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)

km m

k k

x1 x2

km m

k 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

mk,L0y

22

2 nd y y Cdt

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 springsk1

k2

Parallel: stiffness 1 2k k k

k1 k2

Series: stiffness

k1

k2

mk1 +k2

m

1 2

1 1 1k 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!

ng

Linearizing EOM

2

2 ( )d y f y Cdt

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 Cdt dy

d y df C fydt 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 02n n n

d x c dx km xm dt mdt

d x dx k cxdt 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 1k k k

c2c1

Parallel: coefficient1 2

1 1 1c 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 Cdtdt

d x dx Ax Bdtdt

F a

Solution to EOM for damped vibrations

s=L0+xk, L0

mc

22

22 0

2n n nd x dx k cx

dt m kmdt

Initial conditions: 0 0 0dxx 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

Calculating natural frequency and damping factor from a measured vibration response

Displacement

time

t0 t1 t2 t3

T

x(t0) x(t1) x(t2) x(t3)

t4

0( )1 log( )n

x tn x t

Measure log decrement:

2 2

2 2

4

4n T

Measure period: T

Then