9.2 DAMPED SINGLE-DEGREE-OF-FREEDOM (SDOF) SYSTEM

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All mechanical systems have some amount of damping or energy dissipation

associated with the motion of the system, so we need to examine the effects

of damping on the system vibration. It should be noted that a large amount

of damping is not always a good thing to have, especially if one wishes to

reduce the force transmitted at frequencies much above the undamped natural

frequency for the system.

Let us consider the system shown in Fig. 9-2, in which a mass M is

connected to a support through a spring (spring constant KSЮ and a viscous

damper. The force produced by the viscous damper is proportional to the

velocity difference across the damper:

FdрtЮ ј RMvрtЮ ј RM

dy

dt

(9-18)

The quantity RM is the coefficient of viscous damping or the mechanical

resistance, which has units of N-s/m. This combination of units is sometimes

called a mechanical ohm, in analogy with the electrical system, i.e., 1 mech

ohm ј 1 N-s/m.

410 Chapter 9

FIGURE 9-2 Damped SDOF vibrating system.

Copyright © 2003 Marcel Dekker, Inc.

We obtain the following result by applying Newton’s second law of

motion to the mass shown in Fig. 9-2:

M

d2y

dt2 ю RM

dy

dt ю KSy ј 0 (9-19)

If we divide through by the mass M and introduce the undamped natural

frequency from Eq. (9-2), we obtain the following result:

d2y

dt2 ю

RM

M

dy

dt ю !2

ny ј 0 (9-20)

The general solution for Eq. (9-20) is as follows:

yрtЮ ј Aes1t ю B es2t (9-21)

The quantities s1 and s2 are given by:

s1 ј _рRM=2MЮ ю ЅрRM=2MЮ2 _ !2

n_1=2 (9-22)

s2 ј _рRM=2MЮ _ ЅрRM=2MЮ2 _ !2

n_1=2 (9-23)

There are three difference cases as far as the vibratory motion of the

mass with damping is concerned. These cases depend on the nature of the

second term in Eqs (9-22) and (9-23).

9.2.1 Critically Damped System, (RM/2M)ј xn

The value of the damping coefficient in this case is called the critical damping

coefficient RM;cr:

RM;cr ј 2M!n ј 2MрKS=MЮ1=2 ј 2рKSMЮ1=2 (9-24)

For any condition, the damping ratio _ is defined by the following ratio:

_ ј

RM

RM;cr ј

RM

2рKSMЮ1=2 (9-25)

The factors from Eqs (9-22) and (9-23) may be expressed in terms of the

damping ratio:

s1 ј _Ѕ_ _ р_2 _ 1Ю1=2_!n (9-26)

s2 ј _Ѕ_ ю р_2 _ 1Ю1=2_!n (9-27)

For the case of critical damping, _ ј 1, we obtain repeated solutions

for the differential equation, Eq. (9-20), or s1 ј s2 ј _!n. For this situation,

the general solution has the following form:

yрtЮ ј рC1 ю C2tЮ e_!nt (9-28)

Vibration Isolation for Noise Control 411

Copyright © 2003 Marcel Dekker, Inc.

In particular, if the initial displacement is yр0Ю ј yo and the initial velocity is

vр0Ю ј vo, we may evaluate the constants of integration:

C1 ј yo and C2 ј vo ю yo!n (9-29)

The motion of the mass is described by the following relationship:

yрtЮ ј Ѕyo ю рvo ю yo!nЮt_ e_!nt (9-30)

Equation (9-30) shows that, for the case of a free critically-damped system,

there is no oscillatory motion. The system simply moves somewhat slowly

back to its static equilibrium position.

9.2.2 Over-Damped System, f > 1

In this case, the second term in Eqs (9-22) and (9-23) is real and negative,

and not imaginary, so there is no free oscillation. The general solution for

the motion of the over-damped system may be written in terms of the

damping ratio as follows:

yрtЮ ј Aexpf_Ѕ_ _ р_2 _ 1Ю1=2_!ntg ю Bexpf_Ѕ_ ю р_2 _ 1Ю1=2_!ntg

(9-31)

If the initial displacement is yр0Ю ј yo and the initial velocity is

vр0Ю ј vo, we may find the following expressions for the constants of integration

for the over-damped system, _ > 1:

A ј _

vo ю yoЅ_ _ р_2 _ 1Ю1=2_!n

Ѕ2р_2 _ 1Ю1=2_!n

(9-32)

B ј

vo ю yoЅ_ ю р_2 _ 1Ю1=2_!n

Ѕ2р_2 _ 1Ю1=2_!n

(9-33)

The motion of the over-damped system is not oscillatory, and the system

moves toward the equilibrium position more slowly than is the case for the

critically damped system.

9.2.3 Under-Damped System, f < 1

For this case, the second term in Eqs (9-22) and (9-23) is imaginary. The

factors may be written in the following form for the under-damped system:

s1 ј _Ѕ_ _ jр1 _ _2Ю1=2_!n (9-34)

s2 ј _Ѕ_ ю jр1 _ _2Ю1=2_!n (9-35)

Let us define the damped natural frequency !d by the following expression:

!d ј !nр1 _ _2Ю1=2 (9-36)

412 Chapter 9

Copyright © 2003 Marcel Dekker, Inc.

If we make this substitution into Eqs (9-34) and (9-35), we obtain the

following result:

s1 ј __!n _j!d (9-37)

s2 ј __!n юj!d (9-38)

The general solution for the motion of the mass for the under-damped

case may be written in the following form:

yрtЮ ј e__!ntрAcos !dtюBsin!dtЮ (9-39)

The constants of integration may be written in terms of the initial displacement

and initial velocity as follows:

A ј yo and B ј рvo ю_!nyoЮ=!d (9-40)

The motion of the under-damped system may also be written in the

following form:

yрtЮ ј Ce__!nt cosр!dtю       Ю (9-41)

The constant C and the phase angle      may be expressed in terms of the

initial displacement yo and initial velocity vo:

C ј yoр1юtan2            Ю1=2 (9-42)

tan       ј _

vo ю_!nyo

!dyo

(9-43)

We note from Eq. (9-41) that the amplitude of the vibratory motion for the

under-damped system is not constant but decays exponentially with time.

The motion for the three cases is illustrated in Fig. 9-3.

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