Nano

In-extensional vibration of rotating thin ring considering thermoelastic damping effect Thermo-elastic damping and initial stress in nano-structures Sun–Bae Kim and Ji–Hwan Kim School of Mechanical and Aerospace Engineering, College of Engineering Seoul National University, Seoul, 151-742, Korea Abstract In this work, vibration of nano-tubes under thermo-elastic damping and initial stress is investigated. Donnell-Mushtari-Vlasov (DMV) approach is adopted to simplify the shell equations. Using the stress function, the equati-ons for deflection, compatibility and heat conduction can be derived. Then the simultaneous equations can be sol-ved, and then natural frequency and Q-factor are obtained. Finally, the influences of the dimensions of the shell, mode numbers and initial stress on the natural frequency and Q-factor are discussed in detail. 1. Introduction It is well known that decreasing the size of a device, quality factors (Q-factors) greatly decreases. Thus, for a wide range of application fields of MEMS and NEMS, the aim of design for the structures is to maximize the factor. The effect of thermo-elastic damping on natural frequency and Q-factor has been investigated by nu-merous works. Also, initial stress can be generated due to various reasons; this originated from manu-facturing process and inertial motion of the struc-ture. It may bring about either fatal problems or positive effects on the device in each situation. Therefore, it is significantly necessary to compre-hend the characteristics of pre-stress. Also, resear-ch was performed the effect on the hemispherical re-sonator. In this work, thermo-elastic damping anal-ysis of nano-tube with initial stress is performed. Effects of dimensions, mode numbers and initial stress on the natural frequency and Q-factor are widely examined. 2. Heat Conduction and Structural Model Temperature distribution due to the thermal flow as follows. In here , and are the thermal conductivity, the heat capacity coefficient at constant pressure, and the coefficient of thermal expansion, respectively. Meanwhile, is the dila-tation strain. (1) While, the shell equations can be simplified by Donnell- Mushtari-Vlasov(DMV) method. Based on the method, the in-plane displacements are neglected in the bending strains but not in the membrane strains. Thus, the equations of motion and compatibility coupled with temperature are examined. Assuming appropriate mode shapes, natural frequency with the thermo-elastic damping and the initial stress can be obtained. Then, the Q-factor of the model can be obtained as . (2) 3. Results and Discussions For estimation of the Q-factor, thermo-elastic damping in a nano-tubular shell model with initial stress is examined. In here, a mode shape for stress is examined. In here, a mode shape for simply supported boundary condition is adopted. At first, the effect of initial stress for a Single-Walled Carbon Nano-tube (SWCN) for fundamental mode is investigated. As in Fig.1, Q-factor of the nano-tube is proportional to the tensile axial force resultant of the residual stress for a given geometry. Meanwhile, Fig. 2 shows that increasing the tensile axial force, iso-thermal natural frequency of the model also linearly increases. If the compressive axial force is applied to the structure, opposite effect can be expected. From these results, therefore, it can definitely be known that the tensile force leads to the improvement of performance for the oscillating structures. As another example, the variation of thermo-elastic damping with various dimensions ( , and ) in Fig. 3. Furthermore, there is no axial force and mode numbers are . As the radius and the axial length of the tube increases, inverse of Q-factor pa-rabolicaly decreases. Also, it is very important that thermo- elastic damping is extremely small in certain dimensions. These results can be helpfully used in design of resonators. 4. Conclusions In this paper, Q-factor for thermo-elastic damping is investigated in nano-structures with initial stress. In order to simplify the equations of motion, Donnell-Mushtari-Vlasov (DMV) approach is applied. Using the stress function, compatibility and heat conduction equations with the simplified equation of motion can be solved. Natural frequency and Q-factor are widely examined. Fig. 1 Variation of inverse Q-factor with tensile axial force . ( ) Fig. 2 Variation of iso-thermal natural frequency with tensile axial force . ( ) Fig. 3 Variation of inverse Q-factor with radius . ( )