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32.8 Passive Control of Vibration

Back

The techniques discussed in this chapter for reduction of the effects of mechanical vibration fall into the

categories of vibration isolation and design for vibration. The third category, vibration control, is

addressed now. Characteristic of vibration control is the use of a sensing device to detect the level of

vibration in a system, and an actuation (forcing) device to apply a forcing function to the system to

counteract the effects of vibration. In some such devices, the sensing and forcing functions are implicit

and integrated together.

Vibration control may be subdivided into the following two broad categories:

1. Passive control

2. Active control

Passive control of vibration employs passive controllers. By definition, passive devices do not require

external power for their operation. The two passive controllers of vibration discussed in the present

section are vibration absorbers (or dynamic absorbers or Frahm absorbers, named after H. Frahm, who

first employed the technique for controlling ship oscillations) and dampers. In both types of devices,

sensing is implicit and control is achieved through a force generated by the device from its response to the

vibration excitation. A dynamic absorber is a mass – spring-type mechanism with little or no damping,

Vibration Design and Control 32-45

which can “absorb” the vibration excitation through energy transfer into it, thereby reducing the

vibrations of the primary system. The energy received by the absorber will be slowly dissipated due to its

own damping. A damper is a purely dissipative device which, unlike a dynamic absorber, directly

dissipates the energy received from the system rather than storing it. Hence, it is a more wasteful device,

which also may exhibit problems related to wear and thermal effects. However, it has advantages over an

absorber, having, for example, a wider frequency of operation.

32.8.1 Undamped Vibration Absorber

A dynamic vibration absorber (or a dynamic absorber, vibration absorber, or Frahm absorber) is a simple

mass – spring oscillator with very low damping. An absorber that is tuned to a frequency of vibration of a

mechanical system and is able to receive a significant portion of the vibration energy from the primary

system at that frequency. In effect, the resulting vibration of the absorber applies an oscillatory force

opposing the vibration excitation of the primary system and thereby virtually cancels the effect. In theory,

the vibration of the system can be completely removed while the absorber itself undergoes vibratory

motion. Since damping is quite low in practical vibration absorbers, we will first consider the case of an

undamped absorber.

A vibration absorber may be used for vibration control in two common types of situations, as shown in

Figure 32.23. Here, the primary system whose vibration needs to be controlled is modeled as an

Box 32.6

TEST-BASED DESIGN APPROACHES FOR

VIBRATION

1. Component modification:

Modify a component (mass, spring, damper) and determine modal parameters (natural

frequencies, damping ratios, mode shapes).

* Can determine sensitivity to component changes.

* Can check whether a particular change is satisfactory.

2. Modal response specification:

Specify a desired modal response (natural frequencies, damping ratios, mode shapes) and

determine the “best” component changes (mass, spring, damper) that will realize the modal

specs.

* Can be accomplished by first performing a sensitivity study (as in item 1).

3. Substructuring:

(i) Design subsystems to meet specs (analytically, experimentally, or by a mixed approach).

(ii) Establish interconnections between subsystems, and obtain continuity (force balance)

and compatibility (motion consistency) at assembly locations.

(iii) Assemble the overall system by eliminating unknown variables at interconnections.

(iv) Analyze or test the overall system. If satisfactory, stop. Otherwise, make changes to the

subsystems or interconnections, and repeat the above steps.

32-46 Vibration and Shock Handbook

undamped, single-DoF mass – spring system (denoted by the subscript p). An undamped vibration

absorber is also a single-DoF mass – spring system (denoted by the subscript a). In the application shown

in Figure 32.23(a), the objective of the absorber is to reduce the vibratory response yp of the primary

system as a result of a vibration excitation f ðtÞ: The force fs that is transmitted to the support structure

due to the vibratory response of the system is given by

fs ¼ kpyp ð32:98Þ

Therefore, the objective of reducing yp may also be interpreted as one of reducing this transmitted

vibratory force (a goal of vibration isolation). In the second type of application, represented in

Figure 32.23(b), the primary system is excited by a vibratory support motion and the objective of the

absorber is to reduce the resulting vibratory motions yp of the primary system. Note that, in both classes

of application, the purpose is to reduce the vibratory responses. Hence, static loads (e.g., gravity) are not

considered in the analysis.

Table 32.1 shows the development of the equations of motion for the two systems shown in

Figure 32.23. Because we are interested mainly in the control of oscillatory responses to oscillatory

excitations, the frequency-domain model is particularly useful. Note from Table 32.1 that the transfer

function fs=f of System a is simply k times the transfer function yp=f ; and is in fact identical to the transfer

function yp=u of System b. The two problems are essentially identical and, thus, we need only address only

one of them.

Before investigating the common transfer function for the two types of problems, let us look closely at

the frequency-domain equations for the system shown in Figure 32.23(a). We have

ðkp þ ka 2 v2mpÞyp 2 kaya ¼ f ð32:99Þ

ðka 2 v2maÞya ¼ kayp ð32:100Þ

along with Equation 32.98. Here, mp and kp are the mass and the stiffness of the primary system, ma and

ka are the mass and the stiffness of the absorber, f is the excitation amplitude, v is the excitation

frequency, yp is the primary mass response, and ya is the absorber response. Now note from Equation

32.100 that if v ¼

ffiffiffiffiffiffiffi

ka=ma

then yp ¼ 0: This means that if the absorber is tuned so that its natural

frequency is equal to the excitation frequency (drive frequency), the primary system will not (ideally)

undergo any vibratory motion, and is perfectly controlled. The reason for this should be clear from

Equation 32.99 which, when yp ¼ 0 is substituted, gives kaya ¼ 2f : In other words, a tuned absorber

applies to the primary system a spring force that is exactly equal and opposite to the excitation force,

thereby neutralizing the effect. The absorber mass moves, albeit 1808 out of phase with the excitation.

Primary

System

Force

Transmitted

to Support

Vibration

Absorber

f(t)

(a)

Primary

System

Vibration

Absorber

Support

Motion

u(t)

(b)

FIGURE 32.23 Two types of applications of a vibration absorber: (a) reduction of the response to forcing excitation

(or reducing the force transmitted to support structure); (b) reduction of the response to support motion.

Vibration Design and Control 32-47

The frequency of these motions will be v (the same as that of the excitation) and the amplitude is

proportional to that of the excitation ( f ) and inversely proportional to the stiffness of the absorber

spring. It follows that a vibration absorber “absorbs” vibration energy from the primary system.

Furthermore, note from Equation 32.98 that with a tuned absorber the vibration force transmitted to the

support structure is (ideally) zero as well. All this information is observed without any mathematical

manipulation of the equations of motion.

Note that we are dealing with vibratory excitations and responses. Therefore, static loading (such as

gravity and spring preloads) is not considered (we investigate responses with respect to the static

equilibrium configuration of the system). In summary, we are now able to state the characteristics of a

vibration absorber (undamped) as follows:

1. It is effective only for a single excitation frequency (i.e., a sinusoidal excitation).

2. For the best effect, it should be “tuned” such that its natural frequency

ffiffiffiffiffiffiffi

ka=ma

is equal to the

excitation frequency.

3. In the case of forcing vibration excitation, a tuned absorber can (ideally) make the vibratory

response of the primary system and the vibratory force transmitted to the support structure zero.

4. In the case of a vibratory support motion, a tuned absorber can make the resulting response of the

primary system zero.

5. It functions by acquiring vibration energy from the primary system and storing it (as kinetic

energy of the mass or potential energy of the spring) rather than by directly dissipating the energy.

6. It functions by applying a vibration force to the primary system that is equal and opposite to the

excitation force, thereby neutralizing the excitation.

7. The amplitude of motion of the vibration absorber is proportional to the excitation amplitude

and is inversely proportional to the absorber stiffness. The frequency of the absorber motion is the

same as the excitation frequency.

Now, consider the transfer function ð fs =f or yp=uÞ of an undamped vibration absorber, as given in

Table 32.1. We have

GðvÞ ¼

kpðka 2 v2maÞ

mpmav4 2 ½kaðmp þ maÞ þ kpma􀀉v2 þ kpka ð32:101Þ

TABLE 32.1 Equations for the Two Types of Absorber Applications

Absorber Application for the Reduction of Response to a:

Forcing Excitation Support Motion

Time-domain mpy€p ¼ 2kpyp 2 kaðyp 2 yaÞ þ f ðtÞ mpy€p ¼ kpðuðtÞ 2 ypÞ 2 kaðyp 2 yaÞ equations

may€a ¼ kaðyp 2 yaÞ may€a ¼ kaðyp 2 yaÞ

Frequency-domain ð2v2mp þ kp þ ka Þyp ¼ ka ya þ f ð2v2mp þ kp þ ka Þyp ¼ ka ya þ kp u

equations ð2v2ma þ ka Þya ¼ ka yp ð2v2ma þ ka Þya ¼ ka yp

Matrix form

kp þ ka 2 v2mp 2ka

2ka ka 2 v2ma

5 yp

" #

kp þ ka 2 v2mp 2ka

2ka ka 2 v2ma

5 yp

" #

¼ kp

" #

Transfer-function

matrix form

" #

ka 2 v2 ma ka

ka kp ka 2 v2 mp

5 f

" #

ka 2 v2 ma ka

ka kp ka 2 v2 mp

5 u

" #

Vibration-control

transfer function

f ¼

kp yp

f ¼

D ðka 2 v2 ma Þ

u ¼

D ðka 2 v2 maÞ

Characteristic

polynomial

D ¼ ðkp þ ka 2 v2 mp Þðka 2 v2 maÞ 2 k2

¼ mp mav4 2 ½kaðmp þ maÞ þ kp ma􀀉v2 þ kp ka

32-48 Vibration and Shock Handbook

It is convenient to use a nondimensional form in analyzing this frequency-transfer function. To that end,

we define the following nondimensional parameters and frequency variable:

Fractional mass of the absorber m ¼ ma=mp

Nondimensional natural frequency of the absorber a ¼ va=vp

Nondimensional excitation (drive) frequency r ¼ v=vp

where

va ¼

ffiffiffiffiffiffiffi

ka=ma

¼ natural frequency of the absorber

vp ¼

ffiffiffiffiffiffiffiffiffi

kp=mp

¼ natural frequency of the primary system

It is straightforward to divide the numerator and the denominator by kpka and then carry out simple

algebraic manipulations to express the transfer function of Equation 32.101 in the nondimensional form

GðrÞ ¼

a2 2 r2

r4 2 ½a2ð1 þ mÞ þ 1􀀉r2 þ a2 ð32:102Þ

For this undamped system, there is no difference between the resonant frequencies (where the magnitude

of the transfer function peaks) and the natural frequencies (roots of the characteristic equation that

correspond to the “natural” or free time response oscillations). These are obtained by solving the equation

r4 2 ½a2ð1 þ mÞ þ 1􀀉r2 þ a2 ¼ 0 ð32:103Þ

which gives

1 ; r2

2 ¼

2 ½a2ð1 þ mÞ þ 1􀀉 7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

½a2ð1 þ mÞ þ 1􀀉2 2 4a2

ð32:104Þ

These are squared frequencies, both of which are positive as clear from Equation 32.104. The actual,

nondimensional natural frequencies are their square roots. The magnitude of the transfer function

becomes infinite at either of these two natural/resonant frequencies. Furthermore, it is clear from

Equation 32.102 that the transfer function magnitude becomes zero at r ¼ a; where the excitation

frequency (v) is equal to the natural frequency of the absorber (va) as noted above. In the present

undamped case, the transfer function GðrÞ is real but it can be either positive or negative. The

magnitude is thus the absolute value of GðrÞ; which is positive. The magnitude plot given in

Figure 32.24 shows the resonant and control characteristics of a system with an undamped vibration

absorber. Originally, the primary system had a resonance at r ¼ 1 (i.e., v ¼ vp). When the absorber,

which also has a resonance at r ¼ 1; is added, the original resonance becomes an antiresonance with a

zero response. Here, however, two new resonances are created, one at r ¼ 0:854 and the other at

r ¼ 1:171; which are on either side of the tuned frequency ðr ¼ 1Þ of the absorber.

Owing to these two resonances, the effective region of the absorber is limited to a narrow frequency

band centering its tuned frequency. Specifically, the absorber is not effective unless lGl , 1: The effective

frequency band of a vibration absorber may be determined using this condition.

Example 32.8

A high-precision, yet high-power positioning system uses a hydraulic actuator and a valve. The

pressurized oil to this hydraulic servo system is provided by a gear-type rotary pump. The pump and the

positioning system are mounted on the same workbench. The mass of the pump is 25 kg. The normal

operating speed of the pump is 3600 rpm. During operation, it was observed that the pump exhibits a

vertical resonance at this speed and it affects the accuracy of the position servo system. To control the

vibrations of the pump at its operating speed, a vibration absorber of mass 1.25 kg tuned to the normal

operating speed of the pump is attached, as shown schematically in Figure 32.25. Because the speed of the

pump normally fluctuates during operation, we must determine the speed range within which the

vibration absorber is effective. What are the new resonant frequencies of the system? (Neglect damping.)

Vibration Design and Control 32-49

0.5 1.0 1.5

0.0

5.0

10.0

Nondimensional Excitation Frequency r = w /wp

Amplification

of the Primary

System Response

Absorber Mass

Absorber

Frequency Ratio:

a = 1.0

Fraction:

m = 0.1

FIGURE 32.24 The effect of an undamped vibration absorber on the vibration response of a primary system.

Vibration

Absorber Spring

Mount

Hydraulic

Positioning

System

Hydraulic

Oil Pump

FIGURE 32.25 A hydraulic positioning system with a gear pump.

32-50 Vibration and Shock Handbook

Solution

For this problem, the fractional mass m ¼ 1:25=25:0 ¼ 0:05: Since the absorber is tuned to the resonant

frequency of the pump, a ¼ 1:0: From Equation 32.103, the characteristic equation of the modified

system becomes

r4 2 2:05r2 þ 1 ¼ 0

which has roots r1 ¼ 0:854 and r2 ¼ 1:171: It follows that the new resonances are at 0.854 £ 3600 and

1.171 £ 3600 rpm. These are 3074.4 and 4215.6 rpm, which should be avoided. From Equation 32.102,

the system transfer function is

GðrÞ ¼

1 2 r2

ðr4 2 2:05r2 þ 1Þ

The effective frequency band of the absorber corresponds to lGðrÞl , 1:0: Since a sign reversal of GðrÞ

occurs at r ¼ 1; we need to solve both

1 2 r2

ðr4 2 2:05r2 þ 1Þ ¼ 1 and 21

The first equation gives the roots r ¼ 0 and 1.025. The second equation gives the roots r ¼ 0:977

and 1.45.

Hence, the effective frequency band corresponds to Dr ¼ ½0:977; 1:025􀀉: In terms of the operating

speed of the pump, we have an effective band of 3517.2 to 3690 rpm. Thus, a speed fluctuation of about

^ 80 rpm is acceptable.

Finally, recall that the presence of the absorber generates two new resonances on either side of the

resonance of the original system (to which the absorber is normally tuned). It is also clear from Equation

32.104 that these two resonances become farther and farther apart as the fractional mass m of the

vibration absorber is increased.

32.8.2 Damped Vibration Absorber

Damping is not the primary means by which

vibration control is achieved in a vibration

absorber. As noted above, the absorber acquires

vibration energy from the primary system (and in

turn, exerts a force on the system that is equal and

opposite to the vibration excitation), thereby

suppressing the vibratory motion. The energy

received by the absorber has to be dissipated

gradually and, hence, some damping should be

present in the absorber. Furthermore, the two

resonances that are created by adding the absorber

have an infinite magnitude in the absence of

damping. Hence, damping has the added benefit

of lowering these resonant peaks.

The analysis of a vibratory system with a

damped absorber is similar to but somewhat

more complex than, that involving an undamped

absorber. Furthermore, an extra design parameter

— the damping ratio of the absorber — enters into

the scene. Consider the model shown in Figure

32.26. Another version of application of a damped

absorber, which corresponds to Figure 32.23(b), may also be presented. However, because the two

types of application have the same transfer function, it is sufficient to consider Figure 32.26 alone.

Primary

System

Transmitted

Force

Vibration

Absorber

f(t)

FIGURE 32.26 Primary system with a damped

vibration absorber.

Vibration Design and Control 32-51

Again, the transfer function of vibration control may be taken as either ya=f or fs=f ; the latter being simply

kp times the former. Although we will consider the dimensionless case of fs=f ; the results are equally valid

for yp=f ; except that the responses must be converted from force to displacement by dividing by kp.

There is no need to derive the transfer function anew for the damped system. Simply replace ka in

Equation 32.101 by the complex stiffness ka þ jvba; which incorporates the viscous damping constant ba

and the excitation frequency v: Hence, the transfer function of the damped system is

GðvÞ ¼

kpðka þ jvba 2 v2maÞ

mpmav4 2 ½ðka þ jvbaÞðmp þ maÞ þ kpma􀀉v2 þ kpðka þ jvbaÞ ð32:105Þ

With the parameters defined as before, the nondimensional form of this transfer function is obtained by

dividing throughout by kpka and then substituting the appropriate parameters. In particular, we use the

fact that

ka ¼

2ba

ffiffiffiffiffiffi

kama

ffiffiffiffiffi

2za

va ¼

2za

va ¼

2za

avp

ð32:106Þ

where the damping ratio za of the absorber is given by

za ¼

ffiffiffiffiffiffi

kama

p ð32:107Þ

as usual. Then, we follow the same procedure that used to derive Equation 32.102 from Equation 32.101

to get

GðrÞ ¼

a2 2 r2 þ 2jzaar

r4 2 ½ða2 þ 2jzaarÞð1 þ mÞ þ 1􀀉r2 þ ða2 þ 2jzaarÞ ð32:108Þ

Note that this result is equivalent to simply replacing a 2 by a2 þ 2jzaar in Equation 32.102.

It is important to note that the undamped natural frequencies are obtained by solving the

characteristics equation with za ¼ 0: These are the same as before and given by the square roots of

Equation 32.104. The damped natural frequencies are obtained by first setting jr ¼ l (hence, r2 ¼ 2l2

and r4 ¼ l4) and then solving the resulting characteristics equation (see the denominator of Equation

32.108).

l4 þ 2zaað1 þ mÞl3 þ ða2 þ a2m þ 1Þl2 þ 2zaal þ a2 ¼ 0 ð32:109Þ

and then taking the imaginary parts of the roots of l: These depend on za and are different from those

obtained from Equation 32.104. The resonant frequencies correspond to the r values where the

magnitude of GðrÞ will peak. Generally, these are not the same as the undamped or damped natural

frequencies. However, for low damping (small za compared with 1), these three types of system

characteristics frequencies are almost identical.

The magnitude of the transfer function (Equation 32.108) is plotted in Figure 32.27 for the case

m ¼ 1:0 and a ¼ 1:0; as in Figure 32.24, but for damping ratios za ¼ 0:01; 0.1, and 0.5. Note that the

curve for za ¼ 0:01 is very close to that in Figure 32.24 for the undamped case. When za is large, as shown

in the case of za ¼ 0:5; the two masses mp and ma tend to become locked together and appear to behave

like a single mass. Then, the system tends to act like a single-DoF one, and the primary system is modified

only in its mass (which increases). Consequently, only one resonant frequency is produced, which is

smaller than that of the original primary system. Furthermore, as expected in this high-damping case, the

effect of a vibration absorber is no longer present.

All three curves in Figure 32.27 pass through the two common points A and B, as shown. This is true for

all curves corresponding to all values of za; and particularly for the extreme cases of za ¼ 0 and za ! 1:

Hence, these points can be determined as the points of intersection of the transfer function magnitude

curves for the limiting cases za ¼ 0 and za ! 1:

32-52 Vibration and Shock Handbook

Equation 32.102 gives GðrÞ for za ¼ 0: Next, from Equation 32.108, we note that, as za ! 1; all the

terms not containing za can be neglected. Hence,

GðrÞ ¼

2jzaar

2 2jzaarð1 þ mÞr2 þ 2jzaar

Cancel the common term and we get (for r – 0Þ

GðrÞ ¼

1 2 ð1 þ mÞr2 for za ! 1 ð32:110Þ

Note that this is the normalized transfer function of a single-DoF system of natural frequency 1=

ffiffiffiffiffiffiffiffi

p1 þm:

This result confirms the fact that as za ! 1; the two masses mp and ma become locked together and act

as a single mass ðmp þ maÞ supported on a spring of stiffness kp: Its natural frequency is

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kp=ðmp þ maÞ

which, when normalized with respect to

ffiffiffiffiffiffiffiffi

kp=mp

, becomes

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðmp þ maÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðmp þ maÞ

1 ffiffiffiffiffiffiffiffi

p1 þm

In determining the points of intersection between the functions (see Equation 32.102 and Equation

32.110), we should first note that at the first point of intersection (A), the function in Equation 32.102 is

negative and positive in Equation 32.110, while the reverse is true for the second point of intersection (B).

For either point, this means that the sign of one of the functions should be reversed before equating them.

Thus,

a2 2 r2

r4 2 ½a2ð1 þ mÞ þ 1􀀉r2 þ a2 ¼ 2

1 2 ð1 þ mÞr2

which gives

ð2 þ mÞr4 2 2½a2ð1 þ mÞ þ 1􀀉r2 þ 2a2 ¼ 0 ð32:111Þ

Nondimensional Excitation Frequency r = w /wp

0.5 1.0 1.5

0.0

5.0

10.0

Amplification

of the Primary

System Response

A B

za = 0.1

za = 0.5

za = 0.01

FIGURE 32.27 Vibration amplification (transfer function magnitude) curves for damped vibration absorbers

(absorber mass m ¼ 0:1; absorber resonant frequency a ¼ 1:0Þ:

Vibration Design and Control 32-53

This is the equation whose roots (e.g., r1 and r2) give the points A and B. Then, we have the sum of the

squared roots equal to the negative coefficient of r 2 in the quadratic (in r 2) Equation 32.111. Thus,

1 þ r2

2 ¼

2½a2ð1 þ mÞ þ 1􀀉

ð2 þ mÞ ð32:112Þ

In addition, the product of the squared roots is equal to the constant term in the quadratic ðr2Þ in

Equation 32.111. Hence,

1 r2

2 ¼

2a2

ð2 þ mÞ ð32:113Þ

32.8.2.1 Optimal Absorber Design

It has been pointed out (primarily by J.P. Den Hartog) that an optimal absorber design should not only

have equal response magnitudes at the common points of intersection (i.e., equal ordinates of points A

and B in Figure 32.27), but also that the resonances should occur at these points to achieve some balance

and uniformity in the response amplification in the region surrounding the tuned frequency of the

absorber. It is expected that these (intuitive) design conditions would give relations between the

parameters a; m; and za; corresponding to an optimal absorber.

Consider the first requirement of equal transfer function magnitudes at A and B. As noted earlier,

because these two points do not depend on za; we use Equation 32.110 to satisfy the requirement. Thus,

keeping in mind the sign reversal of the transfer function between A and B (i.e., as the transfer function

passes through the resonance), we have

1 2 ð1 þ mÞr2

1 ¼ 2

1 2 ð1 þ mÞr2

which gives

1 þ r2

2 ¼

1 þ m ð32:114Þ

Substituting this result (for equal ordinates) in the intersection-point condition (see Equation 32.112),

we have

1 þ m ¼

2½a2ð1 þ mÞ þ 1􀀉

ð2 þ mÞ

On simplification, we get the simple result

a ¼

1 þ m ð32:115Þ

Next, we turn to the task of achieving peak magnitudes of the transfer function at the points of

intersection (A and B). Generally, when one point peaks the other does not. As reported by Den Hartog,

with straightforward but lengthy analysis, we obtain

a ¼

m½3 2

ffiffiffiffiffiffiffiffiffiffiffiffi

pm=ðm þ 2Þ􀀉

8ð1 þ mÞ3 ð32:116Þ

for peak at the first intersection point, and

a ¼

m½3 þ

ffiffiffiffiffiffiffiffiffiffiffiffi

pm=ðm þ 2Þ􀀉

8ð1 þ mÞ3 ð32:117Þ

for peak at the second intersection point.

32-54 Vibration and Shock Handbook

So, for design purposes, a balance is obtained by taking the average value of the results of Equation 32.116

and Equation 32.117 as

a ¼

8ð1 þ mÞ3 ð32:118Þ

Thus, Equation 32.115 and Equation 32.118 correspond to an optimal vibration absorber. In addition,

practical requirements and limitations need to be addressed in any design procedure. In particular, since

m is considerably less than unity (i.e., absorber mass is a small fraction of the primary mass), the absorber

mass should undergo relatively large amplitudes at the operating frequency in order to receive the energy

of the primary system. The absorber spring must be designed accordingly, while meeting the tuning

frequency conditions that determine the ratio ma=ka:

Example 32.9

The air compressor of a wind tunnel weighs 48 kg and normally operates at 2400 rpm. The first major

resonance of the compressor unit occurs at 2640 rpm, with severe vibration amplitudes that are quite

dangerous. Design a vibration absorber (damped) for installation on the mounting base of the

compressor. What are the vibration amplifications of the compressor unit at the new resonances of the

modified system? Compare these with the vibration amplitude of the original system in normal

operation.

Solution

As usual, we will tune the absorber to the normal operating speed (2400 rpm). Then, we have the

nondimensional resonant frequency of the absorber:

a ¼

vp ¼

2400

2640 ¼

Now, for an optimal absorber, from Equation 32.115

m ¼

2 1 ¼

Hence, the absorber mass

ma ¼ 48 £

kg ¼ 4:0 kg

Then, from Equation 32.118, the damping ratio of the absorber is

za ¼

3=12

8ð1 þ 1=12Þ3

􀀒 􀀓1=2

¼ 0:157

Now,

va ¼

ffiffiffiffiffi

4:0

2400

60 £ 2p rad=sec ¼ 88p rad=sec

Hence,

ka ¼ ð88pÞ2 £ 4:0 N=m ¼ 2:527 £ 105 N=m

Also,

za ¼

ffibffiffiaffiffiffi

maka

Vibration Design and Control 32-55

Then, we have

ba ¼ 2 £ 0:157

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4:0 £ 2:527 £ 105

N sec=m ¼ 315:7 N sec=m

This gives us the damped absorber. Now, let us check its performance. We know that, in theory, the

vibration amplitude at the operating speed should be almost zero now. However, two resonances are

created around the operating point. Since damping is small, we use the undamped characteristic

Equation 32.103 to compute these resonances:

r4 2

122

132 1 þ

􀀏 􀀐

þ 1

" #

r2 þ

122

132 ¼ 0

which gives

r4 2

r2 þ

122

132 ¼ 0

The roots of r2 are 0.692 and 1.231. The (positive) roots of r are 0.832 and 1.109.

These correspond to compressor speeds of (multiply r by 2640 rpm) 2196 and 2929 rpm. Although

they are approximately at 2 10% and þ20% of the operating speed, the first resonance will be

encountered during startup and shutdown conditions. To determine the corresponding vibration

amplifications (force/force), use Equation 32.108 which, when the undamped characteristic equation is

substituted into the denominator, becomes

GðrÞ ¼

a2 2 r2 þ 2jzaar

½22jzaarð1 þ mÞr2 þ 2jzaar􀀉 ¼

1 2 jða2 2 r2Þ=ð2zaarÞ

1 2 ð1 þ mÞr2 ð32:119Þ

Substitute the resonant frequencies r1 ¼ 0:832 and r2 ¼ 1:109: We get lGðr1Þl ¼ 4:223 and

lGðr2Þl ¼ 4:634:

Without the absorber, we approximate the system by a simple undamped oscillator with transfer

function

GpðrÞ ¼

1 2 r2

The corresponding vibration amplification at the operating speed is

lGpðr0Þl ¼

l1 2 122=132l ¼ 6:76

It is observed that after adding the absorber, the resonant vibrations are smaller than even the operating

vibrations of the original system. Hence, the design is satisfactory. Note that we used the force/force

transfer functions. To get the displacement/force transfer functions we divide by kp: However, we have

ffiffiffiffiffi

2640

60 £ 2p rad=sec ¼ 88p rad=sec

Hence,

kp ¼ ð88pÞ2 £ 48 N=m ¼ 3:67 £ 106 N=m ¼ 3:67 £ 103 N=mm

Thus, the amplitude of operating vibrations of the original system is

6:76

3:67 £ 103 mm=N ¼ 1:84 £ 1023 mm=N

The amplitudes of the resonant vibrations of the modified system are

4:223

3:67 £ 103 and

4:634

3:67 £ 103 mm=N or 1:15 £ 1023 and 1:26 £ 1023 mm=N

32-56 Vibration and Shock Handbook

Vibration absorbers are simple and passive

devices, which are commonly used in the control

of narrowband vibrations (limited to a very small

interval of frequencies). Applications are found in

vibration suppression of transmission wires (e.g., a

stockbridge damper, which simply consists of a

piece of cable carrying two masses at its ends),

consumer appliances, automobile engines, and

industrial machinery. It should be noted that the

concepts presented for a rectilinear vibration

absorber may be directly extended to a rotary

vibration absorber. Figure 32.28 provides a schematic

representation of a rotary vibration absorber.

This model corresponds to vibration force

excitations (compare with Figure 32.23(a)). The

case of rotational support-motion excitations (see

Figure 32.23(b)), which has essentially the same transfer function, may also be addressed. Approaches of

vibration control are summarized in Box 32.7.

32.8.3 Vibration Dampers

As discussed above, vibration absorbers are simple and effective passive devices, which are used in

vibration control. They have the added advantage of being primarily nondissipative. The main

disadvantage of a vibration absorber is that it is only effective over a very narrow band of frequencies

enclosing its resonant frequency (tuned frequency). When passive vibration control over a wide band of

frequencies is required, a damper would be a preferable choice.

Vibration dampers are dissipative devices. They control vibration through direct dissipation of the

vibration energy of the primary (vibrating) system. As a result, however, there will be substantial heat

generation, and associated thermal problems and component wear. Consequently, methods of cooling

(e.g., use of a fan, coolant circulation, and thermal conduction blocks) may be required in some special

situations.

Consider a vibrating system modeled as an undamped single-DoF mass – spring system (simple

oscillator). The magnitude of the excitation-response transfer function will have a resonance with a

theoretically infinite magnitude in this case. Operation in the immediate neighborhood of such a

resonance would be destructive. Adding a simple viscous damper, as shown in Figure 32.29(a),

will correct the situation. The equation of motion (about the static equilibrium position) is

my€ þ by_ þ ky ¼ f ðtÞ ð32:120Þ

with the dynamic force that is transmitted through the support base ðfsÞ given by

fs ¼ ky þ by_ ð32:121Þ

Hence, the transfer function between the forcing excitation f and the vibration response y is

f ¼

k 2 v2m þ jvb ð32:122Þ

and that between the forcing excitation and the force transmitted to the support structure is

f ¼

k þ jvb

k 2 v2m þ jvb ð32:123Þ

Using the nondimensional frequency variable r ¼ v=vn where vn ¼

ffiffiffiffiffi

k=m p is the undamped natural

frequency of the system and the damping ratio z ¼ b=ð2

ffiffiffiffi

km p Þ; we can express Equation 32.122 and

k ka p

Primary

System

Vibration

Absorber

b a

t(t)

qp qa

FIGURE 32.28 The application of a rotary vibration

absorber.

Vibration Design and Control 32-57

Equation 32.123 in the form

f ¼

kð1 2 r2 þ 2jzrÞ ð32:124Þ

f ¼

1 þ 2jzr

ð1 2 r2 þ 2jzrÞ ð32:125Þ

When vibration control of the primary system is desired, we use the transfer function in Equation 32.124.

However, when force transmissibility is the primary consideration, we use Equation 32.125.

Box 32.7

VIBRATION CONTROL

Passive control (no external power):

1. Dampers

* A dissipative approach (thermal problems, degradation)

* Useful over a wide frequency band

2. Vibration absorbers (dynamic absorbers, Frahm absorbers)

* Absorbs energy from vibrating system and applies counteracting force

* Useful over a very narrow frequency band (near the tuned frequency)

* Absorber executes large motions

Undamped absorber design:

Transfer function of system with absorber ¼

a2 2 r2

r4 2 ½a2ð1 þ mÞ þ 1􀀉r2 þ a2

where

m ¼ absorber mass/primary system mass

a ¼ absorber natural frequency/primary system natural frequency

r ¼ excitation frequency/primary system natural frequency

The most effective operating frequency rop ¼ a:

Avoid the two resonances.

Optimal damped absorber design:

Mass ratio

m ¼

2 1

Damping ratio

za ¼

8ð1 þ mÞ3

Active control (needs external power):

1. Measure vibration response using sensors/transducers.

2. Apply control forces to vibrating system through actuators, according to a suitable control

algorithm.

32-58 Vibration and Shock Handbook

Furthermore, it is convenient to use the transfer function in Equation 32.124 in the nondimensional form:

f ¼ GðrÞ ¼

ð1 2 r2 þ 2jzrÞ ð32:126Þ

The magnitude of this transfer function is plotted in Figure 32.30 for several values of damping ratio.

Note that the addition of significant levels of damping considerably lowers the resonant peak and flattens

the overall response. This example illustrates the broadband nature of the effect of a damper. However,

unlike a vibration absorber, it is not possible with a simple damper to bring the vibration levels to a

theoretical zero. However, a damper is able to bring the response uniformly close to the static value (unity

in Figure 32.30).

K θ

Shaft

Vibration

Excitation

Torque

Damper

Inertia

(Jd)

System

Inertia

(J)

Damping

Liquid

(B)

(a) fs (b)

Primary

System

Vibration

Forcing

Excitation

f (t)

Vibration

Response

Transmitted

Force

Damper b

t(t)

FIGURE 32.29 (a) A system with a linear viscous damper; (b) a rotary system with a Houdaille damper.

Nondimensional Excitation Frequency r = w /wn

0.1 1.0 10.0

0.0

5.0

25.0

Normalized

Response

Magnitude

G(r)

z = 0.0

0.05

0.025

10.0

15.0

20.0

0.1

0.25

FIGURE 32.30 Frequency response of a system containing a linear damper.

Vibration Design and Control 32-59

Another common application of damper is connecting it through a free inertia element. For a

rotational system, such an arrangement is know as the Houdaille damper, and is modeled as in

Figure 32.30(b). The equations of motion are

Ju€þ Bð u_ 2u_dÞ þ Ku ¼ tðtÞ ð32:127Þ

u€d þ Bð u_d 2 uÞ ¼ 0 ð32:128Þ

In this case, the transfer function between the vibratory excitation torque t and the response angle u is

given by

t ¼

B þ Jdjv

KB 2 BðJ þ JdÞv2 2 JdJjv3 þ KJdjv ð32:129Þ

Again, we use the normalized form of Ku=t: Then, we obtain

t ¼ GðrÞ ¼

2z þ jrm

2z ½1 2 ð1 þ mÞr2􀀉 þ jrmð1 2 r2Þ ð32:130Þ

where r ¼ v=vn; z ¼ B=ð2

ffiffiffiffi

pKJÞ; m ¼ Jd=J; and vn ¼

ffiffiffiffiffi

pK=J:

Note the two extreme cases. When z ¼ 0, the system becomes the original undamped system, as

expected. When z ! 1; the system becomes an undamped simple oscillator, but with a lower natural

frequency of r ¼ 1=

ffiffiffiffiffiffiffiffi

p1 þm; instead of r ¼ 1 that was present in the original system. This is to be expected

because as z ! 1; the two inertia elements become locked together and act as a single combined inertia

J þ Jd: Clearly, in these two extreme systems, the effect of damping is not present. Optimal damping

occurs somewhere in between, as is clear from the curves of response magnitude shown in Figure 32.31

for the case of m ¼ 0:2:

Proper selection of the nature and values of damping is crucial for the effective use of a damper in

vibration control. Damping in physical systems is known to be nonlinear and frequency dependent, as

well as time-variant and dependent on the environment (e.g., temperature). Various models are available

for different types of damping, but these are only approximate representations. In practice, such

considerations as the type of damper used, the nature of the system, the specific application, and the

Nondimensional Excitation Frequency r = w /wn

0.1 1.0 10.0

0.0

10.0

40.0

Normalized

Response

Magnitude

z = 0.0

20.0

30.0

z = 10.0

z = 0.2

z = 0.5

G(r)

FIGURE 32.31 Response curves for a rotary system with Houdaille damper of inertia ratio m ¼ 0:2:

32-60 Vibration and Shock Handbook

speed of operation, determine which particular model (linear viscous, hysteric, Coulomb, Stribeck,

quadratic aerodynamic, etc.) is suitable. In addition to the simple linear theory of viscous damper,

specific properties of physical damping should be taken into consideration in practical designs.

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