# Linear Dynamics – Four Methods Worth Knowing

SOLIDWORKS Simulation Premium offers a great set of tools dealing with linear dynamics employing the method of modal analysis. While almost everyone is aware of what a frequency analysis can do, it is less popular that those extracted eigenmodes can be further processed – and give highly valuable insight into advanced tasks as e.g. responses due to harmonic or transient excitations, random loads or response spectra.

Typical applications are:

- Dynamic load input
- Stress on moving components
- Impact/shock loading
- Simulate shaker test
- Base isolator design of mounted machinery
- Earthquake simulation
- and many more…

There are four different methods we want to have a look at: Time History, Harmonic, Random Vibration and Response Spectrum.

Let’s start with a short recapitulation of the basic concept:

**Preparation**

There are some important things one should take care of when starting to work with linear dynamics, e.g. to include enough eigenfrequencies to have a representative model or to define a proper damping:

- Check Cumulative Effective Mass Participation Factor (CEMPF, in general > 80%)
- At minimum include frequencies of at least twice max. operating frequency
- Define Damping (Modal or Rayleigh, check SW Help for recommendations)

**Example:** Steel comb of a music box

With this simple model we will apply all four kinds of linear dynamic techniques:

**1. Time History**

This study type is used for transient problems to analyze e.g. the response during and after a short-time load.

__Input:__ Time-dependent excitation, e.g. impact load (see standards e.g. MIL-STD-810 G, Method 516.5)

__Output:__ Time-dependent response e.g. accelerations, velocities, displacements, …

**2. Harmonic**

A harmonic study reveals the response when a system is exposed to a certain range of excitation frequencies (think e.g. of a motor working at different speed).

** **__Input:__ Frequency range to be evaluated and damping

__Output:__ Max. response for each of the excitation frequencies

__Fatigue:__ Results can be used for a fatigue assessment (select excitation frequency and number of cycles).

**3. Random Vibration**

There are cases where the load is too complex to be captured with a regular transient study. Therefore, the loading is transformed into a so-called PSD-curve (Power Spectral Density) which serves as input for a random vibration analysis.

__Input:__ PSD-curve to catch the statistical characteristics of the influence of complex loading (provided in codes/standards e.g. MIL-STD-810)

__Output:__ RMS (Root Mean Square) and PSD values of stresses, displacements, etc.

__Fatigue:__ Employing Basquin’s equation the results from a random vibration study can be used to analyze lifetime.

**4. Response Spectrum**

Another method to determine the characteristics of complex loads is to record only the peak responses (often accelerations) vs. frequencies producing a response spectrum.

__Input:__ Response spectrum = max. response of a SDOF oscillator plotted vs. natural frequency for a certain transient loading; also often as a spectrum from transient shock load called SRS (Shock Response Spectrum)

__Output:__ Peak responses of stresses, displacements, etc. resulting from mode combination methods as e.g. SRSS (Square Root Sum of Squares) or CQC (Complete Quadratic Combination)

With this admittedly quite short compendium, you should already have a rough idea of the possibilities working with SOLIDWORKS Simulation and linear dynamics. If you should realize that such analysis could improve your design decisions, please ask your local reseller for further information.