Single Degree of Freedom Systems

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What is a single degree of freedom (SDOF) system ?
Hoe to write and solve the equations of motion?
How does damping affect the response?

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https://wikicourses.wikispaces.com/Lect01+Single+Degree+of+Freedom+Systems
https://eau-esa.wikispaces.com/Topic+Single+Degree+of+Freedom+%28SDOF%29+Systems

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Single Degree of Freedom Systems

  1. 1. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Introduction to Vibrations of Structures
  2. 2. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik References • M. Bismarck-Nasr, "Structural Dynamics in Aeronautical Engineering," AIAA Educational Series, 1999 • D. Inman and E. Austin, “Engineering Vibration,” 2nd edition, Prentice Hall, 2001 • A. A. Shabana, "Vibration of Discrete and Continuous Systems," 2nd edition, Springer, 1997 • D. Thorby, “Structural Dynamics and Vibration in Practice” Elsevier, 2008 • A. G. Ambekar, “Mechanical Vibrations and Noise Engineering” Prentice Hall – India, 2006 • Leonard Meirovitch, “Fundamentals of Vibrations,” 1st edition, McGraw Hill, 2001
  3. 3. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Single degree of freedom systems
  4. 4. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Objectives • Recognize a SDOF system • Be able to solve the free vibration equation of a SDOF system with and without damping • Understand the effect of damping on the system vibration • Apply numerical tools to obtain the time response of a SDOF system
  5. 5. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Single degree of freedom systems • When one variable can describe the motion of a structure or a system of bodies, then we may call the system a 1-D system or a single degree of freedom (SDOF) system. e.g. x(t), q(t) Z(t), y(x).
  6. 6. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Stiffness • From strength of materials recall:
  7. 7. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Newton’s Law • Newton’s Law: 00 )0(,)0( 0)()( )()( vxxx tkxtxm tkxtxm      
  8. 8. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Solving the ODE • The ODE is • The proposed solution: • Into the ODE you get the characteristic equation: • Giving: 0)()(  tkxtxm  t aetx  )( 02  tt ae m k ae   m k 2  m k j
  9. 9. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Solving the ODE (cont’d) • The proposed solution becomes: • For simplicity, let’s define: • Giving: t m k jt m k j eaeatx   21)( m k  tjtj eaeatx    21)(
  10. 10. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Let’s manipulate the solution!
  11. 11. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Recall    ajSinaCose ja           bSinaCosbCosaSinbaSin 
  12. 12. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Manipulating the solution • The solution we have: • Rewriting: tjtj eaeatx    21)(          tjSintCosa tjSintCosatx     2 1)(        tSinaajtCosaatx  2121)(     tSinAtCosAtx  21)( 
  13. 13. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Further manipulation    tSinAtCosAtx  21)(  2 2 2 1 AAA      A A Sin A A Cos 12 &           tSinCostCosSinAtx  )(    tASintx )(
  14. 14. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Different forms of the solution tjtj eaeatx tCosAtSinAtx tASintx        21 21 )( )( )()(
  15. 15. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik NOTE!
  16. 16. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Natural Frequency of Oscillation • In the previously obtained solution: • The frequency of oscillation is  • It depends only on the characteristics of the oscillating system. That is why it is called the natural frequency of oscillation    tASintx )( m k n 
  17. 17. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Frequency periodtheiss 2 Hz 2s2 cycles rad/cycle2 rad/s frequencynaturalthecalledisrad/sinis n nnn n n T f            We often speak of frequency in Hertz or RPM, but we need rad/s in the arguments of the trigonometric functions.
  18. 18. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Recall: Initial Conditions
  19. 19. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Amplitude & Phase from the ICs      Phase 0 01 Amplitude 2 02 0 0 0 tan, yieldsSolving cos)0cos( sin)0sin(           v xv xA AAv AAx n n nnn n     
  20. 20. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Some useful quantities peak valueA   T T dttx T x 0 valueaverage=)( 1 lim valuesquaremeanroot=2 xxrms  valuesquare-mean=)( 1 lim 0 22   T T dttx T x
  21. 21. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Peak Values Ax Ax Ax 2 max max max :onaccelerati :velocity :ntdisplaceme        Maximum or peak (amplitude) values:
  22. 22. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik
  23. 23. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Samples of Vibrating Systems • Deflection of continuum (beams, plates, bars, etc) such as airplane wings, truck chassis, disc drives, circuit boards… • Shaft rotation • Rolling ships
  24. 24. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Wing Vibration
  25. 25. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Ship Vibration
  26. 26. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Effective Stiffness of Structures
  27. 27. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Bars • Longitudinal motion • A is the cross sectional area (m2) • E is the elastic modulus (Pa=N/m2) • l is the length (m) • k is the stiffness (N/m)x(t) m  EA k  l
  28. 28. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Rods • Jp is the polar moment of inertia of the rod • J is the mass moment of inertia of the disk • G is the shear modulus, l is the length Jp J qt) 0  pGJ k 
  29. 29. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Helical Spring 2R x(t) d = diameter of wire 2R= diameter of turns n = number of turns x(t)= end deflection G= shear modulus of spring material 3 4 64nR Gd k 
  30. 30. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Beams f m x • Strength of materials and experiments yield: 3 3 3 3   m EI EI k n   
  31. 31. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Equivalent Stiffness
  32. 32. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Summary • Write down the equation of motion using Newton’s law • Solve the equation of motion for a SDOF • Use initial conditions to determine the amplitude and phase of vibration for a SDOF • Evaluate the effective stiffness of structural members
  33. 33. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik 1. The amplitude of vibration of an undamped system is measured to be 1 mm. the phase shift is measured to be 2 rad and the frequency 5 rad/sec. Calculate the initial conditions. 2. Using the equation: evaluate the constant A1 and A2 in terms of the initial conditions HW #1    tSinAtCosAtx  21)( 
  34. 34. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik HW #1 (cont’d) 3. An automobile is modeled as 1000 kg mass supported by a stiffness k=400000 N/m. When it oscillates, the maximum deflection is 10 cm. when loaded with the passengers, the mass becomes 1300 kg. calculate the change in the frequency, velocity amplitude, and acceleration if the maximum deflection remain 10 cm.
  35. 35. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Adding Damping
  36. 36. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Objectives • Understand the damping as a force resisting motion • Adding viscous damping to the equation of motion of a SDOF • Understand the difference in the responses of different systems depending on the value of the damping
  37. 37. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Damping • Damping is some form of friction! • In solids, friction between molecules result in damping • In fluids, viscosity is the form of damping that is most observed • In this course, we will use the viscous damping model; i.e. damping proportional to velocity
  38. 38. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Viscous Damping • A mathematical form called a dashpot or viscous damper somewhat like a shock absorber the constant c has units: Ns/m or kg/s )(txcfc 
  39. 39. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Shock Absorbers
  40. 40. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Spring-mass-damper systems • From Newton’s law: 00 )0(,)0( 0)()()( )()()( vxxx tkxtxctxm tkxtxctxm      
  41. 41. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Solution (dates to 1743 by Euler) 0)()(2)( 2  txtxtx nn   km c 2 = Where the damping Ratio is given by: (dimensionless) Divide the equation of motion by m
  42. 42. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik        rootstheofnaturethe determines,1ntdiscriminatheHere equationquadraticaofrootsthefrom 1 :inequationalgebraicannowiswhich 02 motionofeq.intosubsitute&)(Let 2 2 2,1 22     nn t n t n t t aeeaea aetx
  43. 43. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Three possibilities: 00201 21 , :conditionsinitialtheUsing )( 221= dampedcriticallycalled repeated&equalareroots1)1 xvaxa teaeatx mkmcc n tt ncr nn         
  44. 44. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Critical damping cont’d • No oscillation occurs t n n etxvxtx     ])([)( 000
  45. 45. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik 12 )1( 12 )1( where )()( 1 :rootsrealdistincttwo-damping-overcalled,1)2 2 0 2 0 2 2 0 2 0 1 1 2 1 1 2 2,1 22                  n n n n ttt nn xv a xv a eaeaetx nnn
  46. 46. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik The over-damped response
  47. 47. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Most interesting Case! 2 2,1 1 :asformcomplexinrootswrite pairsconjugateasrootscomplexTwo commonmost-motiondunderdampecalled,1)3     jnn
  48. 48. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Under-damping                00 01 2 0 2 00 2 1 2 1 1 tan )()( 1 frequencynaturaldamped,1 )sin( )()( 22 xv x xxvA tAe eaeaetx n d dn d nd d t tjtjt n nnn        
  49. 49. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Under-damped-oscillation • Gives an oscillating response with exponential decay • Most natural systems vibrate with an under-damped response
  50. 50. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik Summary • Modeling viscous damping • Solving the equation of motion involving viscous damping • Recognizing the different types of response based on the level of damping
  51. 51. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik 1. Use the given data to plot the response of the SDOF system 2. Solve the equation And plot the response HW #1 (cont’d) 8.0,6.0,4.0,2.0,1.0,01.0 /0,1sec,/2 00     smvmmxradn 0,1 0 00   vx xxx 
  52. 52. #WikiCourses http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems Mohammad Tawfik HW #1 (cont’d) • Homework is due next week: 26/9/2010

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