Elementary Tutorial

1. Elementary Tutorial Fundamentals of Linear Vibrations Prepared by Dr. An Tran in collaboration with Professor P. R. Heyliger Department of Civil Engineering Colorado State University Fort Collins, Colorado June 2003 Developed as part of the Research Experiences of Undergraduates Program on “Studies of Vibration and Sound” , sponsored by National Science Foundation and Army Research Office (Award # EEC-0241979). This support is gratefully acknowledged.

2. Fundamentals of Linear Vibrations Single Degree-of-Freedom Systems Two Degree-of-Freedom Systems Multi-DOF Systems Continuous Systems

3. Single Degree-of-Freedom Systems • A spring-mass system • General solution for any simple oscillator • General approach • Examples • Equivalent springs • Spring in series and in parallel • Examples • Energy Methods • Strain energy & kinetic energy • Work-energy statement • Conservation of energy and example

4. A spring-mass system General solution for any simple oscillator: Governing equation of motion: where:

5. Any simple oscillator General approach: • Select coordinate system • Apply small displacement • Draw FBD • Apply Newton’s Laws:

6. + Simple oscillator – Example 1

7. + Simple oscillator – Example 2

8. + Simple oscillator – Example 3

9. + Simple oscillator – Example 4

10. Springs in series: same force - flexibilities add Springs in parallel: same displacement - stiffnesses add Equivalent springs

11. Equivalent springs – Example 1

12. + Equivalent springs – Example 2 Consider: ka2 > Wln2 is positive - vibration is stable ka2 = Wl statics - stays in stable equilibrium ka2 < Wl unstable - collapses

13. + Equivalent springs – Example 3 We cannot definen since we have sin term If  < < 1, sin   :

14. Strain energy U: energy in spring = work done Kinetic energy T: Energy methods Conservation of energy: work done = energy stored

15. Work done = Change in kinetic energy Conservation of energy for conservative systems Work-Energy principles E = total energy = T + U = constant

16. Energy methods – Example Work-energy principles have many uses, but one of the most useful is to derive the equations of motion. Conservation of energy: E = const. Same as vector mechanics

17. Two Degree-of-Freedom Systems • Model problem • Matrix form of governing equation • Special case: Undamped free vibrations • Examples • Transformation of coordinates • Inertially & elastically coupled/uncoupled • General approach: Modal equations • Example • Response to harmonic forces • Model equation • Special case: Undamped system

18. Two-DOF model problem Matrix form of governing equation: where: [M] = mass matrix; [C] = damping matrix; [K] = stiffness matrix; {P} = force vector Note: Matrices have positive diagonals and are symmetric.

19. Undamped free vibrations • Assumed general solutions: Zero damping matrix [C] and force vector {P} • Characteristic equation: • Characteristic polynomial (for det[ ]=0): • Eigenvalues (characteristic values):

20. Undamped free vibrations • Eigenvalues and frequencies: Special case when k1=k2=k and m1=m2=m • Two mode shapes (relative participation of each mass in the motion): • The two eigenvectors are orthogonal: Eigenvector (1) = Eigenvector (2) =

21. Undamped free vibrations (UFV) Single-DOF: For any set of initial conditions: • We know {A}(1) and {A}(2), 1 and 2 • Must find C1, C2, 1, and 2 – Need 4 I.C.’s For two-DOF:

22. UFV – Example 1 Given: No phase angle since initial velocity is 0: From the initial displacement:

23. UFV – Example 2 Now both modes are involved: From the given initial displacement: Solve for C1 and C2: Hence, or Note: More contribution from mode 1

24. Transformation of coordinates UFV model problem: “inertially uncoupled” Introduce a new pair of coordinates that represents spring stretch: “elastically coupled” z1(t) = x1(t) = stretch of spring 1 z2(t) = x2(t) - x1(t) = stretch of spring 2 or x1(t) = z1(t) x2(t) = z1(t) + z2(t) Substituting maintains symmetry: “inertially coupled” “elastically uncoupled”

25. We have found that we can select coordinates so that: Inertially coupled, elastically uncoupled, or Inertially uncoupled, elastically coupled. Big question: Can we select coordinates so that both are uncoupled? Notes in natural coordinates: Transformation of coordinates • The eigenvectors are orthogonal w.r.t [M]: • The modal vectors are orthogonal w.r.t [K]: • Algebraic eigenvalue problem:

26. Transformation of coordinates Governing equation: General approach for solution Let or We were calling “A” - Change to u to match Meirovitch Substitution: Modal equations: Known solutions Solve for these using initial conditions then substitute into (**).

27. Transformation - Example Model problem with: 1) Solve eigenvalue problem: 2) Transformation: So As we had before. More general procedure: “Modal analysis” – do a bit later.

28. Response to harmonic forces Model equation: {F} not function of time [M], [C], and [K] are full but symmetric. Assume: Substituting gives: Hence:

29. Special case: Undamped system • Entries of impedance matrix [Z]: Zero damping matrix [C] • Substituting for X1 and X2: • For our model problem (k1=k2=k and m1=m2=m), let F2 =0: Notes: 1) Denominator originally (-)(-) = (+). As it passes through w1, changes sign. 2) The plots give both amplitude and phase angle (either 0o or 180o)

30. Multi-DOF Systems • Model Equation • Notes on matrices • Undamped free vibration: the eigenvalue problem • Normalization of modal matrix [U] • General solution procedure • Initial conditions • Applied harmonic force

31. Model equation: Notes on matrices: Multi-DOF model equation Multi-DOF systems are so similar to two-DOF. • Vector mechanics (Newton or D’ Alembert) • Hamilton's principles • Lagrange's equations We derive using: • They are square and symmetric. • [M] is positive definite (since T is always positive) • [K] is positive semi-definite: • all positive eigenvalues, except for some potentially 0-eigenvalues which occur during a rigid-body motion. • If restrained/tied down  positive-definite. All positive.

32. UFV: the eigenvalue problem Equation of motion: Matrix eigenvalue problem in terms of the generalized D.O.F. qi Substitution of leads to • For more than 2x2, we usually solve using computational techniques. • Total motion for any problem is a linear combination of the natural modes contained in {u} (i.e. the eigenvectors).

33. Do this a row at a time to form [U]. This is a common technique for us to use after we have solved the eigenvalue problem. Normalization of modal matrix [U] • We know that:  Let the 1st entry be 1 • So far, we pick our eigenvectors to look like: • Instead, let us try to pick so that: • Then: and

34. Consider the cases of: • Initial excitation • Harmonic applied force • Arbitrary applied force General solution procedure For all 3 problems: • Form [K]{u} = w2 [M]{u} (nxn system) Solve for all w2 and {u} [U]. • Normalize the eigenvectors w.r.t. mass matrix (optional).

35. Initial conditions General solution for any D.O.F.: 2n constants that we need to determine by 2n conditions Alternative: modal analysis • Displacement vectors: • UFV model equation: • n modal equations:  Need initial conditions on h, not q.

36. Initial conditions - Modal analysis • Using displacement vectors: • As a result, initial conditions: is: • Since the solution of hence we can easily solve for • And then solve

37. Applied harmonic force Equation of motion: Driving force {Q} = {Qo}cos(wt) Substitution of leads to Hence, then

38. Continuous Systems • The axial bar • Displacement field • Energy approach • Equation of motion • Examples • General solution - Free vibration • Initial conditions • Applied force • Motion of the base • Ritz method – Free vibration • Approximate solution • One-term Ritz approximation • Two-term Ritz approximation

39. Displacement field: u(x, y, z) = u(x, t) v(x, y, z) = 0 w(x, y, z) = 0 The axial bar Main objectives: • Use Hamilton’s Principle to derive the equations of motion. • Use HP to construct variational methods of solution. A = cross-sectional area = uniform E = modulus of elasticity (MOE) u = axial displacement r = mass per volume

40. Energy approach • For the axial bar: • Hamilton’s principle:

41. Axial bar - Equation of motion • Hamilton’s principle leads to: • If area A = constant  Since x and t are independent, must have both sides equal to a constant. • Separation of variables: Hence

42. Fixed-free bar – General solution Free vibration: = wave speed EBC: NBC: General solution: EBC  NBC  For any time dependent problem:

43. Fixed-free bar – Free vibration For free vibration: General solution: Hence are the frequencies (eigenvalues) are the eigenfunctions

44. Fixed-free bar – Initial conditions Give entire bar an initial stretch. Release and compute u(x, t). Initial conditions: Initial velocity: Initial displacement: or Hence

45. Fixed-free bar – Applied force Now, B.C’s: From we assume: Substituting: B.C. at x = 0: B.C. at x = L: or Hence

46. Fixed-free bar – Motion of the base From Using our approach from before: B.C. at x = 0: B.C. at x = L: Hence Resonance at:

47. Ritz method – Free vibration • Start with Hamilton’s principle after I.B.P. in time: • Seek an approximate solution to u(x, t): • In time: harmonic function cos(wt) (w = wn) • In space: X(x) = a1f1(x) where: a1 = constant to be determined f1(x) = known function of position f1(x) must satisfy the following: • Satisfy the homogeneous form of the EBC. u(0) = 0 in this case. • Be sufficiently differentiable as required by HP.

48. One-term Ritz approximation 1 Substituting: Hence • Ritz estimate is higher than the exact • Only get one frequency • If we pick a different basis/trial/approximation function f1, we would get a different result.

49. One-term Ritz approximation 2 Substituting: Hence • Both mode shape and natural frequency are exact. • But all other functions we pick will never give us a frequency lower than the exact.

50. Two-term Ritz approximation In matrix form: where: