Local coordinate systems-based method to analyze high-order modes of n-step Timoshenko beam

Abstract

High-frequency transverse vibration of stepped beams has attracted increasing attention in various industrial areas. For an n-step Timoshenko beam, the governing differential equations of transverse vibration have been well established in the literature on the basis of assembling classic Timoshenko beam equations for uniform beam segments. However, solving the governing differential equation has not been resolved well to date, manifested by a computational bottleneck: only the first k modes (k 12) are solvable for i-step (i ! 0) Timoshenko beams. This bottleneck diminishes the completeness of stepped Timoshenko beam theory. To address this problem, this study first reveals the root cause of the bottleneck in solving the governing differential equations for high-order modes, and then creates a sophisticated method, based on local coordinate systems, that can overcome the bottleneck to accomplish high-order mode shapes of an n-step Timoshenko beam. The proposed method uses a set of local coordinate systems in place of the conventional global coordinate system to characterize the transverse vibration of an n-step Timoshenko beam. With the method, the local coordinate systems can simplify the frequency equation for the vibration of an n-step Timoshenko beam, making it possible to obtain high-order modes of the beam. The accuracy, capacity, and efficiency of the method based on local coordinate systems in acquiring high-order modes are corroborated using the well-known exact dynamic stiffness method underpinned by the Wittrick-Williams algorithm as a reference. Removal of the bottlenecks in solving the governing differential equations for high-order modes contributes usefully to the completeness of stepped Timoshenko beam theory.

Exact and numerically stable expressions for Euler-Bernoulli and Timoshenko beam modes

“…However, while performing the Gaussian elimination, it is necessary to perform column pivoting since otherwise the Gaussian elimination may not be stable [24]. Assume that after column pivoting the linear system reads B ã = 0, (21) and that we have a bijective permutation map σ from the set {1, 2, 3, 4} onto itself such that ãi = a σ(i) where i ∈ {1, 2, 3, 4}; and Bi,j = B i,σ(j) where j ∈ {1, 2, 3, 4}. (22) Note that the map σ is invertible, i.e., we can write ãσ −1 (i) = a i and Bi,σ −1 (j) = B i,j .…”

“…The issue of numerical stability of modes is even more prominent for stepped beams, both Euler-Bernoulli and Timoshenko, where fewer than 10 modes can be predicted reliably [20,21]. Xu et al [20] and Cao et al [21] have mitigated the issue by defining a set of local coordinates that decrease the growth rate of the hyperbolic sine and cosine terms in Euler-Bernoulli and Timoshenko beams, respectively. However, their approach does not solve the core problem and numerical stabilities continue to be a problem in their formulation.…”

Crack Identification in CFRP Laminated Beams Using Multi-Resolution Modal Teager–Kaiser Energy under Noisy Environments

“…Solving Equation (39) produces a sequence of natural frequencies [ 69 ]; provided with , the corresponding coefficient vector can be derived from Equation (38); substituting and into Equation (34a), the m -th mode shape can be obtained.…”

Evaluation of high-order modes and damage effects of multi-crack beams using enhanced spectral element method

“…Xu et al. (2014) and Cao et al. (2017) solved the bottlenecks in analytically evaluating high-order modes of stepped beams by introducing local coordinate systems to depict stepped beams.…”