Nonlinear Vibration Problems Research Articles

Non-linear vibration of elastic truncated conical moderately thick shells in large overall motion

Based on Hamilton's principle and the Timoshenko–Mindlin hypotheses on moderately thick plates, a set of non-linear governing equations of motion for truncated conical moderately thick shells is set up in large overall motion. By using the Galerkin method and the harmonic balance method, the problem of non-linear vibration of the truncated conical moderately thick shells is analyzed. The present results are compared with the available data for special cases.

A non‐linear inverse vibration problem of estimating the time‐dependent stiffness coefficients by conjugate gradient method

AbstractAn iterative regularization method, i.e. the conjugate gradient method (CGM) is applied to an inverse non‐linear force vibration problem to estimate the unknown time‐dependent stiffness coefficients (or spring constants) in a damped system by using the measured system displacement. It is assumed that no prior information is available on the functional form of the unknown stiffness coefficients in the present study, thus, it is classified as the function estimation in inverse calculation. The accuracy of the inverse analysis is examined by using the simulated exact and inexact displacement measurements. The numerical simulations are performed to test the validity of the present algorithm by using different types of stiffness coefficients and measurement errors. Results show that an excellent estimation on the time‐dependent spring constants can be obtained with any arbitrary initial guesses within a couple of seconds of CPU time at Pentium III‐500 MHz PC. Copyright © 2001 John Wiley & Sons, Ltd.

Reduction of finite-element models of planar frames using non-linear normal modes

This paper is an extended version of Mazzilli et al. (Mazzilli, C.E.N., Soares, M.E.S., Baracho Neto, O.G.P., 1999. Proceedings of the American Congress of Applied Mechanics, PACAM VI, vol. 8, pp. 1589–1592, Rio de Janeiro, Brazil) which presents a powerful reduction technique in non-linear dynamics based on the combination of finite element procedures with a “non-linear” Galerkin method (Zemann, J., Steindl, A., 1996. Proceedings of the 19th International Congress of Theoretical and Applied Mechanics, Kyoto, Japan) and non-linear normal modes (Shaw, S.W., Pierre, C., 1993. Journal of Sound and Vibration 164 (1), 85–124). Its implementation, in the form of a symbolic computation code, was carried out for planar framed structures under assumptions of linear elasticity and geometrical non-linearity, according to the Bernoulli–Euler rod theory (Brasil, R.M.L.R.F., Mazzilli, C.E.N., 1993. Applied Mechanics Reviews 46 (11), S110–S117). To obtain the desired drastic reduction of degrees of freedom and the corresponding set of differential equations of motion in explicit form, it is necessary to supply as input data the displacement components of the pre-selected non-linear normal modes. Validation tests for non-linear free-vibration problems are shown, considering reduced models of higher hierarchy and their ability to supply accurate regenerated non-linear normal modes. For non-linear forced vibration problems, a brief outlook of what is intended to be done is presented.

A computerised implementation of the multiple scales perturbation method using Mathematica

One of the well-established analytical techniques for solving engineering vibration problems, which are represented by ordinary differential equations, is the method of multiple scales (MS). This method can be applied to find approximate solutions to a wide range of nonlinear problems. The main idea of the MS method is to split up the single independent variable into several new independent variables. The method allows the construction of a set of perturbation equations that can be solved under the condition of removal of secular terms. The main emphasis in this paper is on how to generalise a computer implementation of the MS method and its application to nonlinear vibration problems. The necessary macro-steps that are used for the development of the computational system are formulated and the practical ways of encoding these steps using Mathematica are discussed. The Mathematica package “MultipleScale.m” has been developed as a deliverable in this research. This package is capable of performing perturbation analysis on a wide class of multi-degree-of-freedom vibration systems. An example is given to illustrate the concepts discussed.

NON-LINEAR VIBRATION OF CABLE–DAMPER SYSTEMS PART II: APPLICATION AND VERIFICATION

The formulation derived in Part I of this paper is applied to investigate non-linear vibration behavior of inclined sag cables with and without oil dampers in long-span cable-stayed bridges. The inclined sag cables used in computation are also tested in the laboratory with scaled cables and oil dampers. The experimental results are then compared with the theoretical results to verify the suggested approach. Both theoretical analyses and experiments display some typical non-linear behavior of cable vibration, including response hardening, response bifurcation, multiple solutions, jump, and internal resonance. They also demonstrate that an oil damper with an appropriate damping coefficient being selected can effectively suppress large vibration amplitudes of the cable so that in some cases, non-linear cable vibration problem may become a linear vibration problem. The agreement between the experimental and theoretical results is found to be satisfactory.

Application of Maple V to the nonlinear vibration analysis of circular plate with variable thickness

This paper is concerned with the application of Maple V to the nonlinear vibration problems of circular plates with variable thickness. In this paper, the nonlinear equations of plates of variable thickness to the dynamic case can be solved by using the computer algebra systems method. Details of solution expressions and numerical results are given in computer algebra systems forms, for two kinds of boundary conditions, which are the clamped edge and the supported edge. The numerical results show that the solutions of the paper contain other cases when the plates are of uniform thickness. The effect of various thickness parameters has been investigated in detail. In addition, a Runge–Kutta method is used to solve the free vibration and the maximum deflection response to a uniformly distributed step load of plates with variable thickness. It is shown that the adoption of variable thickness plate would be useful in engineering design.

Moment closure by Monte Carlo simulation and moment sensitivity factors

A new closure method is developed to calculate moments of the state vector X of a non-linear system subjected to Gaussian white noise. The method can be applied to a broad class of non-linear random vibration problems and uses two sources of information: (1) samples of X generated by Monte Carlo simulation to estimate lower order moments of this process and (2) moment equations corresponding to a specified closure level to find higher order moments of X. No heuristic arguments are used to close the moment equations. Error bounds, second moment properties of the estimated moments of X, and confidence intervals on these moments are developed to measure the performance of the proposed method. Two examples are presented to illustrate the use of the proposed method and evaluate its performance relative to Monte Carlo simulation and classical closure solution.

STOCHASTIC NON-LINEAR VIBRATIONS OF HIGHWAY SUSPENSION BRIDGE UNDER INERTIAL SPRUNG MOVING LOAD

The problem of vertical non-linear vibrations of a single-span suspension bridge due to a random stream of moving vehicles is considered. The inertia forces of car bodies and vehicle suspensions are taken into account because each vehicle is idealized by a set of viscoelastic oscillators. Vehicle arrivals at the bridge as well as the vehicle types and the masses of the truckloads are assumed to be random variables. The bridge model is defined as flat and geometrically non-linear. The numerical simulation method is developed to determine expected values and standard deviations of the bridge response. Solutions for the bridge deflections, the tension of cables and bending moments at the girder are presented and compared with results which have been calculated in simplified cases when a train of concentrated forces moving in the dynamic or static conditions idealizes vehicular traffic loads.

Fast Fourier nonlinear vibration analysis

We present an implementation of the multi-harmonic balance method (MHB) where intensive use of the Fast Fourier Transform algorithm (FFT) is made at all stages of calculations. The MHB method is not modified in essence, but computations are organized to obtain a very attractive method that can be applied systematically on general nonlinear vibration problems. The resulting nonlinear algebraic problem is solved by a particular implementation of a continuation method. Nonlinear vibration results are analyzed a posteriori by a Floquet method to determine their stability. The technique is applied on a series of problems of different nature, demonstrating the robustness and flexibility of the approach.

Adaptive polynomial chaos expansions applied to statistics of extremes in nonlinear random vibration

This paper presents a new module towards the development of efficient computational stochastic mechanics. Specifically, the possibility of an adaptive polynomial chaos expansion is investigated. Adaptivity in this context refers to retaining, through an iterative procedure, only those terms in a representation of the solution process that are significant to the numerical evaluation of the solution. The technique can be applied to the calculation of statistics of extremes for nongaussian processes. The only assumption involved is that these processes be the response of a nonlinear oscillator excited by a general stochastic process. The proposed technique is an extension of a technique developed by the second author for the solution of general nonlinear random vibration problems. Accordingly, the response process is represented using its Karhunen-Loeve expansion. This expansion allows for the optimal encapsulation of the information contained in the stochastic process into a set of discrete random variables. The response process is then expanded using the polynomial chaos basis, which is a complete orthogonal set in the space of second-order random variables. The time dependent coefficients in this expansion are then computed by using a Galerkin projection scheme which minimizes the approximation error involved in using a finite-dimensional subspace. These coefficients completely characterize the solution process, and the accuracy of the approximation can be assessed by comparing the contribution of successive coefficients. A significant contribution of this paper is the development and implimentation of adaptive schemes for the polynomial chaos expansion. These schemes permit the inclusion of only those terms in the expansion that have a significant contribution.

Nonlinear vibration of circular plate with variable thickness

This paper is concerned with the nonlinear vibration problems of circular plates with variable thickness. The nonlinear equations of plates with variable thickness are extended to the dynamic case. The resulting equations can be solved by using an iterative method, a Galerkin’s approach and a perturbation method. Detailed solutions and numerical results are given for two kinds of boundary conditions, the clamped edge and the supported edge. The results show that the solutions for the case of the plates with uniform thickness can be included in the solution herin as a special case. The effect of various thickness parameters is investigated in detail. Also, a Runge-Kutta method is used to solve the free and forced vibrations of plates with variable thickness, and the results are obtained firstly. It has shown that the adoption of variable thickness plate would be useful in engineering design.

POST-CRITICAL BEHAVIOUR OF ORTHOTROPIC CIRCULAR CYLINDRICAL SHELLS UNDER TIME DEPENDENT AXIAL COMPRESSION

A non-shallow non-linear shell theory is used to analyze the parametric resonance of orthotropic circular cylindrical shells under harmonically varying axial compression. As special cases, post-buckling and non-linear vibration problems are also studied. In the analysis the non-linear terms and the inertias contributed by both normal displacementwand circumferential displacementvare included. Therefore the final dynamic system includes two equations inwandv. The transverse shear deformation is taken into account by a first order theory. The spatial variables in the governing equations are eliminated by the Galerkin procedure. The final ordinary differential equations are solved by an asymptotic method. Numerical results show the dependence of the post-critical behaviour on the properties of material, geometry and excitation.

A Generalized Approach to Coupled Nonlinear Vibrations of Continuous Systems

A mathematical model covering many practical vibration problems of continuous systems has been proposed. The equations of motion consist of two nonlinearly coupled partial differential equations. The quadratic and cubic nonlinearities as well as the linear part of the equations are represented by arbitrary operators. A perturbation approach (method of multiple time scales) has been applied directly to the partial differential equations. The responses as well as the amplitude and phase modulation equations are found for the case of primary resonances of the external excitation and one-to-one internal resonances between the natural frequencies of the equations. The coefficients of the amplitude and phase modulation equations are calculated in their most general form. Results are then applied to a nonlinear cable vibration problem having small sag-to span ratios.

An analytical and numerical method for solving linear and nonlinear vibration problems

A direct analytical and numerical method independent of the existing classical methods for solving linear and nonlinear vibration problems is given with the introduction of a piecewise constant argument [Nt]/N. A new numerical method which produces sufficiently accurate results with good convergence is introduced. Development of the formulae for numerical calculations is based on the original governing differential equations.

THE GALERKIN METHOD FOR INITIAL VALUE PROBLEMS BASED ON THE PRINCIPLE OF TOTAL VIRTUAL ACTION

In this paper is presented the derivation of the unconstrained variational statement for initial value problems from the viewpoint of the Principle of Total Virtual Action. Based on the hybrid form of the variational equation, both the Galerkin and the recurrent-Galerkin procedures are developed. They are used to obtain approximate analytical/semi-analytical solutions. Linear and non-linear vibration problems are used to demonstrate the applications. Comparison between the results obtained with initial conditions unconstrained and those with initial conditions constrained is made. Different weighting functions are tried out to confirm the validity of the “hybrid” form of the unconstrained variational statement.

Nonlinear free vibration of orthotropic shallow shells of revolution under the static loads

In this paper, the axisymmetric nonlinear free vibration problems of cylindrically orthotropic shallow thin spherical and conical shells under uniformly distributed static loads are studied by using MWR and Lindstedt-Poincare perturbation method, from which, the characteristic relation between frequency ratio and amplitude is obtained. The effects of static loads, geometric and material parameters on vibrational behavior of shells are also discussed.

Comparing quasi-conservative averaging and dissipation energy balancing methods in non-linear random vibration

The approximate solution techniques of quasi-conservative averaging and dissipation energy balancing for non-linear random vibration problems are based on totally different physical concepts. In most cases, they produce the same result for the stationary probability density of the energy envelope of the system response. The exception can be traced to the so-called Wong-Zakai correction terms which change the effective stiffness in the system. This discrepancy can be removed by obtaining the Wong-Zakai correction terms on the basis of the original equation. The case of a single degree-of-freedom oscillator is treated in detail.

Free Vibration Analysis of Rotating Nonlinearly Elastic Structures with Symmetry: An Efficient Group-Equivariance Approach

We show in this work that use of group-theoretic ideas can greatly simplify the procedure for finding both linear vibration modes and frequencies and locating solution branches for nonlinear vibration problems. These methods can be employed in both analytic and numerical settings, and they lead to dramatic increases in solving efficiency via block diagonalization. For the class of problems studied here (in which gyroscopic effects are present), we demonstrate that the original structural symmetry group can be employed to analyze the dynamics of the system despite the loss of reflection symmetry. This allows for a simpler and more intuitive implementation of the group representation techniques. In addition, we explain how ideas from bifurcation theory can be included with symmetry analysis to prove the existence of solution branches for nonlinear vibration problems. These examples provide great insight to the many advantages of using group-theoretic methods in structural vibrations problems.

NUMERICAL SOLUTIONS OF SECOND ORDER IMPLICIT NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS

The existing literature usually assumes that second order ordinary differential equations can be put in first order form, and this assumption is the starting point of most treatments of ordinary differential equations. This paper examines numerical schemes for solving second order implicitnon-linear differential equations. Based on the literature review, three specific methods have been selected here. Two of them are based on series expansions: Picard iteration using Chebyshev series and Incremental Harmonic Balance (IHB), in which the non-linear differential equation is transformed into a set of algebraic ones that are solved iteratively. The third method is based on a fourth order (Houbolt's scheme) and an eighth order backward finite difference method (FDM). Each method is presented first, and then applied to specific examples. It is shown: (1) that the Picard method is not valid for solving implicit equations containing large non-linear inertial terms; (2) that the IHB method yields accurate periodic solutions, together with the frequency of oscillation, and the dynamical stability of the system may be assessed very easily; and (3) that both Houbolt's and the eighth order scheme can be used to compute time histories of initial value problems, if the time step is properly chosen. Finally, it is also illustrated how the combination of IHB and FDM can be a powerful tool for the analysis of non-linear vibration problems defined by implicit differential equations (including also explicit ones), since bifurcation diagrams of stable and unstable periodic solutions can be computed easily with IHB, while periodic and non-periodic stable oscillations may be obtained with FDM.

Approximate solutions for non-linear random vibration problems

Two methods are described for obtaining approximate solutions to non-linear random vibration problems. The first method approximates the non-linear system with the linear system whose corresponding probability density function best solves the Fokker-Planck equation associated with the non-linear system. In the second method, a class of non-linear systems with known solutions to the Fokker-Planck equation is used to best approximate the non-linear system of interest. Two illustrative examples are presented and the results are compared with existing methods.