Next 1. Introduction 1. Introduction (cont'd) 1. Introduction (cont'd) 1. Introduction (cont'd) 1. Introduction (cont'd) 1. Introduction (cont'd) 1. Introduction (cont'd) 1. Introduction (cont'd) 2. Rayleigh-Ritz procedure 2. Rayleigh-Ritz procedure (cont'd) 2. Rayleigh-Ritz procedure (cont'd) 2. Rayleigh-Ritz procedure (cont'd) 2. Rayleigh-Ritz procedure (cont'd) 2. Rayleigh-Ritz procedure (cont'd) 2. Rayleigh-Ritz procedure (cont'd) 2. Rayleigh-Ritz procedure (cont'd) 2. Rayleigh-Ritz procedure (cont'd) 3. A simple element analysis 3. A simple element analysis (cont'd) 3. A simple element analysis (cont'd) 3. A simple element analysis (cont'd) 3. A simple element analysis (cont'd) 3. A simple element analysis (cont'd) 3. A simple element analysis (cont'd) 3. A simple element analysis (cont'd) 3. A simple element analysis (cont'd) 3. A simple element analysis (cont'd) 4. Higher-order elements 4. Higher-order elements (cont'd) 5. Assembly of system equation 5. Assembly of system equation (cont'd) 5. Assembly of system equation (cont'd) 5. Assembly of system equation (cont'd) 5. Assembly of system equation (cont'd) 6.1 Static problem 36 of 47 6.2 Undamped dynamic problem (cont'd) 6.2 Undamped dynamic problem (cont'd) 6.2 Undamped dynamic problem (cont'd) 6.3.1 Modal analysis 6.3.1 Modal analysis (cont'd) 6.3.2 Time-domain analysis 6.3.2 Time-domain analysis (cont'd) 6.3.2 Time-domain analysis (cont'd) 6.3.2 Time-domain analysis (cont'd) 6.3.2 Time-domain analysis (cont'd) 6.3.3 Complex-valued analysis

6.2 Undamped dynamic problem

\[ \mathbf K \mathbf u + \mathbf M \ddot {\mathbf u} = \mathbf f \]
where \(\ddot {\mathbf u}\) is defined as \(\text d ^2 \mathbf u / \text d t^2\).
\[\begin{align} \frac {\text d} { \text d t } \sin {\omega t} & = \omega \cos \omega t \\ \frac {\text d} { \text d t } \cos {\omega t} & = - \omega \sin \omega t \end{align}\]

If we consider harmonic vibrations, \(\mathbf M \ddot {\mathbf u}\) becomes \(- \omega^2 \mathbf M \mathbf u\). Because of the lack of damping (or energy dissipation), forcing functions at certain frequencies will lead to infinite displacements, so we consider the unforced problem: \[\mathbf K \mathbf u - \omega^2 \mathbf M \mathbf u = 0\]

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R. Funnell
Last modified: 2018-11-07 12:52:06

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