Nonlinear primary resonance analysis for a coupled thermo-piezoelectric-mechanical model of piezoelectric rectangular thin plates

A model of piezoelectric rectangular thin plates with the consideration of the coupled thermo-piezoelectric-mechanical effect is established. Based on the von Karman large deflection theory, the nonlinear vibration governing equation is obtained by using Hamilton's principle and the Rayleigh-Ritz method. The harmonic balance method(HBM) is used to analyze the first-order approximate response and obtain the frequency response function. The system shows non-linear phenomena such as hardening nonlinearity, multiple coexistence solutions, and jumps. The effects of the temperature difference,the damping coefficient, the plate thickness, the excited charge, and the mode on the primary resonance response are theoretically analyzed. With the increase in the temperature difference, the corresponding frequency jumping increases, while the resonant amplitude decreases gradually. Finally, numerical verifications are carried out by the Runge-Kutta method, and the results agree very well with the theoretical results.     

Experimental and theoretical study on large amplitude vibrations of clamped rubber plates

In this paper, the large amplitude forced vibrations of thin rectangular plates made of different types of rubbers are investigated both experimentally and theoretically. The excitation is provided by a concentrated transversal harmonic load. Clamped boundary conditions at the edges are considered, while rotary inertia, geometric imperfections and shear deformation are neglected since they are negligible for the studied cases. The von Kármán nonlinear strain-displacement relationships are used in the theoretical study; the viscoelastic behaviour of the material is modelled using the Kelvin-Voigt model, which introduces nonlinear damping. An equivalent viscous damping model has also been created for comparison. In-plane pre-loads applied during the assembly of the plate to the frame are taken into account. In the experimental study, two rubber plates with different material and thicknesses have been considered; a silicone plate and a neoprene plate. The plates have been fixed to a heavy rectangular metal frame with an initial stretching. The large amplitude vibrations of the plates in the spectral neighbourhood of the first resonance have been measured at various harmonic force levels. A laser Doppler vibrometer has been used to measure the plate response. Maximum vibration amplitude larger than three times the thickness of the plate has been achieved, corresponding to a hardening type nonlinear response. Experimental frequency-response curves have been very satisfactorily compared to numerical results. Results show that the identified retardation time increases when the excitation level is increased, similar to the equivalent viscous damping but to a lesser extent due to its nonlinear nature. The nonlinearity introduced by the Kelvin-Voigt viscoelasticity model is found to be not sufficient to capture the dissipation present in the rubber plates during large amplitude vibrations.     

Nonlinear dynamic analysis of a quarter vehicle system with external periodic excitation

In this paper, a nonlinear dynamic model of a quarter vehicle with nonlinear spring and damping is established. The dynamic characteristics of the vehicle system with external periodic excitation are theoretically investigated by the incremental harmonic balance method and Newmark method, and the accuracy of the incremental harmonic balance method is verified by comparing with the result of Newmark method. The influences of the damping coefficient, excitation amplitude and excitation frequency on the dynamic responses are analyzed. The results show that the vibration behaviors of the vehicle system can be control by adjusting appropriately system parameters with the damping coefficient, excitation amplitude and excitation frequency. The multi-valued properties, spur-harmonic response and hardening type nonlinear behavior are revealed in the presented amplitude-frequency curves. With the changing parameters, the transformation of chaotic motion, quasi-periodic motion and periodic motion is also observed. The conclusions can provide some available evidences for the design and improvement of the vehicle system.     

Nonlinear dynamic behavior of a bio-inspired embedded X-shaped vibration isolation system

To improve the vibration isolation performance and bandwidth, loading capacity and supporting stability of passive vibration isolation system by utilizing nonlinearity, a bio-inspired embedded X-shaped vibration isolation (BIE-XVI) structure is proposed considering muscle/tendon contractile functions, joint rotational friction and connecting rod mass simultaneously. Furthermore, the dynamic model with pure linear elements and geometric relationship are established and the nonlinear variation properties are investigated. The effects of the key parameters of the BIE-XVI structure on frequency response characteristics and vibration isolation range are analyzed thoroughly by incremental harmonic balance method in various working conditions. From the parametric investigations, it can be found that the sensitivities of the nonlinear resonance properties are markedly different with respect to the different structure parameters. For longer rod length, larger assembly angle and higher stiffnesses, the hardening nonlinearity is weakened, but the resonance peak does not necessarily decrease. Besides, the softening nonlinearity and hardening nonlinearity can be interconverted with changing isolated mass and excitation amplitude. The BIE-XVI structure can widen the isolation frequency range and reduce the resonance peak to improve the vibration isolation properties by adjusting/designing the structural parameters, which could realize quasi-zero-stiffness property for vibration isolation.

     

Limit cycle oscillations of rectangular cantilever wings containing cubic nonlinearity in an incompressible flow

Limit cycle oscillations (LCO) as well as nonlinear aeroelastic analysis of rectangular cantilever wings with a cubic nonlinearity are investigated. Aeroelastic equations of a rectangular cantilever wing with two degrees of freedom in an incompressible potential flow are presented in the time domain. The harmonic balance method is modified to calculate the LCO frequency and amplitude for rectangular wings. In order to verify the derived formulation, flutter boundaries are obtained via a linear analysis of the derived system of equations for five different cases and compared with experimental data. Satisfactory results are gained through this comparison. The problem of finding the LCO frequency and amplitude is solved via applying the two methods discussed for two different cases with hardening cubic nonlinearities. The results from first-, third- and fifth-order harmonic balance methods are compared with the results of an exact numerical solution. A close agreement is obtained between these harmonic balance methods and the exact numerical solution of the governing aeroelastic equations. Finally, the nonlinear aeroelastic analysis of a rectangular cantilever wing with a softening nonlinearity is studied.     

Tuning methods for tuned inerter dampers coupled to nonlinear primary systems

This study develops displacement- and kinetic energy-based tuning methods for the design of the tuned inerter dampers (TIDs) coupled to both linear and nonlinear primary systems. For the linear primary system, the design of the TID is obtained analytically. The steady-state frequency–response relationship of the nonlinear primary system with a softening or hardening stiffness nonlinearity is obtained using the harmonic balance (HB) method. Analytical and numerical tuning approaches based on HB results are proposed for optimal designs of the TID to achieve equal peaks in the response curves of the displacement and the kinetic energy of the primary system. Via the developed approaches, the optimal stiffness of the TID can be obtained according to the stiffness nonlinearity of the primary system and the inertance of the absorber. Unlike the linear primary oscillator case, for a nonlinear primary oscillator the shape of its resonant peaks is mainly affected by the damping ratio of the TID, while the peak values depend more on the stiffness ratio. The proposed designs are shown to be effective in a wide range of stiffness nonlinearities and inertances. This study demonstrates the benefits of using inerters in vibration suppression devices, and the adopted methods are directly applicable for nonlinear systems with different types of nonlinearities.

     

Shear effects on large amplitude forced vibration of symmetrically laminated rectilinearly orthotropic circular plates

I.IntroductionPlatesandshells'consistingoflaminatedcompositematerialsareimportantstructuralmembersinmodernengineering.Thustheresearchfornonlinearproblemsoftheseplatesandshellshasbecomeincreasinglyimportantl'~II].Especially.therehasalreadybeensomeresearchworktlz--lslonthenonlinearproblemoflaminatedcircularplates.Buttheeffectsoftransversestiearonthenonlinearvibrationoftheplateshavenotbeeddiscussedbecauseofthedifficultyofnonlinearmathematics.However,itisexpectedthatthesheareffectsonlaminatedcom…     

Adaptive tuning of elasto-plastic damper

Hysteretic dampers are frequency independent, and thereby potentially effective for several structural vibration modes, provided that the inherent amplitude dependence can be controlled. An adaptive tuning procedure is proposed, aiming at elimination of the amplitude dependence by adjusting the damper parameter(s) with respect to the magnitude of the damper motion. The procedure is demonstrated in terms of the bilinear elasto-plastic damper model, and optimality corresponds to maximum modal damping. A parametric solution for the damping ratio is obtained by a two-component system reduction technique, and maximization leads to an amplitude dependent expression for the optimal yield level. The amplitude is predicted from the most recent extremum of the damper response, and simultaneously used to adjust the yield level. Numerical examples demonstrate that the adaptive tuning procedure succeeds in controlling the amplitude dependence, resulting in equal damping for the first vibration modes.     

Frequency lock-in during nonlinear vibration of an airfoil coupled with van der Pol Oscillator

The frequency lock-in during the nonlinear vibration of a turbomachinery blade is modeled using a spring-mounted airfoil coupled with a van der Pol Oscillator (VDP) oscillator. The proposed reduced-order model uses the nonlinear VDP oscillator to represent the oscillatory nature of wake dynamics caused by the vortex shedding. The damping term in the VDP oscillator is assumed to be nonlinear. The coupled equations governing the pitch and plunge motion of an airfoil are used to approximate the vibration of a turbomachinery blade. Springs having cubic-order nonlinearity for their stiffnesses are used to mount the airfoil. The unsteady lift acting on the blade is modeled using a self-excited nonlinear wake oscillator. The model for wake dynamics takes into account the influence of blade inertia. The nonlinear coupled three degrees of freedom (dof) aeroelastic system is studied for instability resulting in the frequency lock-in phenomenon. The equations are transformed into non-dimensional form, and then the frequencies of the coupled system are plotted to demonstrate the frequency lock-in. Further, the method of multiple scales is used to derive modulation equations which represent the amplitude and phase of the oscillation. The results obtained using the method of multiple scales are compared with direct numerical solutions to verify the present modeling method. The steady-state amplitudes of the response are plotted against the detuning parameter, which represents the frequency response curve. Further, the sensitivity of non-dimensional parameters such as coupling coefficients, mass ratio, reduced velocity, static unbalance, structural damping coefficient and the ratio of uncoupled pitch and plunge natural frequencies on the frequency response is investigated. The study revealed that parameters such as mass ratio, reduced velocity, structural damping coefficient, and coupling coefficients have a stronger influence in suppressing the amplitude of vibration. Meanwhile, parameters such as the frequency ratio, static unbalance, reduced velocity, and mass ratio significantly affect the range of frequency in which the lock-in phenomenon happens. Further, linear perturbation analysis is done to understand the qualitative effect of the system parameters such as coupling coefficients, mass ratio, frequency ratio, and static unbalance on the range of lock-in.     

Amplitude reduction of primary resonance of nonlinear oscillator by a dynamic vibration absorber using nonlinear coupling

The paper presents the characteristics of a new type of nonlinear dynamic vibration absorber for a main system subjected to a nonlinear restoring force under primary resonance. The absorber is connected to the main system by a link in order to be excited with twice the frequency of the motion of the main system. The natural frequency of the absorber is tuned to be twice the natural frequency of the main system, in contrast to autoparametric vibration absorber, whose natural frequency is tuned to be one-half the natural frequency of the main system. The presented absorber is not excited through the autoparametric resonance, i.e., no trivial equilibrium state exists. Therefore, the absorber always oscillates because of the motion of the main system and cannot be trapped by Coulomb friction acting on the absorber, in contrast to the autoparametric vibration absorber. Under small excitation amplitude, this absorber does not produce an overhang in the frequency response curve, which occurs because of the use of the conventional autoparametric vibration absorber; the overhang renders the response amplitude larger than that in the case without an absorber. In addition, the absorber removes the hysteresis in the frequency response curve caused by the nonlinearity of the restoring force acting on the main system. Regarding large excitation amplitude, the response amplitude in the main system can be decreased by increasing the damping of the absorber, but that decrease is limited by the nonlinearity in the restoring force acting on the main system. This paper also describes experimental validation of the absorber under small excitation amplitude using a simple apparatus.     

具有组合非线性阻尼的非线性能量阱振动抑制响应分析

非线性能量阱是一种振动能量吸收装置,其在结构振动抑制中具有十分重要的作用.文章对具有组合非线性阻尼非线性能量阱的系统进行振动抑制相关的分析.首先对具有组合非线性阻尼非线性能量阱的系统进行理论模型的描述,对系统模型的运动方程利用复变量平均法进行推导,得到系统的慢变方程.其次对系统的慢变方程运用多尺度法进行强调制响应的分析,通过对系统进行慢不变流形和相轨迹的研究,描述系统强调制响应发生的条件基础.此外,还利用一维映射对系统进行分析,揭示外激励幅值对强调制响应存在时频率失谐系数取值区间的影响规律.最后利用能量谱、时间响应和庞加莱映射对耦合组合非线性阻尼非线性能量阱系统进行了振动抑制的相关研究,揭示组合非线性阻尼的非线性能量阱不同阻尼比、阻尼和刚度对其振动抑制效果的影响规律,得出组合非线性阻尼非线性能量阱和主结构响应存在一致性的现象,并验证所提出的组合非线性阻尼非线性能量阱模型具有较好的振动抑制能力.     

一类分段光滑隔振系统的非线性动力学设计方法

分段光滑隔振系统是一类具备分段刚度或阻尼的非线性动力学系统,在振动控制领域中具有广泛代表性,诸如限位隔振系统、分级汽车悬挂等. 分段光滑的刚度或阻尼特性能够实现隔振系统的特定动力学性能及提升隔振性能,如抑制共振响应、提升共振区隔振性能等,但是亦会给隔振系统的动力学行为带来诸多不利影响. 以分段双线性分段光滑隔振系统为理论模型,系统研究了摒除不利于隔振的非线性动力学现象设计方法,包括幅值跳跃、周期运动的倍周期分岔等. 首先,利用平均法与奇异性理论给出了主共振频响曲线拓扑特征的完整拼图. 研究结果表明,参数空间分为4 个区域,其中2 个区域存在幅值跳跃,而其产生跳跃原因分别由鞍结分岔与擦边分岔所导致;基于此提出避免主共振跳跃的设计方法. 其次,建立了隔振有效区内周期运动的庞加莱映射,通过特征值分析给出了避免倍周期分岔发生的条件,证实增大阻尼可以抑制倍周期分岔的发生. 最后通过数值仿真分析了噪声对多稳态运动的影响. 研究结果发现在噪声影响下,分段光滑隔振系统的响应会在不同稳态间跃迁,非常不利于隔振. 因此,在完成跳跃与倍周期分岔的防治设计后,应采用数值仿真校验系统是否存在多稳态运动.      

Forced response of quadratic nonlinear oscillator: comparison of various approaches

The primary resonances of a quadratic nonlinear system under weak and strong external excitations are investigated with the emphasis on the comparison of different analytical approximate approaches. The forced vibration of snap-through mechanism is treated as a quadratic nonlinear oscillator. The Lindstedt-Poincaré method, the multiple-scale method, the averaging method, and the harmonic balance method are used to determine the amplitude-frequency response relationships of the steady-state responses. It is demonstrated that the zeroth-order harmonic components should be accounted in the application of the harmonic balance method. The analytical approximations are compared with the numerical integrations in terms of the frequency response curves and the phase portraits. Supported by the numerical results, the harmonic balance method predicts that the quadratic nonlinearity bends the frequency response curves to the left. If the excitation amplitude is a second-order small quantity of the bookkeeping parameter, the steady-state responses predicted by the second-order approximation of the LindstedtPoincaré method and the multiple-scale method agree qualitatively with the numerical results. It is demonstrated that the quadratic nonlinear system implies softening type nonlinearity for any quadratic nonlinear coefficients.     

Study on the force transmissibility of vibration isolators with geometric nonlinear damping

By a special layout of the damper in a vibration isolation system, the velocity-nth power damping of the damper can be transformed into the velocity-displacement dependent damping in the moving direction due to geometric nonlinearity. This study is mainly concerned with the mechanism of this geometric nonlinear damping and tries to find some guidelines for designing isolators with high performance, regardless of the isolator is passive or active. The model used in this study is an unconstrained two-degree-of-freedom isolation system, which is suitable for investigating vibration isolation in space vehicles. The motion equation is solved by the averaging method to obtain the amplitude–frequency equation. The influence of damping coefficients on the force transmissibility is analyzed. Results show that this kind of damping has some advantages in improving isolation performance at both the resonance and higher frequencies. Meanwhile, it can also degrade the isolation performance if unreasonable parameters are chosen.     

Experimental investigation of vibration attenuation using nonlinear tuned mass damper and pendulum tuned mass damper in parallel

The present paper experimentally and numerically explores the response attenuation of a hardening Düffing oscillator using a nonlinear tuned mass damper (NTMD) and an adaptive-length pendulum tuned mass damper (APTMD). The three degrees-of-freedom system is excited by harmonic ground motions. The cubic nonlinearity of the primary structure is obtained using an adaptive passive stiffness (APS) device. When an NTMD is used alone, a high amplitude detached resonance branch in the lower frequency range is identified in the experiment, which validates the results reported in earlier numerical research. In order to attenuate this high amplitude resonance branch, an APTMD with an adaptive frequency realized by means of a variable pendulum length is used in parallel with the NTMD. In the experiment, length of the APTMD is adjusted such that its natural frequency matches the dominant frequency of the harmonic ground motions. Results indicate that the high amplitude resonance branch in the case of an NTMD alone can be greatly attenuated using the APTMD, and significant attenuation of the structural responses over a large frequency range can be obtained. In addition, the APTMD can prevent the occurrence of the “jump phenomenon” existing in the forcing response curve of the nonlinear dynamic system, thereby protecting the primary nonlinear structure effectively when the forcing amplitude varies. Therefore, the present paper provides an effective and viable solution to control the hazardous bifurcations in a Düffing oscillator-NTMD dynamic system.     

Theoretical and experimental studies on nonlinear oscillations of symmetric cross-ply composite laminated plates

The nonlinear oscillations and resonant responses of the symmetric cross-ply composite laminated plates are investigated theoretically and experimentally. The governing equations of motion for the composite laminated plate are derived by using the von Karman type equation, Reddy’s third-order shear deformation plate theory, and Galerkin method with the geometric nonlinearity. The four-dimensional averaged equation is obtained by using the method of multiple scales. The frequency-response functions are analyzed under the consideration of strongly coupled of two modes. The influences of the resonance case on the softening and hardening type of nonlinearity are analyzed with different parameters for the composite laminated plates. The numerical results indicate that there exist the hardening and softening types of the composite laminated plate in the specific resonant case. The variation of the response amplitudes is studied for the composite laminated plate under combined the transverse and in-plane excitations. A sweep frequency experiment is performed to obtain the hardening and softening nonlinearities of a composite laminated plate. The experimental results coincide with the numerical results qualitatively. The influences of the excitation amplitudes on the softening and hardening types of nonlinearity are also analyzed for the composite laminated plate. The amplitude spectrums of the test plate also demonstrate that the change of the nonlinear dynamic responses may be caused by the subharmonic resonance.     

Circuit implementation of a piezoelectric buckled beam and its response under fractional components considerations

In this paper, an analog testing circuit and determinist averaging method for a vibration energy harvesting system with fractional derivative and nonlinear damping under a sinusoidal vibration source is proposed in order to predict the system response and its stability. The objective of this paper is to show that there is a possibility to make a pre-experimental design of the structure by using analog circuit and discussing the performance of a system with fractional derivative. Bifurcation diagram, poincaré maps and power spectral density are provided to deeply characterize the dynamic of the system. These results are corroborated by using 0–1 test. By using the Melnikov method, we find the necessary condition for which homoclinic bifurcation occurs. Understanding and predicting this bifurcation is very judicious in the energy harvesting field because it may lead to different types of motion in the perturbed system. The appearance of chaotic vibrations increases the frequency’s bandwidth of the harvester thereby, allowing to harvest more energy. The pre-experimental investigation is carried out through appropriate software electronic circuit (Multisim^®). The corresponding electronic circuit is designed exhibiting transient to chaos in accord with numerical simulations. The impact of fractional derivatives is presented upon the power generated by the system. In addition, by combining the harmonic force and a random excitation, the stochastic resonance appears, giving rise to large amplitude of vibration and consequently, enhancing the performance of the system. The results obtained in this work show the interest of using the electronic circuit to make the experiment analysis of the physical structure and also, the effects of the use of piezoelectric material exhibiting fractional properties in this research field.     

Reduced-order models for nonlinear vibrations of fluid-filled circular cylindrical shells: Comparison of POD and asymptotic nonlinear normal modes methods

The aim of the present paper is to compare two different methods available for reducing the complicated dynamics exhibited by large amplitude, geometrically nonlinear vibrations of a thin shell. The two methods are: the proper orthogonal decomposition (POD), and an asymptotic approximation of the nonlinear normal modes (NNMs) of the system. The structure used to perform comparisons is a water-filled, simply supported circular cylindrical shell subjected to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency. A reference solution is obtained by discretizing the partial differential equations (PDEs) of motion with a Galerkin expansion containing 16 eigenmodes. The POD model is built by using responses computed with the Galerkin model; the NNM model is built by using the discretized equations of motion obtained with the Galerkin method, and taking into account also the transformation of damping terms. Both the POD and NNMs allow to reduce significantly the dimension of the original Galerkin model. The computed nonlinear responses are compared in order to verify the accuracy and the limits of these two methods. For vibration amplitudes equal to 1.5 times the shell thickness, the two methods give very close results to the original Galerkin model. By increasing the excitation and vibration amplitude, significant differences are observed and discussed. The response is investigated also for a fixed excitation frequency by using the excitation amplitude as bifurcation parameter for a wide range of variation. Bifurcation diagrams of Poincaré maps obtained from direct time integration and calculation of the maximum Lyapunov exponent have been used to characterize the system.     

A quasi-zero-stiffness isolator with a shear-thinning viscous damper

Quasi-zero-stiffness(QZS) vibration isolators have been widely studied,because they show excellent high static and low dynamic stiffnesses and can effectively solve low-frequency and ultralow-frequency vibration. However, traditional QZS(T-QZS)vibration isolators usually adopt linear damping, owing to which achieving good isolation performance at both low and high frequencies is difficult. T-QZS isolators exhibit hardening stiffness characteristics, and their vibration isolation performance is e...     

简谐激励下黏弹性非线性能量阱的研究

黏弹性材料作为一种良好的减振材料, 广泛应用于机械、航空和土木等领域. 本文用黏弹性Maxwell器件代替传统非线性能量阱中的阻尼元件, 提出一种新型的黏弹性非线性能量阱, 并对该模型在简谐激励下的减振性能进行分析. 首先, 根据牛顿第二定律建立系统的动力学方程, 采用谐波平衡法求解系统的幅频响应曲线, 并利用MATLAB中的Runge-Kutta数值方法验证解析解的正确性, 结果吻合良好. 然后, 分析黏弹性非线性能量阱的减振性能和参数的影响. 最后, 分析了不同质量比下非线性刚度比和阻尼比同时变化时减振效果的变化趋势, 并讨论了黏弹性非线性能量阱的最佳取值范围. 研究结果表明: 主系统的最大振幅随着非线性刚度的增加先减小后增大; 当参数选取恰当时, 黏弹性非线性能量阱比传统非线性能量阱的减振效果更优; 另外, 随着质量比的增加, 主系统最大振幅的最小值出现先减小后趋于不变的现象, 且非线性刚度比和阻尼比的最佳取值范围有所增大. 以上结论对黏弹性非线性能量阱的实际应用提供了一定的理论依据.