Research on a nonlinear quasi-zero stiffness vibration isolator with a vibration absorber

Abstract

A negative stiffness mechanism consisting of a spring and cylinder is proposed, and a grounded dynamic vibration absorber is designed based on a quasi-zero stiffness vibration isolator to constitute the vibration isolator with a vibration absorber system. The range of parameters for attaining zero stiffness is derived from static analysis. The dynamic analysis of the vibration isolator with a vibration absorber system is carried out by a multiscale method, and the amplitude–frequency response equation of the system is obtained. The influence of different system parameters on the amplitude–frequency response is analyzed. The amplitude–frequency response of the quasi-zero stiffness vibration isolator is compared with that of the vibration isolator with a vibration absorber, and the linear and nonlinear analytical solutions of the vibration isolator with a vibration absorber system are also compared. The results show that the designed vibration isolator with a vibration absorber is an ideal choice for low-frequency vibration isolation, with no large resonance peak throughout the system and significantly improved reliability of the system.

Keywords

Vibration isolator with a vibration absorber system nonlinear quasi-zero stiffness multiscale method

Introduction

With the improvement of living standards and material quality, people attach increasing importance to energy conservation and vibration reduction. The riding comfort of vehicles is associated with mechanical vibration, and precise instruments require the support device to have good vibration attenuation ability. In order to solve the above-mentioned vibration problems, vibration isolators or absorbers are usually used. In order to isolate vibration at low frequency, a quasi-zero stiffness vibration isolator is designed, but it still shows an extremely high resonance peak at the low-frequency band. If the excitation frequency is close to such a resonance frequency, the isolator system will be quickly unstable. In addition, a dynamic vibration absorber reduces the response amplitude greatly, but the effective frequency bandwidth is extremely narrow because of two new resonance peaks.

In recent years, in the field of vibration isolation, scholars have done a lot of research on quasi-zero stiffness vibration isolators. The negative stiffness mechanism composed of a tension spring and a connecting rod was designed and researched.^1–3 Chai et al.⁴ used a spring-cam-roller mechanism as the negative stiffness mechanism. Meng et al.⁵ utilized a Belleville spring to compose a nonlinear quasi-zero stiffness vibration isolator. Meng et al.⁶ studied the effect of increase of nonlinear damping on a quasi-zero stiffness vibration isolator. Wang et al.⁷ proposed a two-stage quasi-zero stiffness vibration isolation system and built a segmented nonlinear dynamic model. Liu and Yu⁸ added an auxiliary system to the vibration isolation system to reduce the transmissibility and jump phenomenon. Li et al.⁹ proposed a negative stiffness mechanism based on hybrid bistable composite laminates. Liu et al.¹⁰ studied the impact of Coulomb friction on a Euler buckling beam isolator. However, the issue of significant resonance peaks in the quasi-zero stiffness vibration isolator is not yet effectively resolved after years of development.

In the field of vibration absorber, a lot of research studies have been done from damping dynamic absorbers to grounded vibration absorbers with negative stiffness springs.¹¹ Zheng et al.¹² studied a nonlinear vibration energy absorber-based strategy for automotive active suspension. Zhao and Xu¹³ designed a time-delay nonlinear dynamic vibration absorber for a vibration absorber in the controlled system. Dong et al.¹⁴ applied a vibration absorber to a hub motor. Shen et al.¹⁵ added inertial elements to traditional vibration absorbers and used them for automotive shock absorbers. Wang et al.¹⁶ created a grounded three-element vibration absorber from viscous-elastic material. Javidialesaadi and Wierschem¹⁷ proposed a three-element vibration absorber composed of two kinds of current passive devices. Shen et al.¹⁸ brought forward a novel dynamic vibration absorber with a grounded stiffness unit and an amplifying mechanism.

There are many analysis methods for solving a nonlinear system. Nayfeh¹⁹ and Nayfeh and Mook²⁰ proposed a perturbation method to analyze the nonlinear vibration problem and used the multiscale method to analyze the multi degree of freedom problem. Chen and Wu²¹ simplified the analysis process by applying the averaging method to the nonlinear multi degree of freedom field. Li and Yang²² used the multiscale method to analyze the nonlinear multi degree of freedom vibration of vehicle suspension. Kovacic et al.²³ used the nonlinear method to analyze the vibration isolator composed of a tension spring. Li et al.²⁴ used the mode superposition multiscale method to analyze the nonlinear dynamics. Mahdi and Bardaweel²⁵ used the harmonic balance method to analyze the dynamic performance of the vibration isolator.

In this work, the negative stiffness mechanism is designed by using the tension spring to make the vibration isolator achieve the purpose of low-frequency vibration isolation, but the strong resonance problem still exists. Therefore, a quasi-zero stiffness vibration isolator incorporating a grounded dynamic vibration absorber is designed to effectively reduce the resonance peak. By making good for deficiencies, the main system is kept away from high resonance peaks and shows reliability at any excitation frequency.

Static analysis of a mechanical system

As shown in Figure 1, the quasi-zero stiffness vibration isolator is composed of a negative stiffness part and a positive stiffness part. The negative stiffness part consists of the cylinder body 2, piston rod 3, and tension spring 4. The left end of the tension spring 4 is connected to the piston rod 3, while the right side is connected to the cylinder body 2. The tensile quantity $δ$ of the tensile spring 4 at the static equilibrium position has the stiffness $k_{h}$ . $F_{μ}$ is the Coulomb friction force produced by the cylinder block 2 moving relative to the piston rod 3 in the negative stiffness mechanism, and $l$ is the single-side length of the system. The positive stiffness system consists of a vertical spring 6, column 1, and base 7, where the stiffness is $k_{1}$ and the damping is $c_{1}$ . The quality of mass block 5 is $m_{1}$ , and it moves up and down along column 1 under the action of simple harmonic force $F_{0} \cos (ω t)$ . The moving distance is expressed by $X_{1}$ , and the origin of $X_{1}$ is the static balance position of the vibration isolation system.

Figure 1.

Schematic diagram of a tension spring quasi-zero stiffness isolator. (a) Three-dimensional modeling of a vibration isolator. (b) Schematic diagram of a vibration isolator.

Under the exciting force, the mass block $m_{1}$ makes a displacement $X_{1}$ , and the vertical restoring force of a quasi-zero stiffness vibration isolator is

F (X_{1}) = X_{1} k_{1} - \frac{2 X_{1}}{\sqrt{l^{2} + X_{1}^{2}}} {k_{h} [δ - (\sqrt{l^{2} + X_{1}^{2}} - l)] - F_{μ}}

(1)

Let $F (x_{1}) = ((F (X_{1})) / (k_{1} δ))$ , $x_{1} = (X_{1} / l)$ , $δ_{1} = (δ / l)$ , $f_{μ} = ((F_{μ}) / (k_{1} δ))$ , and $k = (k_{h} / k_{1})$ , and when substituting it in equation (1), the dimensionless force of a quasi-zero stiffness vibration isolator can be obtained

F (x_{1}) = \frac{x_{1}}{δ_{1}} + \frac{2 x_{1} f_{μ}}{\sqrt{1 + x_{1}^{2}}} - \frac{2 x_{1} δ_{1} k - 2 x_{1} k \sqrt{1 + x_{1}^{2}} + 2 x_{1} k}{δ_{1} \sqrt{1 + x_{1}^{2}}}

(2)

When equation (2) is divided by x₁, one can get the dimensionless stiffness of the isolator

k (x_{1}) = \frac{1}{δ_{1}} + \frac{2 f_{μ} δ_{1} - 2 δ_{1} k - 2 k}{δ_{1} {(x_{1}^{2} + 1)}^{\frac{1}{2}}} + \frac{2 k}{δ_{1}} + \frac{2 x_{1}^{2} δ_{1} k - 2 x_{1}^{2} f_{μ} δ_{1} + 2 x_{1}^{2} k}{δ_{1} {(x_{1}^{2} + 1)}^{\frac{3}{2}}}

(3)

Let $x_{1} = 0$ ; by substituting it in equation (3), one can determine the quasi-zero stiffness condition of a vibration isolator

2 k = \frac{1}{δ_{1}} + 2 f_{μ}

(4)

When condition (4) is met, the vibration isolation system has quasi-zero stiffness.

From equations (2) and (3), the restoring force–displacement curves of an isolator can be obtained with different Coulomb frictions $f_{μ}$ , pre-tensions $δ_{1}$ , and stiffness ratio $k$ , as shown in Figures 2 –4. It can be seen from Figure 2 that with the increase of Coulomb friction in the negative stiffness mechanism, the positive stiffness characteristics of a system near static equilibrium constantly get enhanced. According to Figure 3, when the pre-stretched length of an extension spring in the negative stiffness mechanism decreases constantly, the positive stiffness property of a quasi-zero stiffness vibration isolation system exhibits a growing trend. It can be seen from Figure 4 with the increase of $k$ , the system shows a positive stiffness characteristic. It is observed from each stiffness–displacement curve that the system exhibits quasi-zero stiffness characteristics in the static equilibrium position when system parameters satisfy equation (4).

Figure 2.

The dimensionless restoring force and stiffness diagram of a negative stiffness mechanism with different Coulomb frictional forces. (a) Dimensionless restoring force displacement. (b) Dimensionless stiffness displacement.

Figure 3.

The dimensionless restoring force and stiffness diagram under different pre-tension quantities. (a) Dimensionless restoring force displacement. (b) Dimensionless stiffness displacement.

Figure 4.

The dimensionless restoring force and stiffness diagram for stiffness ratios. (a) Dimensionless restoring force displacement. (b) Dimensionless stiffness displacement.

The simplified form²⁶ of equations (2) and (3) can be obtained by the Taylor expansion

F (x_{1}) \approx β x_{1} + α x_{1}^{3}

(5)

k (x_{1}) \approx β + 3 α x_{1}^{2}

(6)

where $α = (((δ_{1} + 1) k) / (δ_{1})) - f_{μ}$ , $β = (1 / δ_{1}) + 2 f_{μ} - 2 k$ .

From Figures 2 –4 and equations (5) and (6), it can be noted that the dimensionless force of the system is an odd function, while the dimensionless stiffness is an even function.

Figure 5 gives a comparison between the Taylor expansion and the original expression. Figure 5(a) shows the restoring force property obtained from equations (5) and (2). Figure 5(b) shows the stiffness property obtained from equations (6) and (3). It is clear that the result of the Taylor expansion coincides with that of the original expression when the displacement is small, so the Taylor expansion in (5) and (6) is able to respect the quasi-zero stiffness property of the isolator.

Figure 5.

A comparison between the Taylor expansion and the original expression.

Dynamic analysis for a vibration isolatorwith a vibration absorber

A physical model of a grounded dynamic vibration absorber is designed to weaken the strong resonance peak in a vibration isolator so that the system is stable at any excitation frequency. The absorber is composed of a shock-absorbing spring 8, a negative stiffness rod 9,²⁷ and a shock-absorbing mass block 10. The proposed vibration isolator with a vibration absorber (VIVA) is shown in Figure 6, where $X_{1}, m_{1}$ are the displacement and mass of isolator, respectively; $X_{2}, m_{2}$ are the displacement and mass of the vibration absorber, respectively; $k_{2}$ is the spring stiffness of a vibration absorber; $c_{2}$ is the damping of a vibration absorber; $k_{3}$ is the stiffness of the negative stiffness rod 9 on the vibration absorber.

Figure 6.

Quasi-zero stiffness vibration isolator with a vibration absorber. (a) Three-dimensional modeling of a vibration isolator with a vibration absorber. (b) Schematic diagram of a VIVA.

Nonlinear dynamics analysis

The nonlinear dynamic equations of the VIVA system is established

{\begin{matrix} m_{1} {\overset{\cdot\cdot}{X}}_{1} + c_{1} {\overset{\cdot}{X}}_{1} + β k_{1} X_{1} + α k_{1} \frac{X_{1}^{3}}{l^{2}} + c_{2} ({\overset{\cdot}{X}}_{1} - {\overset{\cdot}{X}}_{2}) + k_{2} (X_{1} - X_{2}) = F_{0} \cos (ω t), \\ m_{2} {\overset{\cdot\cdot}{X}}_{2} + c_{2} ({\overset{\cdot}{X}}_{2} - {\overset{\cdot}{X}}_{1}) + k_{2} (X_{2} - X_{1}) + k_{3} X_{2} = 0 \end{matrix}

(7)

Assuming $ω_{1} = \sqrt{β k_{1} / m_{1}}$ , $ω_{2} = \sqrt{k_{2} / m_{2}}$ , $ξ_{1} = (c_{1} / 2 m_{1} ω_{1})$ , $ξ_{2} = (c_{2} / (2 m_{2} ω_{2}))$ , $γ = (α / β)$ , $μ = (m_{2} / m_{1})$ , $v = (ω_{2} / ω_{1})$ , $f_{0} = (F_{0} / l β k_{1})$ , $Ω = (ω / ω_{1})$ , $τ = ω_{1} t$ , and $k_{4} = (k_{3} / k_{2})$ and substituting them in equation (7)

{\begin{matrix} {\overset{\cdot\cdot}{x}}_{1} + 2 ξ_{1} {\overset{\cdot}{x}}_{1} + x_{1} + γ x_{1}^{3} + 2 v μ ξ_{2} ({\overset{\cdot}{x}}_{1} - {\overset{\cdot}{x}}_{2}) + v^{2} μ (x_{1} - x_{2}) = f_{0} \cos (Ω τ) \\ {\overset{\cdot\cdot}{x}}_{2} + 2 v ξ_{2} ({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{1}) + v^{2} (x_{2} - x_{1}) + v^{2} k_{4} x_{2} = 0 \end{matrix}

(8)

In view of the low damping, small excitation amplitude, and weak nonlinearity,²⁸ we assume that $ξ_{1} = O (ε)$ , $γ = O (ε)$ , $v ξ_{2} = O (ε)$ , $v^{2} = O (ε)$ , $Ω^{2} = 1 + ε σ$ , $σ = O (1)$ , and $f_{0} = O (ε)$ . Here, $ε$ is a small parameter, and $σ$ represents tuning frequency

{\begin{matrix} {\overset{\cdot\cdot}{x}}_{1} + 2 ε ξ_{1} {\overset{\cdot}{x}}_{1} + x_{1} + ε γ x_{1}^{3} + 2 ε v ξ_{2} μ ({\overset{\cdot}{x}}_{1} - {\overset{\cdot}{x}}_{2}) + ε v^{2} μ (x_{1} - x_{2}) = ε f_{0} \cos (Ω τ) \\ {\overset{\cdot\cdot}{x}}_{2} + 2 ε v ξ_{2} ({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{1}) + ε v^{2} (x_{2} - x_{1}) + ε v^{2} k_{4} x_{2} = 0 \end{matrix}

(9)

Let

{\begin{matrix} x_{1} = x_{11} (τ_{0}, τ_{1}, τ_{2}) + ε x_{12} (τ_{0}, τ_{1}, τ_{2}) + ε^{2} x_{13} (τ_{0}, τ_{1}, τ_{2}), \\ x_{2} = x_{21} (τ_{0}, τ_{1}, τ_{2}) + ε x_{22} (τ_{0}, τ_{1}, τ_{2}) + ε^{2} x_{23} (τ_{0}, τ_{1}, τ_{2}) \end{matrix}

(10)

Where $τ_{0} = τ$ , $τ_{1} = ε τ$ , and $τ_{2} = ε^{2} τ$ .

The partial derivative operator is defined as

{\begin{matrix} \frac{d}{d τ} = D_{0} + ε D_{1} + ε^{2} D_{2} + O (ε^{2}) \\ \frac{d^{2}}{d τ^{2}} = D_{0}^{2} + 2 ε D_{0} D_{1} + ε^{2} (2 D_{0} D_{2} + D_{1}^{2}) + O (ε^{2}) \end{matrix}

(11)

Substituting equations (10) and (11) in equation (9), one may get

{\begin{matrix} D_{0}^{2} x_{11} + Ω^{2} x_{11} = 0 \\ D_{0}^{2} x_{21} = 0 \end{matrix}

(12)

{\begin{matrix} D_{0}^{2} x_{12} + Ω^{2} x_{12} = - 2 D_{0} D_{1} x_{11} - 2 ξ_{1} D_{0} x_{11} + σ x_{11} - γ x_{11}^{3} - 2 v ξ_{2} μ D_{0} x_{11} \\ + 2 v ξ_{2} μ D_{0} x_{21} - v^{2} μ x_{11} + v^{2} μ x_{21} + f_{0} \cos (Ω τ_{0}) \\ D_{0}^{2} x_{22} = - 2 D_{0} D_{1} x_{21} - 2 v ξ_{2} D_{0} x_{21} + 2 v ξ_{2} D_{0} x_{11} - v^{2} x_{21} + v^{2} x_{11} + v^{2} k_{4} x_{21} \end{matrix}

(13)

{\begin{matrix} D_{0}^{2} x_{13} + Ω^{2} x_{13} = - 2 D_{0} D_{1} x_{12} - 2 D_{0} D_{2} x_{11} - D_{1}^{2} x_{11} - 2 ξ_{1} D_{1} x_{11} - 2 ξ_{1} D_{0} x_{12} \\ + σ x_{12} - γ x_{12}^{3} - 2 v ξ_{2} μ D_{0} x_{12} - 2 v ξ_{2} μ D_{1} x_{11} + 2 v ξ_{2} μ D_{0} x_{22} \\ + 2 v ξ_{2} μ D_{1} x_{21} - v^{2} μ x_{12} + v^{2} μ x_{22}, \\ D_{0}^{2} x_{23} = - 2 D_{0} D_{1} x_{22} - 2 D_{0} D_{2} x_{21} - D_{1}^{2} x_{21} - 2 v ξ_{2} D_{0} x_{22} - 2 v ξ_{2} D_{1} x_{21} \\ + 2 v ξ_{2} D_{0} x_{12} + 2 v ξ_{2} D_{1} x_{11} - v^{2} x_{22} + v^{2} x_{12} - v^{2} k_{4} x_{22} \end{matrix}

(14)

Assuming the solution to the equation in equation (12) is

{\begin{matrix} x_{11} = A_{1} (τ_{1}, τ_{2}) e^{j Ω τ_{0}} + cc \\ x_{21} = A_{2} (τ_{1}, τ_{2}) + cc \end{matrix}

(15)

Where $cc$ indicates the conjugation of previous terms.

Substituting equation (15) in the equation (13), one can get

{\begin{matrix} D_{0}^{2} x_{12} + Ω^{2} x_{12} = (- 2 D_{1} A_{1} i Ω - 2 ξ_{1} A_{1} i Ω + σ A_{1} - 3 A_{1}^{2} {\bar{A}}_{1} γ - 2 v ξ_{2} μ A_{1} i Ω \\ - v^{2} μ A_{1} + \frac{f_{0}}{2}) e^{i Ω τ_{0}} - γ A_{1}^{3} e^{3 i Ω τ_{0}} + v^{2} μ A_{2} + cc, \\ D_{0}^{2} x_{22} = 2 v ξ_{2} A_{1} i Ω e^{i Ω τ_{0}} + v^{2} A_{1} e^{i Ω τ_{0}} - v^{2} A_{2} - v^{2} k_{4} A_{2} + cc \end{matrix}

(16)

In this system, only the dominant frequency is considered, so the perpetual term condition of the first formula in the elimination equation (16) is obtained as

- 2 D_{1} A_{1} i Ω - 2 ξ_{1} A_{1} i Ω + σ A_{1} - 3 γ A_{1}^{2} {\bar{A}}_{1} - 2 v ξ_{2} μ A_{1} i Ω - v^{2} μ A_{1} + \frac{f_{0}}{2} = 0

(17)

The second formula in equation (16) does not have a perpetual term.

The special solutions $x_{12}$ and $x_{22}$ are

{\begin{matrix} x_{12} = \frac{γ A_{1}^{3}}{8 Ω^{2}} e^{3 i Ω τ_{0}} + \frac{v^{2} μ A_{2}}{Ω^{2}} + cc \\ x_{22} = - \frac{2 v ξ_{2} A_{1} i Ω + v^{2} A_{1}}{Ω^{2}} e^{i Ω τ_{0}} - \frac{(1 + k_{4}) v^{2} A_{2} τ_{0}^{2}}{2} + cc \end{matrix}

(18)

Substituting equations (17) and (18) in equation (14) yields

{\begin{matrix} D_{0}^{2} x_{13} + Ω^{2} x_{13} = B_{1} e^{i Ω τ_{0}} + (\frac{5 A_{1}^{3} σ γ}{4 Ω^{2}} - \frac{3 A_{1}^{6} {\bar{A}}_{1}^{3} γ^{4}}{512 Ω^{6}} - \frac{27 A_{1}^{4} {\bar{A}}_{1} γ^{2}}{8 Ω^{2}} + \frac{9 A_{1}^{2} f_{0} γ}{16 Ω^{2}} \\ - \frac{3 {iA}_{1}^{3} ξ_{1} γ}{Ω} - \frac{5 A_{1}^{3} γ μ v^{2}}{4 Ω^{2}} - \frac{3 {iA}_{1}^{3} ξ_{2} γ μ v}{Ω} - \frac{3 A_{1}^{3} γ^{2} μ^{2} v^{4} {(A_{2} + {\bar{A}}_{2})}^{2}}{8 Ω^{6}}) e^{3 i Ω τ_{0}} \\ + \frac{3 A_{1}^{6} γ^{3} μ v^{2} (A_{2} + {\bar{A}}_{2})}{64 Ω^{6}} e^{6 i Ω τ_{0}} + \frac{A_{1}^{9} γ^{4}}{512 Ω^{6}} e^{9 i Ω τ_{0}} + \frac{σ v^{2} μ A_{2}}{Ω^{2}} \\ - 2 v^{3} ξ_{2} μ A_{2} τ_{0} (1 + k_{4}) - \frac{v^{4} μ^{2} A_{2}}{Ω^{2}} - \frac{v^{4} μ A_{2} τ_{0}^{2} (1 + k_{4})}{2} + \frac{v^{6} μ^{3} γ A_{2}^{3}}{Ω^{6}} \\ + \frac{3 A_{2} {\bar{A}}_{2}^{2} v^{6} μ^{3} γ}{Ω^{6}} + \frac{3 A_{1}^{3} A_{2} {\bar{A}}_{1}^{3} v^{2} μ γ^{3}}{32 Ω^{6}} + cc, \\ D_{0}^{2} x_{23} = B_{2} e^{i Ω τ_{0}} + (\frac{6 i γ A_{1}^{3} v ξ_{2}}{8 Ω} + \frac{γ A_{1}^{3} v^{2}}{8 Ω^{2}}) e^{3 i Ω τ_{0}} + \frac{v^{4} A_{2} τ_{0}^{2} {(1 + k_{4})}^{2}}{2} + \frac{v^{4} μ A_{2}}{Ω^{2}} \\ + v^{2} A_{2} τ_{0} (1 + k_{4}) + cc \end{matrix}

(19)

where

\begin{matrix} B_{1} = - 2 D_{2} A_{1} i Ω - \frac{i f_{0} ξ_{1}}{2 Ω} + \frac{σ^{2} A_{1}}{4 Ω^{2}} - 3 A_{1} ξ_{2}^{2} μ^{2} v^{2} + \frac{σ f_{0}}{8 Ω^{2}} - 2 A_{1} ξ_{1}^{2} + \frac{A_{1} μ^{2} v^{4}}{4 Ω^{2}} \\ - \frac{3 i A_{1} σ ξ_{1}}{2 Ω} + \frac{27 A_{1}^{3} {\bar{A}}_{1}^{2} γ^{2}}{4 Ω^{2}} - 5 A_{1} ξ_{1} ξ_{2} μ v - \frac{3 A_{1}^{2} f_{0} γ}{8 Ω^{2}} - \frac{f_{0} μ v^{2}}{8 Ω^{2}} + \frac{15 {iA}_{1}^{2} {\bar{A}}_{1} γ ξ_{1}}{2 Ω} \\ - \frac{A_{1} σ μ v^{2}}{2 Ω^{2}} + \frac{3 i A_{1} ξ_{1} μ v^{2}}{2 Ω} + \frac{2 i A_{1} ξ_{2} μ^{2} v^{3}}{Ω} - \frac{3 A_{1} {\bar{A}}_{1} f_{0} γ}{4 Ω^{2}} - \frac{3 i f_{0} ξ_{2} μ v}{4 Ω} - \frac{3 A_{1}^{2} {\bar{A}}_{1} σ γ}{Ω^{2}} \\ - \frac{2 i A_{1} σ ξ_{2} μ v}{Ω} + \frac{3 A_{1}^{2} {\bar{A}}_{1} γ μ v^{2}}{Ω^{2}} + \frac{9 {iA}_{1}^{2} {\bar{A}}_{1} γ ξ_{2} μ v}{Ω} - \frac{4 i A_{1} ξ_{2} μ v^{3}}{Ω} + 4 A_{1} ξ_{2}^{2} μ v^{2} + \frac{v^{4} A_{1} μ}{Ω^{2}} \end{matrix}

\begin{matrix} B_{2} = \frac{A_{1} v^{4}}{Ω^{2}} - \frac{f_{0} v^{2}}{2 Ω^{2}} - 2 A_{1} ξ_{1} ξ_{2} v - 4 A_{1} ξ_{2}^{2} v^{2} - \frac{A_{1} σ v^{2}}{Ω^{2}} + \frac{A_{1} k_{4} v^{4}}{Ω^{2}} - 2 A_{1} ξ_{2}^{2} μ v^{2} + \frac{A_{1} μ v^{4}}{Ω^{2}} \\ + \frac{2 i A_{1} ξ_{1} v^{2}}{Ω} + \frac{4 i A_{1} ξ_{2} v^{3}}{Ω} - \frac{i f_{0} ξ_{2} v}{2 Ω} + \frac{3 A_{1}^{2} {\bar{A}}_{1} γ v^{2}}{Ω^{2}} + \frac{i A_{1} ξ_{2} μ v^{3}}{Ω} - \frac{i A_{1} σ ξ_{2} v}{Ω} + \frac{2 i A_{1} k_{4} ξ_{2} v^{3}}{Ω} \\ + \frac{2 i A_{1} ξ_{2} μ v^{3}}{Ω} + \frac{3 {iA}_{1}^{2} {\bar{A}}_{1} γ ξ_{2} v}{Ω} \end{matrix}

The perpetual term condition of the first formula in the elimination formula (19) is

B_{1} = 0

(20)

The second formula in equation (19) does not have a perpetual term.

The complex function $A_{1} (ε τ, ε^{2} τ)$ is expressed as

A_{1} (ε τ, ε^{2} τ) = \frac{a_{1} (τ)}{2} e^{i Φ_{1} (τ)}

(21)

Substituting equation (21) in equations (17) and (20) and separating the real part from the imaginary part, one can get

{\begin{matrix} {\overset{\cdot}{a}}_{1} (τ) = \frac{1}{2 i Ω} (\frac{f_{0} ξ_{1}}{2 i Ω} - ξ_{1} i Ω a_{1} - v ξ_{2} μ a_{1} i Ω - \frac{f_{0}}{2} \sin Φ_{1} + \frac{3 a_{1} σ ξ_{1}}{4 i Ω} - \frac{15 a_{1}^{3} γ ξ_{1}}{16 i Ω} \\ - \frac{3 a_{1} ξ_{1} μ v^{2}}{4 i Ω} - \frac{a_{1} ξ_{2} μ^{2} v^{3}}{i Ω} + \frac{3 f_{0} ξ_{2} μ v}{4 i Ω} + \frac{a_{1} σ ξ_{2} μ v}{i Ω} - \frac{9 a_{1}^{3} γ ξ_{2} μ v}{8 i Ω} + \frac{2 a_{1} ξ_{2} μ v^{3}}{i Ω}) \\ a_{1} {\overset{\cdot}{Φ}}_{1} (τ) = - \frac{1}{2 Ω} (\frac{σ a_{1}}{2} - \frac{3 a_{1}^{3} γ}{8} - \frac{a_{1} μ v^{2}}{2} + \frac{f_{0}}{2} \cos Φ_{1} + \frac{a_{1} σ^{2}}{8 Ω^{2}} - \frac{3 a_{1} ξ_{2}^{2} μ^{2} v^{2}}{2} + \frac{f_{0} σ}{8 Ω^{2}} \\ + a_{1} ξ_{1}^{2} + \frac{a_{1} μ^{2} v^{4}}{8 Ω^{2}} + \frac{27 a_{1}^{5} γ^{2}}{128 Ω^{2}} - \frac{5 a_{1} ξ_{1} ξ_{2} μ v}{2} - \frac{9 a_{1}^{2} f_{0} γ}{32 Ω^{2}} - \frac{f_{0} μ v^{2}}{8 Ω^{2}} - \frac{a_{1} σ μ v^{2}}{4 Ω^{2}} \\ - \frac{3 a_{1}^{3} σ γ}{8 Ω^{2}} + \frac{3 a_{1}^{3} γ μ v^{2}}{8 Ω^{2}} + 2 a_{1} ξ_{2}^{2} μ v^{2} + \frac{a_{1} μ v^{4}}{2 Ω^{2}}) \end{matrix}

(22)

Where the amplitude $a_{1}$ and phase $Φ_{1}$ are real functions of graded time scale $τ_{2}$ . Thus, the amplitude–frequency response equation can be gained

\begin{matrix} (4 a_{1} Ω^{4} - 4 a_{1} Ω^{2} - 3 a_{1}^{3} γ Ω^{2} - 4 a_{1} μ v^{2} Ω^{2} + Ω^{4} a_{1} - 2 Ω^{2} a_{1} + a_{1} - 12 a_{1} ξ_{2}^{2} μ^{2} v^{2} Ω^{2} \\ + Ω^{2} f_{0} - f_{0} - 8 a_{1} ξ_{1}^{2} Ω^{2} + a_{1} μ^{2} v^{4} + \frac{27 a_{1}^{5} γ^{2}}{16} - 20 a_{1} ξ_{1} ξ_{2} μ v Ω^{2} - \frac{9 a_{1}^{2} f_{0} γ}{4} \\ - f_{0} μ v^{2} - 2 a_{1} Ω^{2} μ v^{2} + 2 a_{1} μ v^{2} - 3 a_{1}^{3} Ω^{2} γ + 3 a_{1}^{3} γ + 3 a_{1}^{3} γ μ v^{2} \\ + 16 a_{1} ξ_{2}^{2} μ v^{2} Ω^{2} + {4 a_{1} μ v^{4})}^{2} + (2 f_{0} ξ_{1} + 4 ξ_{1} Ω^{2} a_{1} + 4 v ξ_{2} μ a_{1} Ω^{2} \\ + 3 a_{1} ξ_{1} Ω^{2} - 3 a_{1} ξ_{1} - \frac{15 a_{1}^{3} γ ξ_{1}}{4} - 3 a_{1} ξ_{1} μ v^{2} - 4 a_{1} ξ_{2} μ^{2} v^{3} + 3 f_{0} ξ_{2} μ v \\ + 4 a_{1} ξ_{2} μ v Ω^{2} - 4 a_{1} ξ_{2} μ v - \frac{9 a_{1}^{3} γ ξ_{2} μ v}{2} + {8 a_{1} ξ_{2} μ v^{3})}^{2} = 16 f_{0}^{2} Ω^{4} \end{matrix}

(23)

Linear dynamics analysis

The classical method^29,30 is used to ignore the nonlinear term of the VIVA system to establish the dynamic equation

{\begin{matrix} m_{1} {\overset{\cdot\cdot}{X}}_{1} + c_{1} {\overset{\cdot}{X}}_{1} + β k_{1} X_{1} + c_{2} ({\overset{\cdot}{X}}_{1} - {\overset{\cdot}{X}}_{2}) + k_{2} (X_{1} - X_{2}) = F_{0} \cos (ω t), \\ m_{2} {\overset{\cdot\cdot}{X}}_{2} + c_{2} ({\overset{\cdot}{X}}_{2} - {\overset{\cdot}{X}}_{1}) + k_{2} (X_{2} - X_{1}) + k_{3} X_{2} = 0 \end{matrix}

(24)

Let $f_{1} = (F_{0} / m_{1})$ and $f_{n} = (F_{0} / k_{1})$ . Assuming the response $X_{1} = X_{11} e^{i ω t}$ and $X_{2} = X_{21} e^{i ω t}$ and substituting them in equation (24), one can get

{\begin{matrix} - ω^{2} X_{11} + 2 ω_{1} ξ_{1} j ω X_{11} + ω_{1}^{2} X_{11} + 2 μ ω_{2} ξ_{2} j ω (X_{11} - X_{21}) + μ ω_{2}^{2} (X_{11} - X_{21}) = f_{1}, \\ - ω^{2} X_{21} + 2 ω_{2} ξ_{2} j ω (X_{21} - X_{11}) + ω_{2}^{2} (X_{21} - X_{11}) + k_{4} ω_{2}^{2} X_{21} = 0 \end{matrix}

(25)

According to equation (25), the amplitude–frequency response formula of the VIVA system under a linear condition is obtained

a_{1}^{2} = {| \frac{X_{11}}{f_{n}} |}^{2} = \frac{E_{1}^{2} + E_{2}^{2}}{E_{3}^{2} + E_{4}^{2}}

(26)

where

E_{1} = 2 ξ_{2} Ω v

E_{2} = k_{4} v^{2} - Ω^{2} + v^{2}

E_{3} = 2 ξ_{1} Ω v^{2} - 2 ξ_{1} Ω^{3} - 2 ξ_{2} μ Ω^{3} v - 2 ξ_{2} Ω^{3} v + 2 ξ_{2} Ω v + 2 k_{4} ξ_{1} Ω v^{2} + 2 k_{4} ξ_{2} μ Ω v^{3}

E_{4} = v^{2} - Ω^{2} - Ω^{2} v^{2} + Ω^{4} + k_{4} μ v^{4} - k_{4} Ω^{2} v^{2} + k_{4} v^{2} - μ Ω^{2} v^{2} - 4 ξ_{1} ξ_{2} Ω^{2} v

Parameter analysis for a VIVA

The nonlinear analytical solution of the amplitude–frequency response of a quasi-zero stiffness VIVA system is obtained with equation (23), and the linear analytical solution can be obtained with equation (26). In the nonlinear analytical equation, let $m_{1} = 1 kg$ , $m_{2} = 0.1 kg$ , $δ = 0.1 m$ , $l = 0.2 m$ , $k_{1} = 1000 N / m$ , $k_{h} = 800 N / m$ , $k_{4} = - 0.5$ , $f_{0} = 0.07$ , $f_{μ} = 0$ , $v = 0.58$ , $ξ_{1} = 0.1$ , and $ξ_{2} = 0.2$ , while in the linear analytical equation, let $v = 1.1$ . The effects of parameters, including the ratio of natural frequencies $v$ , the damper ratio $ξ_{1}$ of the isolator, the damping ratio $ξ_{2}$ of the ground absorber, and the excitation amplitude $f_{0}$ , are analyzed in this section.

In this system, only increasing the ratio of natural frequencies $v$ is equivalent to increasing the stiffness $k_{2}$ of the VIVA system. As shown in Figure 7(A), with the increase of the ratio of natural frequencies $v$ , the response amplitude $a_{1}$ increases at the low-frequency resonance section, while it decreases at the high frequency resonance peak. Because the increase of the ratio of natural frequencies $v$ means the increase of $k_{2}$ , the ground absorber can absorb more energy, and the peak value of resonance at low frequency increases, which indicates that the peak value is caused by the ground absorber. It can be seen from Figure 7(A) and (B) that with the change of $v$ , both linear³¹ and nonlinear analytical solutions have this rule.

Figure 7.

Effect of the ratio of natural frequencies: (a) $v = 0.68$ , (b) $v = 0.63$ , (c) $v = 0.58$ , (d) $v = 1$ , (e) $v = 1.1$ , and (f) $v = 1.2$ . (A) Nonlinear analytical solution. (B) Linear analytic solution.

There is damping in the VIVA isolation system, so there is no anti resonance peak in the amplitude–frequency response diagram. As shown in Figure 8(A) and (B), with the increase of the damper ratio $ξ_{1}$ of the isolator, the two resonance peaks of the response amplitude $a_{1}$ decrease most significantly, compared with the other frequency band. The difference is that in Figure 8(A), the nonlinearity is obviously reduced.

Figure 8.

Effect of the vibration isolator damping ratio: (a) $ξ_{1} = 0.085$ , (b) $ξ_{1} = 0.1$ , and (c) $ξ_{1} = 0.14$ . (A) Nonlinear analytical solution. (B) Linear analytic solution.

As shown in Figure 9(A), when only the damping ratio $ξ_{2}$ of the ground absorber is changed, the response amplitude $a_{1}$ at the low-frequency resonance increases, while the response amplitude $a_{1}$ at the high frequency shortness decreases with the increase of $ξ_{2}$ . At the same time, only the amplitude $a_{1}$ at the resonant peak frequency segment changes, while the amplitude $a_{1}$ at other frequency segments almost coincides. This is the rule of obvious difference and linear analytic solution. As shown in Figure 9(B), with the increase of $ξ_{2}$ , the response amplitudes $a_{1}$ of both resonance peaks decrease, while the response amplitudes $a_{1}$ between the two resonance peaks increase.

Figure 9.

Effect of the vibration absorber damping ratio: (a) $ξ_{2} = 0.1$ , (b) $ξ_{2} = 0.2$ , and (c) $ξ_{2} = 0.4$ . (A) Nonlinear analytical solution. (B) Linear analytic solution.

As shown in Figure 10, there are two nonlinear resonance peaks. When the excitation amplitude $f_{0}$ increases, the nonlinearity of the system increases, and the response amplitude $a_{1}$ increases accordingly, which is the most obvious difference from the linear vibration absorption system.

Figure 10.

Effect of excitation amplitude: (a) $f_{0} = 0.07$ , (b) $f_{0} = 0.05$ , and (c) $f_{0} = 0.03$ .

Under the condition of constant system parameters, as shown in the amplitude–frequency response curve a in Figure 10, the VIVA system resonates at $Ω = 0.5$ and $Ω = 1.1$ . Therefore, it can be calculated from equations $Ω = (ω / ω_{1})$ , $ω_{1} = \sqrt{(β k_{1} / m_{1})}$ , and $f = (ω / 2 π)$ that the two frequencies of resonance of the VIVA system are $f = 0.159 Hz$ and $f = 3.503 Hz$ . However, if the isolator is a linear isolator containing only mass block 5 and vertical spring 6, the natural frequency is calculated as $f_{n} = 5.036 Hz$ by equations $ω_{n} = \sqrt{(k_{1} / m_{1})}$ and $f_{n} = (ω_{n} / 2 π)$ . Therefore, the resonance frequency of the VIVA system is less than that of the linear vibration isolation system, which can realize the purpose of low-frequency vibration isolation.

Selection and verification of optimal parameters

Selection of optimal parameters

The VIVA system is nonlinear and the vibration isolation system has damping, so the optimal parameters of the system cannot be obtained by using the set-point theory. Therefore, the iterative algorithm and the influence of parameters in Figures 7 –10 on the amplitude–frequency response are used to obtain the optimal parameter, which is $m_{1} = 1 kg$ , $m_{2} = 0.1 kg$ , $δ = 0.1 m$ , $l = 0.2 m$ , $k_{1} = 1000 N / m$ , $k_{h} = 800 N / m$ , $f_{0} = 0.07$ , $f_{μ} = 0$ , $v = 0.66$ , $ξ_{1} = 0.1$ , and $ξ_{2} = 0.2$ . Under this condition, the VIVA system is compared with the isolator, and the nonlinear analytical solution is compared with the linear analytical solution.

Through equation (23), the comparison diagram of amplitude–frequency response curves of the grounded vibration absorber and the quasi-zero stiffness isolator under nonlinear conditions can be obtained, as shown in Figure 11(A). The peak value of response amplitude of the VIVA system is $a_{1} = 0.63$ , while that of the quasi-zero stiffness isolator is $a_{1} = 0.84$ , which is equivalent to a 25% decrease in response amplitude compared with that without an absorber. As shown in Figure 11(B), the comparison of nonlinear and linear amplitude–frequency response curves under the optimal parameters is shown, in which the linear amplitude–frequency curve is obtained through equation (26). The peak value of the linear response is $a_{1} = 0.26$ , which is 58% lower than the nonlinear peak value. Therefore, in the dynamic analysis of nonlinear two degree of freedom model, the nonlinear term cannot be ignored.

Figure 11.

Comparison diagram under optimal parameters: (a) Nonlinear analytical solution of the vibration isolator with a vibration absorber system, (b) nonlinear analytical solution of the quasi-zero stiffness isolator, and (c) linear analytical solution of the vibration isolator with a vibration absorber system. (A) Comparison with a vibration isolator. (B) Comparison with linear analytical solution.

Numerical verification

The correctness of the analytical results of equation (23) is proved by using fourth order Runge–Kutta method. Numerical simulation is carried out for equation (8), in which $m_{1} = 1 kg$ , $m_{2} = 0.1 kg$ , $δ = 0.1 m$ , $l = 0.2 m$ , $k_{1} = 1000 N / m$ , $k_{h} = 800 N / m$ , $f_{0} = 0.07$ , $f_{μ} = 0$ , $v = 0.66$ , $ξ_{1} = 0.1$ , and $ξ_{2} = 0.2$ . As shown in Figure 12(a), the frequency ratio at the low-frequency resonance peak is $Ω = 0.67$ . As shown in Figure 12(b), the frequency ratio at the high frequency resonance peak is $Ω = 1.22$ . The fitting degree of numerical solution and analytical solution at two resonance peaks is very high, so the correctness of analytical results is proved.

Figure 12.

Comparison between numerical solution and analytical solution. (a) Low-frequency resonance peak. (b) High frequency resonance peak.

The numerical solution of the quasi-zero stiffness isolator at resonance is compared with that of the VIVA system, where $Ω = 1.34$ , as shown in Figure 13(a). At the same time, the numerical solution of the VIVA system under a linear condition is compared with that under a nonlinear condition, where $Ω = 1.01$ , as shown in Figure 13(b). The numerical solutions in Figure 13 are almost equal to the analytical solutions in Figure 11, so the correctness of the analysis results is proved.

Figure 13.

Comparison diagram under optimal parameters. (a) Comparison with a vibration isolator. (b) Comparison with linear analytical solution.

Conclusion

Based on static and dynamic analysis of a nonlinear quasi-zero stiffness VIVA system, a novel VIVA system is proposed to achieve low-frequency vibration isolation to avoid strong resonance, and the system parameters are optimized, with the correctness of the analytical solution proved by numerical verification.

According to the static analysis of the quasi-zero stiffness isolator, it is found that the quasi-zero stiffness is related to the Coulomb friction, pre-tension, and stiffness ratio. Only if these three system parameters satisfy equation (4), the quasi-zero stiffness can be achieved.

The proposed VIVA system is able to solve the problem of strong resonance effectively and realize vibration isolation in low frequency. Under the optimal parameters, the peak value of the response amplitude of the isolator is reduced by 25% compared with that of the quasi-zero stiffness isolator.

The nonlinear analytical solution of the VIVA system is compared with the linear analytical solution. The influence of system parameters on the two solutions is basically the same, but the nonlinear term cannot be ignored; otherwise, the error between them can reach 58%.

Footnotes

Availability of data and material

The manuscript data used to support the findings of this study are included within the article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by the National Natural Science Foundation of China (11972238).

ORCID iDs

Shao-Hua Li

Nan Liu

Hu Ding

Author biographies

Shao-Hua Li received PhD degree in Vehicle Engineering from Beijing Jiaotong University in 2009. Now she works at Shijiazhuang Tiedao University as a professor. Her current research interests include vehicle-road interaction dynamics and vibration control.

Nan Liu is a graduate student of Shijiazhuang Tiedao University. His research interest is vibration control.

Hu Ding received PhD degree in general mechanics and mechanics foundation from Shanghai University in 2008. Now he is a professor of Shanghai University. His research interest is structural dynamics and vibration control.

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