Free Vibrations – concept checklist
You should be able to:
1. Understand simple harmonic motion (amplitude, period, frequency,
phase)
2. Identify # DOF (and hence # vibration modes) for a system
3. Understand (qualitatively) meaning of ‘natural frequency’ and
‘Vibration mode’ of a system
4. Calculate natural frequency of a 1DOF system (linear and nonlinear)
5. Write the EOM for simple spring-mass systems by inspection
6. Understand natural frequency, damped natural frequency, and
‘Damping factor’ for a dissipative 1DOF vibrating system
7. Know formulas for nat freq, damped nat freq and ‘damping factor’ for
spring-mass system in terms of k,m,c
8. Understand underdamped, critically damped, and overdamped motion
of a dissipative 1DOF vibrating system
9. Be able to determine damping factor from a measured free vibration
response
10. Be able to predict motion of a freely vibrating 1DOF system given its
initial velocity and position, and apply this to design-type problems
Number of DOF (and vibration modes)
If masses are particles:
Expected # vibration modes = # of masses x # of directions
masses can move independently
If masses are rigid bodies (can rotate, and have inertia)
Expected # vibration modes = # of masses x (# of directions
masses can move + # possible axes of rotation)
x1
x2
k
k
m
k
m
Vibration modes and natural frequencies
• A system usually has the same # natural freqs as degrees of
freedom
•Vibration modes: special initial deflections that cause entire
system to vibrate harmonically
•Natural Frequencies are the corresponding vibration frequencies
x1
x2
k
k
m
k
m
Calculating nat freqs for 1DOF systems – the basics
m
y
k,L0
EOM for small vibration of any 1DOF
undamped system has form
d2 y
2


n y  C
2
dt
n is the natural frequency
1. Get EOM (F=ma or energy)
2. Linearize (sometimes)
3. Arrange in standard form
4. Read off nat freq.
Useful shortcut for combining springs
k1
k1
k2
k2
Parallel: stiffness k  k1  k2
Series: stiffness
k1
m
k1 +k2
m
k2
k1
Are these in series on parallel?
m
1 1
1
 
k k1 k2
A useful relation
Suppose that static deflection 
(caused by earths gravity) of a
system can be measured.
k,L0
L0+ 
Then natural frequency is
n 
m
Prove this!
g

Linearizing EOM
d2y
 f ( y)  C
2
dt
Sometimes EOM has form
We cant solve this in general…
Instead, assume y is small
d2y
df
m 2  f (0) 
dt
dy
d 2 y 1 df

2
dt
m dy
y 0
y  ...  C
y 0
C  f (0)
y
m
There are short-cuts to doing the Taylor expansion
Writing down EOM for spring-mass systems
Commit this to memory! (or be able to derive it…)
s=L0+x
k, L0
F  ma 
m
c
d2x
dt 2
d2x
dt 2
 2n

c dx k
 x 0
m dt m
dx
 n2 x  0
dt
n 
k
m
 
c
2 km
x(t) is the ‘dynamic variable’ (deflection from static equilibrium)
k1
k1
k2
Parallel: stiffness k  k1  k2
c1
c2
Parallel: coefficient c  c1  c2
k2
1 1
1


Series: stiffness k k
k2
1
c1
c2
1 1
1


Parallel: coefficient c c c
1
2
Examples – write down EOM for
k1
k1
k2
m
m
c
k2
k
c1
m
c2
If in doubt – do F=ma, and
arrange in ‘standard form’
F  ma 
d2 y
dt
2
d2x
dt 2
A
dy
dt
 2n
 By  C
dx
 n2 x  0
dt
n  B
 
A
2n
Solution to EOM for damped vibrations
s=L0+x
k, L0
d2x
m
dt 2
c
x  x0
Initial conditions:
Underdamped:
 1
Critically damped:
 1
Overdamped:
1
 2n
dx
 n2 x  0
dt
dx
 v0
dt
n 
k
m

c
2 km
t 0


v0  n x0
x(t )  exp(n t )  x0 cos d t 
sin d t 

d


x(t )  x0  v0  n x0 texp(nt )
 v  (n  d ) x0

v  (n  d ) x0
x(t )  exp(nt )  0
exp(d t )  0
exp(d t ) 
2d
2d


Critically damped gives fastest return to equilibrium
Calculating natural frequency and damping
factor from a measured vibration response
Displacement
x(t0)
x(t1)
x(t2)
x(t3)
time
t0
t1
t2
t3
t4
T
Measure log decrement:
Measure period:
Then
1
n
 x(t0 ) 

x
(
t
)
 n 
  log 
T


4 2   2
4 2   2
n 
T