free vibrations – concept checklist you should be able to: 1.understand simple harmonic motion...
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Free Vibrations – concept checklistYou should be able to:1.Understand simple harmonic motion (amplitude, period, frequency, phase)2.Identify # DOF (and hence # vibration modes) for a system3.Understand (qualitatively) meaning of ‘natural frequency’ and ‘Vibration mode’ of a system4.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 system7.Know formulas for nat freq, damped nat freq and ‘damping factor’ for spring-mass system in terms of k,m,c8.Understand underdamped, critically damped, and overdamped motion of a dissipative 1DOF vibrating system9.Be able to determine damping factor from a measured free vibration response10.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)
k
m mk k
x1 x2
k
m mk k
x1 x2
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
Calculating nat freqs for 1DOF systems – the basics
EOM for small vibration of any 1DOF undamped system has form
m
k,L0y
22
2 n
d yy C
dt
1. Get EOM (F=ma or energy)2. Linearize (sometimes)3. Arrange in standard form4. Read off nat freq.
n is the natural frequency
Useful shortcut for combining springs
k1
k2
Parallel: stiffness 1 2k k k
k1 k2
Series: stiffness
k1
k2
mk1 +k2
m
1 2
1 1 1
k k k
k1
mAre these in series on parallel?
A useful relation
k,L0
m
L0+
Suppose that static deflection (caused by earths gravity) of a system can be measured.
Then natural frequency is
Prove this!
n
g
Linearizing EOM
2
2( )
d yf y C
dt Sometimes EOM has form
We cant solve this in general… Instead, assume y is small
2
20
2
20
(0) ...
1 (0)
y
y
d y dfm f y C
dt dy
d y df C fy
dt m dy m
There are short-cuts to doing the Taylor expansion
Writing down EOM for spring-mass systems
s=L0+xk, L0
mc
2
2
22
2
0
2 02
n n n
d x c dx km x
m dt mdt
d x dx k cx
dt m kmdt
F a
k1
k2
Commit this to memory! (or be able to derive it…)
x(t) is the ‘dynamic variable’ (deflection from static equilibrium)
Parallel: stiffness 1 2k k k
c2
c1
Parallel: coefficient 1 2c c c
k1 k2
Series: stiffness 1 2
1 1 1
k k k
c2c1
Parallel: coefficient1 2
1 1 1
c c c
k1
k2
m
Examples – write down EOM for
c2
c1 m
2kk1
mc
If in doubt – do F=ma, andarrange in ‘standard form’
k
2
2
22
22 0
2n n nn
d y dym A By C
dtdt
d x dx Ax B
dtdt
F a
Solution to EOM for damped vibrations
s=L0+xk, L0
mc
22
22 0
2n n n
d x dx k cx
dt m kmdt
Initial conditions: 0 0 0dx
x x v tdt
0 00( ) exp( ) cos sinn
n d dd
v xx t t x t t
1 Underdamped:
Critically damped: 1 0 0 0( ) exp( )n nx t x v x t t
0 0 0 0( ) ( )( ) exp( ) exp( ) exp( )
2 2n d n d
n d dd d
v x v xx t t t t
Overdamped: 1
Critically damped gives fastest return to equilibrium