A methodology is presented to study the resonance and stability for a single-degree-of-freedom (SDOF) system with a piecewise linear-nonlinear stiffness term (i.e., one piece is linear and the other is weakly nonlinear). Firstly, the exact response of the linear governing equation is obtained, and a modified perturbation method is applied to finding the approximate solution of the weakly nonlinear equation. Then, the primary and 1/2 subharmonic resonances are obtained by imposing continuity conditions and periodicity conditions. Furthermore, Jacobian matrix is derived to investigate the stability of resonance responses. Finally, the results of theoretical study are compared with numerical results, and a good agreement is observed.

Vibration isolation and shock absorbing have always been a hot research topic in engineering practice. A solid and liquid mixture (SALiM) vibration isolator was developed to isolate vibrations and shocks induced by heavy machines [

For nonsmooth stiffness systems, most approaches finding their steady state responses could be sorted into three groups including harmonic balance method (HBM) and its modified form, increment harmonic balance method (IHBM) [

Both approaches mentioned above can be applied to approximate solutions, but more substantial and sophisticated dynamic properties induced by nonsmooth nonlinearity cannot be explored thoroughly. Matching method provides the exact form of periodic solution, but some unknown coefficients have to be determined by the numerical method. The return map technique associating with matching method is preferable to explore more detailed nonlinear phenomena involving stability analysis, bifurcations, and chaos [

The present investigation intends to extend the previous studies by developing an analytical procedure which determines the single-crossing periodic responses of a class of harmonically excited piecewise linear-nonlinear oscillators. And also, the developed procedure can determine the crossing time as well as the stability of located periodic orbits.

The organization of this paper proceeds as follows. Firstly, the mathematical model is abstracted from a real bellows type SALiM isolator. And primary and subharmonic resonances are evaluated with matching method in association with perturbation method which is utilized to seek approximate solution of weakly nonlinear equation. Based on the preceding discussion, the Jacobian matrix is established with return map method and thus the stability of located periodic resonance orbits is determined, respectively, using the two different approaches. Finally, one or two examples will be introduced to demonstrate the validity of the approaches.

For the bellows type SALiM isolator, as the isolator is stretched at a quasi-static speed, there will come a point where elastic restoring force of the solid elements filled in the container is completely balanced by its internal pressure, which can be referred to as discontinuity point in a sense of stiffness [

A single-degree-of-freedom mass-spring system.

The equation of motion of the system in Figure

Equation (

Obviously, there exist two kinds of solutions for the piecewise linear-nonlinear system, namely, a small amplitude vibration (which means that the displacement does not exceed the critical displacement point) and a large amplitude vibration which can be beyond the critical point. The main work of this paper is to seek the large amplitude response of the overall system.

This section presents a detailed analytical procedure to find approximate solutions for the overall system under primary and subharmonic resonances. The analysis is based on the assumption that there exists an asymptotic expansion solution for the nonlinear differential equation with a weak nonlinearity [

Before further discussion, (

The single-crossing steady state response is depicted in Figure

Periodic large amplitude response of the overall system.

For the first segment of the motion, an approximate solution for the primary resonance response can be determined using an asymptotic expansion method. In interval

It is assumed that an approximate periodic solution to (

Substituting the above relating parameters and asymptotic solution into (

The solution of (

The displacement response

Till now, both of displacement histories

Substituting (

The analytical procedures for seeking an approximate solution of 1/2 subharmonic resonance are implemented similarly. The transform of

In order to study 1/2 subharmonic resonance of the examined system, the following variables are introduced:

Abstracting the coefficients of

Substituting (

Although (

Next, the solution of (

Again, the six constants,

After obtaining

In this section, the return map based on the Poincare section is employed to study the stability of periodic response. Generally, in order to establish the Poincare section for a single-degree-of-freedom periodically forced smooth oscillator, one frequently extends the two-dimensional phase plane

The three-dimensional space and the Poincare section represented by (

The three-dimensional trajectory and the Poincare section.

And as can be seen in Figure

Next, the detailed procedures for seeking the Jacobian matrixes of both submaps will be studied. Firstly, considering that the state parameters (

Similarly, taking derivatives of (

Similarly, the Jacobian matrix of submap

In fact,

The previous analytical procedure is implemented based on the Poincare section defined by (

As described before, the whole space could be divided into two separated subspaces

In fact, there are two intersection points

The two-dimensional periodic orbit and the Poincare map.

Obviously, this section overlays that defined by (

Then how to seek the Jacobian matrixes of the four submaps will be explained. In subspace

Having established the Jacobian matrices of submaps

The original and disturbed orbits.

When the orbit goes across the boundary from

To validate the approximate approaches, the periodic solutions determined by the developed analysis method are compared with the solutions from the numerical integration method. The classical fourth-order fixed-step Runge-Kutta algorithm is employed for the numerical integration. It is found that the approximate solutions and the numerical solutions are in excellent agreement for the cases of primary resonance and 1/2 subharmonic resonance.

As an example, the system’s parameters are given as follows:

Comparison between the displacement amplitudes from approximation analysis and numerical method.

Comparison of the periodic orbits from approximate analysis and numerical methods at

To examine the 1/2 subharmonic resonance, consider a system with parameters

Comparison of the periodic orbits of 1/2 subharmonic resonance response between the approximate and numerical solutions at

Comparison between frequency spectrums from approximate analysis and numerical integration (○: numerical, □: analytical).

Actually, the 1/2 subharmonic motion can also be observed by the bifurcation diagram. The period-doubling bifurcation indicates the existence of 1/2 subharmonic motion. The fourth-order Runge-Kutta numerical integration is performed to plot the bifurcation diagram, and the Poincare section used here is defined by (

The bifurcation diagrams.

The appearance or disappearance of period-doubling motion may be explained by the change of eigenvalues of the Jacobian matrix of period-one motion. For the inverse period-doubling bifurcation near 9.98 Hz, Figure

Unit circle and eigenvalues of period-one and period-doubling motions. The eigenvalue with larger magnitude passes by −1 into unit circle from −1.022 to −0.9647.

Phase orbits at different excitation frequencies: (a) 9.98 Hz and (b) 10.00 Hz.

In the first part of this paper, the matching method in association with the perturbation technique has been proven an efficient approach to find primary resonance and 1/2 subharmonic resonance responses. And in the second part, the return map based on the Poincare sections defined by

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Natural Science Foundation of China under Grant no. 11272145, Funding of Jiangsu Innovation Program for Graduate Education under Grant no. CXLX12_0135, and the Fundamental Research Funds for the Central Universities.