David, N.V. 1

Damped system: Free vibration

Equation of motion (well you should be able to show this):

0 kxxcxm

The solution for the 2nd

order ODE above is

The natural frequency n is the same as the undamped system,

m

kn .

The damped natural frequency is

21d n .

The damping ratio, 2cr n

c c

c m

where c is the damping coefficient of the dashpot, and the critical damping coefficient,

ccr = nm2 .

Under-damped: < 1 (see Figure 2 on p.4)

Critically-damped: = 1

Over-damped: > 1

Logarithmic decrement, which is characterizes the

continuous reduction of amplitudes in an under-

damped system, is

1

22

2ln

1n d

x

x

x

c k

m

David, N.V. 2

Damped system: Forced vibration with Direct Force

The forcing function F(t) is harmonic and is applied at

an excitation frequency of .

The mass oscillates with a frequency identical to the

forcing functions frequency.

There is a phase shift between the mass and the forcing function (see Figure 5 on p.8 and on p.12).

The amplitude of the mass X peaks near the forcing

frequency.

The smaller the damping ratio , the larger the amplitude.

Characteristic equation (see p.7):

Solution (see p.7):

where the amplitude of vibration X is

0

2 2 2

/

(1 ) (2 )

F kX

r r

(see Figure 6 on p.13 for response curves)

where n

r

, and what happens when r = 1?

2

2tan .

1

r

r

what happens when = 0, no damping?

F(t) = F0sin(t)

x

c k

m

David, N.V. 3

Damped system: Forced vibration with Imbalance Mass

Imbalance rotating mass creates inertially-induced forcing function of sinusoidal form:

0 sin( )F F t

where F0 = Me

2, is the centrifugal force.

The imbalance mass will be situated at an eccentric position e.

Characteristic equation (see p.21):

Solution (see p. 22):

2

2 2 2

/

(1 ) (2 )

Me kX

r r

(see Figure 11 on p.22 for response curves)

where n

r

, and what happens when r = 1?

What will be the maximum force exerted onto the machine m due to imbalance?

How about the force transmitted to the ground due to applied force or imbalance mass?

x

c k

m

M e

F(t) = F0sin(t)

ground

David, N.V. 4

Damped system: Forced transmissibility (Direct and Imbalance)

The force applied or induced by rotating

imbalance will be transmitted to the

foundation (ground) where the machine is

installed.

The force transmitted (Ft) to the foundation is

the sum of the spring force and the damping

force.

Force transmissibility, TR is defined as

0

tFTRF

.

In any of the two cases of loading (direct or induced), TR is given as

2

2 2 2

1 (2 )

(1 ) (2 )

rTR

r r

. .(see Figure 13 on p.28 for response curves)

ground

F(t) = F0sin(t)

x

c k

m

M e

Ft

David, N.V. 5

Damped system: Forced vibration via Base Excitation

This is a case where force is NOT directly applied

to the mass.

Excitation is applied to the mass at the base via the

latters motion s defined as

( ) sins t S t .

where S is the amplitude of base motion.

A new variable that relates x and s, called relative

displacement is introduced as z where

z = x s.

Equation of motion (see p.33)

Solution (see p. 34):

2

2 2 2(1 ) (2 )

Srz

r r

what happens when r = 1?

and

2

2tan .

1

r

r

(same as for the case of direct force)

The mass motion amplitude can be determined from 2

2 2 2

1 (2 ).

(1 ) (2 )

rX

S r r

Force transmitted to the base in base vibration is

22

2 2 2

1 (2 ).

(1 ) (2 )

t rFTR rkS r r

s

base

x

c k

m