber under stationary frequency of the absorber is then tuned to be the same as distributed force. A closed form was developed by utilizing the troublesome operating frequency of the host structure to change of variables and modal analysis to decouple and create counter force, which in return reduces the vibration solve diﬀerential equations. Oniszczuk [3] studied the free of the structure. As beams are important structures in civil vibrations of two identical parallel simply supported beams or mechanical engineering, several works have also been continuously joined by an elastic layer. The eigen frequencies established to investigate the performance of the absorbing and mode shapes of vibration of the double-beam system beam which is attached also to a beam structure. were found using the classical assumed mode summation. Among the earliest studies of the double-beam system Another theoretical study for ﬁnding the optimum is one proposed by Yamaguchi [1], which investigated the design of beam-type absorber was presented by Aida et al. eﬀectiveness of the dynamic vibration absorber consisting [4]. A uniform absorbing beam with the same boundary 2 Advances in Acoustics and Vibration iωt F e L/2 iωt F e K = E I 1 1 1 Primary beam c k Linear spring-damper ... Main beam element m Absorbing beam Visco-elastic layer element k x L μm Absorber beam element 1 K = eK 2 1 (a) (b) Figure 1: (a) Schematic diagram of a visco-elastically connected simply supported double-beam system and (b) the two-degree-of-freedom system of distributed lumped parameter model. condition is connected to the main beam by spring and 2. Mathematical Modeling damper components. In the study, an optimum tuning The schematic diagram of a beam connected with an method to reduce the main beam vibration was proposed. absorbing beam having the same length L and simply Another structural analysis and optimum design of a supported is shown in Figure 1(a). Here distributed lumped dynamic absorbing beam with free edges was studied by system of two degrees of freedom is assumed, where the Chen and Lin [5]. The eﬀect of mass ratio and the stiﬀness visco-elastic element between the beams consists of parallel layer on the vibration response was discussed. distributed springs and dampers as shown in Figure 1(b). The eﬀect of diﬀerent forcing types on the double-beam The equation of motion of the dynamic model can system has also been discussed by several authors. Zhang therefore be written as et al. [6] studied vibration characteristics of the double- beam system under axial compressive load. The studies were ∂ w iωt E I + m w ¨ + c(w˙ − w˙ ) + k(w − w ) = Pe , 1 1 1 1 1 2 1 2 limited for two identical simply supported double-beam 4 ∂x system and the eﬀect of the axial load on the beams vibration (1) amplitude was reported. Abu-Hilal [7] studied the eﬀect of ∂ w a moving constant load on the dynamic response of the (2) E I + m w ¨ + c(w˙ − w˙ ) + k(w − w ) = 0, 2 2 2 2 2 1 2 1 beams. This was done for diﬀerent values of speed parameter, ∂x damping ratio, and stiﬀness parameter. where P = F δ(x − L/2) is the external point force with Several works focus on the development of mathematical frequency ω at the midspan of the beam, F is the force model to provide solutions of the vibration response. De magnitude, K= EIis the ﬂexural rigidity, cis the damping Rosa and Lippiello [8] studied free vibration of double- constant, kis the stiﬀness constant of the viscous layer, and identical beam system using the Diﬀerential Quadrature mis the mass of the beam. The subscripts 1 and 2 refer to Method (DQM). Sadek et al. [9] presented a computational the main beam and the absorber beam, respectively. The method for solving optimal control of transverse vibration damping of the beam can be introduced by replacing the of a two-parallel-beam system based on legendry wavelets ﬂexural rigidity in (1)and (2)by K(1 + iη), where η is the approach. It is found here that the reduction of the beam damping loss factor. The vertical displacement of the main vibration depends on the spring location on the beam. beam w and the absorber w can be expressed as a series 1 2 This paper investigates the eﬀect of structural param- expansion in terms of mode shape function X (x) for the nth eters, namely, the rigidity ratio, mass ratio, damping loss mode.The amplitudes in generalized time coordinates q are 1n factor, the stiﬀness, as well as the damping ratio of the layer given by on the dynamic response of simply supported double-beam system to provide a thorough analysis. The discussion is w (x, t) = X (x) q (t), (3) 1 n 1n limited on controlling the fundamental mode of a single- n=1 beam structure using a dynamic absorbing beam. The fol- lowing section discusses the derivation of the mathematical w (x, t) = X (x) q (t), (4) 2 n 2n model. n=1 Advances in Acoustics and Vibration 3 −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 −5 −5 10 10 −6 −6 10 10 −7 −7 10 10 10 20 30 40 50 60 10 20 30 40 50 60 Frequency (Hz) Frequency (Hz) e = 0.25 e = 1 e = 0.25 e = 1 e = 0.5 e = 0.5 Single beam Single beam e = 0.75 e = 0.75 (a) (b) −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 −5 −5 10 10 −6 −6 10 10 −7 −7 10 10 10 20 30 40 50 60 10 20 30 40 50 60 Frequency (Hz) Frequency (Hz) e = 0.25 e = 1 e = 0.25 e = 1 e = 0.5 e = 0.5 Single beam Single beam e = 0.75 e = 0.75 (c) (d) Figure 2: Eﬀect of the rigidity ratio e and the mass ratio μ on the vibration amplitude at the midspan location of the primary beam: (a) μ = 0.1, (b) μ = 0.2, (c) μ = 0.4, and (d) μ = 0.8. where x is the position on the beam at which the load is The orthogonality conditions can be used for simplifying the applied. For the simply supported boundary condition, the equations of motion which is represented in the following nth mode shape function can be written as form [1]: X (x) = sin(σ x), (5) n n (7) X (x)X (x)dx = 0, n= m. n m / where σ is the eigenvalue of the mode shape which can be expressed as: Equations (3)and (4)can be substitutedto(1)and (2). Multiplying both sides of the equations by mode shape nπ function X then integrating through the beam length σ = . (6) L and applying the orthogonality condition as in (7)yield Amplitude (m/N) Amplitude (m/N) Amplitude (m/N) Amplitude (m/N) 4 Advances in Acoustics and Vibration −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 −5 −5 10 10 −6 −6 10 10 −7 −7 10 10 10 20 30 40 50 60 10 20 30 40 50 60 Frequency (Hz) Frequency (Hz) Primary beam Primary beam Absorbing beam Absorbing beam Single beam Single beam (a) (b) −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 −5 −5 10 10 −6 −6 10 10 −7 −7 10 10 10 20 30 40 50 60 10 20 30 40 50 60 Frequency (Hz) Frequency (Hz) Primary beam Primary beam Absorbing beam Absorbing beam Single beam Single beam (c) (d) Figure 3: The vibration amplitude at the midspan location of the double-beam system (e = 0.25): (a) μ = 0.1, (b) μ = 0.2, (c) μ = 0.4, and (d) μ = 0.8. a matrix form for the equations of motion expressed as (see vibration absorber [10]. The damping ratio of the visco- Appendix) elastic layer can be deﬁned as c c ¨ ˙ m 0 q c −c q 1 1 1 ξ = = , (9) 2m ω 2μm ω 2 n n 0 μm q¨ −cc q˙ 1 2 2 (8) where ω is the original natural frequency of the primary K σ + k −k q F 2 nπ 1 1 0 iωt beam (without the absorber attached) which is given by + = sin e , −keK σ + k q 0 1 2 L 2 E I σ 1 1 (10) where μ = m /m is the mass ratio and e = E I /E I is the ω = . 2 1 2 2 1 1 n rigidity ratio. The damping of the layer between the main beam and the absorbing beam can be approached by using For a stationary harmonic load, it is therefore necessary to the concept of “mixed damping ratio” in discrete dynamic analyze the system performance in the frequency domain Amplitude (m/N) Amplitude (m/N) Amplitude (m/N) Amplitude (m/N) Advances in Acoustics and Vibration 5 −2 10 to clearly identify the distinct responses in the resonant frequencies. By deﬁning the amplitude q in terms of the complex exponential notation it gives −3 −4 q Q 1 1 iωt = Re e , (11) q Q 2 2 −5 −6 10 where Q is the complex amplitude of q. Substituting (11) into (8) yields 4 2 K σ + k + iωc − ω m −iωc − k Q 1 1 1 −7 n 4 2 −iωc − k eK σ + k + iωc − ω μm Q 1 2 −8 F 2 nπ 10 20 30 40 50 60 = sin , L 2 Frequency (Hz) (12) Primary beam; ζ = 0.01 Absorbing beam; ζ = 0.1 Primary beam; ζ = 0.1 Single beam Absorbing beam; ζ = 0.01 where the solutions for the complex amplitude of the frequency response function in terms of the receptance, that Figure 4: Eﬀect of visco-elastic layer damping on the vibration is, Q/F can then be obtained for each mode n which are amplitude at the midspan location of the double-beam system (e given by = 0.25, μ = 0.4, η = η = 0.01). 1 2 4 2 Q 2 eK σ +k+iωc − ω μm sin(nπ/2) 1 1 1 n = , (13) F 4 2 4 2 K σ +k+iωc − ω m eK σ +k+iωc − ω μm −(iωc+k) L 1 n 1 1 n 1 Q 2(iωc + k) sin(nπ/2) = . (14) 4 2 4 2 K σ +k+iωc − ω m eK σ +k+iωc − ω μm − (iωc+k) L 1 1 1 1 n n As observed from (13)and (14), the double-beam structure damping loss factor of the beams are assumed very small, that is expected to have two natural frequencies for each mode is, ζ = η = 0.01. It can be seen that the double-beam system successfully of vibration, as the system is modeled using two-degree- suppresses the amplitude at the fundamental frequency of of-freedom system. It can also be seen in (13), where for the single beam (at 12 Hz) as a result of countering some of the lumped parameter system with K = 0and foran the energy force from the main beam. However, the system undamped case where c = 0, the magnitude of the numerator now behaves as a two-degree-of-freedom system which, in is proportional to k − ω m . The main beam amplitude can consequence, creates new resonances. Thus, a signiﬁcant therefore be suppressed to zero by tuning the layer stiﬀness level of vibration amplitude appears at lower frequency and the mass of the absorbing beam to be equal to the forcing around 8–10 Hz and much lower amplitude level for the frequency, that is, ω = k/m . The subsequent sections second resonance around 25–30 Hz. Increasing the elasticity discuss the eﬀect of structural parameters on the response ratio gives insigniﬁcant eﬀect to reduce the amplitude at the in (13)and (14). ﬁrst resonance. Instead, this shifts both resonant frequencies to higher frequency, which causes the ﬁrst resonance to approach that of the single beam; a situation which should 3. Effects of Rigidity and Mass Ratio be avoided especially if forcing frequency from the primary beam is not stable, for example, a nonsynchronous machine In this investigation, the rigidity ratio and mass ratio are having certain range of operating frequency. Meanwhile, varied to observe their eﬀects on the main beam response. increasing the mass ratio of the double-beam is shown to This is calculated for a steel beam (Young’s modulus E = 11 3 decrease the frequency of both ﬁrst and second resonances, 2.1 × 10 Pa, density ρ = 7800 kg/m ) having length 2 m, although it has less eﬀect on the former. This hence reduces width 0.065 m, and thickness 0.02 m. The stiﬀness of the the frequency gap of the two resonances. However, one layer is assumed 100 kN/m. The calculation here is done for can observe that the amplitude of the primary beam is the ﬁrst mode of vibration (n = 1) for each beam. Figure 2 reduced with the increasing mass ratio (note the dips shows the vibration amplitude for the ﬁrst two resonances just before the second resonance which approaches the of the primary beam plotted in logarithmic scale. Here the fundamental frequency of the single primary beam). It can damping ratio of the visco-elastic layer and the structural Amplitude (m/N) 6 Advances in Acoustics and Vibration −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 −5 −5 10 10 −6 −6 10 10 −7 −7 10 10 −8 −8 10 20 30 40 50 60 10 20 30 40 50 60 Frequency (Hz) Frequency (Hz) Primary beam; η = 0.01 Primary beam; η = 0.1 1 1 Absorbing beam; η = 0.1 2 Absorbing beam; η = 0.01 Single beam; η = 0.01 Single beam; η = 0.1 1 1 (a) (b) −2 −2 10 10 −3 −3 10 10 −4 −4 −5 −5 10 10 −6 −6 10 10 −7 −7 −8 −8 10 10 10 20 30 40 50 60 10 20 30 40 50 60 Frequency (Hz) Frequency (Hz) Primary beam; η = 0.1 Primary beam; η = 0.1 1 1 Absorbing beam; η = 0.1 Absorbing beam; η = 0.1 2 2 Single beam; η = 0.1 Single beam; η = 0.1 1 1 (c) (d) Figure 5: Eﬀect of structural damping of the beam on the vibration amplitude at the midspan location of the double-beam system (e = 0.25, μ = 0.4): (a)–(c) ζ = 0.01 and (d) ζ = 0.1. also be observed that the amplitude at the second resonance amplitude of the absorber at the second resonance can also be increases as the mass ratio is increased. observed to have negligible eﬀect due to the change of mass ratio, and it steeply decreases above this resonant frequency. Figure 3 presents the amplitudes of the primary beam and the absorbing beam. Both amplitudes at the ﬁrst reso- nance can be seen not aﬀected by the mass ratio. However, 4. Effect of Damping and Layer Stiffness the response of the primary beam at the second resonance is considerably lower than that of the absorbing beam for Previous results in Figures 2 and 3 assume very small low mass ratio, but both amplitudes become comparable damping of the visco-elastic layer as well as the structural as the mass ratio increases. As also seen in Figure 2, the damping loss factor of the beam. Figure 4 plots the eﬀect compromised situation is that the vibration amplitude of the of the layer damping, which yields reduction of vibration primary beam becomes lower as the mass ratio increases. The amplitude only at the second resonance of both the primary Amplitude (m/N) Amplitude (m/N) Amplitude (m/N) Amplitude (m/N) Advances in Acoustics and Vibration 7 −2 −2 −3 −3 −4 −4 −5 −5 −6 −6 −7 −7 −8 −8 10 10 10 20 30 40 50 60 10 20 30 40 50 60 Frequency (Hz) Frequency (Hz) Primary beam; β = 50 Absorbing beam; β = 350 Primary beam; β = 50 Absorbing beam; β = 350 Primary beam; β = 350 Single beam Primary beam; β = 350 Single beam Absorbing beam; β = 50 Absorbing beam; β = 50 (a) (b) Figure 6: Eﬀect of visco-elastic stiﬀness on the vibration amplitude at the midspan location of the double-beam system (e = 0.25): (a) μ = 0.1 and (b) μ = 0.4. DAQ (data acquisition) Signal analyzer Shaker Accelerometer Power ampliﬁer Primary 00:00 beam Shaker Rubber Force Main beam transducer Accelerometer Absorbing beam Rubber pieces Absorber beam Supporting columns (b) (a) Figure 7: (a) Diagram of the experimental setup and (b) laboratory test. beam and the absorbing beam as the damping is increased. Figure 5 shows the eﬀect of the structural damping. It At this frequency, the primary beam and the absorber move can be seen in Figure 5(a) that increasing the damping of the out of phase for which the role of the visco-elastic layer absorbing beam does not give signiﬁcant eﬀect to reduce the damping is important to absorb the vibration energy. At the vibration amplitude. Increasing only the damping of the pri- ﬁrst resonance, as the two beams move in-phase, the layer mary beam reduces only the amplitude at the ﬁrst resonance, damping can therefore be seen to have negligible eﬀect on but not at the second resonance as shown in Figure 5(b). the vibration amplitude. The same applies if the damping of the absorbing beam is Amplitude (m/N) Amplitude (m/N) 8 Advances in Acoustics and Vibration −3 −3 ×10 ×10 2 2 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 5 101520 5 10 15 20 Frequency (Hz) Frequency (Hz) Experiment Experiment Theory Theory (a) (b) −3 ×10 1.8 1.6 1.4 1.2 0.8 0.6 0.4 0.2 5 101520 Frequency (Hz) Experiment Theory (c) Figure 8: Comparison of experimental and theoretical results of the vibration amplitude of the double-beam system for the case of (a) ST-ST, (b) ST-WD, and (c) ST-AL. also increased (see Figure 5(c)). Figure 5(d) shows that the ratio increases, this frequency shift becomes greater, which amplitude of the second resonance is mainly controlled by also reduces the frequency gap of the two resonances. the damping of the visco-elastic layer. Figure 6 plots the results for the eﬀect of stiﬀness of the visco-elastic layer. Here the stiﬀness is presented using a 5. Experimental Validation non-dimensional parameter as a function of the rigidity and length of the main beam, that is, β = kL /K . By reducing Figure 7 shows the schematic diagram and the laboratory the stiﬀness of the layer, the second resonance shifts to low setup of the experiment. As in the simulation, the primary frequency, while no eﬀect is given to the ﬁrst resonance. beam is placed at the upper side of the absorbing beam. Again same for the case of the layer damping, the in-phase However, it was diﬃcult to realize a simply supported motion of both beams at the ﬁrst resonance will not be boundary condition. Both beams were therefore clamped aﬀected by the hardness (stiﬀness) of the layer. As the mass to the supporting columns. As the measurement point is at Amplitude (m/N) Amplitude (m/N) Amplitude (m/N) Advances in Acoustics and Vibration 9 Table 1: Physical parameters of the materials used in the experiment. Material Cross section Mass per unit length (kg/m) Young’s modulus (GPa) Steel Solid 10.27 210 Wood Solid 1.12 10.3 Aluminum Hollow 0.77 69 the midspan of the primary beam, the edge condition has frequencies. However, this increases the ﬁrst resonance to negligible eﬀect on the vibration amplitude. approach the troublesome resonant frequency of the original The primary beam was excited by broadband pseudo- single beam but further reduces the vibration of the primary random signal at location close to the midspan of the beam, a compromised situation which should be taken into primary beam using TIRA electromagnetic shaker type account in the absorber design. The same phenomena apply TV50018. The input force was measured by a Dytran for the absorbing beam, but it does not aﬀect its vibration force gauge type 1051V1. A stud was tightly bolted on amplitude. Adding more damping to the layer has been the transducer surface and then glued to the beam surface shown to reduce only the second resonant amplitude. The using an epoxy glue. A 200 mm long stinger was used to amplitude at the ﬁrst resonance can be reduced by increasing connect the force gauge and the shaker to minimize the the damping of the primary beam. Meanwhile, increasing eﬀect of moments transmitted from the shaker. A Dytran the layer elasticity (reducing stiﬀness) reduces the second accelerometer type 3225F1 was attached exactly at the mid resonance. The theoretical results have been validated by point to measure the vibration amplitude. experimental data with a reasonable agreement. The experimental test was conducted with the primary beam made of steel having thickness 20 mm, width 60 mm, and length 2 m. Four measurement cases were made with Appendix diﬀerent conﬁguration and material of the absorber beam as Substituting (3)and (4) into (1)and (2)gives follows: (a) steel single beam without absorber (ST), ∂ X (x) (b) steel main beam with steel absorber beam; double- E I q (t) + m X (x)q¨ (t) 1 1 1n 1 n 1n ∂x identical beams (ST-ST), (c) steel main beam with wood absorber beam (ST-WD), + cX (x) q˙ (t) − q˙ (t) + kX (x) q (t) − q (t) n 1n 2n n 1n 2n and iωt = Pe , (d) steel main beam with aluminum absorber beam (ST- (A.1) AL). ∂ X (x) A rubber material was used as the viscoelastic layer between E I q (t) + m X (x)q¨ (t) 2 2 2n 2 n 2n the beams having stiﬀness of 131 kN/m. ∂x The physical parameters of the beam materials are listed ˙ ˙ + cX (x) q (t) − q (t) + kX (x) q (t) − q (t) = 0. n 2n 1n n 1n 2n in Table 1. (A.2) Figure 8 presents the experimental results for the double- beam structure. It can be seen that it demonstrates good agreement with the theory especially around the resonance, Substituting (5) into (A.1) and (A.2) with P = F δ(x − although for each case, broader frequency response from L/2) yields the measured results can also be observed, which slightly overestimated the model. E I σ X (x) q (t) + m X (x)q¨ (t) 1 1 n 1n 1 n 1n 6. Conclusions ˙ ˙ + cX (x) q (t) − q (t) + kX (x) q (t) − q (t) n 1n 2n n 1n 2n The eﬀectofstructuralparametersonthe dynamicresponse iωt of a beam structure attached with a beam vibration absorber = F δ x − e , through a visco-elastic layer under a stationary harmonic (A.3) load has been studied. The amplitude of the original primary single beam at the fundamental frequency can be consider- 4 E I σ X (x) q (t) + m X (x)q¨ (t) 2 2 n 2n 2 n 2n ably reduced. It is found that increasing the rigidity ratio shifts the resulting resonances of the double-system to higher + cX (x) q˙ (t) − q˙ (t) + kX (x) q (t) − q (t) = 0. n 1n 2n n 1n 2n frequency but gives small eﬀect on reducing the vibration (A.4) amplitude of the resonances. Reducing the mass ratio reduces considerable level of the second resonance of the primary beam as well as widens the gap between the resonant Equations (A.3) and (A.4) can be simpliﬁed to 10 Advances in Acoustics and Vibration m q¨ (t) + cq˙ (t) − cq˙ (t) 1 1n 1n 2n Nomenclature 4 2 E : Modulus of elasticity of the beam (N/m ) + E I σ + k q (t) − kq (t) X (x) 1 1 1n 2n (A.5) I: Area moment of inertia of the beam (m ) iωt w: Vertical displacement of the beam (m) = F δ x − e , x: Position coordinate (m) t:Time(s) m q¨ (t) + cq˙ (t) − cq˙ (t) 2 2n 2n 1n m: Mass per unit length (kg/m) (A.6) k: Layer stiﬀness (N/m ) + E I σ + k q (t) − kq (t) X (x) = 0. 2 2 2n 1n n β: Non-dimensional layer stiﬀness parameter Multiplying both sides of (A.5) and (A.6) by mode shape ω: Radial frequency (rad/s) function X ω : Natural frequency at the nth mode (rad/s) m n μ: Mass ratio of absorbing beam to primary beam m q¨ (t) + cq˙ (t) − cq˙ (t) 1 1n 1n 2n ζ: Damping ratio of layer e: Rigidity ratio of absorbing beam to primary beam + E I σ + k q (t) − kq (t) X (x)X (x) 1 1 1n 2n n m (A.7) F : Magnitude of the external load (N) σ: Eigenvalue of the mode shape function iωt = F δ x − X (x) e , 0 m X: Mode shape function q: Generalized time function of amplitude (m) m q¨ (t) + cq˙ (t) − cq˙ (t) 2 2n 2n 1n Q: Complex displacement amplitude (m) (A.8) δ: Dirac delta function + E I σ + k q (t) − kq (t) X (x)X (x) = 0 2 2 2n 1n n m L:Lengthofbeam(m) N: Newton, unit of force. Integrating (A.7) and (A.8) through the beam length ¨ ( ) ˙ ( ) ˙ ( ) m q t + cq t − cq t 1 1n 1n 2n References [1] H. Yamaguchi, “Vibrations of a beam with an absorber con- + E I σ + k q (t) − kq (t) X (x)X (x) 1 1 1n 2n n m (A.9) sisting of a viscoelastic beam and a spring-viscous damper,” Journal of Sound and Vibration, vol. 103, no. 3, pp. 417–425, L nπ 1985. iωt iωt = F e X (x)δ x − = F e sin 0 m 0 [2] H.V.Vu, A. M. Ordo´ nez, ˜ and B. H. Karnopp, “Vibration of 2 2 a double-beam system,” Journal of Sound and Vibration, vol. m q¨ (t) + cq˙ (t) − cq˙ (t) 2 2n 2n 1n 229, no. 4, pp. 807–822, 2000. [3] Z. Oniszczuk, “Free transverse vibrations of elastically con- nected simply supported double-beam complex system,” + E I σ + k q (t) − kq (t) X (x)X (x) = 0. 2 2 n 2n 1n n m Journal of Sound and Vibration, vol. 232, no. 2, pp. 387–403, ( ) 2000. A.10 [4] T.Aida,S.Toda, N. Ogawa, andY.Imada,“Vibrationcontrol Applying the orthogonality conditions of beams by beam-type dynamic vibration absorbers,” Journal of Engineering Mechanics, vol. 118, no. 2, pp. 248–258, 1992. [5] Y. H. Chen and C. Y. Lin, “Structural analysis and optimal (A.11) X (x)X (x)dx = 0, n= m, n m / design of a dynamic absorbing beam,” Journal of Sound and Vibration, vol. 212, no. 5, pp. 759–769, 1998. and for n=m, (A-11) gives [6] Y. Q. Zhang, Y. Lu, S. L. Wang, and X. Liu, “Vibration and buckling of a double-beam system under compressive axial L L x sin(2σ x) L loading,” JournalofSound andVibration, vol. 318, no. 1-2, pp. 2 2 X (x)dx = sin (σ x)dx = − = . 341–352, 2008. 2 4σ 2 0 0 n 2 [7] M. Abu-Hilal, “Dynamic response of a double Euler-Bernoulli (A.12) beam due to a moving constant load,” Journal of Sound and Vibration, vol. 297, no. 3–5, pp. 477–491, 2006. Equations (A-9) and (A-10) can therefore be expressed [8] M. A. De Rosa and M. Lippiello, “Non-classical boundary in a matrix form in terms of mass ratio (μ = m /m )and 2 1 conditions and DQM for double-beams,” Mechanics Research rigidity ratio (e = E I /E I ), 1 1 2 2 Communications, vol. 34, no. 7-8, pp. 538–544, 2007. [9] I. Sadek, T. Abualrub, and M. Abukhaled, “A computational m 0 q¨ c −c q˙ 1 1 1 method for solving optimal control of a system of parallel ¨ ˙ 0 μm q −cc q 2 2 beams using Legendre wavelets,” Mathematical and Computer Modelling, vol. 45, no. 9-10, pp. 1253–1264, 2007. K σ + k −k q 1 1 [10] D. J. Inman, Engineering Vibrations, Pearson Prentice Hall, + (A.13) −keK σ + k q 1 2 F 2 nπ iωt = sin e L 2 International Journal of Rotating Machinery International Journal of Journal of The Scientific Journal of Distributed Engineering World Journal Sensors Sensor Networks Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Volume 2014 Journal of Control Science and Engineering Advances in Civil Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Submit your manuscripts at http://www.hindawi.com Journal of Journal of Electrical and Computer Robotics Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 VLSI Design Advances in OptoElectronics International Journal of Modelling & Aerospace International Journal of Simulation Navigation and in Engineering Engineering Observation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2010 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com http://www.hindawi.com Volume 2014 International Journal of Active and Passive International Journal of Antennas and Advances in Chemical Engineering Propagation Electronic Components Shock and Vibration Acoustics and Vibration Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014
Advances in Acoustics and Vibration – Hindawi Publishing Corporation
Published: Nov 7, 2012
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