Class 19-20 - Free Vibrations (Damped and Undamped)
-
Upload
api-26676616 -
Category
Documents
-
view
4.170 -
download
5
Transcript of Class 19-20 - Free Vibrations (Damped and Undamped)
![Page 1: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/1.jpg)
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
![Page 2: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/2.jpg)
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.
![Page 3: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/3.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 3
Contents
•
Modeling of free vibrations–
Free vibrations
–
Forced vibrations–
Characteristics of response
•
Numerical on the derivations
![Page 4: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/4.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 4
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.
![Page 5: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/5.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 5
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.
![Page 6: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/6.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 6
Equations of motion
•
Free body diagram
![Page 7: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/7.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 7
Solutions to this equation
•
The solution to this equation is given by
![Page 8: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/8.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 8
Output graphs
![Page 9: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/9.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 9
Most important case
![Page 10: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/10.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 10
Output graph
![Page 11: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/11.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 11
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.
![Page 12: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/12.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 12
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.
![Page 13: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/13.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 13
Case (1): Over damped system with ζ
= 2
•
The displacement is given by•
Differentiating with respect to time
![Page 14: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/14.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 14
Output for over damped system with ζ
= 2
![Page 15: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/15.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 15
Case (2): Critically damped system, ζ
= 1
![Page 16: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/16.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 16
Case (3): Under damped system with ζ
= 0.05
![Page 17: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/17.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 17
Case (3): Under damped system with ζ
= 0.05
![Page 18: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/18.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 18
Summary
![Page 19: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/19.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 19
Summary
![Page 20: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/20.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 20
Numerical -
1
![Page 21: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/21.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 21
Solution to Numerical 1
![Page 22: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/22.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 22
Numerical -
2
![Page 23: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/23.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 23
Numerical -
2
![Page 24: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/24.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 24
Numerical 3
![Page 25: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/25.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 25
Solution to Numerical 3
![Page 26: Class 19-20 - Free Vibrations (Damped and Undamped)](https://reader034.fdocuments.net/reader034/viewer/2022042607/5528a3f655034666588b48a2/html5/thumbnails/26.jpg)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 26
And, before we break…
•
Reality is often a matter of personal perception as much as objective fact.
Thanks for listening…