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9.9 DYNAMIC VIBRATION ISOLATOR

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There are some vibration isolation situations in which the machine may

operate at or near the resonant frequency of the system. This may occur

if the support system is a flexible or resilient floor. Other cases may arise

such that conventional vibration isolation techniques, such as those discussed

in previous sections, are not practical when the system operates

near the resonant frequency. In these problems, one solution may be to

add an additional mass connected through a spring and damper such that

the additional mass opposes the motion and practically cancels out the

motion of the main mass. The additional mass, spring, and damper system

is called a dynamic absorber. A typical system is shown in Fig. 9-11.

Let us denote the mass, spring constant, and damping coefficient of

the main mass by M, KS, and RM, respectively. The corresponding proper-

Vibration Isolation for Noise Control 439

ties of the dynamic absorber will be denoted by Ma, KSa, and RMa, respectively.

If we apply Newton’s second law of motion to each mass, we obtain

the following equations:

d2y

dt2 ю RM

dt ю RMa

dt _

dya

_ _

ю KSy ю KSaрy _ yaЮ ј FрtЮ

(9-132)

d2ya

dt2 ю RMa

dya

dt _

_ _

ю KSaр ya _ yЮ ј 0 (9-133)

The variable yрtЮ is the displacement of the main mass and yaрtЮ is the

displacement of the additional mass.

We may introduce the following variables into Eqs (9-132) and (9-

133):

n ј KS=M and !2a

ј KSa=Ma (9-134)

2_!n ј RM=M and 2_a!a ј RMa=Ma (9-135)

440 Chapter 9

FIGURE 9-11 Dynamic vibration absorber system.

The following result is obtained:

d2y

dt2 ю 2_!n

dt ю 2_a!a

dt _

dya

_ _

рMa=MЮ

ю !2

ny ю !2a

рMa=MЮр y _ yaЮ ј

nFрtЮ

(9-136)

d2ya

dt2 ю 2_a!a

dya

dt _

_ _

ю !2a

р ya _ yЮ ј 0 (9-137)

The steady-state solution for Eqs (9-136) and (9-137) is quite lengthy

(Reynolds, 1981). Dynamic absorbers are often designed such that the following

relationships are valid:

!n ј !a and _ ј _a (9-138)

We may define the frequency ratio r ј !=!n, where ! is the frequency of the

applied force and the magnification factor for the main mass

MF ј ymax=рFo=KSЮ. If we have a sinusoidal force applied to the main

mass and the conditions of Eq. (9-138) are valid, we may solve for the

magnification factor for the main mass:

MF ј

ymax

Fo=KS ј Ѕр1 _ r2Ю2 ю р2_rЮ2_1=2

рA2 ю 4B2Ю1=2 (9-139)

The quantities A and B are defined as follows:

A ј r4 _ р2 ю ю 4_2Юr2 ю 1 (9-140)

B ј 2_r _ _р2 ю Юr3 (9-141)

The quantity is the ratio of the absorber mass to the main mass,

ј Ma=M.

A similar expression may be obtained for the maximum amplitude of

motion for the absorber mass, ya;max:

MFa ј

ya;max

Fo=KSa ј Ѕ1 ю р2_rЮ2_1=2

рA2 ю 4B2Ю1=2 (9-142)

Note that Eq. (9-142) is valid only under the conditions given by Eq. (9-

138).

The transmissivity for the dynamic absorber with _ ј _a and !n ј !a

may be determined from the displacement expression and the fact that the

force transmitted is given by FTрtЮ ј KSy1рtЮ ю RMрdy1=dtЮ.

Tr ј

Fo ј fЅ1 ю р2_rЮ2_Ѕр1 _ r2Ю2 ю р2_rЮ2_g1=2

рA2 ю 4B2Ю1=2 (9-143)

Vibration Isolation for Noise Control 441

The addition of the dynamic absorber to an SDOF system results in a

two-degree-of-freedom system, so there are two resonant frequencies for the

combination at which the displacements are large, even though the displacement

may be small at the resonant frequency for the SDOF system. For the

special case of an undamped dynamic absorber р_ ј _a ј 0Ю and for _ ј !a=!n not necessarily equal to unity, the magnification factor for the main

mass has been determined (Rao, 1986):

MF ј

ymax

рFo=KSЮ ј

_2 _ r2

р1 ю _2 _ r2Юр_2 _ r2Ю _ _4

(9-144)

The magnification factor for the absorber mass has also been determined for

this case:

MFa ј

ya;max

рFo=KSaЮ ј р1=Ю

р1 ю _2 _ r2Юр_2 _ r2Ю _ _4

(9-145)

We note that the expressions for magnification factor, Eqs (9-144) and

(9-145), become infinite when the denominator achieves a value of zero:

р1 ю _2 _ r2Юр_2 _ r2Ю _ _4 ј 0

r4 _ Ѕ1 ю _2р1 ю Ю_r2 ю _2 ј 0 (9-146)

The solution for the two frequencies from Eq. (9-146) is as follows:

r21

ј 1

2 Ѕ1 ю _2р1 ю Ю_ _ f1

4 Ѕ1 ю _2р1 ю Ю_2 _ _2g1=2 (9-147a)

r22

ј 1

2 Ѕ1 ю _2р1 ю Ю_ ю f1

4 Ѕ1 ю _2р1 ю Ю_2 _ _2g1=2 (91-47b)

The operating frequency of the system should not be equal to either of the

frequencies f1 ј fnr1 or f2 ј fnr2. These frequencies are functions of the

mass ratio, ј Ma=M, and the natural frequency ratio,

_ ј !a=!n ј fa=fn ј ЅрKSa=KSЮрM=MaЮ_1=2.

Example 9-8. A large electric motor has an effective mass of 300 kg

(661 lbm). The frequency of the driving force causing vibration of the

motor is 100 Hz, and the effective force is 250N (56.2 lbf ). The motor is

attached to a concrete floor having dimensions of 3.00m (9.843 ft) _ 3.00m

_150mm (5.91 in) thick. The damping ratio for the support is _ ј 0:06. It is

desired to attach a dynamic absorber to the motor to limit the vibratory

motion of the motor. The undamped natural frequency for the absorber and

motor are equal, and the damping ratios are the same for the absorber

support and the motor support. Determine the required absorber mass

and the maximum amplitude of motion for the motor.

442 Chapter 9

The spring constant for the support of the motor (the concrete floor)

may be found from the following expression, which is valid for a force

applied at the center of a rectangular plate rigidly attached (clamped)

along all four edges (Timoshenko and Woinowsky-Krieger, 1959). The

plate dimensions are a (shorter length), b (longer length), and h (thickenss).

KS ј

CKEh3

р1_2Юa2 (9-148)

The quantities E and are the Young’s modulus and Poisson’s ratio for the

plate material, respectively. The coefficient CK depends on the aspect ratio

рb=aЮ for the plate. Numerical values for CK are listed in Table 9-4. The

aspect ratio for the floor slab in this problem is b=a ј 1. From Appendix C,

Vibration Isolation for Noise Control 443

TABLE 9-4 Coefficients CK in the

Spring Constant Expression for a

Rectangular Plate Having

Dimensions a (Shorter Length), b

(Longer Length), and h

(thickness)a

Aspect ratio, b=a Coefficient, CK

1.0 14.88

1.1 13.66

1.2 12.88

1.3 12.39

1.4 12.06

1.5 11.85

1.6 11.70

1.8 11.57

2.0 11.54

3.0 11.53

4.0 11.52

1 11.49

aThe plate has the driving force

applied at the center of the plate, and

all four edges of the plate are fixed or

clamped. E is Young’s modulus, and

is Poisson’s ratio for the plate material.

KS ј

CKEh3

р1 _ 2Юa2

we find Young’s modulus E ј 20:7GPaр3:00 _ 106 psi) and Poisson’s ratio

ј 0:13 for concrete. Therefore:

KS ј р14:88Юр20:7Юр109Юр0:150Ю3

р1 _ 0:132Юр3:00Ю2

KS ј 117:49 _ 106 N=m ј 116:49MN=m р670,900 lbf=inЮ

The undamped natural frequency for the motor system is as follows:

!n ј рKS=MЮ1=2 ј Ѕр117:49Юр106Ю=р300Ю_1=2 ј 625:8 rad=s

fn ј р625:8Ю=р2_Ю ј 99:6Hz ј fa

The frequency ratio is practically unity in this case:

r ј f =fn ј р100Ю=р99:6Ю ј 1:004

The magnification ratio for the absorber is given by Eq. (9-142):

MFa ј

ya;max

рFo=KSaЮ ј

ya;maxMa!2a

Fo ј

ya;maxM!2

MFa ј р0:003ЮрЮр300Юр625:8Ю2

р250Ю ј 1409:9

From Eqs (9-140) and (9-141), we find the following values:

A ј р1:004Ю4 _ Ѕ1 ю ю р4Юр0:06Ю2_р1:004Ю2 ю 1 ј _2:01445 _ 1:00802

B ј р2Юр0:06Юр1:004Ю _ р0:06Юр2 ю Юр1:004Ю3 ј _0:0009658 _ 0:06072

The magnification ratio for the absorber may also be written as follows:

MFa ј f1 ю Ѕр2Юр0:06Юр1:004Ю_2g1=2

рA2 ю 4B2Ю1=2 ј

1:00723

рA2 ю 4B2Ю1=2 ј 1409:9

The expression for the magnification factor of the absorber is a function

of the mass ratio . By iteration, we find the mass ratio as follows:

ј Ma=M ј 0:0203

The corresponding mass of the absorber may be found:

Ma ј р0:0203Юр300Ю ј 6:09 kg р13:43 lbmЮ

The magnification factor for the absorber is as follows:

MFa ј р1409:9Юр0:0203Ю ј 28:62 ј

ya;max

рFo=KSaЮ

444 Chapter 9

The required spring constant for the absorber is as follows:

KSa ј р28:62Юр250Ю

р0:0030Ю ј 2:385 _ 106 N=m ј 2:385MN=m р13,620 lbf=inЮ

The magnification factor for the motor may be determined from Eq.

(9-139):

MF ј fр1 _ 1:0042Ю2 ю Ѕр2Юр0:06Юр1:004Ю_2g1=2

Ѕр_0:03491Ю2 ю р4Юр_0:002198Ю2_1=2 ј 3:431

The maximum amplitude of motion of the motor may be found from the

definition of the magnification factor:

ymax ј

FoMF

KS ј р250Юр3:431Ю

р117:49Юр106Ю ј 7:30 _ 10_6 m ј 0:0073mm

The deflection of the floor under the weight рMgЮ for the motor may be

found as follows:

dst ј

KS ј р300Юр9:806Ю

р117:49Юр106Ю ј 25 _ 10_6 m ј 0:025mm

The amplitude of the vibration is (0:0073=0:025Ю ј 0:292 or 29% of the

static deflection.

The undamped resonant frequencies for the motor–absorber system

may be found from Eq. (9-147) with _ ј 1 and ј 0:0203:

r21

ј 1

2 р2 ю Ю _ Ѕ1

4 р2 ю Ю2 _ 1_1=2 ј 0:8673 and r1 ј 0:9313

r22

ј 1

2 р2 ю Ю ю Ѕ1

4 р2 ю Ю2 _ 1_1=2 ј 1:1530 and r2 ј 1:0738

f1 ј fnr1 ј р99:6Юр0:9313Ю ј 92:8Hz

f2 ј fnr2 ј р99:6Юр1:0738Ю ј 106:9Hz

The second harmonic of the forcing frequency р2Юр100Ю ј 200 Hz is far

removed from these two frequencies.

Let us determine the vibration amplitude for the case in which no

dynamic absorber is used. The magnification factor for the basic vibrating

system is given by Eq. (9-81):

MF ј

fЅ1 _ 1:0042_2 ю Ѕр2Юр0:060Юр1:004Ю_2g1=2 ј 8:282

Vibration Isolation for Noise Control 445

The maximum amplitude of vibration of the motor without the dynamic

absorber attached is as follows:

ymax ј

MFFo

KS ј р8:282Юр250Ю

р117:49Юр106Ю ј 17:6 _ 10_6 m

ј 0:0176mm р0:00069 inЮ

The use of the dynamic absorber reduces the amplitude of vibration of the

motor in this example by a factor of р0:0073=0:0176Ю ј 0:414 ј 1=2:41.

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