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# Vibration Absorber

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### Vibration Absorber

1. 1. #WikiCourses http://WikiCourses.WikiSpaces.com Vibration Absorber Mohammad Tawfik Vibration Absorber The first passive damping technique we will learn!
2. 2. #WikiCourses http://WikiCourses.WikiSpaces.com Vibration Absorber Mohammad Tawfik For a 2-DOF System • For the shown 2-DOF system, the equations of motion may be written as: • Where: fxx  KM         2 1 f f f
3. 3. #WikiCourses http://WikiCourses.WikiSpaces.com Vibration Absorber Mohammad Tawfik For Harmonic Excitation • We may write the equation for each of the excitation frequency in the form of: • Then we may add both solutions!          0 11 tCosf KM  xx         tCosf KM 22 0  xx
4. 4. #WikiCourses http://WikiCourses.WikiSpaces.com Vibration Absorber Mohammad Tawfik Consider the first force • We may write the equation in the form: • And the solution in the form: • Which will give:  tCosfKM 1 0 1        xx  tCos x x         2 1 x   xx 2 2 12          tCos x x 
5. 5. #WikiCourses http://WikiCourses.WikiSpaces.com Vibration Absorber Mohammad Tawfik The equation of motion becomes • Get x1() and find out when does it equal to zero!                                      00 0 1 2 1 22 221 2 2 1 2 f x x kk kkk m m  
6. 6. #WikiCourses http://WikiCourses.WikiSpaces.com Vibration Absorber Mohammad Tawfik Using the Dynamic Stiffness Matrix • Writing down the dynamic stiffness matrix: Getting the inverse:                     0 1 2 1 22 2 2 2211 2 f x x KmK KKKm                           0 1 2 222 2 211 2 211 2 2 222 2 2 1 f KKmKKm KKmK KKm x x   
7. 7. #WikiCourses http://WikiCourses.WikiSpaces.com Vibration Absorber Mohammad Tawfik Obtaining the Solution • Multiply the inverse by the right-hand-side • For the first degree of freedom:                    12 122 2 21 2 21212 4 212 1 1 fK fKm KKKmKKmmmx x        0 21 2 21212 4 21 122 2 1     KKKmKKmmm fKm x  
8. 8. #WikiCourses http://WikiCourses.WikiSpaces.com Vibration Absorber Mohammad Tawfik Vibration Absorber • For the first degree of freedom to be stationary, i.e. x1=0 • The excitation frequency have to satisfy: • Note that this frequency is equal to the natural frequency of the auxiliary spring- mass system alone 2 2 m K 
9. 9. #WikiCourses http://WikiCourses.WikiSpaces.com Vibration Absorber Mohammad Tawfik Vibration absorber
10. 10. #WikiCourses http://WikiCourses.WikiSpaces.com Vibration Absorber Mohammad Tawfik Vibration absorber
11. 11. #WikiCourses http://WikiCourses.WikiSpaces.com Vibration Absorber Mohammad Tawfik Homework #2 • Repeat the example of this lecture using f2=f3=0 and f1=1 AND f1=f2=0 and f3=1 • Plot the response of each mass for each of the excitation functions • Comment on the results in the lights of your understanding of the concept of vibration absorber
12. 12. #WikiCourses http://WikiCourses.WikiSpaces.com Vibration Absorber Mohammad Tawfik Homework #2 (cont’d) • Use modal decomposition (diagonalization) to obtain the same results.