Experimental And Theoretical Studies Of Vibration On Size-dependent Microbeams

Microplates and microbeams have been widely applied in micro/nano-electromechanical-systems(MEMS/NEMS).It is approved by experiments that the mechanical performances are different from their behavior in the macro scale when the size of a material is on micro or nanoscale.And the smaller the size is,the greater the differences are.It's called elastic size effect.Nonclassical continuum mechanical theories such as the modified couple stress theory are developed in the elastic regime.Containing length scale parameter,the phenomenon that the equivalent bending rigidity changes with the scale of the material are explained.Due to the lack of experiments especially dynamical experiments in the literature,dynamical experiments as well as theoretical study are conducted on elastic size effects in this thesis.The main research results of this article are as follows:1.An improved experimental method for determining the intrinsic material length scale of the modified couple stress theory is proposed.Based on a non-contact laser Doppler vibration measurement system,the first-order resonant frequencies of two different cantilever microbeams made of copper and titanium with thickness ranging from 13-2.8 and 15 to 2 ?m are measured,respectively.The dimensionless frequency(the ratio of experimental data to theoretical data)increases with the decrease of thickness,exhibiting size effects.The length scale parameters of Cu and Ti are obtained by fitting the experimental data with the modified couple stress theory.2.The higher-order mode vibration of nickel microbeams is studied for the first time and the modified couple stress theory is tested for the suitability.The vibration of the second and third mode manifests a strong size effect.The normalized natural frequencies of the second and third mode nearly double as the thickness of the microbeam decreases to 2.1 ?m.Similarly the normalized bending rigidity increases to about 4 times.The theoretical prediction from the first mode also fits the experimental results of higher modes.Hence it is confirmed that the modified couple stress theory is valid for the vibration of higher modes.3.The nonlinear vibration of cantilever nickel microbeams with thickness of 2.1 ?m is investigated through experiment and theory.The nonlinear amplitude-frequency curves near the primary resonance are obtained,exhibiting a marked softening-type behavior.A modified couple stress-based Euler-Bernoulli beam model incorporating geometric nonlinearity is established.And the multiple scales method is applied to solve the model.A good agreement is observed in analytical and experimental results.The effects of length scale parameter,damping and excitation force on the nonlinear system are studied.Besides,a p-version Ritz method is employed and the frequencies of nonlinear free vibration are obtained,demonstrating the size-dependency in the nonlinear regime.A fourth-order Runge–Kutta algorithm is applied.It is illustrated that the nonlinear vibration turns into chaotic motion when the excitation is relatively large.4.The superharmonic resonance of size-dependent cantilever microbeams is investigated experimentally and analytically.Nickel cantilever microbeams with thickness of 3.2 ?m are employed,linear and nonlinear vibration experiments are then conducted.The superharmonic resonance of order two,three,four,five and six are observed.The frequency-response curves near the superharmonic resonance frequencies are obtained.Furthermore,the nonlinear model is then solved by multiple scales method and the frequency-response curve is obtained.Analytical results are in good agreement with experimental results in superharmonic resonance of order three.It is found that the length scale parameter increases the frequency of superharmonic resonance as well as decreases the amplitude in the A-F curve.5.The nonlinear forced vibrations of size-dependent clamped-hinged microbeams to fundamental excitations of respectively the first mode and second mode in the presence of three-to-one internal resonance are investigated.The equation of motion is established based on Euler-Bernoulli beam theory and the modified couple stress theory,and then solved by multiple scales method.The frequency-response relationships in the fundamental resonance for the first mode and the second mode are presented.Results reveal that the amplitude of the size-dependent model is higher than the classical model.Internal resonance is observed,the amplitude of the first mode is higher than the second one when the excitation frequency is at the second mode,revealing that energy is transferred from the second mode to the first mode.A fourth-order Runge–Kutta algorithm is employed to verify the analytical results.Numerical simulations indicate the appearance of chaos under relatively large excitation force whether it is the vibration of the first mode or the second mode.