Prof Bower`s free vibration summary slides
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Transcript Prof Bower`s free vibration summary slides
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
2n
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
2n
By C
dx
n2 x 0
dt
n B
A
2n
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
2n
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 texp(nt )
v (n d ) x0
v (n d ) x0
x(t ) exp(nt ) 0
exp(d t ) 0
exp(d t )
2d
2d
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