The decoupling of damped linear systems in free or forced vibration

The purpose of this paper is to extend classical modal analysis to decouple any viscously damped linear system in non-oscillatory free vibration or in forced vibration. Based upon an exposition of how exponential decay in a system can be regarded as imaginary oscillations, the concept of damped modes of imaginary vibration is introduced. By phase synchronization of these real and physically excitable modes, a time-varying transformation is constructed to decouple non-oscillatory free vibration. When time drifts caused by viscous damping and by external excitation are both accounted for, a time-varying decoupling transformation for forced vibration is derived. The decoupling procedure devised herein reduces to classical modal analysis for systems that are undamped or classically damped. This paper constitutes the second and final part of a solution to the “classical decoupling problem.” Together with an earlier paper, a general methodology that requires only the solution of a quadratic eigenvalue problem is developed to decouple any damped linear system.     

ON THE FREQUENCY RESPONSE FUNCTION OF A DAMPED CANTILEVER SIMPLY SUPPORTED IN-SPAN AND CARRYING A TIP MASS

This paper deals with the determination of the frequency response function of a cantilevered Bernoulli-Euler beam which is viscously damped by a single damper. The beam is simply supported in-span and carries a tip mass. The frequency response function is obtained through a formula that was established for the receptance matrix of discrete linear systems subjected to linear constraint equations, by considering the simple support as a linear constraint imposed on generalized co-ordinates. The comparison of the numerical results obtained via a boundary value problem formulation justifies the approach used here.     

Frequency-dependent damping in structural vibration analysis by use of complex series expansion of transfer functions and numerical Fourier transformation

The response of damped linear finite systems (discrete and continuous) to harmonic, stationary random, and transient excitations is studied. The assumed damping may be light or heavy, viscous and/or hysteretic (the latter being frequency-independent or frequency-dependent), and proportionally or non-proportionally distributed over the structure. Closed-form analytic transfer functions are derived for beam systems. In order to rationalize subsequent numerical calculations, the transfer functions are approximated by using truncated series. Complex eigenfrequencies of the structure and complex residues of the actual transfer function are used. Special interest is paid to the mathematical modelling of experimentally measured damping. Causality requirements are considered. Numerical examples are given. An efficient method (by which aliasing is eliminated) for numerical Fourier transformation has been developed and applied.     

Bending vibration of axially loaded Timoshenko beams with locally distributed Kelvin-Voigt damping

Utilizing the Timoshenko beam theory and applying Hamilton's principle, the bending vibration equations of an axially loaded beam with locally distributed internal damping of the Kelvin-Voigt type are established. The partial differential equations of motion are then discretized into linear second-order ordinary differential equations based on a finite element method. A quadratic eigenvalue problem of a damped system is formed to determine the eigenfrequencies of the damped beams. The effects of the internal damping, sizes and locations of damped segment, axial load and restraint types on the damping and oscillating parts of the damped natural frequency are investigated. It is believed that the present study is valuable for better understanding the influence of various parameters of the damped beam on its vibration characteristics.     

The effect of a viscously damped dynamic absorber on a linear multi-degree-of-freedom system

The effect of a viscously damped dynamic absorber on the dynamic behaviour of a linear vibration system with many degrees of freedom is investigated. The dynamic absorber is connected to the roof of the primary system. In the sequence, optimal values for the parameters describing the behaviour of the dynamic absorber are determined, in order that the transmissibility of the composite system be minimized over the whole frequency range. As an application, respective types of anti-vibration mountings are proposed.     

The vibration characteristics of a beam with an axial force

Equations of motion are found for a non-uniform damped Timoshenko beam with a distributed axial force. Principal modes may be extracted by numerical means when the boundary conditions are specified, and the appropriate orthogonality conditions are given. The theory of linear forced vibration can thus be derived. It is an implicit requirement that all axial forces are conservative. That is to say, tangential, follower and partial follower axial forces (whether applied at an extremity or distributed along the beam) are excluded.     

Stationariness provided by filtration to a periodic non-stationary random process

The concept of generalized spectral density is used for evaluating the variance of the response of a simple linear oscillator excited by periodic non-stationary random functions. The generalized Wiener-Khintchine relations are slightly reshaped, so that the generalized spectral density of the excitation function could be expressed as the sum of Dirac delta functions, whose coefficients could be given in the form of closed analytical expressions. The results show that, apart from a few exceptions, the response of a lightly damped system is practically stationary. It is noted in general that filtration brings about stationarization.     

Sustained high-frequency energy harvesting through a strongly nonlinear electromechanical system under single and repeated impulsive excitations

This work investigates a vibration-based energy harvesting system composed of two oscillators coupled with essential (nonlinearizable) stiffness nonlinearity and subject to impulsive loading of the mechanical component. The oscillators in the system consist of one grounded, weakly damped linear oscillator mass (primary system), which is coupled to a second light-weight, weakly damped oscillating mass attachment (the harvesting element) through a piezoelastic cable. Due to geometric/kinematic mechanical effects the piezoelastic cable generates a nonlinearizable cubic stiffness nonlinearity, whereas electromechanical coupling simply sees a resistive load. Under single and repeated impulsive inputs the transient damped dynamics of this system exhibit transient resonance captures (TRCs) causing high-frequency ‘bursts’ or instabilities in the response of the harvesting element. In turn, these high-frequency dynamic instabilities result in strong and sustained energy transfers from the directly excited primary system to the lightweight harvester, which, through the piezoelastic element, are harvested by the electrical component of the system or, in the present case, dissipated across a resistive element in the circuit. The primary goal of this work is to demonstrate the efficacy of employing this type of high-frequency dynamic instability to achieve enhanced nonlinear vibration energy harvesting under impulsive excitations.     

关于线性阻尼振子第一积分的研究

通过引入一维线性阻尼振子基本积分来构造其他第一积分, 包括不含时的积分. 将这种方法推广到多维情形, 构造二维和n维线性阻尼振子不同形式的第一积分; 证明不同类型的二维线性阻尼振子都存在三个独立的不含时的第一积分, n维线性阻尼振子存在2n-1个独立的不含时的第一积分. 利用变量变换将线性阻尼振子的第一积分变换成简谐振子形式的第一积分. 关键词： 线性阻尼振子 第一积分 基本积分 简谐振子     

The decoupling of damped linear systems in configuration and state spaces

It has recently been reported that any viscously damped linear system can be decoupled in the configuration space by a real, nonlinear, time-dependent transformation. The purpose of this rapid communication is to provide a few clarifying remarks about the decoupling operation. It is shown that, for homogeneous systems, the time-dependent configuration-space decoupling transformation is real, linear and time-invariant when cast in state space. In addition, the configuration-space transformation generates a diagonalizing structure-preserving transformation. In non-homogeneous systems, both the configuration and associated state transformations are nonlinear and depend continuously on the excitation. An example is given of a linear system that can be decoupled in configuration but not in state space.     

THE SPATIAL AVERAGE MEAN SQUARE MOTION AS AN OBJECTIVE FUNCTION FOR OPTIMIZING DAMPING IN DAMPED MODIFIED SYSTEMS

This work considers the addition of damping to a beam system in order to limit the response due to random-in-time forcing functions. Here the optimization process for such dissipative attachments will use as an objective function the spatial average of the mean square response over the extent of the beam. It is shown here that optimal dampers and damped vibration absorbers can be found which minimize this objective function.     

Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibro-impact attachments

We study complex damped and undamped dynamics and targeted energy transfers (TETs) in systems of coupled oscillators, consisting of single-degree-of-freedom primary linear oscillators (LOs) with vibro-impact attachments, acting, in essence, as vibro-impact nonlinear energy sinks (VI NESs). First, the complicated dynamics of such VI systems is demonstrated by computing the VI periodic orbits of underlying Hamiltonian systems and depicting them in appropriate frequency–energy plots (FEPs). Then, VI damped transitions and distinct ways of passive TETs from the linear oscillators to the VI attachments for various parameter ranges and initial conditions are investigated. As in the case of smooth stiffness nonlinearity [Y. Lee, G. Kerschen, A. Vakakis, P. Panagopoulos, L. Bergman, D.M. McFarland, Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment, Physica D 204 (1–2) (2005) 41–69], both fundamental and subharmonic TET can be realized in the VI systems under consideration. It is found that the most efficient mechanism for VI TET is through the excitation of highly energetic VI impulsive orbits (IOs), i.e., of periodic or quasiperiodic orbits corresponding to zero initial conditions except for the initial velocities of the linear oscillators. In contrast to NESs with smooth essential nonlinearities considered in previous works, VI NESs are capable of passively absorbing and locally dissipating significant portions of the energies of the primary systems to which they are attached, at fast time scale. This renders such devices suitable for applications, like seismic mitigation, where dissipation of vibration energy in the early, highly energetic regime of the motion is a critical requirement.     

Forced transverse vibrations of an elastically connected complex simply supported double-beam system

The present paper is devoted to analyzing undamped forced transverse vibrations of an elastically connected complex double-beam system. The problem is formulated and solved in the case of simply supported beams. The classical modal expansion method is applied to ascertain dynamic responses of beams due to arbitrarily distributed continuous loads. Several cases of particularly interesting excitation loadings are investigated. The action of stationary harmonic loads and moving forces is considered. In discussing vibrations caused by exciting harmonic forces, conditions of resonance and dynamic vibration absorption are determined. The beam-type dynamic absorber is a new concept of a continuous dynamic vibration absorber (CDVA), which can be applied to suppress excessive vibrations of corresponding beam systems. A numerical example is presented to illustrate the theoretical analysis.     

Estimation of interfacial forces in time domain for linear systems

A structural path rank ordering process under transient excitations requires a good knowledge of the interfacial path forces, which are difficult to directly measure. Four time domain methods to estimate the interfacial forces are proposed and comparatively evaluated with application to linear time-invariant, proportionally damped discrete systems. First, the transient response is derived by modal analysis and a direct time domain technique to calculate the interfacial forces is outlined. Next, the frequency domain estimation methods, based on the sub-system concept are reviewed, and an inverse Fourier transform scheme is introduced. An indirect method of estimating interfacial force in transient state is then developed through an inverse procedure of modal analysis. The sub-system approach is employed to obtain the interfacial forces based on the forced vibration response of the original system and modal data of the sub-system. Finally, an approximate time domain scheme is suggested that could be used only if the system properties are known or precisely estimated. Although the proposed indirect methods are designed for eventual experimental applications, this article provides numerical feasibility studies via a simple source-path-receiver system (with parallel vibration paths) that has five translational degrees of freedom. The proposed methods are compared under ideal impulse force excitation input and a periodic sawtooth load (without and with Gaussian noise) to observe the starting transients as well as subsequent motions and interfacial forces. Preliminary comparisons with a laboratory experiment are very promising.     

STOCHASTIC AVERAGING OF STRONGLY NON-LINEAR OSCILLATORS UNDER BOUNDED NOISE EXCITATION

A stochastic averaging method for strongly non-linear oscillators under external and/or parametric excitation of bounded noise is proposed by using the so-called generalized harmonics functions. The method is then applied to study the primary resonance of Duffing oscillator with hardening spring under external excitation of bounded noise. The stochastic jump and its bifurcation of the system are observed and explained by using the stationary probability density of amplitude and phase. Subsequently, the method is applied to study the dynamical instability and parametric resonance of Duffing oscillator with hardening spring under parametric excitation of bounded noise. The primary unstable region is delineated by evaluating the Lyapunov exponent of linearized system, and the response and jump of non-linear system around the unstable region are examined by using the sample functions and stationary probability density of amplitude and phase.     

Model independent control of lightly damped noise/vibration systems

Feedforward control is a popular strategy of active noise/vibration control. In well-damped noise/vibration systems, path transfer functions from actuators to sensors can be modeled by finite impulse response (FIR) filters with negligible errors. It is possible to implement noninvasive model independent feedforward control by a recently proposed method called orthogonal adaptation. In lightly damped noise/vibration systems, however, path transfer functions have infinite impulse responses (IIRs) that cause difficulties in design and implementation of broadband feedforward controllers. A major source of difficulties is model error if IIR path transfer functions are approximated by FIR filters. In general, active control performance deteriorates as model error increases. In this study, a new method is proposed to design and implement model independent feedforward controllers for broadband in lightly damped noise/vibration systems. It is shown analytically that the proposed method is able to drive the convergence of a noninvasive model independent feedforward controller to improve broadband control in lightly damped noise/vibration systems. The controller is optimized in the minimum H2 norm sense. Experiment results are presented to verify the analytical results.     

Chaos after period-doubling bifurcations in the resonance of an impact oscillator

A linear, viscously damped oscillator that rebounds elastically whenever the displacement drops to zero is a well-defined quasi-linear deterministic system arising throughout the physical sciences. It exhibits a family of sub-harmonic resonant peaks between which we delineate cascades of period-doubling bifurcations leading to chaotic regimes typical of a strange attractor.     

Predicting optimal drive sweep rates for autoresonance in Duffing-type oscillators: A beat method using Teager-Kaiser instantaneous frequency

Sustained resonance in a linear oscillator is achievable with a drive whose constant frequency matches the resonant frequency of the oscillator. But in oscillators with nonlinear restoring forces such as the pendulum, Duffing and Duffing-Van der Pol oscillator, the resonant frequency changes as the amplitude changes, so a constant frequency drive results in a beat oscillation instead of sustained resonance. Duffing-type nonlinear oscillators can be driven into sustained resonance, called autoresonance, when the drive frequency is swept in time to match the changing resonant frequency of the oscillator. We find that near-optimal drive linear sweep rates for autoresonance can be estimated from the beat oscillation resulting from constant frequency excitation. Specifically, a least squares estimate of the Teager-Kaiser instantaneous frequency versus time for the beat response to a stationary drive provides a near-optimal estimate of the nonstationary drive linear sweep rate needed to sustain resonance in the pendulum, Duffing and Duffing-Van der Pol oscillators. We confirm these predictions with model-based numerical simulations. An advantage of the beat method of estimating optimal drive sweep rates for maximal autoresonant response is that no model is required so experimentally generated beat oscillation data can be used for systems where no model is available.     

Optimum design of Lanchester damper for a viscously damped single degree of freedom system by using minimum force transmissibility criterion

The problem of optimum design of a Lanchester damper for minimum force transmission from a viscously damped single degree of freedom system subjected to harmonic excitation is investigated. Explicit expressions are developed for determining the optimum absorber parameters. It is shown that for the particular case of the undamped single degree of freedom system the results reduce to the classical ones obtained by using the concept of a fixed point on the transmissibility curves.     

Optimum design of a Lanchester damper for a viscously damped single degree of freedom system subjected to inertial excitation

The problem of designing an optimum Lanchester damper for a viscously damped single degree of freedom system subjected to inertial harmonic excitation is investigated. Two criteria are used for optimizing the performance of the damper: (i) minimum motion transmissibility; (ii) minimum force transmissibility. Explicit expressions are developed for determining the absorber parameters.