A study on the forced vibrations of a class of nonlinear systems,with application to the Duffing equation Part II: Numerical treatment

Summary Numerical results on the wave form, the amplitude and the phase of the steady state solutions of the Duffing equation are obtained here. The steady state vibrations are studied:i) by a procedure of direct numerical integration based on the Runge-Kutta method;ii) by calculating the approximate solution of the equation which is obtained, as a particular case, from the theory developed in Part I. The results obtained by the two methods are compared and tested by the results of other Authors. From the comparisons it appears that the theory of Part I gives satisfactory results also in the cases of strong nonlinearity of the system.

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Sommario Nel presente lavoro sono ricavati ed illustrati alcuni risultati numerici relativi alle forme d'onda, le ampiezze e le fasi delle soluzioni a regime dell'equazione di Duffing. I moti vibratori a regime vengono ricercati:a) mediante un procedimento di integrazione numerica diretta, basato sul metodo di Runge-Kutta;b) adottando la soluzione approssimata dell'equazione stessa, che si ottiene come caso particolare della teoria sviluppata nella Parte I. I risultati ottenuti vengono confrontati fra di loro e con i risultati di altri Autori. Dai confronti appare che la teoria sviluppata nella Parte I dà risultati soddisfacenti anche quando la nonlinearità del sistema è molto elevata.

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This paper is an elaboration of a paper [10] presented at the 2nd National Congress of AIMETA (Naples, 6 October, 1974).     

A new type of non-linear approximation with application to the duffing equation

A new type of trial solution which differs from the usual linear combination of approximating functions is considered. It involves modifying the approximating functions with “form functions;” functions containing undetermined parameters appearing non-linearly, the proper choice of which provide a closer approximation to the large local curvatures which appear in some non-linear problems. In this paper the “form function” approximation is demonstrated for steady-state solutions of the Duffing equation. This equation arises in the problem of non-linear vibration of buckled beams and plates. It is shown that the stability behavior of these steady-state solutions is governed by a Hill equation. It is found that the “form function” approximation gives noticeably better numerical results than, for example, those given by the harmonic balance method. The method also provides additional insight into the non-linear behavior, particularly in the low frequency response region.     

A note on vibrations of damped linear systems

In a damped linear system, for the free vibrations to be oscillatory with decaying amplitude, it is proven that the mass and damping matrices must be positive definite. Also, a certain matrix product involving the mass, damping and stiffness matrices must be commutative.     

Bounds on the forced response of damped linear systems

Bounds are obtained on the magnitude of the displacement vector representing the response of a damped linear mechanical system to an oscillatory forcing function. Several restrictions used in previous treatments are avoided, particularly to the effect that the system modes must satisfy a decoupling condition.     

Induced vibrations and dynamic instabilities of a non linear structural model due to pulse loading

Summary In this paper we present some results from a model of Shanley type, used as a tool to investigate anomalous computed response behavior of a pin-ended beam subjected to a short pulse of transverse force. The nature of critical load values is here studied, using both exact solutions and an approximate step-by-step numerical integration of standard type. It is shown that convergence with decreasing time step may fail, near a critical value. The significance of these results for the original beam computational problem, and for other physical structures, is indicated briefly.

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Sommario Il presente lavoro illustra alcuni risultati ottenuti mediante un modello alla Shanley, usato per analizzare anomalie ottenute numericamente nel comportamento di una trave incernierata soggetta a un carico trasversale, applicato nella forma di un breve impulso. Viene studiata la natura di alcuni valori critici del carico, facendo uso sia di soluzioni esatte che di metodi approssimati di integrazione numerica. Si mostra che al crescere della accuratezza può non esservi convergenza dei risultati, e che vi è anzi divergenza per alcuni valori critici del passo di integrazione. Si discute brevemente l'interesse di tali risultati nei confronti delle difficoltà numeriche ottenute studiando il problema originale di trave incernierata.

     

A differential equation for the minors of the state-transition matrix of linear systems

Summary A differential equation for the minors of the state transition matrix of linear time-invariant systems is established.An example is then presented, where the differential equation is employed in order to check the existence of the extremal control of a linear system with a performance index given by the integral of a quadratic form.

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Sommario Si stabilisce un'equazione differenziale per i minori della matrice di transizione dei sistemi lineari stazionari.Si illustra quindi un esempio, in cui tale equazione differenziale viene impiegata per verificare l'esistenza del controllo estremale di un sistema lineare con indice di comportamento dato dall'integrale di una forma quadratica.

     

On the stability of the solutions to an integral evolution equation of the non linear particle transport theory

Summary In this paper we study the stability, with respect to the initial data, of the solutions to a non linear integral equation arising in particles transport theory.

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Sommario In questo lavoro si prova la stabilità, rispetto ai dati iniziali, delle soluzioni di un'equazione integrale non lineare che si incontra in teoria del trasporto di particelle.

     

Analytical and numerical investigation on the tempered time-fractional operator with application to the Bloch equation and the two-layered problem

Nonlinear Dynamics - In the continuous-time random walk model, the time-fractional operator usually expresses an infinite waiting time probability density. Different from that usual setting, this...     

Improved averaging method for turbulent flow simulation. Part I: Theoretical development and application to Burgers' transport equation

This is the first of two articles intended to develop, apply and verify a new method for averaging the momentum and mass transport equations for turbulence. The new method is based on Gaussian filtering in both the spatial and temporal domains. Application is made to the problem of momentum and scalar transport in a one-dimensional transient Burgers' flow field. No actual calculations, with the averaged equations, are presented in this paper. However, an ‘exact’ solution of the one-dimensional flow situation is presented as an economical tool for verifying the performance of the different turbulence models. In the second paper calculations are performed with the averaged one-dimensional equations on coarse grids, and the results are compared to the exact or fully simulated data with a statistical verification procedure.     

A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam

This paper delineates a class of time-periodically perturbed evolution equations in a Banach space whose associated Poincaré map contains a Smale horseshoe. This implies that such systems possess periodic orbits with arbitrarily high period. The method uses techniques originally due to Melnikov and applies to systems of the form x=f _o(X)+f ₁(X,t), where f _o(X) is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics: sinusoidally forced vibrations of a buckled beam.     

Critical conditions for a class of switched linear systems based on harmonic balance: applications to DC-DC converters

Critical conditions of period-doubling bifurcation and saddle-node bifurcation are derived for a class of switched linear systems. They can be applied to other similar hybrid, Filippov, or piecewise-smooth systems. Such switched linear systems are common in DC-DC converters. Those previously known critical conditions in the past research become special cases in the generalized framework. Given an arbitrary control scheme, a systematic procedure is proposed to derive the critical condition, which shows the stable range of a system parameter and the minimum stabilizing ramp slope. Two different systems with similar loop gains at high frequency have similar forms of critical conditions. The effect of switching delay is also analyzed.     

Numerical study of the noninertial systems: application to train coupler systems

Car coupler forces have a significant effect on the longitudinal train dynamics and stability. Because the coupler inertia is relatively small in comparison with the car inertia; the high stiffness associated with the coupler components can lead to high frequencies that adversely impact the computational efficiency of train models. The objective of this investigation is to study the effect of the coupler inertia on the train dynamics and on the computational efficiency as measured by the simulation time. To this end, two different models are developed for the car couplers; one model, called the inertial coupler model, includes the effect of the coupler inertia, while in the other model, called the noninertial model, the effect of the coupler inertia is neglected. Both inertial and noninertial coupler models used in this investigation are assumed to have the same coupler kinematic degrees of freedom that capture geometric nonlinearities and allow for the relative translation of the draft gears and end of car cushioning (EOC) devices as well as the relative rotation of the coupler shank. In both models, the coupler kinematic equations are expressed in terms of the car body and coupler coordinates. Both the inertial and noninertial models used in this study lead to a system of differential and algebraic equations that are solved simultaneously in order to determine the coordinates of the cars and couplers. In the case of the inertial model, the coupler kinematics is described using the absolute Cartesian coordinates, and the algebraic equations describe the kinematic constraints imposed on the motion of the system. In this case of the inertial model, the constraint equations are satisfied at the position, velocity, and acceleration levels. In the case of the noninertial model, the equations of motion are developed using the relative joint coordinates, thereby eliminating systematically the algebraic equations that represent the kinematic constraints. A quasistatic force analysis is used to determine a set of coupler nonlinear force algebraic equations for a given car configuration. These nonlinear force algebraic equations are solved iteratively to determine the coupler noninertial coordinates which enter into the formulation of the equations of motion of the train cars. The results obtained in this study showed that the neglect of the coupler inertia eliminates high frequency oscillations that can negatively impact the computational efficiency. The effect of these high frequencies that are attributed to the coupler inertia on the simulation time is examined using frequency and eigenvalue analyses. While the neglect of the coupler inertia leads, as demonstrated in this investigation, to a much more efficient model, the results obtained using the inertial and noninertial coupler models show good agreement, demonstrating that the coupler inertia can be neglected without having an adverse effect on the accuracy of the solution.     

On the boundedness in the mean square of the forced oscillations of linear systems with stochastic coefficients

Systems of linear differential equations with parameters that are nonwhite, Gaussian random functions are studied. Conditions sufficient for the boundedness in the mean and in the mean square of the solutions are derived. A second order system is given as an example to illustrate the methods used.     

The application of green's multi-dimensional function to investigate the stochastic vibrations of dynamical systems

Summary In the paper is presented the application of Green's multi-dimensional function to determine the probabilistic characteristics of the solutions of stochastic linear equations with time-variable coefficients, with random initial conditions and random excitations.The method is applied to calculate the variances of solutions for the vibrations of a vehicle model (or suspension) accelerated over a random profile.

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Anwendung der Greenschen Funktion zur Untersuchung von stochastischen Schwingungen dynamischer Systeme

Übersicht Die Greensche Funktion wird angewendet zur Ermittlung des Lösungsverhaltens stochastischer, linearer Differentialgleichungen mit zeitabhängigen Koeffizienten, mit Zufalls-Anfangsbedingungen und mit Zufalls-Erregung. Die Methode wird in einem Anwendungsbeispiel (Schwingungen eines beschleunigten Fahrzeugmodells) demonstriert.