# Dynamics

Mathematically, a linear dynamic system can be discretized in matrix form as

$\begin{matrix} \mathbf{M}\ddot{\mathbf{X}}(t)+ \mathbf{D}\dot{\mathbf{X}}(t)+\mathbf{K}\mathbf{X}(t)=\mathbf{F}(t), \end{matrix}$

where $\mathbf{M}$, $\mathbf{D}$ and $\mathbf{K}$ denote the structural mass, damping and stiffness matrix of size $N \times N$ and $\mathbf{F}(t)$ is the time-varying applied load of size $N \times 1$. The primary objective of dynamic analysis is the evaluation of the time histories of displacement $\mathbf{X}$, velocity $\dot{\mathbf{X}}$ and acceleration $\ddot{\mathbf{X}}$, which can be further processed for the determination of e.g. stresses, strains, etc. The mathematical expression defining the dynamic displacements is called equation of motion. The equation of motion can be derived by expressing the equilibrium of the forces associated with each of the $N$ degrees of freedom. If written in vector form, one obtains

$\begin{matrix} \mathbf{f}_I(t)+ \mathbf{f}_D(t)+\mathbf{f}_S(t)=\mathbf{F}(t), \end{matrix}$

where $\mathbf{f}_I(t)$ is the inertial force given by a relationship between the mass coefficients and the accelerations of the corresponding degrees of freedom, $\mathbf{f}_D(t)$ denotes the viscous damping force based on the assumption that the damping force depends on the velocity, and $\mathbf{f}_S(t)$ is the vector of the elastic forces. In this way, the above equation of motion is obtained.

## Evaluation of the equation of Motion

The evaluation of the equation of motion can be performed in time domain or in frequency domain:

• Time domain analysis is the most general type of dynamic analysis where the response due to a time-varying excitation of arbitrary nature is investigated. In time domain analysis, the equation of motion can be numerically integrated or the modal superposition method can be used which is discussed in the chapter Time domain analysis.
• Frequency domain analysis determines the steady-state response of a linear structure due to sinusoidal loading. In frequency domain analysis, the force-response transfer function for either the spatial or the modal model is required for the calculation of the system responses which is discussed in the chapter Frequency domain analysis.

Comprehensive overviews of structural dynamics can be found in e.g.[1][2][3][4][5].

## References

1. R.W. Clough and J. Penzien. Dynamics of Structures . McGraw-Hill, 1993
2. A.K. Chopra. Dynamics of Structures: Theory and Application to Earthquake Engineering . Pretinence Hall, 1995
3. R. Roy, Jr. Craig. Structural Dyamics: An Introduction to Computer Methods . John Wiley & Sons, 1981
4. A. Girard, N. Roy. Structural Dynamics in Industry . John Wiley & Sons, 2008
5. G.C. Hart, K. Wong. Structural Dynamics for Structural Engineers . John Wiley & Sons, 2000