| With advances in technology,vibration control has become more and more important in the high-new technology and people's lives.Mechanical vibration restricts the stability of satellites,the ship stealth,missile strike accuracy and influences the stability of car.Metamaterials are expected to be the promising avenue to isolate or suppress the vibration effectively.This thesis focuses on the hard issue of metamaterials,namely how to open a band gap in the ultra-low frequency range.This work proposed a quasi-zero-stiffness(QZS)resonator with high-static-low-dynamic stiffness(HSLDS)characteristic by introducing the negative stiffness mechanism into the traditional linear resonator,which shifts the band gap from a high frequency range to a low one.Through the dynamic analysis of the resonator,the fundamental principle of the decreasing of the band gap created by QZS meta-structures(meta-rod,meta-beam,meta-shaft and meta-plate)is explored thoroughly.In addition,wave characteristics along the QZS met-structure is also carried out by modelling the QZS meta-structures and solving it with a numerical method.Finally,a prototype of the QZS resonator is fabricated by connecting a negative stiffness mechanism(a pair of magnet rings)and two linear springs.Connecting QZS resonators with rubber rings periodically,a prototype of the QZS meta-rod is obtained.This work provides a promising avenue for the tough issue that it is hard to create an ultra-low frequency band gap by traditional locally resonant metamaterials,and extends the application of metamaterials for manipulating the elastic wave in the low,even ultra-low frequency range.The main results of this thesis are listed as follows.(1)A QZS resonator is put forward by connecting a negative stiffness mechanism(two inclined springs)with two linear springs.The static analysis of the QZS resonator is carried out,and the parametric condition of the QZS resonator for a desired stiffness feature is obtained.Then,a QZS meta-rod is constructed by connecting such QZS resonators with rubber blocks,and both the linearized and nonlinear dispersion relations are derived by the harmonic balance method.By solving the dispersion relations,the band structures of the QZS meta-rod are acquired.In addition,the wave propagation along the QZS meta-rod is also analyzed by solving the equations of motion of the lumped-mass model of the meta-rod.Both the theoretical and numerical results show that introducing the negative stiffness mechanism moves the band gap from a high frequency range to a low one effectively.(2)A compact QZS meta-rod with resonators containing negative-stiffness mechanisms is proposed and fabricated for generating very low-frequency bandgaps.The underlying principle employs the negative-stiffness mechanism(a pair of mutual repelling permanent magnet rings)to partially or totally neutralize the stiffness of the positive-stiffness element(two coil springs)of the resonator and thus to achieve an ultralow,even zero,stiffness,which enables a significant shift of the bandgap from a high frequency range to a low one.Experiments on the restoring force feature of the resonator and the bandgap of the QZS meta-rod are carried out,which provide sufficient evidence to validate the proposed concept for substantially lowering bandgaps in meta-structures.(3)A QZS torsional resonator is proposed by connecting a cam-roller-spring mechanism and a rubber ring.Attaching resonators onto a shaft periodically,a QZS meta-shaft is created.The QZS condition of the torsional resonator is derived firstly,and then the dispersion relation of the QZS meta-shaft is acquired through the transfer matrix method.The band structures of the QZS meta-shaft are presented according to the dispersion relation,and validated by numerical simulation s.The theoretical and numerical results show that the negative stiffness mechanism is useful to shift the torsional wave band gap to an ultra-low frequency range,which provides a theoretical guideline for the control of ultra-low torsional wave using meta-structures.(4)By attaching QZS resonators onto a thin plate periodically,a QZS meta-plate is obtained.Linearizing the nonlinear stiffness of the resonator by the stiffness at the equilibrium position,and utilizing both the plane wave expansion method and the extended plane wave expansion method,dispersion relations of the QZS meta-plane are obtained.Solving dispersion relations with given frequencies,the real and complex band structures are exhibited.More importantly,by setting up the dynamic model of the QZS meta-plate and solving the equation of motion with Galerkin method,the wave propagation analysis along the QZS meta-plate is carried out.The numerical result keeps in agreement with the theoretical one,which means that the linearization of the nonlinear stiffness and the analysis of the band structure are reasonable.In addition,the effect of the resonator mass,and the geometrical asymmetric of the resonator configuration on the band structure of the meta-plate is also performed.The theoretical results show that the negative stiffness mechanism is useful for lowering the bending wave band gap.(5)A semi-active resonator containing a negative-stiffness mechanism(a pair of mutual repelling permanent magnet rings)and a band regulatory mechanism(a permanent magnet ring and a charged coil)is proposed.Attaching semi-active resonators onto a beam periodically,a semi-active QZS meta-beam is constructed.The static analysis for the semi-active resonator is conducted to demonstrate the influence of the control current on the stiffness of the resonator,and then the dispersion relation of the semi-active QZS meta-beam is obtained by the transfer matrix method.The wave propagation characteristics along the semi-active QZS meta-beam is revealed through solving the equations of motion numerically.Both the theoretical and numerical results show that the band regulatory mechanism could broaden the band gap effectively at ultra-low frequency,which is useful to extend the application of metamaterials into ultra-low frequency range. |
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