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# How To Perform Linear Dynamic Analysis?

The intended readership includes all practicing analysts although the book is written with the fresh graduate engineer in mind. The author expects that a new practitioner will attain sufficient knowledge to independently carry out analysis of challenging problems after fully understanding the example problems. This guide assumes some basic knowledge of Finite Element Analysis (FEA), typically through static analysis of structures. This book is complementary to an important earlier NAFEMS publication, namely “A Dynamics Primer” by Dennis Hitchings.

Dynamic analysis covers a range of analysis types. These include solving for the response of a structure,subjected to an external loading or an applied motion. The most intuitive type of loading is applied as a function of time. An analogy is with a real test; hitting a structure and recording strains or displacements, accelerations, stresses etc. through time. A more complex dynamic event is a vehicle crash test.Instead of a real vehicle and the onboard dummies, we can use FEA simulation.

Alternatively, we can solve for varying frequency input. This is not so intuitive, but a simple analogy is given by a test on a structure mounted on a shaker table (a rigid table that we will vibrate at known input frequencies). We measure the peak deflections,accelerations and stresses etc. of the structure at each driving frequency in turn, to build up a picture of how the structure responds across a range of input frequencies. It is important to understand how a building, vehicle,washing machine etc. will respond to a spectrum of steady state vibration input.

More advanced dynamic response analysis types include shock spectra analysis and random vibration analysis. These are not covered in this booklet.

Dynamic response analysis is often the goal in FEA. However, to understand the nature of the response it is vital to survey the physical characteristics of the resonant frequencies inherent in each structure.  The fundamental building block of all dynamic analyses is the natural frequency or normal modes analysis stage. This first step is important and it is vital that it is conducted thoroughly. Some dynamic response methods can bypass this step in their computational process. That is efficient in many respects, but as engineers we cannot bypass the exploration and understanding of the natural frequency and mode shapes of a structure.

## Contents

### 2. Introduction to Dynamics

• 2.1 Background
• 2.2 Natural Frequency or Normal Modes Analysis

### 3. Introduction to Equation of Motion

• 3.1 Undamped Free Vibration
• 3.2 Example with 2 DOF Undamped Free Motion – Method 1
• 3.3 Example with 2 DOF Undamped Free Motion – Method 2
• 3.4 Eigenvector Normalization
• 3.5 Mode Identification and Correlation
• 3.6 Modal Assurance Criteria

### 6. Importance of Mass Modeling

• 6.1 Element Lumped Mass Terms
• 6.2 Overall Influence of Mass
• 6.3 Mass Moments of Inertia

### 7. Accurate Idealization

• 7.1 Boundary Conditions
• 7.2 Joints, Flanges and Other Features
• 7.3 Mesh Density

### 8. Quality Assurance for Normal Modes Analysis

• 8.1 Other Modelling Issues in Normal Modes Analysis
• 8.2 Checklist for Normal Modes Analysis

### 9. A Modal Based Approach

• 9.1 Modal Coordinates
• 9.2 Modal Effective Mass

### 10. Damping

• 10.1 Damped Free Vibration
• 10.2 Simple Example Problem
• 10.3 Practical Damping

### 11. Forcing Functions – Response Analysis

• 11.1 Overview of Response Analysis
• 11.2 Modal Coordinates in Response Analysis
• 11.3 Damping with a Forcing Function

### 12. Transient Response Analysis

• 12.1 Direct Transient
• 12.2 Example Structure
• 12.3 Damping in Direct Transient
• 12.4 Modal Transient Analysis
• 12.5 Transient Base Motion
• 12.6 Check list for Transient Response Analysis

### 13. Frequency Response Analysis

• 13.1 Direct Frequency Response
• 13.2 Modal Based Frequency Response Analysis
• 13.3 Practical Frequency Response Analysis
• 13.4 Modal Frequency Response Considerations
• 13.5 Direct Frequency Response Considerations
• 13.6 Check List for Frequency Response Analysis

### 14. Bibliography 15. Abbreviations and Acronyms16. Appendix A: Modal Effective Mass

The form below can be used to peorder "How To Perform Linear Dynamic Analysis" by Tony Abbey. A 33% discount is available to individuals attending Tony's Dynamics Basic & Advanced Dynamic Finite Element Analysis Training Courses.

Members Price
£30 | \$40 | €35

Non-Members Price
£120 | \$158 | €140
Order Ref: HT47

T. Abbey
Hardback - Approximately  194 pages

# Discount Details

Attendees on the NAFEMS Dynamics courses run by Tony Abbey can purchase this publication at a discounted rate.

Member price £30 | \$40 | €35
Non-member price £120 | \$160 | €135

Discounted member price £20 | \$26 | €23
Discounted non-member price £80 | \$106 | €90

Complete the form at the bottom of this page to obtain the book at a discounted rate.