Design And Characteristics Analysis Of The Novel Quasi-zero Stiffness Isolator

Based on the summarization of recent theoretical research and physical design about the quasi- zero stiffness(QZS) isolator, this dissertation presents two novel QZS isolators to meet demand of good isolation performance in low frequency range and space limitation in the on-board precision instrument. By using the approximate analytical and numerical methods in nonlinear vibration, the static characteristics of the two Q ZS isolators, the dynamic characteristics and isolation performance of the corresponding Q ZS vibration isolation system are investigated. The force transmissibility of two degree-of- freedom QZS vibration isolation system is explored. After the QZS isolator prototype is assembled, its static and dynamic experiments are conducted. The main work and conclusions of this dissertation are as follows:(1) According to the vibration isolation mechanism of combining positive and negative stiffness, a QZS isolator is designed by combining a negative stiffness disk spring of constant thickness in parallel with a positive stiffness linear coil spring. At first, the static characteristics are analyzed. The restoring force and stiffness expressions for the disk spring of constant thickness and the Q ZS isolator are established. The parameters conditions of owning negative stiffness for the disk spring and owning QZS characteristics at the equilibrium position for the isolator are achieved. It is found that the stiffness of the disk spring of constant thickness can reach the minimum value at the flatten state. The minimum value and range of the negative stiffness are related with the equivalent ratio between height and thickness as well as the distance ratio between the two supporting points. Increasing the equivalent ratio between height and thickness and decreasing the distance ratio between the two supporting points can decrease the minimum value and enlarge the range of the negative stiffness. The QZS characteristics of the isolator are influenced by the equivalent ratio between height and thickness, the distance ratio between the two supporting point and the stiffness ratio between the disk spring of constant stiffness and the linear coil spring. As the equivalent ratio between height and thickness increases and the distance ratio between the two supporting point decreases, a smaller stiffness ratio is demanded to achieve zero stiffness at the equilibrium position. Meanwhile, increasing the equivalent ratio between height and thickness and decreasing the distance ratio between the two supporting point, the QZS isolator can own a smaller stiffness and a wider range in which the stiffness gets a lower value in the neighborhood of the equilibrium position. There fore, a wider displacement range of QZS characteristics for the QZS isolator is achieved.In the following dynamic analysis, the dynamic equations of the QZS vibration isolation system under harmonic force and displacement excitations are established. By using the average method, the steady-state responses are obtained. Compared with the equivalent linear system(ELS), effects of the system parameters and excitation amplitude on the QZS vibration isolation system are studied. Applying the Mathieu equation criterion, the boundary between the stable and unstable regions of the steady-state solution is achieved. The results show that the dynamic equations of the QZS vibration isolation system under the two types of harmonic excitations are the Duffing equation under symmetric excitation. As a nonlinear system, the QZS vibration isolation system owns hardening stiffness, unstable region and jump phenomenon. If decreasing the excitation amplitude and increasing the damping ratio properly, the unstable region and the starting isolation frequency will decrease. And the steady-state solution of the system under harmonic displacement excitation will not appear the unbounded value. Besides, the Q ZS vibration isolation system excited by a proper amplitude can possess a better isolation performance in low frequency range compared with the ELS, if there is a proper larger damping ratio and a smaller nonlinear term achieved by increasing the equivalent ratio between height and thickness or decreasing the distance ratio between the two supporting point.(2) Due to a small deflection for the disk spring of constant thickness, the Q ZS isolator using it as negative stiffness element has a small displacement range of QZS characteristics. To solve the problem, a new Q ZS isolator is designed by taken a disk spring of variable thickness as the new negative stiffness element. At first, the static characteristics are analyzed. The restoring force and stiffness expressions for the disk spring of variable thickness and the new QZS isolato r are established. The parameters conditions of owning negative stiffness for the disk spring and owning QZS characteristics at the equilibrium position for the isolator are achieved. The displacement ranges of Q ZS characteristics for the two isolators separately constituted by disk springs of constant and variable thickness are compared under the same parameters value. To get the maximum displacement range of QZS characteristics for the isolator constituted by the disk spring of variable spring, the configurative parameters is optomized. Then, the following conclusions can be drawn. The stiffness of the disk spring of variable thickness and the corresponding QZS isolator both reach the minimum value when the disk spring is flatten. Compared with using a disk spring of constant thickness, the disk spring of variable thickness can result in a smaller stiffness at the same displacement. Meanwhile, larger displacement from the equilibrium position and value of the thickness variation parameter k lead to larger reduction of the stiffness. Hence the QZS isolator using a disk spring of variable thickness as negative stiffness element can own a smaller stiffness and a wider range in which the stiffness gets a lower value in the neighborhood of the equilibrium position compared with that using a disk spring of constant thickness. Thus, a wider displacement range of QZS characteristics for the QZS isolator is achieved.By considering the overloaded and underloaded conditions in the practical applications, the dynamic analysis is finished. The dynamic equations of the overloaded and underloaded system under harmonic force and displacement excitations are established. The steady-state approximate solution is obtained by using the Harmonic Balance Method(HBM) and confirmed by the numerical simulation using fourth order Runge-Kutta method. Applying Floquet theory, the boundary between the stable and unstable regions of the steady-state solution is obtained. Effects of the offset displacement, excitation amplitude and damping ratio on the overloaded system, ideal system and ELS are studied. The results show that the dynamic equations of the overloaded and underloaded system under the two types of harmonic excitations are the Helmoholtz-Duffing equation. And they can be recast in the form of a Duffing equation under asymmetric excitation. The steady-state approximate and exact solutions obtained by the HBM and fourth order Runge-Kutta method respectively agree well, which also confirms the accuracy of the appropriate solutions obtained by the HBM. Meanwhile, the steady-state solution of overloaded system has a maximum number of one, three, or five values for different combinations of offset displacement and excitation amplitude. The overloaded system can exhibit purely linear, purely softening, mixed softening- hardening and purely hardening stiffness in turn as the excitation amplitude increases. Smaller offset displacement and excitation amplitude can lead to smaller starting isolation frequency and wider range of isolation frequency. By increasing the damping ratio, the jump phenomenon and unstable region will not occur. Larger damping ratio can also result in better isolation performance in low frequency range but worse isolation performance in high frequency range. Besides, the isolation performance of the system can be improved by proper overload when the excitation amplitude is large. Nevertheless, the QZS vibration isolation system can possess better isolation performance than its ELS in low frequency range while loaded with an appropriate mass, excited by not too large amplitude and owning a relative larger damp.(3) Since the maximum value of force transmissibility corresponding to the second resonance frequency is large, there is no vibration attenuation in the neighborhood of the second resonance frequency for the undamped or smaller damped two degree-of- freedom linear vibration isolation system. Addressing the problem, a two degree-of- freedom QZS vibration isolation system is constructed by utilizing the optimal parameters of the designed QZS isolator. At first, the dynamic modeling of two degree-of-freedom Q ZS vibration isolation system and the two degree-of- freedom ELS are accomplished. Second, by using the average method, the force transmissibility of the two systems under harmonic force excitation is derived. Finally, effects of the excitation amplitude, mass ratio and damping ratio on the force transmissibility are investigated while a comparison about the isolation performance between the two systems is analyzed. It is found that the maximum value of force transmissibility corresponding to the second resonance frequency for two degree-of- freedom Q ZS vibration isolation system is smaller than 1. Hence vibration attenuation also occurs in the neighborhood of its second resonance frequency. Compared with two degree-of-freedom ELS, two degree-of- freedom QZS vibration isolation system has smaller sta rting isolation frequency and larger range of isolation frequency, thereby owning better isolation performance in low frequency range. Moreover, smaller excitation amplitude results in larger advantage for the isolation performance in low frequency range. Meanwhile, two degree-of- freedom QZS vibration isolation system also has a better isolation performance in high frequency range. But as the mass ratio and damping ratio increases, the advantage for the isolation performance in high frequency range decreases. Besides, smaller starting isolation frequency, lager range of isolation frequency and better isolation performance in low frequency range for two degree-of- freedom Q ZS vibration isolation system can be achieved by increasing the mass ratio and damping ratio properly.(4) The QZS isolator prototype is designed and assembled. Its static and dynamic experiments are conducted by constructing the experimental platform respectively. At first, the static characteristics of the designed disk spring, linear coil spring and the Q ZS isolator are investigated in the static experiment. Second, the dynamic experiment for studying the effect of excitation amplitude on the isolation performance of overloaded system is carried out. Finally, an experimental comparison with the ELS is finished. Based on the experimental results, it is found that the actual force-displacement curves obtained by experiment agree well with theoretical curves both for the designed disk spring and linear coil spring. However, there is a certain error between the actual and theoretical force-displacement curves for existence of the influence from material and manufactur ing factors. The restoring force of the Q ZS isolator when its disk spring is flatten is larger than the summed force of disk spring and linear coil spring at the same position. With the increasing of the excitation amplitude, the resonance frequency and its corresponding peak amplitude of the transmissibility decreases at first and increases later, this is in aggrement with therotica l results. But for the occurrence of large damp from the prototype and experimental platform, the actual transmissibility curves only exhibit linear stiffness and the nonlinear phenomenon does not occur. Nevertheless, the dynamic stiffness of the QZS isolator can be reduced because the negative stiffness of the disk spring can cancel out the positive stiffness of the linear coil spring. Compared with the ELS, overloaded system has smaller starting isolation frequency, wider range of isolation frequency and better isolation performance in low frequency range. And the damp should not be too large for ensusing its advantage in high frequency range.The theoretical and experimental conclusions about the designed QZS isolator in the dissertation have great significance in guaranteeing the security and reliability of on-board precision instrument.