On the jump-up and jump-down frequencies of the Duffing oscillator

doi:10.1016/j.jsv.2008.04.032

Journal of Sound and Vibration

Volume 318, Issues 4–5, 23 December 2008, Pages 1250-1261

https://doi.org/10.1016/j.jsv.2008.04.032 Get rights and content

Abstract

In this paper, simple approximate non-dimensional expressions, and the corresponding displacement amplitudes for the jump-up and jump-down frequencies of a softening and hardening lightly damped Duffing oscillator with linear viscous damping are presented. Although some of these expressions can be found in the literature, this paper presents a full set of expressions determined using the harmonic balance approach. These analytical expressions are validated for a range of parameters by comparing the predictions with calculations from direct numerical integration of the equation of motion. They are also compared with similar expressions derived using a perturbation method. It is shown that the jump-down frequency is dependent on the degree of nonlinearity and the damping in the system, whereas the jump-up frequency is dependent primarily upon the nonlinearity, and is only weakly dependent upon the damping. An expression is also given for the threshold of the excitation force and the nonlinearity that needs to be exceeded for a jump to occur. It is shown that this is only dependent upon the damping in the system.

Introduction

The Duffing oscillator has been studied for many years, as it is representative of many nonlinear systems [1]. For this type of system there are frequencies at which the vibration suddenly jumps-up or down, when it is excited harmonically with slowly changing frequency. The frequencies at which these jumps occur depend upon whether the frequency is increasing or decreasing and whether the nonlinearity is hardening or softening. Between these frequencies, multiple solutions exist for a given frequency of excitation, and the initial conditions determine which of these solutions represents the response of the system.

Although the Duffing oscillator has been studied at length, to the authors’ knowledge, analytical expressions for the jump frequencies and the amplitudes of vibration at these frequencies for both a softening and hardening system do not appear together in a single publication, which is rather surprising since it has been almost 90 years since Duffing's original publication [2]. There have been two approaches to this problem: analytical and numerical. The analytical approach has included the use of classical perturbation methods such as the method of multiple scales, which can be applied to weakly nonlinear systems, for which the jump frequencies are close to the undamped natural frequency of the linear system (nonlinear term set to zero). The other analytical method used is harmonic balance, which can be applied to both weakly and strongly nonlinear systems. Nayfeh and Mook [1] and Kevorkian and Cole [3] used the method of multiple scales to determine the jump-down frequency and the peak amplitude of the response, and additionally Kevorkian and Cole have also determined the jump-up frequency and the corresponding amplitude of vibration for a hardening system. Magnus [4] has used the harmonic balance method (HBM) to determine the jump-down frequency and the peak amplitude of a hardening system, but did not consider the jump-up frequency. Unfortunately, all the publications describing these studies have different notation, so that the results cannot be compared in a straightforward manner.

Perturbation methods are quite accurate in predicting the jump-up frequency, because even when there is strong nonlinearity, the jump-up frequency is close to the undamped natural frequency of the linear system. However, this is not the case for the jump-down frequency and it is shown in this paper that the results derived using these techniques are inaccurate for strong nonlinearity. In particular the maximum response is predicted to be the same as the linear system, which is not the case when there is strong nonlinearity.

Since the advent of computers, researchers working on the Duffing oscillator have mainly used a numerical or a hybrid analytical-numerical approach. Peleg [5], studying a hardening isolation system with friction and viscous damping, has proposed implicit equations for both the jump-down and jump-up frequencies, and the amplitude of the response at these frequencies by locating the loci of the vertical tangents to the response curve. Murata et al. [6] examined a hardening system described by Duffing's equation. Using catastrophe theory, they determined the jump frequencies numerically and presented their results graphically. Friswell and Penny [7], and Worden [8] computed the jump-up and jump-down frequencies of the Duffing oscillator with linear damping. In both studies, the HBM was used to obtain the frequency response curves. Friswell and Penny [7] used a numerical approach based on Newton's method, including terms up to the ninth harmonic, to compute the jump frequencies, while Worden [8], used a first order expansion and set the discriminant of a cubic polynomial in the square of the amplitude to be equal to zero, and solved the resulting equation numerically. Remarkably the difference between Friswell and Penny's results and Worden's first order approximation never exceeded 0.34% for the parameters chosen in his study. Carrella [9] using the HBM to a first order expansion has found closed form expressions that, with parameters used in [7], [8], yield values which differs from those found by Friswell and Worden by less than 1%.

The primary aim of this paper is to re-visit the analytical approach in the determination of the jump frequencies and the corresponding vibration amplitudes of the Duffing oscillator. Using the HBM, in which the response is approximated by a single harmonic, a consistent approach is used to determine these for both softening and hardening systems with linear viscous damping. The results are presented in tabular form for ease of reference. A comparison is made between these results and those in the literature determined using the method of multiple scales. The limitation in the multiple scales method in determining the jump-down frequency is also discussed. The secondary aim of the paper is to provide a link, in terms of a reference trail, between the earlier analytical work and the later numerical studies.

In some recent work, Malatkar and Nayfeh [10] determined the minimum excitation force required for the jump phenomenon to appear. Their key results are also presented in this paper, but in a modified and simplified form, which is consistent with the analysis for the jump frequencies.

Section snippets

Calculation of the jump frequencies

Duffing's equation is given by [1], [2] $m \ddot{x} + c \dot{x} + k_{1} x + k_{3} x^{3} = F \cos (ω t)$ which is the equation of motion of a single degree-of-freedom system of mass m, suspended on a parallel combination of a dashpot with damping coefficient c, and a spring with nonlinear restoring force, k₁x+k₃x³, excited by a harmonic force F cos(ωt). If k₃>0 the system is hardening and if k₃<0 the system is softening. It is convenient to write Eq. (1) in non-dimensional form as $y^{″} + 2 ζ y^{'} + y + α y^{3} = \cos (Ω τ)$ where $y = \frac{x}{x_{0}}, α = \frac{k_{3} x_{0}^{2}}{k_{1}}, ζ = \frac{c}{2 m ω_{0}}, ω_{0}^{2} = k_{1}$

Effect of nonlinearity and damping on the occurrence of the jump phenomena

If the nonlinear parameter |α| is increased from zero, the frequency response curve bends to the left or the right depending on whether the system is softening or hardening, respectively. At a threshold value of |α|, the frequency response curve becomes multi-valued above a certain frequency for a hardening system and below a certain frequency for a softening system. This frequency and threshold value has been determined by Malatkar and Nayfeh [10], who formulated the problem in a slightly

Conclusions

In this paper approximate expressions for the jump frequencies and the corresponding displacement amplitudes for the softening and hardening lightly damped Duffing oscillator have been presented. The expressions derived using the HBM have been compared with those derived using a perturbation method, which are valid at frequencies close to the undamped natural frequency of the corresponding linear system. The fundamental assumption in the derivations is that the dominant response of the

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Frequency response of non-linear single degree-of-freedom systems
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Application of catastrophe theory to the forced vibration of a diaphragm air spring
Journal of Sound and Vibration
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M.I. Friswell et al.
The accuracy of jump frequencies in series solutions of the response of a Duffing oscillator
Journal of Sound and Vibration
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P. Malatkar et al.
Calculation of the jump frequencies in the response of a s.d.o.f. non-linear systems
Journal of Sound and Vibration
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Nonlinear Oscillations
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There are more references available in the full text version of this article.

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On the jump-up and jump-down frequencies of the Duffing oscillator

Abstract

Introduction

Section snippets

Calculation of the jump frequencies

Effect of nonlinearity and damping on the occurrence of the jump phenomena

Conclusions

International Journal of Mechanical Science

Journal of Sound and Vibration

Journal of Sound and Vibration

Journal of Sound and Vibration

Nonlinear Oscillations