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the transient response of constrained damping of laminated cantilever plate is obtained, when the concentrated force is withdrawn suddenly.The theoretical calculations are compared with the experimental data, the results show：both the frequencies and the response time of theoretical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.     

ELASTIC-PLASTIC DYNAMIC RESPONSE OF A CANTILEVER BEAM SUBJECTED TO OBLIQUE IMPACT AT ITS TIP

By employing large deformation governing equations expressed in the form of finite difference, the dynamic responses of an elastic, perfectly plastic cantilever subjected to an oblique impact at its tip was numerically studied. Through analyzing the instantaneous distribution of the yield function （φ = ｜M/Mo｜ ＋ （N/No）^2）, bending moment and axial force during the early stage of the response, the elastic-plastic deformation mechanism and the influence of axial component of an oblique impact on the dynamic response of a cantilever beam were discussed. The present analysis shows that the deformation mechanism of an elastic-plastic cantilever subjected to an oblique impact consists of four phases, i.e. ‘the expanding compressed plastic region＇ mode; the ‘generalized traveling plastic hinge＇ and ‘shrinking plastic region＇ mixed mode; the ‘stationary plastic hinge＇ mode and ‘elastic vibration＇ mode. Compared with the two-phase deformation mode obtained by using the rigid, perfectly plastic approach, the mode of shrinking plastic region that occurred instantly after the oblique impact and the mode of stationary hinge were both confirmed. The primary features of the deformation mechanism are captured by both analysis methods. It has also been found that the beam＇s deformation is mainly controlled by the axial component of the oblique impact in the early phase of the dynamic response, the deformation mechanism is obviously different from the case of a transverse impact. With further development of the response, the axial component attenuates rapidly and gives negligible contribution to the yielding of the beam cross-section. At the same time, the bending moments along the cantilever develop gradually and dominate the beam＇s deformation. The numerical results indicate that the mass, impact speed and oblique angle are the important factors that influence the elastic-plastic dynamic response of a cantilever beam.     

FREE VIBRATION ANALYSIS OF SIMPLY SUPPORTED BEAM WITH BREATHING CRACK USING PERTURBATION METHOD

In this paper, a new analytical method for vibration analysis of a cracked simply supported beam is investigated. By considering a nonlinear model for the fatigue crack, the governing equation of motion of the cracked beam is solved using perturbation method. The solution of the governing equation reveals the superhaxmonics of the fundamental frequency due to the nonlinear effects in the dynamic response of the cracked beam. Furthermore, considering such a solution, an explicit expression is also derived for the system damping changes due to the changes in the crack parameters, geometric dimensions and mechanical properties of the cracked beam. The results show that an increase in the crack severity and approaching the crack location to the middle of the beam increase the system damping. In order to validate the results, changes in the fundamental frequency ratios against the fatigue crack severities are compared with those of experimental results available in the literature. Also, a comparison is made between the free response of the cracked beam with a given crack depth and location obtained by the proposed analytical solution and that of the numerical method. The results of the proposed method agree with the experimental and numerical results.     

Approximate and numerical analysis of nonlinear forced vibration of axially moving viscoelastic beams

Steady-state periodical response is investigated for an axially moving viscoelastic beam with hybrid supports via approximate analysis with numerical confirmation. It is assumed that the excitation is spatially uniform and temporally harmonic. The transverse motion of axially moving beams is governed by a nonlinear partial-differential equation and a nonlinear integro-partial-differential equation. The material time derivative is used in the viscoelastic constitutive relation. The method of multiple scales is applied to the governing equations to investigate primary resonances under general boundary conditions. It is demonstrated that the mode uninvolved in the resonance has no effect on the steady-state response. Numerical examples are presented to demonstrate the effects of the boundary constraint stiffness on the amplitude and the stability of the steady-state response. The results derived for two governing equations are qualitatively the same,but quantitatively different. The differential quadrature schemes are developed to verify those results via the method of multiple scales.     

MODELING AND DYNAMICS ANALYSIS OF SHELLS OF REVOLUTION BY PARTIALLY ACTIVE CONSTRAINED LAYER DAMPING TREATMENT

A new model for a smart shell of revolution treated with active constrained layer damping （ACLD） is developed, and the damping effects of the ACLD treatment are discussed. The motion and electric analytical formulation of the piezoelectric constrained layer are presented first. Based on the authors~ recent research on shells of revolution treated with passive constrained layer damping （PCLD）, the integrated first-order differential matrix equation of a shell of revolution partially treated with ring ACLD blocks is derived in the frequency domain. By virtue of the extended homogeneous capacity precision integration technology, a stable and simple numerical method is further proposed to solve the above equation. Then, the vibration responses of an ACLD shell of revolution are measured by using the present model and method. The results show that the control performance of the ACLD treatment is complicated and frequency-dependent. In a certain frequency range, the ACLD treatment can achieve better damping characteristics compared with the conventional PCLD treatment.     

ON THE NATURAL FREQUENCIES,COMPLEX MODE FUNCTIONS,AND CRITICAL SPEEDS OF AXIALLY TRAVELING LAMINATED BEAMS:PARAMETRIC STUDY

The dynamic response of an axially traveling laminated composite beam is investigated analytically,with special consideration to natural frequencies,complex mode functions and critical speeds of the system.The equation of motion for a symmetrically laminated system,which is in the form of a continuous gyroscopic system,is considered;the equation of motion is not discretized — no spatial mode function is assumed.This leads to analytical expressions for the complex mode functions and critical speeds.A paramet...     

DEPENDENCE OF DAMPING CHARACTERISTICS OF A BEAM WITH DAMPING RUBBER MAGNETIC POWDER ON RUBBER PROPERTY

A cantilever beam with Damping Material Applying Rubber Magnetic Powder （DRM） has been investigated. Two methods are selected to hold DRM to a vibrating steel beam, one is to attach DRM by the magnetic attractive force （called DRM beam） and the other by adhesive bonding （called AB-DRM beam）. Different from the damping property of AB-DRM beam caused by shear deformation of damping material, the damping property of DRM beam is characterized by the sliding frictional loss together with the internal loss of damping material. The authors established a formulation to predict the damping characteristics of DRM beam, which was validated experimentally. It is found that rubber material loss factor β has a decisive influence on damping improvement of DRM beam versus AB-DRM beam. If β is smaller than the critical value around 0.8255, a valid range of vibratory amplitude always exists in which DRM beam can achieve better damping than AB-DRM beam; conversely, if β is bigger than the critical value, the valid range does not exist when slide occurs. Such results are used to determine the merits and limitations of DRM and develop design guidelines.     

Dynamic analysis of beam-cable coupled systems using Chebyshev spectral element method

The dynamic characteristics of a beam–cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a double Timoshenko beam system interconnected by discrete springs. Utilizing Chebyshev series expansion and meshing the system according to the locations of its connections,numerical results of the natural frequencies and mode shapes are obtained using only a few elements, and the results are validated by comparing them with the results of a finiteelement method. Then the effects of the cable parameters and layout of connections on the natural frequencies and mode shapes of a fixed-pinned beam are studied. The results show that the modes of a beam–cable coupled system can be classified into two types, beam mode and cable mode, according to the dominant deformation. To avoid undesirable vibrations of the cable, its parameters should be controlled in a reasonable range, or the layout of the connections should be optimized.     

Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam constituted by standard linear solid model

The weakly forced vibration of an axially moving viscoelastic beam is investigated.The viscoelastic material of the beam is constituted by the standard linear solid model with the material time derivative involved.The nonlinear equations governing the transverse vibration are derived from the dynamical,constitutive,and geometrical relations.The method of multiple scales is used to determine the steady-state response.The modulation equation is derived from the solvability condition of eliminating secular terms.Closed-form expressions of the amplitude and existence condition of nontrivial steady-state response are derived from the modulation equation.The stability of nontrivial steady-state response is examined via the Routh-Hurwitz criterion.     

Closed-form solution of beam on Pasternak foundation under inclined dynamic load

The dynamic response of an infinite Euler–Bernoulli beam resting on Pasternak foundation under inclined harmonic line loads is developed in this study in a closed-form solution.The conventional Pasternak foundation is modeled by two parameters wherein the second parameter can account for the actual shearing effect of soils in the vertical direction.Thus,it is more realistic than the Winkler model,which only represents compressive soil resistance.However,the Pasternak model does not consider the tangential interaction between the bottom of the beam and the foundation;hence,the beam under inclined loads cannot be considered in the model.In this study,a series of horizontal springs is diverted to the face between the bottom of the beam and the foundation to address the limitation of the Pasternak model,which tends to disregard the tangential interaction between the beam and the foundation.The horizontal spring reaction is assumed to be proportional to the relative tangential displacement.The governing equation can be deduced by theory of elasticity and Newton’s laws,combined with the linearly elastic constitutive relation and the geometric equation of the beam body under small deformation condition.Double Fourier transformation is used to simplify the geometric equation into an algebraic equation,thereby conveniently obtaining the analytical solution in the frequency domain for the dynamic response of the beam.Double Fourier inverse transform and residue theorem are also adopted to derive the closed-form solution.The proposed solution is verified by comparing the degraded solution with the known results and comparing the analytical results with numerical results using ANSYS.Numerical computations of distinct cases are provided to investigate the effects of the angle of incidence and shear stiffness on the dynamic response of the beam.Results are realistic and can be used as reference for future engineering designs.     

Development of a generalised active vibration suppression strategy for a cantilever beam using internal resonance

In this paper, the potential to utilise modal coupling effects in the formulation of a generalised vibration suppression algorithm is investigated. The plant, a flexible cantilever beam undergoing first mode oscillation, is modelled by a second order differential equation with a spring constant and damping coefficient that are representative of the first mode flexibility and material damping of the beam, respectively.In order to establish an internal resonance condition, a second equation, designated the supplementary equation or controller, is appended to the plant to render a two-degree-of-freedom system. The objective is to generate an internally resonant pair. Upon successful completion of this task, a suppression technique is implemented whereby energy is removed from the plant via the supplementary system.The introduction of the supplementary system results in a set of design parameters which are employed to realise a state of internal resonance and to establish the desired dynamic response. The choice of 2:1 internal resonance models results in a unidirectional control torque making this technique particularly attractive for systems using thrusters or tendons as actuators. A similar structural configuration regulated under a PD (Proportional-Derivative) control law is compared to the proposed control scheme via simulation.     

Dynamic analysis of a deployable/retractable damped cantilever beam

Deployable/retractable damped cantilever beams are a class of time-varying parametric structures which have attracted considerable research interest due to their many potential applications in the intelligent robot field and aerospace. In the present work, the dynamic characteristics of a deployable/retractable damped cantilever beam are investigated experimentally and theoretically. The time-varying damping, as a function of the beam length, is obtained by both the enveloped fitting method and the period decrement method. Furthermore, the governing equation of the deployable/retractable damped cantilever beam is derived by introducing the time-varying damping parameter,and the corresponding closed-form solution and vibration principles are investigated based on the averaged method. The theoretical predictions for transient dynamic responses are in good agreement with the experimental results. The dynamic mechanism analysis on time-varying damping offers flexible technology in mechanical and aerospace fields.     

磁约束双层夹心悬臂梁的振动分析

采用在约束层端部上设置永磁体的新方法可使阻尼层获得比传统约束阻尼处理方法更高的剪应变，从而增强粘弹层的阻尼耗能，降低共振峰。本文应用Hamilton原理，推导了全覆盖双层约束阻尼悬臂梁的运动方程，对模型进行了实验验证；分析了不同物理和几何参数下该方法的减振效果。研究表明，磁约束能提高阻尼减振效果。随着阻尼层厚度的增加，无磁约束的共振峰降低较少，而有磁约束的共振峰急剧降低，减振效果明显。阻尼层剪切模量G的变化对磁约束减振效果的影响较大；当G小于一定值时，减振效果明显，但当G大于一定值时，减振效果急剧降低。在不同的约束层弹性模量和厚度下，磁约束仍起作用。此外，分析了具有较大控制力的主被动混合磁约束振动控制方法的可行性。     

Enhanced damping of a cantilever beam by axial parametric excitation

A uniform cantilever beam under the effect of a time-periodic axial force is investigated. The beam structure is discretized by a finite-element approach. The linearised equations of motion describing the planar bending vibrations of the beam structure lead to a system with time-periodic stiffness coefficients. The stability of the system is investigated by a numerical method based on Floquet’s theorem and an analytical approach resulting from a first-order perturbation. It is demonstrated that the parametrically excited beam structure exhibits enhanced damping properties, when excited near a specific parametric combination resonance frequency. A certain level of the forcing amplitude has to be exceeded to achieve the damping effect. Upon exceeding this value, the additional artificial damping provided to the beam is significant and works best for suppression of vibrations of the first vibrational mode of the cantilever beam.     

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).     

自感知主被动阻尼悬臂梁动态特性分析

由Ｈａｍｉｌｔｏｎ原理导出了压电层作约束层作约束层的自感知主被动阻尼控制结构的振动控制方程;由自感知电压引入速度负反馈闭环控制,并由假设模态法将位移按模态展开,求解了悬臂梁结构的动态特征;对被动控制、自感知主动控制、自感知主被动控制的控制效果进行了分析比较;分析了粘弹层厚度变化、材料参数变化以及压电层厚度、位置等结构参数变化对控制效果及模态频率的影响;并对自感知主被动阻尼控制结构的特点和设计中应注     

Suppression of multiple modal resonances of a cantilever beam by an impact damper

Impact dampers are usually used to suppress single mode resonance. The goal of this paper is to clarify the difference when the impact damper suppresses the resonances of different modes. A cantilever beam equipped with the impact damper is modeled. The elastic contact of the ball and the cantilever beam is described by using the Hertz contact model. The viscous damper between the ball and the cantilever beam is modeled to consume the vibrational energy of the cantilever beam. A piecewise ordinary differential-partial differential equation of the cantilever beam is established, including equations with and without the impact damper. The vibration responses of the cantilever beam with and without the impact damper are numerically calculated. The effects of the impact absorber parameters on the vibration reduction are examined. The results show that multiple resonance peaks of the cantilever beam can be effectively suppressed by the impact damper. Specifically, all resonance amplitudes can be reduced by a larger weight ball. Moreover, the impacting gap is very effective in suppressing the vibration of the cantilever beam. More importantly, there is an optimal impacting gap for each resonance mode of the cantilever beam, but the optimal gap for each mode is different.     

The composite damping capacity of sandwich cantilever beams

The governing equation of the flexural forced vibration of a cantilever sandwich beam excited by a sinusoidal displacement at the clamped end is developed by utilizing the conventional Hamilton's Principle. The effect of damping of the composite beam is incorporated into the elastic equation of motion by utilizing the Correspondence Principle of the linear viscoelastic theory. Several plots for different values of the composite damping factor are presented. For comparison, an experimental setup was utilized to test different composite beams. The variation of the experimental results with those derived theoretically seem to be in agreement within the frequency range of the first few modes.     

Transient wave analysis of a cantilever Timoshenko beam subjected to impact loading by Laplace transform and normal mode methods

This study applies two analytical approaches, Laplace transform and normal mode methods, to investigate the dynamic transient response of a cantilever Timoshenko beam subjected to impact forces. Explicit solutions for the normal mode method and the Laplace transform method are presented. The Durbin method is used to perform the Laplace inverse transformation, and numerical results based on these two approaches are compared. The comparison indicates that the normal mode method is more efficient than the Laplace transform method in the transient response analysis of a cantilever Timoshenko beam, whereas the Laplace transform method is more appropriate than the normal mode method when analyzing the complicated multi-span Timoshenko beam. Furthermore, a three-dimensional finite element cantilever beam model is implemented. The results are compared with the transient responses for displacement, normal stress, shear stress, and the resonant frequencies of a Timoshenko beam and Bernoulli–Euler beam theories. The transient displacement response for a cantilever beam can be appropriately evaluated using the Timoshenko beam theory if the slender ratio is greater than 10 or using the Bernoulli–Euler beam theory if the slender ratio is greater than 100. Moreover, the resonant frequency of a cantilever beam can be accurately determined by the Timoshenko beam theory if the slender ratio is greater than 100 or by the Bernoulli–Euler beam theory if the slender ratio is greater than 400.