Bifurcation and singularity analysis of HSLDS vibration isolation system with elastic base

1. Introduction

Undesirable vibration can affect human comfort and even the structural safety, which has become an urgent problem to be solved in engineering. It is evident that the bandwidth of vibration isolation is often limited by the mount stiffness element required to support a static load. To overcome this limitation, the high-static-low-dynamic-stiffness (HSLDS) mechanism is put forward, what results in low natural frequency with a small static displacement. Whilst it maintains locally low stiffness near equilibrium and static load bearing, which reduces the natural frequency and extends the frequency isolation region [1]. The isolation system with HSLDS characteristic has been well established both theoretically and experimentally in recent literatures and has recently been the subject of growing interest of both engineers and researchers. Carrella et al. and Wu et al. investigated vibration isolators with HSLDS property via the combination of a mechanical spring and magnets [2, 3], Li et al. presented a device using a magnetic spring combined with rubber membranes to suppress vibration [4], Zhou et al. develop a tunable isolator with HSLDS property by using a pair of electromagnets and a permanent magnet and can act passively or semi-actively[5], Meng et al. concerned the quasi-zero-stiffness by combining a negative disk spring with a linear positive spring [6].

Recently, many methods for bifurcation analysis of high-dimensional dynamical system have been proposed. Among these methods, singularity theory is of much importance and has been widely applied as a quanlative analysis method, which can solve the bifurcation problems uniformly and definitely. Yu et al. studied the local bifurcation of nonlinear vibration isolation system with 1:2 internal resonance [7, 8], Qin et al. investigated two-degree-of-freedom (two-DOF) bifurcation equation for elastic cable with 1:1 internal resonance [9], Wang et al. analyzed the bifurcation models of a class of power system by using C-L method [10], Zhou et al. considered the local bifurcation of a nonlinear system based on MR damper [11], Volkov et al. studied the bifurcation in the system of two identical diffusively coupled Brusselators [12].

In this paper, the local bifurcation and singularity analysis of a HSLDS-VIS with elastic base have been presented, which is organized as follows. In Section 2, the restoring force and the motion equations of HSLDS-VIS were established. In Section 3, the motion equations of HSLDS-VIS with elastic base were established. In Section 4, the averaging method was applied to obtain the bifurcation equation for the system with primary resonance and 1:2 internal resonance. In Section 5, the singularity theory was used to analyze the system local bifurcations to obtain the 4-codimensional universal unfolding, the transition sets and bifurcation diagrams. In Section 6, the transition sets were transferred back to the physical parameter space to obtain the bifurcation behaviors of the amplitude with respect of the external force. Finally, some conclusions were summarized in Section 7.

2. Modeling of the HSLDS-VIS

Consider a simple model of the isolator shown in Fig. 1. Two nonlinear oblique springs are assumed to have nonlinearity with linear stiffness $k_1$ and cubic nonlinear stiffness $k_3$. In addition, they are pre-stressed, i.e. compressed by length $\delta$ and connected at point C with a vertical unstressed linear spring of stiffness $k_2$. The oblique springs are hinged at $A$ and $B$. The geometry of configuration is decided by horizontal distance $a$ from point $A$ to $C$ and initial height $h$, while $x$ denotes the vertical displacement from the initial unloaded position caused by the force $F$.

Fig. 1Schematic representation of an isolator based on HSLDS

a)

b)

The general equation between the force $f$ and the displacement $x$ can be derived as:

Introducing $y=x-h$, $f=F-k_{2}h$, Eq. (1) can been recast in dimensionless form as:

where:

The dimensionless dynamic stiffness can be obtained by differentiating Eq. (2) to $\widehat{y}$:

Using the Maclaurin series expansion, an approximate expression for the stiffness is found to be:

It is clear that the oblique springs can reduce the positive stiffness so that the linear natural frequency is smaller in the isolation range; and they introduce the cubic stiffness so that the peak response bends to higher frequencies, which potentially reduces the frequency region.

For a SDOF system depicted in Fig. 2. It includes a rigid mass $m$ suspended on a three springs mount in parallel with a viscous damper $c$. $x$ denotes the vertical displacement from the equilibrium position caused by a harmonic excitation $f_e=F\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\Omega }T$. The mass moves in the vertical direction through the guide rod and bushing. By applying the Newton’s second law, the motion equation of SDOF system can be expressed as

where:

Fig. 2Structural model of the SDOF system with HSLDS characteristic: 1 – loading platform, 2 – oblique spring, 3 – guide device, 4 – pillar, 5 – vertical spring, 6 – base plate, 7 – linear bearing, 8 – sliding rod

a)

b)

3. Modeling of the HSLDS-VIS with elastic base

On the condition that taking the first order mode of the elastic base, it can be reduced to a rigid mass supported by a spring and damper. Therefore, the HSLDS-VIS with elastic base can be simplified to two-DOF mass-spring system, which is shown in Fig. 3. $M_1$ and $M_2$ denote the isolation equipment and the intermediate mass, respectively. $M_1$ is supported by a linear damper and a nonlinear spring which possesses both linear and cubic stiffness. $M_2$ is connected with a fixed plane using a linear spring and a linear damper. When the origins of coordinates are set at the position where the springs are not compressed, as shown in Fig. 3(a), the equation of the HSLDS-VIS with elastic base can be given by:

where $C_1$ is the damping coefficient of HSLDS vibration isolator; $K_1$ and $K_3$ are the linear and cubic stiffness coefficients of HSLDS vibration isolator, respectively. $C_2$ is the damping coefficient of damper between the $M_2$ and fixed plane; $K_2$ is the stiffness coefficient of the linear stiffness between the $M_2$ and fixed plane. $F$ and $\mathrm{\Omega }$ are the amplitude and frequency of the harmonic excitation, respectively.

Fig. 3HSLDS-VIS with elastic base. a) Coordinates not at the equilibrium state, b) Coordinates at the equilibrium state

Note that the origins are not the equilibrium points of the system in Fig. 3(a), which is inconvenient for further analysis, and hence the coordinate transformation should be carried out. As is shown in Fig. 3(b), the origins of the new coordinates are set at the equilibrium state and the relations between the old and new coordinates are: $Z_1=X_1+h_1$, $Z_2=X_2+h_2$. In the equilibrium state, the gravitation terms in Eq. (6) can be eliminated by the following relations: $M_{1}g+M_{2}g=K_2{h}_2$ and $M_{1}g=K_{1}H+K_3{H}^3$, where $H=h_1-h_2$. The governing equations of Eq. (6) are given by:

Setting $K_0=K_1+3K_3{H}^2\text{,}$$B=\sqrt{K_3/K_0}$ and introducing dimensionless parameters: $x_1=X_1/B$, $x_2=X_2/B$, ${\mathrm{\Omega }}_0=\sqrt{K_0/M_1}$, $t={\mathrm{\Omega }}_{0}T$, $\omega =\mathrm{\Omega }/{\mathrm{\Omega }}_0$, $w=M_1/M_2$, ${\xi }_1=C_1/\left(K_0{M}_1\right)$, ${\xi }_2=C_2/\left(K_0{M}_2\right)\text{,}$$\gamma =-3K_{3}H/\left(K_{0}B\right)\text{,}$$f=FB/K_0\text{,}$$k_2=K_2/K_0\text{,}$ the first-order of the dimensionless motion equations are given by:

Therefore, the HSLDS-VIS with elastic base is a coupled nonautonomous dynamic system with quadratic and cubic nonlinearities.

4. Standard form of the motion equations

Standard form of the motion equations refers to the vector fields are proportional to the small perturbation parameter $\epsilon$. Here, assume the nonlinear stiffness coefficients, damping coefficients and the amplitude of excitation force are small quantities and introduce an additional parameter $\epsilon$ into the equations. Setting $x_1=z_1$, ${\dot{x}}_1=z_2$, $x_2=z_3$, ${\dot{x}}_2=z_4$, Eq. (8) can be rewritten in the general form:

where:

The derivative equation of Eq. (9) is:

No damping and nonlinear force in the matrix $\mathbf{A}$, suppose it has only simple purely imaginary eigenvalues $\pm {\omega }_{1}i,\cdots ,\pm {\omega }_{q}i$. The special solution of the homogeneous part with $\pm {\omega }_i$ are as follows:

where $P_{si}$ and $Q_{si}$ are real constants.

Setting $z_1\left(t\right)=A_1{e}^{I\omega t}$, $z_2\left(t\right)=A_2{e}^{I\omega t}$, $z_3\left(t\right)=A_3{e}^{I\omega t}$, $z_4\left(t\right)=A_4{e}^{I\omega t}$, where $I=\sqrt{-1}$, substitute them into Eq. (11):

Substituting $A_1=\text{1}$ into Eq. (12), one may obtain:

where ${\omega }_{\mathrm{1,2}}^2=\left(1+w+w{k}_2\pm \sqrt{w^2{k}_2^2+2w^2{k}_2+w^2-2w{k}_2+2w+1}\right)/2$ are the natural frequencies of the derived system. Substituting ${\omega }_{\mathrm{1,2}}$ into Eq. (13) and separating the result into real and imaginary parts, yield the special solution of Eq. (10):

The same procedure may be easily adapted to obtain the special solution of the conjugate equation of Eq. (10), that is $d\mathbf{z}/dt=-{\mathbf{A}}^T\mathbf{z}$:

Introducing the coordinate transformation:

where ${\theta }_i={\omega }_{i}t$, $z_s\left(t\right)$ and $z_s^{\mathrm{*}}\left(t\right)$ are the general solution and special solution of Eq. (9), respectively. If $a_i=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ and $d{\theta }_i/dt={\omega }_i$, yield:

Substitute Eq. (16) into Eq. (9) and one may obtain:

Based on the orthogonality [13] between Eq. (14) and Eq. (15):

If $i=j$, ${\delta }_{ij}=1$, else if $i\ne j$, ${\delta }_{ij}=0$. Multiplying Eq. (17) by ${\psi }_{si}$ and ${\psi }_{si}^{\mathrm{*}}$, respectively, and then summing $s$ from 1 to 4, the standard form of Eq. (9) was obtained as follows:

where ${\mathrm{\Delta }}_{\mathrm{1,2}}=({\omega }_{\mathrm{1,2}}^4-2{\omega }_{\mathrm{1,2}}^2+w+1)/w$, $R_i$ and $S_i$ are periodic functions of $t$ and $\theta$ with period $2\pi /\omega$.

5. Single-variable bifurcation equation based on the averaging method

Applying the singularity theory, we can reveal all possible bifurcation behaviors when the original system is subjected to a small perturbation. While the singularity theory is only suitable for autonomous system, it is essential to obtain the single-variable bifurcation equation by using the averaging method. For the system with multiple DOFs, perhaps there exists internal resonance. Introducing the first-order KB transformation:

where ${\omega }_i={\omega }_{i0}+\epsilon {\sigma }_i$, $p_1{\omega }_{10}+p_2{\omega }_{20}=q\omega$, ${\sigma }_i$ is detuning parameters, $p_i\mathrm{}(i=1,\mathrm{}2)$ and $q$ are nonzero integers. $U_i$ and $V_i$ are periodic function of $\upsilon$ and $t$. This paper focuses the second order primary resonance and 1:2 internal resonance, which means $\omega$, ${\omega }_1$ and ${\omega }_2$ should meet $\omega ={\omega }_2+\epsilon {\sigma }_2$ and ${\omega }_2=2{\omega }_1+\epsilon {\sigma }_1$. The derivatives of the new parameters should satisfy the conditions below:

where $Y_i$ and $Z_i$ do not contain $t$, $Y_i^{\mathrm{*}}$ and $Z_i^{\mathrm{*}}$ are periodic function of $\upsilon$ and $t$. Applying the averaging method, substitute Eq. (20) and Eq. (21) into Eq. (19) and collect terms of the first order in $\epsilon$:

where $R_{i0}\left(y\right)$, $R_{i\zeta }\left(y\right)$, $R_{i\zeta }^{\text{'}}\left(y\right)$, $S_{i0}\left(y\right)$, $S_{i\zeta }\left(y\right)$, $S_{i\zeta }^{\text{'}}\left(y\right)$ are generalized Fourier series coefficients of $R_i$ and $S_i$, ${\alpha }_{\zeta }^0=\left(p_1^{\zeta }{\omega }_{10}+p_2^{\zeta }{\omega }_{20}+p_3^{\zeta }\omega \right)t+p_1^{\zeta }{\upsilon }_1+p_2^{\zeta }{\upsilon }_2$. $Y_i$ and $Z_i$ are the slowly changing functions, while $\frac{\partial U_i}{\partial t}+\sum _i^2\frac{\partial U_i}{\partial {\upsilon }_i}\left({\omega }_i-{\omega }_{i0}\right)$ and $\frac{\partial V_i}{\partial t}+\sum _i^2\frac{\partial V_i}{\partial {\upsilon }_i}\left({\omega }_i-{\omega }_{i0}\right)$ are the fast changing functions. The following average equations are obtained:

where $B_{\mathrm{1,2}}=(2{\xi }_2{\omega }_{\mathrm{1,2}}^2-{\xi }_1{\omega }_{\mathrm{1,2}}^4-{\xi }_2{\omega }_{\mathrm{1,2}}^4-{\xi }_2)/{\mathrm{\Delta }}_{\mathrm{1,2}}\text{.}$ Letting $d{y}_1\left(t\right)/dt\text{,}$$d{y}_2\left(t\right)/dt\text{,}$$d{\upsilon }_1\left(t\right)/dt$, $d{\upsilon }_2\left(t\right)/dt$ be zero, one may obtain:

Supposing $y_1=\sqrt{Y_1}$ and $y_2=\sqrt{Y_2}$, the Eq. (24) is further recast as follows:

The governing equations of Eq. (25) are given by:

Substituting $m^2={\omega }_1^6{\omega }_2^4{\gamma }^2{Y}_2-B_1^2{\mathrm{\Delta }}_1^2$ into Eq. (26) results in:

The same procedure may be easily adapted to obtain the following equations:

Substituting Eq. (27) into Eq. (28), the steady-state solution of Eq. (23) is obtained:

where the detailed $b_i$ are shown in Appendix A1. Setting $m=x+4\gamma {\omega }_1^3$, Eq. (29) can be written as follows:

where ${\alpha }_1$~${\alpha }_4$ is the universal unfolding parameters related to the stiffness coefficients, damping coefficients, tune parameters and so on, $\mu$ is the bifurcation parameter related to the amplitude of excitation force. The detailed ${\alpha }_1$~${\alpha }_4$ and $\mu$ are shown in Appendix A2.

6. Bifurcation analysis in the unfolding parameter space

In the singularity theory [14], the bifurcation Eq. (30) is the universal unfolding of the normal form $g=x^6-\mu$, and the codimension is 4. The functions $G_x$, $G_{xx}$ and $G_{\mu }$ can be given by differentiating:

According to the definition of the transition set [15], it consists of bifurcation point set (BS), hysteresis point set (HS), and double limit point set (DLS).

$BS=\left\{\alpha \in R^k\left|\exists (x,\mu ),G=G_{xx}=G_{\mu }=0\mathrm{}\right.\right.$ at the point of $\left.(x,\mu ,\alpha )\right\}$. It is evident that $BS=\mathrm{\Phi }$.

$HS=\left\{\alpha \in R^k\left|\exists (x,\mu ),G=G_{xx}=G_x=0\right.\right.$ at the point of $\left.(x,\mu ,\alpha )\right\}$. Setting $\mu =0$ and ${\alpha }_3={\alpha }_4=0$, $H{S}_0=\left\{{\alpha }_1\in R,{\alpha }_3=0,{\alpha }_4=0,{\alpha }_2\ne 0\right\}$ was obtained. Setting $x\ne 0$ and $\mu \ne 0$, $H{S}_1=\left\{{\alpha }_2=\frac{-10x^6-3{\alpha }_1{x}^4+\mu }{x^3},{\alpha }_3=\frac{3(5x^6+{\alpha }_1{x}^4-\mu )}{x^2},{\alpha }_4=\frac{-6x^6-{\alpha }_1{x}^4-3\mu }{x}\right\}$ was obtained.

$DLS=\left\{\alpha \in R^k\left|\exists (x_i,\mu )(i=\mathrm{1,2}),x_1\ne x_2,G=G_x=0\mathrm{}\right.\right.$at the point of $\left.(x,\mu ,\alpha )\right\}$. Because $(x,\mu )=\left(\mathrm{0,0}\right)$ is a limit point, $DL{S}_0=\left\{{\alpha }_2=-4x^3-2{\alpha }_{1}x,{\alpha }_3=3x^4+{\alpha }_1{x}^2,{\alpha }_4=0,x\ne 0\right\}$ can be obtained. As for $\mu \ne 0$, $DL{S}_1$ will be discussed in the two-dimensions transition set.

However, from the fact that the bifurcation of Eq. (31) is the universal unfolding with 4-codimension, it is very different to analyze the bifurcation behavior in full detail. Therefore, it is necessary to discuss all forms of two parameter unfolding contained in Eq. (31), that is ${\alpha }_1-{\alpha }_2$, ${\alpha }_1-{\alpha }_3$, ${\alpha }_1-{\alpha }_4$, ${\alpha }_2-{\alpha }_3$, ${\alpha }_2-{\alpha }_4$ and ${\alpha }_3-{\alpha }_4$. In this paper, only the transition sets and bifurcation diagrams in the ${\alpha }_1-{\alpha }_2$ plane and ${\alpha }_3-{\alpha }_4$ plane is presented.

Fig. 4Bifurcation diagrams of transition sets and persistent regions in the α1-α2 plane

a) Transition set and its bifurcation diagrams in the ${\alpha }_1-{\alpha }_2$ plane

b) Bifurcation diagrams on different persistent regions in the ${\alpha }_1-{\alpha }_2$ plane

Setting ${\alpha }_3={\alpha }_4=0\text{,}$ one may obtain $H{S}_0=\left\{{\alpha }_1\in R,{\alpha }_2\ne 0\right\}\text{,}$$DL{S}_0=\left\{{\left({\alpha }_2/2\right)}^2={\left(-{\alpha }_1/2\right)}^3\right\}$, $H{S}_1=\left\{{\alpha }_2^2=-128{\alpha }_1^3/729\right\}$, $DL{S}_1=\left\{{\alpha }_1\le 0,{\alpha }_2=0\right\}$. The transition set in the ${\alpha }_1-{\alpha }_2$ plane is ${\mathrm{\Sigma }}_{{\alpha }_1-{\alpha }_2}=H{S}_0\cup DL{S}_0\cup H{S}_1\cup DL{S}_1$, as shown in Fig. 4. The whole parametric plane was divided into six different persistent regions by the transition sets, corresponding to different type of the solutions. When ${\alpha }_1$ and ${\alpha }_2$ change anticlockwise, the bifurcation diagrams are completely symmetric about $\mu$ axis between I-$H{S}_1^{+}$-II-$DL{S}_1^{+}$-III and VI-$H{S}_1^{-}$-V-$DL{S}_1^{-}$-IV.