Vibration Absorber
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This document discusses vibration absorbers for damping vibrations in structures. It presents the equations of motion for a 2-degree of freedom system with an absorber and goes through the process of obtaining the solution. It is shown that for the primary system to be stationary, the excitation frequency must match the natural frequency of the absorber mass-spring system. Homework problems are assigned to further analyze systems with different excitation frequencies using the presented approach and modal decomposition.
Vibration Absorber
- 1. #WikiCourses http://WikiCourses.WikiSpaces.com Vibration Absorber Mohammad Tawfik Vibration Absorber The first passive damping technique we will learn!
- 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. #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. #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. #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. #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. #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. #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. #WikiCourses http://WikiCourses.WikiSpaces.com Vibration Absorber Mohammad Tawfik Vibration absorber
- 10. #WikiCourses http://WikiCourses.WikiSpaces.com Vibration Absorber Mohammad Tawfik Vibration absorber
- 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. #WikiCourses http://WikiCourses.WikiSpaces.com Vibration Absorber Mohammad Tawfik Homework #2 (cont’d) • Use modal decomposition (diagonalization) to obtain the same results.