Influence of excitation amplitude on the characteristics of nonlinear butyl rubber isolators

The purpose of this study is to explore the advantages and characteristics of nonlinear butyl rubber (type IIR) isolators in vibratory shear by comparison with linear isolators. It is known that the mechanical properties of viscoelastic materials exhibit significant frequency and temperature dependence, and in some cases, nonlinear dynamic behavior as well. Nonlinear characteristics in shear deformation are reflected in mechanical properties such as stiffness and damping. Furthermore, even when the excitation amplitude is small the response amplitude may often be large enough that nonlinearities cannot be ignored. The treatment involves developing phenomenological models of the effective storage modulus and effective loss factor of a rubber isolator material as a function of excitation amplitude. The transmissibility of a nonlinear viscoelastic isolator is compared with that of a linear isolator using an equivalent linear damping coefficient. Forced resonance vibration and impedance tests are used to characterize nonlinear parameters and to measure the normalized transmissibility. It is found that as the excitation amplitude of the nonlinear viscoelastic isolator increases, the response amplitude decreases and the transmissibility is improved over that of the linear isolator for excitation frequency that exceeds a particular value governed by the temperature and excitation amplitude. The method of multiple scales and numerical simulations are used to predict the response characteristics of the isolator based on the phenomenological modeling under different values of system parameters.     

Effect of the deformation in the inertia forces on the inverse dynamics of planar flexible mechanical systems

The inverse dynamics problem for articulated structural systems such as robotic manipulators is the problem of the determination of the joint actuator forces and motor torques such that the system components follow specified motion trajectories. In many of the previous investigations, the open loop control law was established using an inverse dynamics procedure in which the centrifugal and Coriolis inertia forces are linearized such that these forces in the flexible model are the same as those in the rigid body model. In some other investigations, the effect of the nonlinear centrifugal and Coriolis forces is neglected in the analysis and control system design of articulated structural systems. It is the objective of this investigation to study the effect of the linearization of the centrifugal and Coriolis forces on the nonlinear dynamics of constrained flexible mechanical systems. The virtual work of the inertia forces is used to define the complete nonlinear centrifugal and Coriolis force model. This nonlinear model that depends on the rate of the finite rotation and the elastic deformation of the deformable bodies is used to obtain the solution of the inverse dynamics problem, thus defining the joint torques that produce the desired motion trajectories. The effect of the linearization of the mass matrix as well as the centrifugal and Coriolis forces on the obtained feedforward control law is examined numerically. The results presented in this investigation are obtained using a slider crank mechanism with a flexible connecting rod.     

An intelligent parameter varying (IPV) approach for non-linear system identification of base excited structures

Health monitoring and damage detection strategies for base-excited structures typically rely on accurate models of the system dynamics. Restoring forces in these structures can exhibit highly non-linear characteristics, thus accurate non-linear system identification is critical. Parametric system identification approaches are commonly used, but require a priori knowledge of restoring force characteristics. Non-parametric approaches do not require this a priori information, but they typically lack direct associations between the model and the system dynamics, providing limited utility for health monitoring and damage detection. In this paper a novel system identification approach, the intelligent parameter varying (IPV) method, is used to identify constitutive non-linearities in structures subject to seismic excitations. IPV overcomes the limitations of traditional parametric and non-parametric approaches, while preserving the unique benefits of each. It uses embedded radial basis function networks to estimate the constitutive characteristics of inelastic and hysteretic restoring forces in a multi-degree-of-freedom structure. Simulation results are compared to those of a traditional parametric approach, the prediction error method. These results demonstrate the effectiveness of IPV in identifying highly non-linear restoring forces, without a priori information, while preserving a direct association with the structural dynamics.     

Transition sets analysis based parametrical design of nonlinear metal rubber isolator

This research proposes the parametrical design of Metal Rubber (MR) isolation platform based on the investigation of nonlinear vibration properties under different types of excitation. Based on the mechanical model established by experiments, the restoring force of the isolation platform is proposed as a nonlinear function in consideration of the stiffness nonlinearity and Coulomb friction of metal wires. Then, the perturbation method is utilized to solve the steady states whose local stability is studied by singularity theory. The main results obtained by singularity theory show that there are five different types of vibration property, and the critical conditions for the transformation of different vibration properties are defined by transition sets. For impact excitation, the optimum structural parameters are obtained based on the vibration dissipation time via nonlinear backbone analysis; for periodic excitation, the optimum structural parameters are determined according to multiple standards including the bandwidth for effective isolation, bandwidth for multi-steady states, resonance peak and displacement transmissibility in high frequency band etc. The vibration performances for optimum structural parameters are verified by dynamical experiments. In conclusion, this paper carries out a novel sight of choosing optimum parameters, and therefore provides the guidance for the utilization of MR isolation platform for different types of excitation in engineering practices.     

Control structure interaction in the nonlinear analysis of flexible mechanical systems

The effect of the control structure interaction on the feedforward control law as well as the dynamics of flexible mechanical systems is examined in this investigation. An inverse dynamics procedure is developed for the analysis of the dynamic motion of interconnected rigid and flexible bodies. This method is used to examine the effect of the elastic deformation on the driving forces in flexible mechanical systems. The driving forces are expressed in terms of the specified motion trajectories and the deformations of the elastic members. The system equations of motion are formulated using Lagrange's equation. A finite element discretization of the flexible bodies is used to define the deformation degrees of freedom. The algebraic constraint equations that describe the motion trajectories and joint constraints between adjacent bodies are adjoined to the system differential equations of motion using the vector of Lagrange multipliers. A unique displacement field is then identified by imposing an appropriate set of reference conditions. The effect of the nonlinear centrifugal and Coriolis forces that depend on the body displacements and velocities are taken into consideration. A direct numerical integration method coupled with a Newton-Raphson algorithm is used to solve the resulting nonlinear differential and algebraic equations of motion. The formulation obtained for the flexible mechanical system is compared with the rigid body dynamic formulation. The effect of the sampling time, number of vibration modes, the viscous damping, and the selection of the constrained modes are examined. The results presented in this numerical study demonstrate that the use of the driving forees obtained using the rigid body analysis can lead to a significant error when these forces are used as the feedforward control law for the flexible mechanical system. The analysis presented in this investigation differs significantly from previously published work in many ways. It includes the effect of the structural flexibility on the centrifugal and Coriolis forces, it accounts for all inertia nonlinearities resulting from the coupling between the rigid body and elastic displacements, it uses a precise definition of the equipollent systems of forces in flexible body dynamics, it demonstrates the use of general purpose multibody computer codes in the feedforward control of flexible mechanical systems, and it demonstrates numerically the effect of the selected set of constrained modes on the feedforward control law.     

Application of subharmonic resonance for the detection of bolted joint looseness

Bolted joint structures are prone to bolt loosening under environmental and operational vibrations, which may severely affect the structural integrity. This paper presents a bolt looseness recognition method based on the subharmonic resonance analysis. The bolted joint structure was simplified to a two-degree-of-freedom nonlinear model, and a multiple timescale method was used to explain the phenomenon of the subharmonic resonance and conditions for the generation of subharmonics. Numerical simulation predictions for the generation of the subharmonics and conditions for the subharmonics can be found with respect to the excitation frequency and the excitation amplitude. Experiments were performed on a bolt-joint aluminum beam, where the damage was simulated by loosening the bolts. Two surface-bonded piezoelectric transducers were utilized to generate continuous sinusoidal excitation and to receive corresponding sensing signals. The experimental results demonstrated that subharmonic components would appear in the response spectrum when the bolted structure was subjected to the excitation of twice its natural frequency. This subharmonic resonance method was found to be effective on bolt looseness detection.     

Nonlinear dynamics and chaos in shape memory alloy systems

Smart material systems and structures have remarkable properties responsible for their application in different fields of human knowledge. Shape memory alloys, piezoelectric ceramics, magnetorheological fluids, and magnetostritive materials constitute the most important materials that belong to the smart materials category. Shape memory alloys (SMAs) are metallic alloys usually employed when large forces and displacements are required. Applications in aerospace structures, rotordynamics and several bioengineering devices are investigated nowadays. In terms of applied dynamics, SMAs are being used in order to exploit adaptive dissipation associated with hysteresis loop and the mechanical property changes due to phase transformations. This paper presents a general overview of nonlinear dynamics and chaos of smart material systems built with SMAs. Oscillators, vibration absorbers, impact systems and structural systems are of concern. Results show several possibilities where SMAs can be employed for dynamical applications.     

Autoresonant homeostat concept for engineering application of nonlinear vibration modes

Analysis of strongly nonlinear (vibro-impact) systems revealed an existence of nonlinear modes of vibration with spatial and temporal concentration of energy. The modes can be realised, for example, through intensification of the vibration process by condensing the vibration into a sequence of collisions for impulsive action of the tools to the media being treated or can be as a result of some discontinuity (slackening of a contact, arrival of crack, etc.) in the structure. The use of the nonlinear modes to develop useful mechanical work leads to necessity of excitation and control of resonance in ill-defined dynamical systems. This is due to the poorly predictable response of the media being treated. Excitation, stabilisation and control of a nonlinear mode at the top intensity in such systems is an engineering challenge and needs a new method of adaptive control for its realisation. Such a control technique was developed with the use of self-exciting mechatronic systems. The excitation of the nonlinear mode in such systems is a result of artificial instability of mechanical system conducted by positive electronic feedback. The instability is controlled by intelligent identification of the mode and active tracing of the optimal relationship between phase shifting and limitation in the feedback circuitry. This method of control is known as autoresonance. Applications of autoresonant control for development of the new machines are described. The paper is a revised and extended version of authors’ presentation at ASME 2004 International Mechanical Engineering Congress, Anaheim, CA, USA. An erratum to this article can be found at     

Energy integrals for mechanical systems with nonlinear nonholonomic constraints

The work analyzes energy relations for nonholonomic systems, whose motion is restricted by nonlinear nonholonomic constraints. For the mechanical systems with linear constraints, the analysis of energy relations was carried out in [1], [2], [3], [4], [5], [6] …. On the basis of corresponding Lagrange’s equations, a general law of the change in energy dε/dt is formulated for mentioned systems by the help of which it is shown that there are two types of the laws of conservation of energy, depending on the structure of elementary work of the forces of constraint reactions. Also, the condition for existing the second type of the law of conservation of energy is formulated in the form of the system of partial differential equations. The obtained results are illustrated by a model of nonholonomic mechanical system.     

Effect of damping and wave parameters on offshore structure under random excitation

This paper describes the linearized and nonlinear dynamic response of a tension leg platform (TLP) to random waves and current forces. The forcing term of the equation of motion is inherently nonlinear due to the nonlinear drag force. Two analysis procedures are used: nonlinear time domain analysis and linear frequency domain analysis. For the nonlinear analysis, the random wave particle velocities and accelerations are simulated for a given wave spectrum. The nonlinear equation of motion is then integrated directly to obtain the system response statistics. For the linear frequency domain analysis, the nonlinear drag force is linearized through an introduction of linearization coefficients. The main objective of this paper is to investigate the effect of the structural damping and wave parameters on both nonlinear and linear dynamic response of the TLP by parametric studies. The results of stochastic nonlinear and linear dynamic response of the TLP, with and without the presence of current, are presented and compared.     

Non-linear output frequency response functions of MDOF systems with multiple non-linear components

In engineering practice, most mechanical and structural systems are modelled as multi-degree-of-freedom (MDOF) systems such as, e.g., the periodic structures. When some components within the systems have non-linear characteristics, the whole system will behave non-linearly. The concept of non-linear output frequency response functions (NOFRFs) was proposed by the authors recently and provides a simple way to investigate non-linear systems in the frequency domain. The present study is concerned with investigating the inherent relationships between the NOFRFs for any two masses of non-linear MDOF systems with multiple non-linear components. The results reveal very important properties of the non-linear systems. These properties clearly indicate how the system linear characteristic parameters govern the propagation of the non-linear effect induced by non-linear components in the system. One potential application of the results is to detect and locate faults in engineering structures which make the structures behave non-linearly.     

On some performance characteristics of base excited vibration isolation systems with a purely nonlinear restoring force

Base excited vibration isolation systems with a purely nonlinear restoring force and a velocity nth power damper are considered. The restoring force has a single-term power form with the exponent that can be any non-negative real number. Approximations for the steady-state response at the frequency of excitation are obtained by using the Jacobi elliptic function with a changeable elliptic parameter and by applying an elliptic averaging method. The relative and absolute displacement transmissibility of this system are analysed. These performance characteristics are expressed in terms of the damping parameters, but they are also determined for an arbitrary non-negative real power of geometric nonlinearity, which represent new and so far unknown results. Some examples illustrating the effect of the system parameters on these performance characteristics are also presented.     

On the use of neural networks for identification of linear and nonlinear systems

A procedure based on neural networks for the classification of linear and nonlinear systems is presented, using excitation and response data under swept sine excitation. Special attention is paid to the classification and identification of linear and bilinear systems, the latter being considered since they exhibit typical characteristics of cracked systems. The computer simulations show that: (1) using the procedure presented in this paper the trained classification network can reliably classify a linear system and different nonlinear systems; (2) the output of the trained identification neural network for a linear system and a bilinear system can be used as a quantitative indicator of characteristics of bilinear systems having different stiffness ratios (k _(x>0)/k _(x<0)) with respect to the bilinear system used in the training stage; (3) for two-degree-of-freedom systems, the trained network can not only determine the existence of a bilinear stiffness and the magnitude of its stiffness ratio, but also specify which stiffness is bilinear, i.e. indicate its position. These results provide a possibility of using the trained neural networks to detect and locate structural cracks which have the characteristics of bilinear systems.Visiting scholar, from People's Republic of China.     

Adaptive fault compensation control for a class of nonlinear systems with unknown time-varying delayed faults

An adaptive approximation design for the fault compensation (FC) control is addressed for a class of nonlinear systems with unknown multiple time-delayed nonlinear faults. The magnitude and occurrence time of the multiple faults with unknown time-varying delays are unknown. The function approximation technique using neural networks is employed to adaptively approximate the unknown nonlinear effects and changes in model dynamics due to the time-delayed faults. We design an adaptive memoryless FC control system with a prescribed performance bound to compensate the faults and to guarantee the transient performance of the tracking error from unexpected changes of system dynamics. The adaptive laws for neural networks and the bound of residual approximation errors are derived using the Lyapunov stability theorem, which are used for proving that the tracking error is preserved within the prescribed performance bound regardless of unknown multiple time-delayed nonlinear faults. Simulation examples are presented for illustrating the effectiveness of the proposed control methodology     

Analysis and design of the force and displacement transmissibility of nonlinear viscous damper based vibration isolation systems

This study is concerned with the analysis and design of the force and displacement transmissibility of nonlinear viscous damper based vibration isolation systems. Analytical algorithms are derived using the Ritz–Galerkin method to evaluate the transmissibility of SDOF displacement vibration isolation and force vibration isolation systems where a nonlinear viscous damper is used as an energy dissipating device. The results reveal that compared to linear dampers, nonlinear viscous dampers can more significantly improve the system vibration isolation performance in a wider frequency range. A procedure is then proposed based on the analysis results to facilitate the design of nonlinear viscous dampers for system vibration isolation purposes. These results have significant implications for the design of vibration isolation systems in many engineering applications.     

Periodic and Non-Periodic Oscillations of a Class of Hysteretic Two Degree of Freedom Systems

The equations governing the response of hysteretic systems to sinusoidal forces, which are memory dependent in the classical phase space, can be given as a vector field over a suitable phase space with increased dimension. Hence, the stationary response can be studied with the aids of classical tools of nonlinear dynamics, as for example the Poincaré map. The particular system studied in the paper, based on hysteretic Masing rules, allows the reduction of the dimension of the phase space and the implementation of efficient algorithms. The paper summarises results on one degree of freedom systems and concentrates on a two degree of freedom system as the prototype of many degree of freedom systems. This system has been chosen to be in 1:3 internal resonance situation. Depending on the energy dissipation of the elements restoring force, the response may be more or less complex. The periodic response, described by frequency response curves for various levels of excitation intensity, is highly complex. The coupling produces a strong modification of the response around the first mode resonance, whereas it is negligible around the second mode. Quasi-periodic motion starts bifurcating for sufficiently high values of the excitation intensity; windows of periodic motions are embedded in the dominion of the quasi-periodic motion, as consequence of a locking frequency phenomenon.     

Interaction of subway LIM vehicle with ballasted track in polygonal wheel wear development

This paper develops a coupled dynamics model for a linear induction motor (LIM) vehicle and a subway track to investigate the influence of polygonal wheels of the vehicle on the dynamic behavior of the system. In the model, the vehicle is modeled as a multi-body system with 35 degrees of freedom. A Timoshenko beam is used to model the rails which are discretely supported by sleepers. The sleepers are modeled as rigid bodies with their vertical, lateral, and rolling motions being considered. In order to simulate the vehicle running along the track, a moving sleeper support model is introduced to simulate the excitation by the discrete sleeper supporters, in which the sleepers are assumed to move backward at a constant speed that is the same as the train speed. The Hertzian contact theory and the Shen- Hedrick-Elkins’ model are utilized to deal with the normal dynamic forces and the tangential forces between wheels and rails, respectively. In order to better characterize the linear metro system (LMS), Euler beam theory based on modal superposition method is used to model LIM and RP. The vertical electric magnetic force and the lateral restoring force between the LIM and RP are also taken into consideration. The former has gap-varying nonlinear characteristics, whilst the latter is considered as a constant restoring force of 1 kN. The numerical analysis considers the effect of the excitation due to polygonal wheels on the dynamic behavior of the system at different wear stages, in which the used data regarding the polygonal wear on the wheel tread are directly measured at the subway site.     

Effects of bracings and motion coupling on resonance features of semi-submersible platform under irregular wave conditions

Resonance is a critical consideration in the design of offshore floating structures. This paper aims at analysing the nonlinear effects of bracings and motion coupling on the resonance features of a semi-submersible platform. An improved mathematical model based on potential theory is proposed to simulate the motion response of a semi-submersible platform under irregular wave conditions, considering both the variations of hydrostatic and hydrodynamic forces induced by the bracings entering and exiting the water and the nonlinear coupling induced by the platform motions. For comparison purposes, numerical simulations are also performed using a mathematical model without considering the aforementioned effects. Validated by results of wave basin tests and numerical simulations, the proposed model performs much better in capturing the characteristic resonance features of pitch motion in low-frequency region. The nonlinear hydrostatic effect of bracings leads to the increase of resonance frequency as the motion amplitude increases, while the hydrodynamic force on the bracings and the nonlinear motion coupling only influence the amplitude of resonance spectral peak. In addition, factors influencing the nonlinear effects such as the vertical position and diameter of bracings and the pitch restoring coefficient are further investigated. It is revealed that the deviation of pitch resonance frequency has evident dependence on the ratio between nonlinear and linear volumetric variations, and an empirical formula estimating the resonance frequency is proposed using the observed dependence. Theoretically, both smaller bracing radius and larger pitch restoring coefficient are beneficial for suppressing the resonance induced by the nonlinear effects. The proposed model can be an effective tool for predicting the motion response, and the understanding of the resonance features is helpful for the design of semi-submersibles.     

Non-linear model updating of a three-dimensional portal frame based on Wiener series

The common metrics used in linear finite element (FE) model updating using vibration test data are generally functions of relationships based on unidimensional convolution, for example, distances involving natural frequencies, frequency response or impulse response functions, modal shapes, etc. When a structure has local elements or geometry, like joints, bolts, gaps, backlash, etc., these approaches can fail once it could to induce non-linear behavior. Thus, the methods for FE model updating, when considering the existence of localized non-linear parameters, have been receiving much attention in the last years. In this sense, the present paper proposes the use of a strategy through objective functions based on multiples convolutions described by the first order and second order discrete-time Volterra kernels. These kernels are effective metrics for a model updating into large FE model with local non-linearity. In order to improve the non-linear coefficient identification, an orthogonal basis involving Kautz filter is used to expand the kernels, called by Wiener kernel. To exemplify in full details the steps of the updating procedure, an FE model of a three-dimensional portal frame with commons non-linearities is simulated with different excitation forces and used to identify the non-linear parameters. These results allow us to characterize the practical applicability and the drawbacks of the proposed method with suggestions and remarks for further use in industrial structures.