Abstract
The frequency of a nonlinear vibration system is nonlinearly related to its amplitude, and this relationship is critical in the design of a packaging system and a microelectromechanical system (MEMS). This paper proposes a straightforward frequency prediction method for nonlinear oscillators with arbitrary initial conditions. The tangent oscillator, the hyperbolic tangent oscillator, a singular oscillator, and a MEMS oscillator are chosen to elucidate the simple solving process. The results, when compared with those obtained by the homotopy perturbation method, exhibit a good agreement. This paper introduces a very convenient procedure for attaining quick and accurate insight into the vibration property of a nonlinear vibration system.
Highlights
Vibration absorption and vibration attenuation are two critical factors in designing a nonlinear vibration system; for example, a low amplitude is always considered in the design of the packaging system [1,2,3,4] and the seismic design of architecture [5]
A nonlinear vibration equation is written as mw + h(w) = 0, w(0) = a, w (0) = b
When h(w) = k tan w, we have the well-known tangent oscillator arising in packaging systems [1,2,3,4]
Summary
Vibration absorption and vibration attenuation are two critical factors in designing a nonlinear vibration system; for example, a low amplitude is always considered in the design of the packaging system [1,2,3,4] and the seismic design of architecture [5]. A nonlinear vibration equation is written as mw + h(w) = 0, w(0) = a, w (0) = b (1). For a linear vibration system, one can choose h(w) = kw, k is the spring coefficient, this is the well-known harmonic oscillation. When h(w) = k tan w, we have the well-known tangent oscillator arising in packaging systems [1,2,3,4]. The amplitude is determined by the initial conditions and the frequency of the system. The goal of this paper is to suggest a simple method [19,20] for gaining a timely and efficient glimpse into the frequency–amplitude relationship of Equation (1) with arbitrary initial conditions. The comparison with the aforementioned existing methods shows that this work would be greatly challenging for nonlinear vibration theory
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