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

Abstract. In order to reveal the bifurcation mechanism and optimize the system design for high-static-low-dynamic-stiffness (HSLDS) vibration isolation system (VIS) with elastic base, the local bifurcation analyses both in unfolding parameter space and physical parameter space were carried out theoretically and numerically. Firstly, the restoring force of the HSLDS-VIS was approximated to linear and cubic stiffness by applying the Maclaurin series expansion and the motion equations of HSLDS-VIS with elastic base were established. Subsequently, the motion equations of HSLDS-VIS with elastic base were formulated to transform the system into a standard form and the averaging method was applied to obtain the single-variable bifurcation equation for the HSLDS-VIS with elastic base in case of primary resonance and 1:2 internal resonance. Furthermore, the transition sets and bifurcation diagrams in the unfolding parameter space were studied by means of singularity theory. Finally, for the engineering application, the transition sets were transferred back to the physical parameter space, thus to obtain the bifurcation diagrams of the amplitude with respect to the external force. The numerical simulation results show that the local bifurcations of HSLDS-VIS with elastic base in case of 1:2 internal resonance are considerable complex and need to be analyzed in six two-parameters spaces, meanwhile, the necessary condition of multiple solutions lies in some physical parameters, which can provide a theoretical basis and reference for design and application of the HSLDS-VIS with elastic base.


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.

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  and cubic nonlinear stiffness  .In addition, they are pre-stressed, i.e. compressed by length  and connected at point C with a vertical unstressed linear spring of stiffness  .The oblique springs are hinged at  and .The geometry of configuration is decided by horizontal distance  from point  to  and initial height ℎ, while  denotes the vertical displacement from the initial unloaded position caused by the force .The general equation between the force  and the displacement  can be derived as: Introducing  =  − ℎ,  =  −  ℎ, Eq. ( 1) can been recast in dimensionless form as: where: The dimensionless dynamic stiffness can be obtained by differentiating Eq. ( 2) to  : 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  suspended on a three springs mount in parallel with a viscous damper  . denotes the vertical displacement from the equilibrium position caused by a harmonic excitation  = cosΩ.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:

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.  and  denote the isolation equipment and the intermediate mass, respectively. is supported by a linear damper and a nonlinear spring which possesses both linear and cubic stiffness. 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  is the damping coefficient of HSLDS vibration isolator;  and  are the linear and cubic stiffness coefficients of HSLDS vibration isolator, respectively. is the damping coefficient of damper between the  and fixed plane;  is the stiffness coefficient of the linear stiffness between the  and fixed plane. and Ω are the amplitude and frequency of the harmonic excitation, respectively.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:  =  + ℎ ,  =  + ℎ .In the equilibrium state, the gravitation terms in Eq. ( 6) can be eliminated by the following relations:   +   =  ℎ and   =   +   , where  = ℎ − ℎ .The governing equations of Eq. ( 6) are given by: Setting  =  + 3  ,  =   ⁄ and introducing dimensionless parameters:  Therefore, the HSLDS-VIS with elastic base is a coupled nonautonomous dynamic system with quadratic and cubic nonlinearities.

Standard form of the motion equations
Standard form of the motion equations refers to the vector fields are proportional to the small perturbation parameter .Here, assume the nonlinear stiffness coefficients, damping coefficients and the amplitude of excitation force are small quantities and introduce an additional parameter  into the equations.Setting  =  ,  =  ,  =  ,  =  , 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 , suppose it has only simple purely imaginary eigenvalues ± , ⋯ , ±  .The special solution of the homogeneous part with ± are as follows: Re ( ) =  ( ) =  sin( ) −  cos( ), Im ( ) =  * ( ) =  sin( ) +  cos( ), ( = 1,2,  = 1,2,3,4), (11) where  and  are real constants.

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 Supposing  =  and  =  , the Eq. ( 24) is further recast as follows: The governing equations of Eq. ( 25) are given by: Substituting  =     −  Δ 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  are shown in Appendix A1.Setting  =  + 4 , Eq. ( 29) can be written as follows: (, ,  ,  ,  ,  ) =  −  +   +   +   +   = 0, where  ~ is the universal unfolding parameters related to the stiffness coefficients, damping coefficients, tune parameters and so on,  is the bifurcation parameter related to the amplitude of excitation force.The detailed  ~ and  are shown in Appendix A2.
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 -5 show that the bifurcation of HSLDS-VIS with elastic base with 1:2 internal resonance are considerable complex.There are qualitative difference of bifurcation diagrams between the different transition sets and persistent regions.If the unfolding parameters lying in the transition sets, a small perturbation is likely to cause a qualitative change in the bifurcation diagrams, which means that are degraded.While the unfolding parameters lying in the persistent regions, any perturbation is unable to cause a qualitative change in the bifurcation diagrams, which means that are universal.

Bifurcation analysis in the physical parameter space
Considering the HSLDS-VIS with elastic base under actual engineering background, the unfolding parameters and bifurcation parameters may be constrained by some conditions, which are called boundary induced transition sets.For example, the force amplitude, damping ratio and stiffness coefficient should be non-negative.Therefore, in order to obtain some results for the engineering application, it is essential that the bifurcation analysis is transferred from the unfolding parameter space to the physical parameter space.In this paper, only the transition sets and bifurcation diagrams in the  −  plane are presented, whose physical parameter space consists of  ,  ,  and  .Setting  =  = 0, one may obtain:   The analysis should be carried out with considering  ≥ 0 and  ≥ 0 in the actual engineering.Therefore, only the  , II, and III are presented.Then, the bifurcation diagrams will also be transferred from the unfolding parameter space to the physical parameter space, whose procedure consists of two steps: first, the bifurcation analysis is transferred from the  −  plane to the  −  plane, and then, the bifurcation analysis is transferred from the  −  plane to the  −  plane.A point lying in  is used as an example to illustrate the above process.Setting  = 0.05 and substituting it into Eq.( 31 7.When ( ,  ) lying in  , all of the bifurcation diagrams have a hysteresis point.When ( ,  ) lying in  and region III, the multiple solutions cannot coexist on the system; while ( ,  ) lying in region II, the same  corresponds to two or three amplitudes in some areas.The system has three different bifurcation behaviors in the physical parameter space, therefore, the beneficial movement mode can be selected by adjusting the parameters, which offers a theoretical guidance for design and application of the HSLD-VIS with elastic base.

Conclusions
In this work, the restoring force of the HSLDS-VIS was approximated to linear and cubic stiffness and the motion equations of HSLDS-VIS with elastic base were established.The averaging method was applied to obtain the single-variable bifurcation equation for the HSLDS-VIS with elastic base in case of primary resonance and 1:2 internal resonance.According to singularity theory, the transition sets and bifurcation diagrams in the unfolding parameter space were studied.For the engineering application, the transition sets were transferred back to the physical parameter space, thus to obtain the bifurcation behaviors of the amplitude with respect to the external force.Analytical results show that local bifurcations of HSLDS-VIS with elastic base in case of 1:2 internal resonance are considerable complex and should be analyzed in engineering with considering the constrained conditions of the physical parameters.Additionally, the excitation force may change the values of unfolding parameters and the types of the bifurcations, which can provide a theoretical guidance for design and application of the HSLDS-VIS with elastic base.

Fig. 1 .
Schematic representation of an isolator based on HSLDS

Fig. 4 . 5 .
Fig. 4. Bifurcation diagrams of transition sets and persistent regions in the  −  plane

7 .
a) ( ,  ) lying in  x -f z -f y -f b) ( ,  ) lying in region III x -f z -f y -f c) ( , ) lying in region II Fig. Bifurcation diagrams in the physical parameter space 2920.BIFURCATION AND SINGULARITY ANALYSIS OF HSLDS VIBRATION ISOLATION SYSTEM WITH ELASTIC BASE.XIANG YU, KAI CHAI, YONGBAO LIU, SHUYONG LIU =  +  ,   +   = ,  is detuning parameters,  ( = 1, 2) and  are nonzero integers. and  are periodic function of  and .This paper focuses the second order primary resonance and 1:2 internal resonance, which means  ,  and  should meet  =  +  and  = 2 +  .The derivatives of the new parameters should satisfy the conditions below: * (, , ),(21)where  and  do not contain  ,  * and  * are periodic function of  and  .Applying the averaging method, substitute Eq. (20) and Eq.(21) into Eq.(19) and collect terms of the first order in :where  (),  (),  (),  (),  (),  () are generalized Fourier series coefficients of  and  ,  =   +   +    +   +   . and  are the slowly changing