Class 19-20 - Free Vibrations (Damped and Undamped)

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System Modeling Coursework

P.R. VENKATESWARANFaculty, Instrumentation and Control Engineering,

Manipal Institute of Technology, ManipalKarnataka 576 104 INDIAPh: 0820 2925154, 2925152

Fax: 0820 2571071Email: pr.venkat@manipal.edu, prv_i@yahoo.com

Web address: http://www.esnips.com/web/SystemModelingClassNotes

Class 19-20: Free (undamped and damped) vibrations

July – December 2008 prv/System Modeling Coursework/MIT-Manipal 2

WARNING!

I claim no originality in all these notes. These are the compilation from various sources for the purpose of delivering lectures. I humbly acknowledge the wonderful help provided by the original sources in this compilation.

For best results, it is always suggested you read the source material.

Contents

Modeling of free vibrations–

Free vibrations

Forced vibrations–

Characteristics of response

Numerical on the derivations

Introduction

In theory, the single degree-of-freedom spring-mass system, once set into motion, would continue to move up and down for ever.

In practice all systems are damped, which means that energy is dissipated, and the amplitude of the motion gradually gets smaller and smaller until it stops altogether.

Damping can be introduced from various sources, and is hard to model accurately.

System with damping

One model, known as viscous damping, is that of a force that is proportional to the velocity of the mass, and which opposes its motion.

This is represented by a dashpot, which is given the symbol shown in Fig.

Equations of motion

Free body diagram

Solutions to this equation

The solution to this equation is given by

Output graphs

Most important case

Output graph

Summary

> 1, the system is heavily damped, and no vibration occurs.

= 1, the system is critically damped, and the damping is only just sufficient to prevent vibration.

< 1, as it is in the majority of cases, there is insufficient damping in the system to prevent vibration and the motion is oscillatory.

Numerical

Let us return to the system in Figure, and, as before, assume that the weight is given an initial displacement, X0

, and then released from rest. Let X0

= 100 mm. Therefore, the initial conditions are that at t = 0, x = 100 mm, and x

= 0 . Find the solution for various values of ζ.

Case (1): Over damped system with ζ

The displacement is given by•

Differentiating with respect to time

Output for over damped system with ζ

Case (2): Critically damped system, ζ

Case (3): Under damped system with ζ

= 0.05

Case (3): Under damped system with ζ

= 0.05

Summary

Numerical -

Solution to Numerical 1

Numerical -

Numerical 3

Solution to Numerical 3

And, before we break…

Reality is often a matter of personal perception as much as objective fact.

Thanks for listening…

Class 19-20 - Free Vibrations (Damped and Undamped)

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