Abstract

This study develops a modified elliptic harmonic balance method (EHBM) and uses it to solve the force and displacement transmissibility of a two-stage geometrically nonlinear vibration isolation system. Geometric damping and stiffness nonlinearities are incorporated in both the upper and lower stages of the isolator. After using the relative displacement of the nonlinear isolator, we can numerically obtain the steady-state response using the first-order harmonic balance method (HBM1). The steady-state harmonic components of the stiffness and damping force are modified using the Jacobi elliptic functions. The developed EHBM can reduce the truncation error in the HBM1. Compared with the HBM1, the EHBM can improve the accuracy of the resonance regimes of the amplitude-frequency curve and transmissibility. The EHBM is simple and straightforward. It can maintain the same form as the balancing equations of the HBM1 but performs better than it.

1. Introduction

Nonlinearity has become the focus of recent research studies on improving vibration isolation performance [1, 2]. Many of them have the configuration of geometrically nonlinear springs [3–5] and dampers [6–8] whose arrangement can obtain low dynamic stiffness, thereby reducing the natural frequency without inducing large static deflection [9,10]. Furthermore, two-degree-of-freedom (2-DOF) vibration isolation systems with quasi-zero-stiffness (QZS) and geometrically nonlinear damping have been studied [11–14]. Lu et al. showed that both the force and displacement transmissibilities are mitigated in the isolation range as the horizontal stiffnesses in both stages are increased [11]. Comparing three kinds of nonlinear 2-DOF vibration isolation models, which are grounded-grounded, bottom-springs grounded, and top-springs grounded isolators, Wang et al. found that the bottom-springs grounded isolator has the best isolation performance when the excitation force amplitude is small [12]. The vibrational power flow method was applied to investigate the performance of a 2-DOF nonlinear isolation system [15]. Recently, Yang et al. illustrated that the addition of a nonlinear inertance mechanism to a QZS isolator could enhance vibration isolation performance [16]. Deng et al. illustrated that a multilayer QZS combined system might give distinguished isolation performance in low-frequency regime without losing supporting stability [17].

The harmonic balance method (HBM) is a technique for solving the equations of motion of strongly nonlinear oscillators in the papers mentioned above. However, when these nonlinear equations of motion are solved using the HBM, the truncation error is induced when the high-order harmonic components are neglected. To improve the results’ accuracy, complex and numerous balance equations should be included [18–20]. Some modified methods, such as an incremental HBM [21], two-timescale HBM [22], and the HBM with prior linearization [23], have been presented. In this respect, Zhou et al. illustrated that an accurate approximation solution for the resonance response of harmonically forced strongly nonlinear oscillator could be obtained by linearizing the governing equation before harmonic balancing [24].

Jacobi elliptic functions are widely used to solve the nonlinear dynamical systems [19, 25], especially for strongly nonlinear problems [26–28]. The exact solutions for certain conservative nonlinear solutions can be directly solved using the Jacobi elliptic functions, which can be equivalent to the Fourier series expansion with high orders [26]. A nonlinear vibration isolation system with a negative stiffness mechanism was investigated using the averaging method [29]. However, when the elliptic harmonic balance method is applied to solve two or multiple DOFs systems, the number of equations obtained by harmonic balancing is not equal to that of unknowns [30]. For this issue, Chen and Liu analysed a 2-DOF self-excited oscillator with strongly cubic nonlinearity by an additional equation prior to harmonic balancing using Jacobi elliptic functions [30]. Using an elliptic balance method, Elias-Zuniga and Beatty obtained the forced response of a 2-DOF undamped system with cubic nonlinearity [31]. Cveticanin proposed an approximate method for solving coupled strongly nonlinear differential equations with small nonlinearities based on the Krylov-Bogolyubov procedure with Jacobi elliptic functions [32]. Recently, Wu and Tang modified the harmonic components of the first-order terms of the damping and stiffness force of a single DOF nonlinear isolation system using Jacobi elliptic functions [33].

Herein, we developed an elliptic harmonic balance method (EHBM) and used it to obtain the transmissibility of a two-stage geometrically nonlinear vibration isolation system. The geometrically nonlinear damping and stiffness are included in both the upper and lower stages to improve the isolation characteristics. The steady-state response of the two-stage vibration isolator is derived using Jacobi elliptic functions and trigonometric functions. Using Jacobi elliptic functions, we modified the first-order harmonic balance method (HBM1) according to the orthogonal relationship between force components. After calculating the amplitude-frequency characteristics, we can obtain the force and displacement transmissibility. The accuracy of the results is analysed.

2. A Two-Stage Geometrically Nonlinear Vibration Isolator

A two-stage nonlinear vibration isolator is shown in Figure 1. In addition to the suspended mass and the vertical springs and dampers, horizontal springs and dampers are inserted in each stage to produce geometric nonlinearity [3–8]. The exciting force is applied in the vertical direction. The relative displacement between each adjacent stage are given by and , respectively, where is the displacement of the mass of the upper stage , is the displacement of the mass of the lower stage , and is the displacement of the base. The force transmitted from to and the force transmitted from to the basement are given bywhere and c_h1 are the damping coefficients of the vertical and horizontal dampers of the upper stage, and c_h2 are the damping coefficients of the vertical and horizontal dampers of the lower stage, and k_h1 are the stiffnesses of the vertical and horizontal springs of the upper stage, and k_h2 are the stiffnesses of vertical and horizontal springs of the lower stages, l₁ and l₂ are the lengths of the upper and lower horizontal springs between the suspended mass and host structure in the state of rest, and l₀₁ and l₀₂ are the free lengths of the horizontal springs of the upper and lower stages, respectively.

Figure 1 

Model of a two-stage nonlinear vibration isolator.

If the relative displacement between each adjacent stage is less than 20 percent of the length of the horizontal spring, the series expansion of the forces f_r and f_m shown in equations (1a) and (1b) can be approximated as [8, 33]where and are the approximated damping forces when and , and are the approximated spring forces when and , where , , , and , and the percentage error is less than four percent [8,34]. It should also be noticed that if the amplitude of the excitation is large, so that the force-deflection characteristic of the two-stage system cannot be approximately described by a series expansion to the third order as in equations (2a) and (2b), the nonlinearity induces undesirable effects [35].

For the force excitation case, where the base displacement , , and , the equations of motion are written aswhere is the excitation force amplitude and is its frequency.

Setting , the nondimensional forms of equations (3a) and (3b) becomewhere , , , , , , , , , , , , , , , , , , , , , , and .

When the excitation force and the base displacement , where and are the base excitation amplitude and frequency, the equations of motion of the base excited system are given bywhere .

3. Elliptic Harmonic Balance Method

3.1. HBM1

The solutions of equations (4a) and (4b) are assumed aswhere is the nondimensional amplitude of the relative displacement between m_s1 and m_s2, is the nondimensional amplitude of the relative displacement between m_s2 and the base, is the phase difference between the relative displacement and the excitation force, and is the phase difference between the relative displacement and the excitation force.

The nondimensional damping and stiffness force can be approximately written as , , , and , where , , , and are the amplitudes of the first order of series expansion of , , , and at the steady-state response, respectively, which can be expressed as follows [18]:

Substituting equations (6a) and (6b) into equations (4a) and (4b) and equating coefficients of the harmonic components , , , and , we obtainwhere , , , , and .

Combining equations (7a)–(7d) and (8a)–(8d), the amplitude-frequency equations of equations (4a) and (4b) are given by

3.2. Jacobi Elliptic Function Solutions

Using Jacobi elliptic functions, we assume that the steady-state solutions of equations (4a) and (4b) are in the following form:where and are the complete phases, and are the squares of the elliptic modulus, and are the Jacobi elliptic frequencies, and and are the Jacobi phases of the corresponding displacement responses, respectively. It should also be noticed that . Following a similar procedure in [33], the relationship between the second derivative and the linear and cubic terms of equations (10a) and (10c) are given by

Substituting equations (10a) and (10c) and equations (11a) and (11b) into equations (8b) and (8d), respectively, the equations of the cosine harmonic component become

Equating the coefficients of and of equation (12a), the coefficients of and of equation (12b) give

When , , , and , the relationship between the excited frequency and the Jacobi elliptic frequencies and can be expressed aswhere and are the complete elliptic integrals of the first kind when .

3.3. Modifying the Stiffness Force Components

Because equations (7a) and (7d) are the first-order harmonic components, both the damping force components and and the stiffness force components and are approximate terms. The stiffness force components and are first modified using Jacobi elliptic functions.

Using equation (14), , and , equations (7b) and (7d) can be rewritten as

Substituting the first Fourier series expansion of and , where and are the approximate phases of the displacement responses and respectively, into equations (15a) and (15b), the modified stiffness components becomewhere , , , and .