Fundamentals of Vibrations

On the Conventional Finite Element Sizes in Pseudostatic Seismic Analyses of 2d Frames

“…The resulting distribution of forces and displacements in the 2D frame (D stands for the dead load; L implies the live load, E denotes the load of earthquake, and W represents the wind load), according to the Iranian codes [4,6].by considering the typical distributions of live and dead loads as displayed in Figure 3, and putting it besides the distribution in Figure 2 and also Eq. (1), and not forgetting the superposition principle in linear analyses [7], it would be apparent that pseudo-static analysis is a static analysis against loads that do not change, unless at structural nodes (see Figure 1). Consequently, in view of the considerations in the simplest column-beam theory [8] and strength of materials [9], it is reasonable to except sufficient accuracies, when locating the finite elements nodes on the structural nodes.…”

“…The 2D frame in Figure 4 with structural members introduced in Table 1, located on a type II soil [4], and designed according to Iranian codes (see [4][5][6][7][10][11][12]) is undergone pseudostatic seismic analysis, considering the beam column members using Euler-Bernoulli beam theory, and elements sized according to Figure 1. Separately, the much more accurate results are obtained using about twenty times smaller elements.…”

Nonlinear dynamics of a spinning shaft with non-constant rotating speed

“…The linearised system is forming decoupled PDEs, whereas considering the B.C.s (Eq. 7-9), it can be shown that the equations are also self-adjoint, since the equations describe the well-known elastic motions in bending and torsion with typical BCs [29]. Therefore, the linear mode shapes are orthogonal to each other and also the linear modal equations decoupled.…”

“…Using Eq. (22a) in (22b), the system can be decoupled easily, and the solution of these equations is a trivial problem given by [29],…”

“…In this case, the linear BVP is similar to this one of a rod in axial vibration with clamped-free B.C.s. The solution of this BVP is having the following mode shapes and natural frequencies of the k-mode [29],…”