1. Fundamentals of Linear Vibrations
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 ). 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
Energy Methods
Strain energy & kinetic energy
Work-energy statement
Conservation of energy and example

4. A spring-mass system Governing equation of motion:
General solution for any simple oscillator:
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. Equivalent springs Springs in series: same force - flexibilities add
Springs in parallel:
same displacement - stiffnesses add

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

15. Work-Energy principles
Work done = Change in kinetic energy
Conservation of energy for conservative systems
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
Zero damping matrix [C] and force vector {P}
Assumed general solutions:
Characteristic equation:
Characteristic polynomial (for det[ ]=0):
Eigenvalues (characteristic values):

20. Undamped free vibrations
Special case when k1=k2=k and m1=m2=m
Eigenvalues and frequencies:
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 two-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

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”
“elastically coupled”
Introduce a new pair of coordinates that represents spring stretch:
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. Transformation of coordinates
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:
The eigenvectors are orthogonal w.r.t [M]:
The modal vectors are orthogonal w.r.t [K]:
Algebraic eigenvalue problem:

26. Transformation of coordinates
General approach for solution
Governing equation:
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:
[M], [C], and [K] are full but symmetric.
{F}
not function of time
Assume:
Substituting gives:
Hence:

29. Special case: Undamped system
Zero damping matrix [C]
Entries of impedance matrix [Z]:
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 General solution procedure
Notes on matrices
Undamped free vibration: the eigenvalue problem
Normalization of modal matrix [U]
General solution procedure
Initial conditions
Applied harmonic force

31. Multi-DOF model equation
Multi-DOF systems are so similar to two-DOF.
Model equation:
We derive using:
Vector mechanics (Newton or D’ Alembert)
Hamilton's principles
Lagrange's equations
Notes on matrices:
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:
in terms of the generalized D.O.F. qi
Substitution of
leads to
Matrix eigenvalue problem
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. 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:
Do this a row at a time to form [U].
Then:
and
This is a common technique
for us to use after we have solved
the eigenvalue problem.

34. General solution procedure
Consider the cases of:
Initial excitation
Harmonic applied force
Arbitrary applied force
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.  Need initial conditions on h,
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:
Since the solution of
is:
hence we can easily solve for
And then solve

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

38. Continuous Systems The axial bar Examples Ritz method – Free vibration
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. 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
Displacement field: u(x, y, z) = u(x, t)
v(x, y, z) = 0
w(x, y, z) = 0

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:

51. Two-term Ritz approximation (cont.)
Substitution of:
leads to
Solving characteristic polynomial (for det[ ]=0) yields 2 frequencies:
Let a1 = 1:
Mode 1:
Mode 2: