On Parameter Study of Cubic Nonlinear Vibration Isolator

The purpose of optimizing isolator parameters is to have a system that is stiff at low frequency but soft in high frequency. However, in this case a small amount of transmitted force will cause a large displacement in isolator. Exceeding the allowable relative displacement may cause serious damage to the system. Therefore a trade off between the transmitted force, which is indicated by absolute acceleration, and the relative displacement should be considered in the design of a mount. The simplest type of mount is considered as a single degree of freedom linear spring-massdashpot. Nakhaie et. al. used the Root Mean Square of absolute acceleration and relative displacement to find the optimal damping ratio and natural frequency of the isolator. They have also obtained the optimal value for damping ratio in order to minimize the absolute acceleration for a step input [1]. Traditional methods of isolation based on linear passive systems with dissipative elements turns out to be insufficient in many situations. They cannot completely satisfy the complex and often contradictory requirements of modern vibration isolation systems. Passive nonlinear mounts have provided some remedy. Examples of nonlinear mounts include hydraulic engine mounts and different types of rubber mounts. Isolators gain their characteristics from their component material and shape. The shape of the isolator and the material affects the performance of the system by influencing the transmitted force from vibration source to the structure. Therefore study of nonlinear isolator is the subject of contribution in vibration isolation studies. The practice of analysis of nonlinear systems started with the expectation of more optimized performance of passive systems. Roberson [2] and Arnold [3] considered vibration absorbers with nonlinear cubic spring and found that a softening-type nonlinear spring improves the performance of the absorber. Hunt and Nissen [4] were the first to implement a practical nonlinear damped absorber, and constructed a softening-type spring device as an assembly of steel Belleville washers. However, nonlinear absorbers may exhibit undesirable secondary resonance. Natsiavas [5] employed an averaging methodology using numerical integration in studying dynamics of a two degree of freedom nonlinear oscillator. The main system is modeled as Van der Pol oscillator under harmonic forcing. The objective is to reduce the amplitude of oscillation of main mass near resonance by attaching to it a damped vibration absorber with a Duffing spring. The method of multiple time scales is employed to obtain the approximate solution of system for small nonlinearities [6]. The effect of variation of parameters in Duffing spring is investigated numerically along with stability and bifurcation analysis. Moreover, Oueini et. al. [8] study the dynamic of the cubic nonlinear vibration absorber. They have considered the cubic nonlinearity for the system and introduced a second-order absorber that is coupled with the plant through user defined cubic nonlinearities. The equations of the motion are developed and an approximate solution to the nonlinear differential equation using the method of multiple scales is obtained. Then, they conduct bifurcation analysis and investigate performance of the control strategy. Additionally, they demonstrate the existence of dynamic solutions that may lead to high-amplitude or chaotic response. Chandra et al. [9] considered a single stage shock isolator comprising a parallel combination of a cubic nonlinearity in spring and damper. Combining straightforward perturbatio n method and Laplace transform they have determined the transient response of the system. Three types of input base excitations were considered: the rounded step, the rounded pulse and the oscillatory step. Although the solution is valid for limited range of nonlinearities, it was shown that the nonlinearity in the damping rather than in the stiffness has a more pronounced effect on performance of a shock isolator. The presence of a nonlinear velocity dependent damping term with a positive coefficient was found to have detrimental effects on the isolator performance. They have considered different alternatives to improve the performance of an isolator having a nonlinear cubic damping over and above the usual viscous damping [10]. Since they concluded nonlinearity in the isolator stiffness does not have any appreciable effect on its performance [9], the stiffness of the isolator is assumed to be constant. Four types of isolators are be ing studied. Where isolator with a Coulomb damper, three-element damper, an isolator along with vibration absorber, and a two-stage isolator are considered and compared. Overall, it is seen that the three-element and twostage isolators are preferable in the presence of a nonlinear cubic damping for their configurations. None of the above investigations performed optimization and parameter study on the system to find the parameters which improves the performance of the isolator. This work extends the previous studies by considering nonlinearity in both the damping and stiffness that are polynomial functions of the velocity and displacement through the isolator. The closed form frequency response of the system is obtained as following ( ) 2 6 6 4 2 4 4 2 2 2 2 2 6 2 4 3 1 3 1 9 3 9 3 A + 6 A A 1 + A 4 A 2A + A +A + A 0 4 2 16 2   ξ ω ζ ξ + − ω − ρ + ζ − ω ρ ρ =     The results from analytical method are compared with numerical solutions to confirm the validity of the method. Characteristics such instability and jump phenomena are identified. Finally ranges of optimized parameters are obtained which improves the performance of the system based on RMS method.