粘弹性-弹性层合微悬臂梁的自由振动

Free Vibration of a Viscoelastic-Elastic Laminated Microcantilever Beam

吸附膜的粘弹性性质对微梁生物传感器的固有频率有显著影响.首先,在欧拉梁假设下,采用线性粘弹性积分型本构关系和拉普拉斯变换方法,建立了动态识别技术中粘弹性-弹性层合微悬臂梁自由振动的基本方程;其次,采用空域分离变量法和时域微分求导法,获得了积分-偏微分系统的固有频率,并采用求解代数方程的卡尔丹公式和不等式的性质,在材料参数和几何参数张成的高维空间获得了齐次通解的结构;最后,研究了微梁的几何尺寸、吸附膜的粘弹性参数、膜基厚度比和模量比对微梁自由振动的影响.结果表明:吸附膜的粘弹性阻尼效应使得微梁的稳态固有频率低于瞬态固有频率;随着吸附膜松弛时间的减小,微梁瞬态固有频率漂移与稳态固有频率漂移之间的差别逐渐增大;通过控制膜基厚度比或模量比等参数可以使微梁振动进入弱阻尼振动区域.

Viscoelastic properties of adsorbed films have a significant influence on the natural frequency of microbeam biosensors. First, under the hypothesis of Euler beam, the integral type constitutive relation and Laplace transformation method were used to establish the fundamental equations for the free vibration of a viscoelastic-elastic laminated microcantilever beam in the technology of dynamic identification. Second, the separation method in the spatial domain and the differentiating method in the temporal domain were applied to obtain the natural frequency of the integral-partial differential system. Then, the Cardan＇s formula for roots of cubic algebra equations and the inequality properties were used to obtain the structure of the homogenous universal solution in the high-dimensional space splayed by material parameters and geometrical parameters. Finally, the effects of the beam geometrical size, viscoelastic parameters of adsorbed films, the film-to-substrate thickness ratio and modulus ratio on the free vibration of the microbeam were studied. Results show that the viscoelastic damping effect of adsorbed films can make the steady-state natural frequency lower than the transient natural frequency; with the decrease of the relaxation time of the adsorption film, the difference between the transient natural frequency shift and the natural frequency shift of the microbeam is gradually increased; and the microbeam vibration can get into the weakly damped domain by controlling the parameters such as the film-to-substrate thickness ratio and the modulus ratio.