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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.

In this study, we investigate the application of Galerkin’s method to evaluate the deflection of stepped beams. The naïve implementation, as usually executed in the literature, is contrasted with the rigorous version of the Galerkin method. Naïve implementation involves the integration extended over the steps of the beam where the cross-sectional area and flexural rigidity remain constant. Rigorous implementation treats flexural rigidity as the generalized function, with attendant additional terms associated with the derivatives of the generalized function. The above additional terms do not appear in the naïve implementation of the Galerkin method.

In this paper the free vibration frequencies of beam on one‐parameter (Winkler) elastic foundation are analyzed via original Timoshenko‐Ehrenfest theory as well as two truncated versions of the Timoshenko‐Ehrenfest beam theory. The traditional version of Timoshenko‐Ehrenfest beam theory is contrasted with the truncated version that lacks the fourth‐order time derivative, as well as with the recently developed version that incorporates slope inertia effect. Advantages and disadvantages of each approach are indicated in the context of free vibrations of the beam on Winkler foundation. In addition an intriguing intermingling phenomenon is discussed in detail, and its implications on three theories.

In this paper we study free vibrations of stepped structures, specifically for longitudinal vibrations of bars and flexural vibrations of rectangular plates, providing two versions of the Galerkin method. Specifically, we first apply the straightforward version of the Galerkin method which stipulates the employment of the Galerkin procedure to be conducted in each subdomain, or step, of the structure. Second, the rigorous realization of the Galerkin method is presented where the structural parameters, like rigidity and mass, are treated as generalized functions over the entire domain. This latter implementation utilizes unit step functions, as well as the Dirac's delta function, and its derivative to treat the changes of the structural parameters across the steps. It turns out that this rigorous implementation leads to additional terms that do not appear in the straightforward (or “naive”) realization of the Galerkin method. Both versions of Galerkin methods are compared with the exact solutions of the considered problems. It turns out that with increase of number of terms in the expansion, the rigorous, generalized-functions based Galerkin method tends to exact solution. In contrast the naive realization of Galerkin's method, which is usually utilized in literature, does not tend to exact solution. This study demonstrates that extreme care must be taken when implementing the Galerkin's method for stepped structures, and only the rigorous version should be employed.