# Introduction to Linear Dynamic Analysis

2018-08-20

Andrew Berndt

## Introduction

An analyst faced with the task of performing a dynamic finite element analysis has numerous choices to make. For someone that is starting out in the field, the range of options available can be daunting. In this blog article, some of the foundational concepts related to linear dynamic analysis will be discussed, with the aim of providing a high-level overview which will make the task of performing an analysis appreciably less daunting.

## Linear Dynamic Analysis Types

There are three types of dynamic analysis:

• ## Normal Modes Analysis

A normal modes or eigenvalue analysis involves the calculation of eigenvalues (natural frequencies) and corresponding eigenvectors (mode shapes) for a structure. An example of a mode shape of a soda can is given in Figure 1. These analyses are relatively simple to perform and provide helpful insights into the characteristics of a structure from both design and analysis perspectives. The eigenvalues and eigenvectors are solely dependent on the structure’s characteristics (stiffness and mass matrices) as no loads are considered when executing this analysis type. A modal analysis is often the first step in the process of performing a linear dynamic analysisIt thus serves as the foundation for a range of engineering investigations.  The interested reader is referred to the following blog article for a detailed discussion on modal analysis – Modal Analysis: What it is and is not. Figure 1: Mode shape of a soda can

• ### Transient Response

Transient response analysis is used when loads are defined as arbitrary functions of time, making this approach generically applicable to linear dynamic structural analysis. Figure 2 depicts a time-varying force signal that could be used as a load input in a transient response analysis. Figure 2: Random force vs. time input

• ### Frequency Response

• Frequency response analysis is intended for use on structures that are subjected to constant sinusoidal excitation. The excitation loads are defined in the frequency domain. Rotating machinery often fit into this class, so this analysis technique is applicable in a fairly wide range of scenarios. It takes some time to understand and work effectively with the output obtained from a frequency response analysis, but the advantages gained in terms of solving time and the size of simulation output files, make it very helpful when solving applicable problems. Figure 3 depicts a sinusoidal time domain signal at the top, with its frequency domain representation below (following from a FFT (Fast Fourier Transform)). Structures subjected to loading of this form can be analysed using frequency response methods. The interested reader is referred to Frequency Response Analysis – What is it? for more information. Figure 3: Sinusoidal time domain signal with the associated frequency domain representation

## Solving Approaches

Two approaches can be used when solving transient and frequency response simulations:

• ### Direct

The direct method employs numerical integration procedures directly on the full set of coupled equations of motion to calculate the structural response.

• ### Modal Methods

The modal method uses the structure’s mode shapes to decouple the equations of motion, resulting in a set of single degree of freedom systems that can be solved much more efficiently. Additionally, it is standard practice to exclude some modes from the analysis set, thereby reducing the size of the analysis. For a comprehensive description of modal methods, please see An introduction to Modal Methods for Dynamic Analysis.

The choice of whether to use the direct or modal approach is highly dependent on the specific model in question. Below are a few guidelines that can be used in the decision-making process:

• The direct approach has a higher level of accuracy because the entire system is analysed whereas the modal approach uses a truncated set of modes.
• Transient analyses that require many time steps tend to be better suited to a solution using the modal approach.
• Structures that undergo high-frequency excitation are likely better suited to the direct method because the modal method will require many modes to accurately approximate the response.
• Small models are usually better solved using the direct method. When the model is large, modal methods hold distinct advantages so long as the time required to compute the eigenvalues and eigenvectors does not become excessive.

This information will help make the first steps performing dynamic analyses simpler. For practical assistance in setting up these types of analyses in Simcenter, please see the Simcenter tutorials page.