Studies On The Nonlinear Mechanical Problems Of The Micro Mechanical Components

Micro-electro-mechanical system(MEMS)has been widely used in aerospace,automotive,biomedical and other fields due to its small size,light weight,fast response,low energy consumption and so on,and formed a new technology field based on microelectronics,micro mechanics and material science gradually.The system size of MEMS is usually less than 10 microns,and the microbeam is one of the most common microstructures.Although the micro mechanics is the geometric miniaturization on the scale of the macro mechanics,it is not merely quantitative miniature in the scale.With the decrease of the size of the microstructure,the characteristics of the material and the mechanical laws are all changed and formed a special rule in microscale.In recent years,the size effects which can not be explained by traditional continuum mechanics were found in some experiments and which implies the limitations and shortcomings of the classical theory.Therefore,it is urgent to explore the elasticity theory suitable for micro scale mechanics.The material particle is continuous and regardless of geometric size in the classical continuum mechanics,while the material is non-uniform and size of the micro particle cannot be ignored in microscale.Then the higher order stress and strain such as the couple stress and strain gradient tensor,the characteristic length scale parameter were introduced in the size effect mechanical model.And the size effect can be described by some nonclassical continuum mechanical modal such as the strain gradient theory,the micropolar elasticity theory and the modified couple stress theory and so on.Among these theories,besides the characteristic scale parameter,the second shear modulus and the micro inertia moment,the independent micro rotation of the particle was considered in the micropolar elastic theory;the high order moment equilibrium is used to constrain in the couple stress theory,the couple stress tensor is symmetric and only one characteristic scale parameter is introduced compared with the classical couple stress theory.Although there have been much research on the size effect on micro and nano structures,however,most of the researchers are focused on the static problems and the squeeze-film damping and time-delay are seldom considered,and literatures on the micropolar elasticity theory are not rich enough now.In this paper,microbeam,the common component in MEMS are mainly analyzed and the size effect models are established based on the micropolar elasticity theory and modified couple stress,and the static bending,free vibration,forced vibration were studied in detail and the mainly attention were paid on the size effects of the microbeam.Main content of our work are as follows:The size-dependent microbeam model including von Karman geometric nonlinearity is developed based on the micropolar elasticity theory,and the governing equation and boundary conditions are obtained using the Hamilton principle.A new dependent micro rotation variable was introduced in the displacement hypothesis.In addition,two new material parameters(characteristic length parameter of material and second shear modulus)are introduced besides the micro rotation quantity in the present model.The model was studied in static and dynamic respectively.The static bending behavior was simulated by the numerical method directly and special attentions are paid on the operating mechanism of the micro-rotation,the effects of the second shear modulus,the size,the nonlinearity and the macro angle of the centroidal axis on the system.Multiple scales method is employed to obtain an approximate analytical solution for nonlinear natural frequency and time response of the free vibration of the beam.And shows that the second modulus,geometric nonlinearity,Poisson ratio and size effect have significant effects on the natural frequency of the beam.At the same time,the micropolar elasticity theory simplified as the classical couple stress theory when the microscopic angle is equal to the macro angle of the centroidal axis,and the classical couple stress theory simplified as the classical elasticity theory further when the material characteristic length scale parameters equals to zero.The nonlinear dynamics of an electrostatic actuated microbeam with von Karman geometric nonlinearity and the squeeze-film damping are studied.The governing equations and the boundary conditions are developed using the modified couple stress theory and the Hamilton's principle.Compared to the classical elasticity theory,a unique characteristic length of material is included in the constitutive relations for the size effects analysis.The frequency-response of the resonance in the forced vibration of the microbeam were obtained using the global Garlerkin method and multiple scales methods,and the behavior of primary resonance,super harmonic resonance and sub harmonic resonance are investigated in detail.The effects of the thickness,width of the beam and the initial gap between electrodes on the frequency-response of the resonance,peak amplitude,nonlinearity of the system are studied,and special attentions are paid on the “softening” and “stiffening” effects of the linear stiffness.A study on the nonlinear dynamics of a MEMS system capacitor actuated by AC voltage loads with von Karman geometric non-linearity,the squeeze-film damping and linear time-delay feedback controller is presented.The governing equations and the boundary conditions are developed applying the modified couple stress theory and the Hamilton's principle,and the size-dependent frequency-response equations of resonance(primary resonance,super harmonic resonance and sub harmonic resonance)were approximate derived using the global Garlerkin method and the multiple scale methods,respectively.As the introduce of the time-delay controller,the effects of the thickness,width of the beam,the initial gap between electrodes on the steady state frequencyresponse of the resonance,the peak amplitude,the squeeze damping force and the nonlinearity of the system are investigated with a certain time-delay feedback;on the other hand,the influence of different values of the time-delay feedback on the frequencyresponse and the peak amplitude with the certain value of the thickness,width and initial gap of the system,and special attention are paid on the comparison between the systems with and without control.The governing equation and the boundary conditions for a simply supported microbeam with von Karman nonlinearity are derived using the modified couple stress theory and the variational principle.The influence of the width,thickness of the beam and the initial gap between the microbeam and the stationary electrode on the electrostatic force were analyzed and the size effects on the pull-in voltage and the deflection of the microbeam were discussed were discussed in detail.The size effect Timoshenko simply supported microbeam with von Karman geometric nonlinearity was establised based on the modified couple stress theory,the governing equations and the boundary conditions are derived by using Hamilton's principle and the variational principle,and the time-delay feedback control was introduced to the present systeam.The stability,bifurcation and chaos of the Timoshenko beam with time-delay feedback control are studied in detail.The results show that it is effective to control the dynamic behavior of the beam effectively by selecting the control gain and the control delay.The above size dependent mechanical models can capture the size effect on the static bending deformation,the pull-in voltage,the natural frequency and the forced vibration in electro-mechanical coupling of the micro components.The research results can provide a theoretical guideline for the optimal design and experimental tests of the microcomponents in MEMS devices.