I want to calculate the eigenfrequencies of a 3D truss using the finite element method. The beams should be modelled as Timoshenko beams and I want to use arbitrary shape functions.

I know how to use the finite element method to solve equations of the form $a(u,\bar{u}) = l(\bar{u}) \quad \forall \bar{u} \in V$ where $V$ is a Sobolev space of all functions fulfiling the boundary conditions: $V \subset L^2(\Omega, \mathbb{R})$ and $a(\cdot,\cdot)$ a bilinear form and $l(\cdot)$ a linear functional.

I simply partition $\Omega$ into "finite elements" $\Omega_i := (x_i,x_{i+1})$ which are connected at "nodes" $x_i$: $\Omega = \bigcup\limits_{i=0}^{n} \Omega_i$ For every node $i$ there will be defined a shape function $\varphi_i \in V$ which holds these conditions:

  1. $\varphi_i(x_j) = \delta_{ij}$

  2. $\mathrm{supp}(\varphi_i) = [x_{i-1}, x_{i+1}]$ (The shape function is almost everywhere zero except around node $i$)

The shape functions will construct a subspace of V: $\mathrm{span} \left( \left\lbrace \varphi_i \right\rbrace_{i=1}^{n} \right) =: V^{*} \subset V$ So my problem becomes $x_i \cdot a(\varphi_j,\varphi_i) = l(\varphi_i) \quad \forall i,j \in \left\lbrace 1,...,n \right\rbrace$ which is basically a linear system of equations. Then $u = \sum\limits_{i=1}^n x_i \cdot \varphi_i$

But how can I apply this to eigenfrequencies of Timoshenko beams? I know that for 1D Timoshenko beams the kinetic energy (T) and the potential energy (U) are defined as following:

$$T = \dfrac{1}{2} \int\limits_0^l \left( \rho A \left( \dfrac{\partial u}{\partial t} \right)^2 + \rho I \left( \dfrac{\partial \phi}{\partial t} \right)^2 \right)dx$$

$$U = \dfrac{1}{2} \int\limits_0^l \left( EI \left( \dfrac{\partial \phi}{\partial x} \right)^2 + k A G \left( \dfrac{\partial u}{\partial x} - \phi \right)^2 \right)dx$$

But what are the next steps? My best guess: Calculating the Lagrangian $L=T-U$, doing a Gâteaux differentiation at $u$ in the direction $\bar{u}$ and set it to zero: $\delta L(u,\bar{u}) = 0 \quad \forall \bar{u} \in V^{*}$.

Edit: I figured out, how to calculate the eigenfrequencies: Assuming $u(x,t) = \hat{u}(x) \cdot e^{i \lambda t}$ one gets $\ddot{u}(x,t) = -\lambda^2 \cdot \hat{u}(x) \cdot e^{i \lambda t}$. Both can be put into $\delta L(u,\bar{u})$ which can be solved for the eigenvalues $\lambda$.

Am I right? But how can I derive/find formulations for T and U respectively δL for 3D beams?

  • $\begingroup$ This might be better asked in engineering or numerical methods in mathematics. My only suggestion would be that using differentiation to solve such a numerical problem may not be a great idea and may lead to significant errors, especially at low spatial frequencies (and maybe even unstable spurious modes in the numerics), but I can't tell you what the correct algorithm for this case is. $\endgroup$ – CuriousOne Feb 1 '16 at 1:12
  • $\begingroup$ Thank you @CuriousOne for your comment! I asked the "engineerig part" of this question also in the engineering community. $\endgroup$ – Ferdi811 Feb 1 '16 at 10:41

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