Multiple Degree of Freedom (MDOF) Systems

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What is a multiple dgree of freedom (MDOF) system?
How to calculate the natural frequencies?
What is a mode shape?
What is the dynamic stiffness matrix approach?

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https://wikicourses.wikispaces.com/Lect04+Multiple+Degree+of+Freedom+Systems
https://eau-esa.wikispaces.com/Topic+Multiple+Degree+of+Freedom+%28MDOF%29+Systems

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Multiple Degree of Freedom (MDOF) Systems

  1. 1. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Multiple Degree of Freedom Systems
  2. 2. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Multiple Degrees of Freedom
  3. 3. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Objectives • What is a multiple degree of freedom system? • Obtaining the natural frequencies of a multiple degree of freedom system • Interpreting the meaning of the eigenvectors of a multiple degree of freedom system • Understanding the mechanism of a vibration absorber
  4. 4. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Two Degrees of Freedom Systems
  5. 5. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Two Degrees of Freedom Systems • When the dynamics of the system can be described by only two independent variables, the system is called a two degree of freedom system
  6. 6. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Two Degrees of Freedom
  7. 7. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Free-Body Diagram
  8. 8. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Equations of Motion    )()()( )()()()( 12222 1221111 txtxktxm txtxktxktxm     0)()()( 0)()()()( 221222 2212111   txktxktxm txktxkktxm   Rearranging:
  9. 9. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Initial Conditions • Two coupled, second -order, ordinary differential equations with constant coefficients • Needs 4 constants of integration to solve • Thus 4 initial conditions on positions and velocities 202202101101 )0(,)0(,)0(,)0( xxxxxxxx  
  10. 10. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik In Matrix Form                    )( )( )(, )( )( )(, )( )( )( 2 1 2 1 2 1 tx tx t tx tx t tx tx t       xxx                22 221 2 1 , 0 0 kk kkk K m m M 0xx KM Where: With initial conditions:              20 10 20 10 )0(,)0( x x x x   xx
  11. 11. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Recall: For SDOF • The ODE is • The proposed solution: • Into the ODE you get the characteristic equation: • Giving: 0)()(  tkxtxm t aetx  )( 02  tt ae m kae   m k2  m kj
  12. 12. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Solving the system • The ODE is • The proposed solution: • Into the ODE you get the characteristic equation: • Giving: tj et  ax )( 02  tjtj ee   KaMa 0xx KM   02  tj e  aKM
  13. 13. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Giving: 0 2 1         a a a Either: Trivial solution; No motion! 02  KMOR:
  14. 14. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Giving: 0 22 2 2 2211 2    kmk kkkm   0)( 21 2 221221 4 21  kkkmkmkmmm  Which can be solved as a quadratic equation in 2.
  15. 15. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik NOTE! • For spring mass systems, the resulting roots are always positive, real, and distinct • Which give two couples of distinct roots. 2 24,3 2 12,1 &  
  16. 16. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Example • m1=9 kg,m2=1kg, k1=24 N/m and k2=3 N/m • In Matrix form: 0 33 327 10 09               xx
  17. 17. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Example (cont’d) • The proposed solution: • Into the ODE you get the characteristic equation: 4-62+8=(2-2)(2-4)=0 • Giving: 2 =2 and 2 =4 tj et  ax )( Each value of 2 yields an expression for a:
  18. 18. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Calculating the corresponding vectors a1 and a2 0a 0a   2 2 2 1 2 1 )( )( KM KM   A vector equation for each square frequency And: 4 equations in the 4 unknowns (each vector has 2 components, but...
  19. 19. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Computing the vectors a let2,=For 12 11 1 2 1        a a a 2 equations, 2 unknowns but DEPENDENT! 03and039 0 0 )2(33 3)2(927 )(- 12111211 12 11 2 1                      aaaa a a KM 0a
  20. 20. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik 0a0a a0a u     1 2 11 2 1 11 2 1 1 2 1 1211 12 11 )()( :arbitrary,doesso,)( satisfiesSupposearbitrary.ismagnitudeThe .0:becauseisThis !determinedbecanmagnitudenot thedirection,only the :equationsbothfrom 3 1 3 1 cKMcKM ccKM KM aa a a    continued
  21. 21. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik For the second value of 2: 3 1aor039 0 0 )4(33 3)4(927 )(- havethen welet4,=For 22212221 22 21 2 1 22 21 2 2 2 aaa a a KM a a                             0a a   Note that the other equation is the same
  22. 22. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik What to do about the magnitude!              1 1 1 1 3 1 222 3 1 112 a a a a Several possibilities, here we just fix one element: Choose: Choose:
  23. 23. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Thus the solution to the algebraic matrix equation is:              1 ,2 1 ,2 3 1 24,2 3 1 13,1 a a  
  24. 24. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Return now to the time response:     nintegratioofconstantsareand,,,where )sin()sin( )( )( ,,,)( 2121 22221111 21 2211 2211 2211 2211 2211      AA tAtA decebeaet edecebeat eeeet tjtjtjtj tjtjtjtj tjtjtjtj aa aax aaaax aaaax        We have four solutions: Since linear we can combine as: determined by initial conditions
  25. 25. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Physical interpretation of all that math! • Each of the TWO masses is oscillating at TWO natural frequencies 1 and 2 • The relative magnitude of each sine term, and hence of the magnitude of oscillation of m1 and m2 is determined by the value of A1a1 and A2a2 • The vectors a1 and a2 are called mode shapes
  26. 26. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik What is a mode shape? • First note that A1,A2, 1 and 2 are determined by the initial conditions • Choose them so that A2 = 1 = 2 =0 • Then: • Thus each mass oscillates at (one) frequency 1 with magnitudes proportional to a1 the1st mode shape t a a A tx tx t 1 12 11 1 2 1 sin )( )( )(             x
  27. 27. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Multiple Degrees of Freedom
  28. 28. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Things to note • Two degrees of freedom implies two natural frequencies • Each mass oscillates at these two frequencies present in the response • Frequencies are not those of two component systems
  29. 29. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Eigenvalues and Eigenvectors • Can connect the vibration problem with the algebraic eigenvalue problem • This will give us some powerful computational skills • And some powerful theory • All the codes have eigensolvers so these painful calculations can be automated
  30. 30. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Compound Pendulum
  31. 31. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Pendulum Video
  32. 32. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Frequency Response
  33. 33. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Frequency Response • Similar to SDOF systems, the frequency response of a MDOF system is obtained by assuming harmonic excitation. • An analytical relation between all the possible input forces and output displacements may be obtained, called transfer function • For our course, we will pay more attention to the plot of the relation.
  34. 34. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Dynamic Stiffness • The system of equations we obtain for an undamped vibrating system is always in the form fKxxM  • For harmonic excitation  harmonic response, we may write   fxKM  2  fxKD 
  35. 35. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Dynamic Stiffness • Now, we have a system of algebraic equations that may be solved for the amplitude of vibration of each DOF as a response to given harmonic excitation at a certain frequency! fKx D 1 
  36. 36. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Example • For the three DOF system given in the sketch, consider all stiffness values to be 2 and m1=2, m2=1, m3=3
  37. 37. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Example • The equations of motion may be written in the form:                                                  3 2 1 3 2 1 3 2 1 420 242 024 300 010 002 f f f x x x x x x    FKxxM 
  38. 38. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Example • Getting the eigenvalues, and frequencies                     796.0 295.1 241.2 , 633.000 0677.10 00023.5 
  39. 39. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Getting the Frequency Response                          300 010 002 420 242 024 2 DK                  0 1 0 3 2 1 f f f
  40. 40. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik
  41. 41. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Notes: • For all degrees of freedom, as the frequency reaches one of the natural frequencies, the amplitudes grows too much • For some frequencies, and some degrees of freedom, the response becomes VERY small. If the system is designed to tune those frequencies to a certain value, vibration is absorbed: “Vibration absorber”
  42. 42. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Vibration Absorber The first passive damping technique we will learn!
  43. 43. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems 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
  44. 44. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems 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
  45. 45. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems 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 
  46. 46. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems 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  
  47. 47. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems 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   
  48. 48. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems 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 fKmx  
  49. 49. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems 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
  50. 50. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Vibration absorber
  51. 51. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Vibration absorber
  52. 52. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems 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
  53. 53. #WikiCourses http://WikiCourses.WikiSpaces.com Multiple Degree of Freedom Systems Mohammad Tawfik Homework #2 (cont’d) • Use modal decomposition (diagonalization) to obtain the same results.

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