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

System Modeling Coursework

P.R. VENKATESWARANFaculty, Instrumentation and Control Engineering,

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

Fax: 0820 2571071Email: [email protected], [email protected]

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

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

mailto:[email protected]

mailto:[email protected]

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

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

•

If ζ

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

•

If ζ

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

•

If ζ

< 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

= 2

•

The displacement is given by•

Differentiating with respect to time


Output for over damped system with ζ

= 2


Case (2): Critically damped system, ζ

= 1


Case (3): Under damped system with ζ

= 0.05


Summary


Numerical -

1


Solution to Numerical 1


Numerical -

2


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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