# Dynamics

## Setup

1. Specify the modal results to which the analysis is linked. The modal solution must exist in the current design study. In SimSolid, the time integration of the equations of motion is extremely fast and all modes are always included in the analysis.
2. For Frequency and Random Response, specify the frequency span upper and lower limits. For Transient response, specify Time span.
3. Specify damping using Rayleigh damping coefficients or Modal damping.
4. Select the Evaluate peak responses during solving check box to evaluate peak responses during solving phase.

See Create Analysis for additional information.

## Damping

Two methods to specify damping are supported.
Rayleigh Damping Coefficient
Assumes the damping matrix is proportional to the mass and stiffness matrices. You need to specify values for Mass (F1) and Stiffness (F2) in the Dynamics creation dialog to use this method.
Modal Damping
Creates critical damping ratio for each mode. You can specify this value in the Dynamics analysis creation dialog.

## Notes for Dynamics Analysis

1. When the base excitation type is displacement, the initial condition for displacement and velocity is always assumed to be zero.
2. In SimSolid, the boundary compatibility is approximately met. The response at the constrained end is not going to be an absolute zero but is relatively small compared to the peak responses.
3. Equivalent radiated power density is calculated as:(1)
$\text{ERP}\text{ }\text{Density}\text{\hspace{0.17em}}\text{=}\text{\hspace{0.17em}}\text{ERPRLF}\text{\hspace{0.17em}}\text{*}\text{\hspace{0.17em}}\text{(0}\text{.5}\text{\hspace{0.17em}}\text{*}\text{\hspace{0.17em}}\text{ERPC}\text{\hspace{0.17em}}\text{*}\text{\hspace{0.17em}}\text{ERPRHO)}\text{\hspace{0.17em}}*\text{\hspace{0.17em}}{v}^{2}$
Where:
$v$
Normal velocity of the picked point
ERPC (Speed of sound in air)
343 m/s
ERPRHO (Density of air)
1.225 Kg/m3
1

Equivalent radiated power is calculated as an integral of ERP density over picked faces as:

(2)
$\text{ERP}\text{\hspace{0.17em}}\text{=}\text{\hspace{0.17em}}\text{ERPRLF}\text{\hspace{0.17em}}\text{*}\text{\hspace{0.17em}}\text{(0}\text{.5}\text{\hspace{0.17em}}\text{*}\text{\hspace{0.17em}}\text{ERPC}\text{\hspace{0.17em}}\text{*}\text{\hspace{0.17em}}\text{ERPRHO)}\text{\hspace{0.17em}}{\int }_{S}\text{ }{v}^{2}\text{ }ds$
4. Phase for Absolute displacement can be queried using Pick Info for frequency dynamics.
5. Relationship between relative and absolute displacement in frequency dynamics is as follows.
Relative motion is calculated as:(3)
$\stackrel{¨}{x}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2\zeta {\omega }_{n}\stackrel{˙}{x}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\omega }_{n}{}^{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\stackrel{¨}{b}$
Where base excitation $b$ is:(4)
$-\stackrel{¨}{b}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{Y}_{0}\left(\omega \right){e}^{i\left(\omega t\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\varphi \right)}$
Solving this differential equation, the relative displacement can be calculated as:(5)
${x}_{r}\left(t\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{Y}_{0}\left(\omega \right){e}^{i\varphi }\text{\hspace{0.17em}}\left(\frac{{\omega }_{n}{}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\omega }^{2}}{{\left({\omega }_{n}{}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\omega }^{2}\right)}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\left(2\zeta {\omega }_{n}\omega \right)}^{2}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}i\frac{2\zeta {\omega }_{n}\omega }{{\left({\omega }_{n}{}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\omega }^{2}\right)}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\left(2\zeta {\omega }_{n}\omega \right)}^{2}}\right){e}^{i\omega t}$
Absolute motion is calculated as:(6)
$\stackrel{¨}{x}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2\zeta {\omega }_{n}\stackrel{˙}{x}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\omega }_{n}{}^{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2\zeta {\omega }_{n}\stackrel{˙}{b}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\omega }_{n}{}^{2}b$
Where, (7)
$b\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{Y}_{0}\left(\omega \right)}{{\omega }^{2}}{e}^{i\left(\omega t\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\varphi \right)}$
Solving this equation, the absolute displacement can be calculated as:(8)
${x}_{a}\left(t\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\left({\left(\frac{{\omega }_{n}}{\omega }\right)}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}i2\zeta \frac{{\omega }_{n}}{\omega }\right){x}_{r}\left(t\right)$
6. In Frequency and Random dynamics, the Complex Function Method is used to solve differential equations. Displacement, velocity, and acceleration results have complex components.
Given complex values of displacement as:(9)
${D}_{x}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{a}_{x}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}i{b}_{x}$
${D}_{y}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{a}_{y}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}i{b}_{y}$
${D}_{z}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{a}_{z}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}i{b}_{z}$
Displacement magnitude is calculated as:(10)
$\left|\sqrt{{D}_{x}{}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{D}_{x}{}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{D}_{x}{}^{2}}\right|\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sqrt{\left|{D}_{x}{}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{D}_{x}{}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{D}_{x}{}^{2}\right|}\text{\hspace{0.17em}}$
7. When a Transient Dynamics analysis is linked to a prestressed modal analysis, a new result type is offered called Total Displacement. Total Displacement is the combination of the prestressed and dynamic displacements. Therefore, displacement magnitude is the displacement caused by the dynamic analysis.