Non-linear vibrations of parametrically excited cantilever beams subjected to non-linear delayed-feedback control

Non-linear feedback control provides an effective methodology for vibration mitigation in non-linear dynamic systems. However, within digital circuits, actuation mechanisms, filters, and controller processing time, intrinsic time-delays unavoidably bring an unacceptable and possibly detrimental delay period between the controller input and real-time system actuation. If not well-studied, these inherent and compounding delays may inadvertently channel energy into or out of a system at incorrect time intervals, producing instabilities and rendering controllers’ performance ineffective. In this work, we present a comprehensive investigation of the effect of time delays on the non-linear control of parametrically excited cantilever beams. More specifically, we examine three non-linear cubic delayed-feedback control methodologies: position, velocity, and acceleration delayed feedback. Utilizing the method of multiple scales, we derive the modulation equations that govern the non-linear dynamics of the beam. These equations are then utilized to investigate the effect of time delays on the stability, amplitude, and frequency–response behavior. We show that, when manifested in the feedback, even the minute amount of delays can completely alter the behavior and stability of the parametrically excited beam, leading to unexpected behavior and responses that could puzzle researchers if not well-understood and documented.     

Vibration control of a transversely excited cantilever beam with tip mass

The problem of controlling the vibration of a transversely excited cantilever beam with tip mass is analyzed within the framework of the Euler–Bernoulli beam theory. A sinusoidally varying transverse excitation is applied at the left end of the cantilever beam, while a payload is attached to the free end of the beam. An active control of the transverse vibration based on cubic velocity is studied. Here, cubic velocity feedback law is proposed as a devise to suppress the vibration of the system subjected to primary and subharmonic resonance conditions. Method of multiple scales as one of the perturbation technique is used to reduce the second-order temporal equation into a set of two first-order differential equations that govern the time variation of the amplitude and phase of the response. Then the stability and bifurcation of the system is investigated. Frequency–response curves are obtained numerically for primary and subharmonic resonance conditions for different values of controller gain. The numerical results portrayed that a significant amount of vibration reduction can be obtained actively by using a suitable value of controller gain. The response obtained using method of multiple scales is compared with those obtained by numerically solving the temporal equation of motion and are found to be in good agreement. Numerical simulation for amplitude is also obtained by integrating the equation of motion in the frequency range between 1 and 3. The developed results can be extensively used to suppress the vibration of a transversely excited cantilever beam with tip mass or similar systems actively.     

Non-linear vibrations of cantilever beams with feedback delays

A comprehensive investigation of the effect of feedback delays on the non-linear vibrations of a piezoelectrically actuated cantilever beam is presented. In the first part of this work, we examine the linear and non-linear free responses of a beam subjected to a delayed-acceleration feedback. We show that the trivial solution loses stability via a Hopf bifurcation leading to limit-cycle oscillations. We analyze the stability of the dynamic response in the postbifurcation, close to the stability boundaries by examining the nature of the Hopf bifurcation and away from the stability boundaries by using the method of harmonic balance and Floquet theory. We find that, increasing the gain for certain feedback delays may culminate in quasiperiodic and chaotic oscillations of the beam.In the second part, we analyze the effect of feedback delays on a beam subjected to a harmonic base excitations. We find that the nature of the forced response is largely defined by the stability of the trivial solutions of the unforced response. For stable trivial solutions (i.e., inside the stability boundaries of the trivial solutions), the homogeneous response emanating from the feedback diminishes, leaving only the particular solution resulting from the external excitation. In this case, delayed feedback acts as a vibration absorber. On the other hand, for unstable trivial solutions, the response contains two co-existing frequencies. Depending on the excitation amplitude and the commensurability of the delayed-response frequency to the excitation frequency, the response is either periodic or quasiperiodic.     

Dynamic characteristics of a cantilever beam with partial self-sensing active constrained layer damping treatment

The equations of motion governing the vibration of a cantilever beam with partially treated self-sensing active constrained layer damping treatment(SACLD) are derived by application of the extended Hamilton principle. The assumed-modes method and closed loop velocity feedback control law are used to analyze and control the flexural vibration of the beam. The influence of the bonding layer and piezoelectric layer thickness, material properties, placements of the piezoelectric patch and feedback control parameters on the actuation ability of the vibration suppression are investigated. Some design considerations for pure passive, pure active control, and self-sensing active constrained layer damping are discussed. The present work is supported by the National Natural Science Foundation of China (No. 59635140).     

Effect of electromagnetic actuations on the dynamics of a harmonically excited cantilever beam

The influence of electromagnetic actuators (EMAs) on the frequency response of a harmonically excited cantilever beam is investigated analytically, numerically and experimentally in this paper. Specifically, the intensity of the current generating the EMAs force is varied and its effect on the dynamic behavior of the system is analyzed. Analytical treatment based on perturbation analysis is performed on a simplified equation modeling the one mode vibration of the cantilever beam. Results indicated that EMAs produce a softening behavior in the system. Further, it is shown that as the current intensity of EMAs increases, the resonance curve shifts toward smaller values of frequency and the non-linear characteristic of the system becomes softer. The analytical predictions have been verified numerically and confirmed experimentally using a test rig.     

Large deflection of flexible tapered functionally graded beam

In this paper the semi-analytical analyses of the flexible cantilever tapered functionally graded beam under combined inclined end loading and intermediate loading are studied.In order to derive the fully non-linear equations governing the non-linear deformation,a curvilinear coordinate system is introduced.A general non-linear second order differential equation that governs the shape of a deflected beam is derived based on the geometric nonlinearities,infinitesimal local displacements and local rotation concepts with remarkable physical properties of functionally graded materials.The solutions obtained from semi-analytical methods are numerically compared with the existing elliptic integral solution for the case of a flexible uniform cantilever functionally graded beam.The effects of taper ratio,inclined end load angle and material property gradient on large deflection of the beam are evaluated.The Adomian decomposition method will be useful toward the design of tapered functionally graded compliant mechanisms driven by smart actuators.     

主动控制压电旋转悬臂梁的参数振动稳定性分析

在工程实际中旋转机械由于制造和加工误差,装配的不均匀性等原因,往往会脉动运行,这将使得机械系统发生参数振动. 当脉动参数满足一定关系时,这种参数振动将会失稳,进而影响机械结构的正常运转. 本文针对这一问题,引入压电材料对 脉动旋转悬臂梁系统的振动进行控制,研究主动控制悬臂梁系统的参数振动优化设计问题,采用 Hamilton 变分原理与一阶 Galerkin 离散相结合的方法,建立了受速度反馈传感器主动控制的压电旋转悬臂梁的一阶近似线性控制方程. 运用多尺度方法,得到了压电旋转悬臂梁系统在发生1/2亚谐波参数共振时稳定性边界的控制方程,并利用直接分析方法验证了解析摄动解的正确性. 将摄动解中临界阻尼比和轮毂角速度脉动幅值的无量纲参数作为评价系统稳定性能的指标. 通过数值算例,分析了轮毂半径、轮毂角速度平均值和脉动幅值、梁长以及速度传感器的反馈增益系数对系统稳定性区域的影响. 研究结果表明,梁长、轮毂半径、脉动幅值会降低系统稳定性,反馈增益系数可以提高系统稳定性,而轮毂角速度平均值与系统稳定性之间有非单调的关系. 为进一步设计压电旋转机械结构提供了理论依据.      

柔性梁受迫振动时滞变结构控制的试验研究

时滞反馈控制是一种利用时滞进行系统控制的策略,目前对该控制策略的研究多是在理论上进行探讨,少有试验研究报道. 以受简谐激励的柔性悬臂梁为对象,开展时滞反馈控制的试验研究,给出了一个多时滞控制律的设计方法. 首先给出悬臂梁系统含有时滞项的控制模态状态方程; 然后对方程进行离散化和一种特殊的状态变量增广,得到形式上不含有时滞项的标准差分方程; 最后使用离散变结构控制的方法设计控制律. 试验中采用压电片作为作动器和外界激励,应变片作为传感器,分别考虑单时滞和双时滞的情况,通过试验验证了时滞反馈控制的可行性和有效性.关键词:柔性悬臂梁;变结构控制;时滞;实验      

Large deflections of cantilever beams of non-linear materials of the Ludwick type subjected to an end moment

This paper deals with the large deflections (finite) of thin cantilever beams of non-linear materials of the Ludwick type. The beam is subjected to an end constant moment. Large deflections of beams induce geometrical non-linearity. Therefore, in formulating the analysis, the exact expression of the curvature is used in the Euler-Bernoulli law. A closed-form solution is presented for the resulting second-order non-linear differential equation. This solution is compared to previous results assuming linear elastic materials. Deflections at the free end of beams of aluminum alloy and annealed copper are obtained.     

Approximate analytical solutions and experimental analysis for transient response of constrained damping cantilever beam

Vibration mode of the constrained damping cantilever is built up according to the mode superposition of the elastic cantilever beam. The control equation of the constrained damping cantilever beam is then derived using Lagrange＇s equation. Dynamic response of the constrained damping cantilever beam is obtained according to the principle of virtual work, when the concentrated force is suddenly unloaded. Frequencies and transient response of a series of constrained damping cantilever beams are calculated and tested. Influence of parameters of the damping layer on the response time is analyzed. Analyitcal and experimental approaches are used for verification. The results show that the method is reliable.     

Non-linear dynamic response of a rotating radial Timoshenko beam with periodic pulse loading at the free-end

Consideration is given to the dynamic response of a Timoshenko beam under repeated pulse loading. Starting with the basic dynamical equations for a rotating radial cantilever Timoshenko beam clamped at the hub in a centrifugal force field, a system of equations are derived for coupled axial and lateral motions which includes the transverse shear and rotary inertia effects, as well. The hyperbolic wave equation governing the axial motion is coupled with the flexural wave equation governing the lateral motion of the beam through the velocity-dependent skew-symmetric Coriolis force terms. In the analytical formulation, Rayleigh-Ritz method with a set of sinusoidal displacement shape functions is used to determine stiffness, mass and gyroscopic matrices of the system. The tip of the rotating beam is subjected to a periodic pulse load due to local rubbing against the outer case introducing Coulomb friction in the system. Transient response of the beam with the tip deforming due to rub is discussed in terms of the frequency shift and non-linear dynamic response of the rotating beam. Numerical results are presented for this vibro-impact problem of hard rub with varying coefficients of friction and the contact-load time. The effects of beam tip rub forces transmitted through the system are considered to analyze the conditions for dynamic stability of a rotating blade with intermittent rub.     

Non-linear dynamics of a cracked cantilever beam under harmonic excitation

The presence of cracks in a structure is usually detected by adopting a linear approach through the monitoring of changes in its dynamic response features, such as natural frequencies and mode shapes. But these linear vibration procedures do not always come up to practical results because of their inherently low sensitivity to defects. Since a crack introduces non-linearities in the system, their use in damage detection merits to be investigated. With this aim the present paper is devoted to analysing the peculiar features of the non-linear response of a cracked beam.The problem of a cantilever beam with an asymmetric edge crack subjected to a harmonic forcing at the tip is considered as a plane problem and is solved by using two-dimensional finite elements; the behaviour of the breathing crack is simulated as a frictionless contact problem. The modification of the response with respect to the linear one is outlined: in particular, excitation of sub- and super-harmonics, period doubling, and quasi-impulsive behaviour at crack interfaces are the main achievements. These response characteristics, strictly due to the presence of a crack, can be used in non-linear techniques of crack identification.     

存在时滞的柔性梁的振动主动控制

本文对控制存在时滞的柔性梁的振动主动控制进行研究,主动控制策略采用独立模态空间控制,模态控制律采用离散变结构控制进行设计,给出了从实际测量中提取模态坐标和将模态控制力转换成实际控制力的方法,以及离散切换面和离散变结构模态控制律的确定方法,由于模态控制律直接通过时滞微分方程而得出,因此所给控制方法易于保证控制系统的稳定性。算例结果显示,文中所述控制方法能够有效地对梁的振动进行控制;若按无时滞时的控制设计对时滞系统进行控制,系统响应有可能出现发散。     

Dynamic analysis of rubber-like material using absolute nodal coordinate formulation based on the non-linear constitutive law

Non-linear constitutive models of the elastic forces for a hyperelastic material are presented. Three elastic force models including Neo-Hookean, Mooney-Riblin 2nd, and Yeoh models are derived based on non-linear continuum mechanics. Elastic forces are applied to the three-dimensional absolute nodal coordinate beam element, and the transient response of the cantilever beam is analyzed. Simulation results are compared to experiment data, and the dynamic characteristics of elastic force models presented in this paper are discussed.     

Non-linear buckling and postbuckling behavior of thin-walled beams considering shear deformation

The static stability of thin-walled composite beams, considering shear deformation and geometrical non-linear coupling, subjected to transverse external force has been investigated in this paper. The theory is formulated in the context of large displacements and rotations, through the adoption of a shear deformable displacement field (accounting for bending and warping shear) considering moderate bending rotations and large twist. This non-linear formulation is used for analyzing the prebuckling and postbuckling behavior of simply supported, cantilever and fixed-end beams subjected to different load condition. Ritz's method is applied in order to discretize the non-linear differential system and the resultant algebraic equations are solved by means of an incremental Newton-Rapshon method. The numerical results show that the beam loses its stability through a stable symmetric bifurcation point and the postbuckling strength is in relation with the buckling load value. Classical predictions of lateral buckling are conservative when the prebuckling displacements are not negligible and the non-linear buckling analysis is required for reliable solutions. The analysis is supplemented by investigating the effects of the variation of load height parameter. In addition, the critical load values and postbuckling response obtained with the present beam model are compared with the results obtained with a shell finite element model (Abaqus).     

压电智能梁振动控制的优化设计

本文研究了压电智能梁振动控制中压电传感作动元件的优化设计,并考虑了控制增益和结构阻尼的影响。采用负速度反馈控制策略及等效阻尼比优化准则。对简支梁和悬臂梁采用遍历搜索的方法计算了不同增益和结构阻尼下的优化,结果对进一步研究优化设计和振动控制有一定的参考价值。     

Large deflection of a non-linear,elastic, asymmetric Ludwick cantilever beam subjected to horizontal force,vertical force and bending torque at the free end

The investigated cantilever beam is characterized by a constant rectangular cross-section and is subjected to a concentrated constant vertical load, to a concentrated constant horizontal load and to a concentrated constant bending torque at the free end. The same beam is made by an elastic non-linear asymmetric Ludwick type material with different behavior in tension and compression. Namely the constitutive law of the proposed material is characterized by two different elastic moduli and two different strain exponential coefficients. The aim of this study is to describe the deformation of the beam neutral surface and particularly the horizontal and vertical displacements of the free end cross-section. The analysis of large deflection is based on the Euler–Bernoulli bending beam theory, for which cross-sections, after the deformation, remain plain and perpendicular to the neutral surface; furthermore their shape and area do not change. On the stress viewpoint, the shear stress effect and the axial force effect are considered negligible in comparison with the bending effect. The mechanical model deduced from the identified hypotheses includes two kind of non-linearity: the first due to the material and the latter due to large deformations. The mathematical problem associated with the mechanical model, i.e. to compute the bending deformations, consists in solving a non-linear algebraic system and a non-liner second order ordinary differential equation. Thus a numerical algorithm is developed and some examples of specific results are shown in this paper.     

SELF-CORRECTED PID VIBRATION CONTROL BASED ON DUAL-FREEDOM DUAL-DRIVE INTELLIGENT CANTILEVER BEAM 1)

随着科技不断进步,智能结构的振动控制在航天航空、机械制造、车辆与船舶等领域得到了广泛应用。由于多输入多输出存在多样性和复杂性,严重威胁系统稳定性。为了解决这一问题,针对两输入单输出的双驱动智能悬臂梁系统提出一种自适应控制策略,首先基于压电线性本构方程,应用假设模态方法建立双驱动智能悬臂梁的力学模型,得到了基于闭环控制系统的状态方程,同时利用递推最小二乘法在线辨识系统参数设计比例积分微分(proportional--integral--derivative, PID)控制器实现自校正PID控制。通过数值仿真对比在有无PID 控制下两输入单输出双驱动智能悬臂梁系统的振动情况,分析自校正PID 控制的控制效果。通过实验验证自校正PID 控制对双输入单输出的双驱动智能悬臂梁系统的控制效果;再设置两组不同的单输入单输出自校正PID控制实验作对比。结果表明：自校正PID 控制方法可以较为有效地抑制智能悬臂梁的自由振动,相比单输入单输出的两组,两输入单输出自校正PID控制的效果更为明显和有效。     

两自由度参数激励摆旋转运动分析和实验

研究具有支撑参数激励摆系统的支撑结构振动对摆旋转的影响,其中支撑结构是受到扭簧约束的刚性悬臂梁,参数激励摆与刚性悬臂梁的悬臂段铰接．首先,通过拉格朗日方程建立了系统两自由度的动力学方程．其次,利用多尺度法对建立的模型进行理论分析,得到悬臂梁的振动与上摆不同运动形式的关系,从而得到上摆不同运动形式下的参数平面分类和悬臂梁在上摆转动时的振动频响．最后,通过建立实验装置,观察理论预测,实验结果验证了理论分析的正确性．实验与理论对照得到,当参数激励频率接近悬臂梁的一阶固有频率时,悬臂梁的振幅变大,会破坏摆的转动稳定性．     

APPROXIMATE SOLUTIONS FOR TRANSIENT RESPONSE OF CONSTRAINED DAMPING LAMINATED CANTILEVER PLATE

The series composed by beam mode function is used to approximate the displacement function of constrained damping of laminated cantilever plates, and the transverse deformation of the plate on which a concentrated force is acted is calculated using the principle of virtual work.By solving Lagrange＇s equation, the frequencies and model loss factors of free vibration of the plate are obtained, then the transient response of constrained damping of laminated cantilever plate is obtained, when the concentrated force is withdrawn suddenly.The theoretical calculations are compared with the experimental data, the results show：both the frequencies and the response time of theoretical calculation and its variational law with the parameters of the damping layer are identical with experimental results.Also, the response time of steel cantilever plate, unconstrained damping cantilever plate and constrained damping cantilever plate are brought into comparison, which shows that the constrained damping structure can effectively suppress the vibration.