Effects of couple stresses on the in-plane vibration of micro-rotating disks

Micro-rotating disks are extensively used in micro-electromechanical systems such as micro-gyroscopes and micro-rotors. Because of the sensitivity of these elements, enough knowledge about the mechanical behavior of these structures is an essential matter for designers and fabricators. The small-scale effects on the in-plane free vibration of such micro-disks present an important aspect of the mechanical behavior of these elements. The small-scale effects on the in-plane free vibration of these micro-disks are investigated in this study using the modified couple stress theory. By using the Hamilton principle, the partial differential equations governing the coupled radial and tangential motion of the disk particles with their corresponding boundary conditions are derived. Then, the solution for the boundary value problem is analytically presented. The effects of the angular speed of the micro-disks and the length scale parameter of the modified couple stress theory on the steady radial and tangential displacements, and on the natural frequencies are investigated. Those results are compared with the ones previously obtained from the classical continuum mechanics analysis.

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Free Vibration Analysis of Shear Deformable Small-Scale Tubes

Academic Perspective Procedia ◽

10.33793/acperpro.03.01.86 ◽

2020 ◽

Vol 3 (1) ◽

pp. 418-427

Author(s):

Reza Aghazadeh

Keyword(s):

Free Vibration ◽

Shear Deformation ◽

Couple Stress ◽

Couple Stress Theory ◽

Free Vibration Analysis ◽

Modified Couple Stress Theory ◽

Small Scale ◽

Tube Model ◽

Stress Theory ◽

Modified Couple Stress

This study presents an approach for investigating free vibration problem of small-scale tubes based on modified couple stress theory in conjunction with higher order shear deformation tube model. The size effect is captured through utilization of a length scale parameter involved in modified couple stress theory. A newly developed refined tube model is employed to satisfy friction-free conditions on inner and outer surfaces of micro-tubes. Hamilton&rsquo;s principle is used as a variational technique for derivation of governing system of equations. For axial vibrations, an analytical procedure is conducted, while for transverse vibrations differential quadrature method is used as a numerical technique. The correctness of numerical results are verified through comparisons made with results which are available in the literature for limiting cases. The analyses reveal the effects of size and transverse shear deformation on the natural frequencies of micro-tubes.

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Dynamic stiffness formulation for a micro beam using Timoshenko–Ehrenfest and modified couple stress theories with applications

Journal of Vibration and Control ◽

10.1177/10775463211048272 ◽

2021 ◽

pp. 107754632110482

Author(s):

J Ranjan Banerjee ◽

Stanislav O Papkov ◽

Thuc P Vo ◽

Isaac Elishakoff

Keyword(s):

Free Vibration ◽

Couple Stress ◽

Scale Parameter ◽

Couple Stress Theory ◽

Dynamic Stiffness ◽

Modified Couple Stress Theory ◽

Length Scale Parameter ◽

Length Scale ◽

Stress Theory ◽

Modified Couple Stress

Several models within the framework of continuum mechanics have been proposed over the years to solve the free vibration problem of micro beams. Foremost amongst these are those based on non-local elasticity, classical couple stress, gradient elasticity and modified couple stress theories. Many of these models retain the basic features of the Bernoulli–Euler or Timoshenko–Ehrenfest theories, but they introduce one or more material scale length parameters to tackle the problem. The work described in this paper deals with the free vibration problems of micro beams based on the dynamic stiffness method, through the implementation of the modified couple stress theory in conjunction with the Timoshenko–Ehrenfest theory. The main advantage of the modified couple stress theory is that unlike other models, it uses only one material length scale parameter to account for the smallness of the structure. The current research is accomplished first by solving the governing differential equations of motion of a Timoshenko–Ehrenfest micro beam in free vibration in closed analytical form. The dynamic stiffness matrix of the beam is then formulated by relating the amplitudes of the forces to those of the corresponding displacements at the ends of the beam. The theory is applied using the Wittrick–Williams algorithm as solution technique to investigate the free vibration characteristics of Timoshenko–Ehrenfest micro beams. Natural frequencies and mode shapes of several examples are presented, and the effects of the length scale parameter on the free vibration characteristics of Timoshenko–Ehrenfest micro beams are demonstrated.

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