94 analysis of vibrational dampers

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ANALYSIS OF VIBRATIONAL DAMPERS FOR TRANSMISSION

LINES VIA SCATTERING THEORY

Technical Note Submitted to IEEE Transactions on Power Systems

August 19, 1994

Dr. Jacob S. Glower, Member ASME, IEEE

Department of Electrical Engineering

North Dakota State University, Fargo, North Dakota 58105

Abstract

In this paper, the problem of adding damping to a vibrating string is considered. Instead of

using feedback control or frequency domain techniques, a wave-analysis approach proposed

by von-Flotow for beams and trusses is used. This technique is based upon the premise that

resonances are caused by waves reflecting off of discontinuities. By eliminating the

reflections, the resonances will dissappear. The purpose of the controller is therefore to

minimize reflections.

This paradigm is used to analyze two different controllers commonly used to dampen

vicrations in high-voltage transmission lines: a mass damper tied to an absolute reference

similar to "winglets" and a mass damper similar to a Stockbridge damper. Using scattering

theory, it is shown that both designs can be effective at minimizing reflections of traveling

waves ad thereby minimize oscillations. Further, this analyzsis approach gives good insight as

to why each controller works and what the effect of varying different parameters will be -

features which are often lacking in other design and analysis approaches. Hence, the wave

analysis approach of von Flotow appears to be highly appropriate to the analysis of such

controllers.

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Introduction

A large number of systems have lightly damped modes that are undesirable. Such systems range

from long bridges during an earthquake to tall buildings swaying in the breeze to lightweight robotic

arms that flex when rapidly slewed to high-voltage transmission lines which vibrate in the wind. In all

these cases, a better system could be obtained if the resonances could be reduced. The objective of

a controller for each of these systems is likewise the same: reduce the resonance.

While the objective for the controller in each case may be similar, the approach used to design the

controller often varies considerably. Often times, the approach used is dictated by the point of view

taken to model the system.

One modeling approach uses modern control techniques along with a state-space formulation (Balas

1979, Joshi 1989). With this approach, the differential equation describing the system is expressed in

terms of an infinite-series whose elements consist of eigenvectors and eigenvalues - often referred to

as mode shapes and modal frequencies. The resonances results in the poles of the system being

lightly damped. The objective of the controller is then to increase the damping of these poles.

An implicit assumption behind using a state-space formulation is that some form of state-feedback

will be used to control the system. Those who take this design approach likewise use similar

controller structures with the gains computed using different algorithms ranging from LQR techniques

(Lim, 1992), root locus techniques (Maghami 1992, Yang 1992), inverse plant and sliding-modes

(Watkins 1992, Yurkovich 1990), and others. This approach is extremely popular and has been

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used on trusses (Watkins, 1992), beams (Ferri, 1992), simulations of space structures, (Balas 1978,

Lim 1992), one-link (Kotnik, 1986) and multilink robotic arms (Yurkovich 1992, Xu 1992).

While these approaches all work well in simulations, they often have problems when implemented

(Lim, 1992). One explanation for this may be the modelling errors introduced by using a finite-order

state-space model of the system rather than the infinite-order model. Gibson (1980) demonstrated

that this alone will destabilize any feedback control law that does not use collocated feedback unless

the system has natural damping. Another possible explanation could be the sensitivity of the model

and likewise the controllers based upon them. Since lightly damped structures often are time-varying

(as the load changes) or are uncertain, this sensitivity may be a problem (Desanghre 1986, Joshi

1989). The design of a controller than is robust to model uncertainty has likewise come into

prominence recently.

A second approach for modeling an oscillatory system is to look at the transfer function from the

applied force to the resulting displacement - termed the input impedance (Snowdon, 1968). Using

this point of view, resonances result in the impedance becoming large. The design of a controller,

therefore, should be to reduce the amplitude of the input impedance.

Ormondroyd (1928) first used this notion to design a mass damper for a flexible beam. By tuning the

mass and spring to the first resonance of the beam, the resonance could be shifted to a higher

frequency. The natural damping of the beam at higher frequencies would then reduce the impedance

at this resonance and likewise reduce the resonance. Brock (1946) extended this work by adding a

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damping to the mass damper and showing how the damper could be tuned to minimize the

impedance of the shifted resonance. Snowdon (1968) followed this by showing how several mass

dampers could be used to dampen several resonances. These techniques have been used to design

dampers for flag poles, power lines, and tall buildings with some degree of success (Hunt 1972).

Unfortunately, these controllers are often difficult to tune and often are not as effective as predicted

(Iwuchukwu 1985). Some adaptive techniques to automatically tune these controllers has likewise

been proposed (Desahghre, 1986).

A third approach as proposed by von Flotow takes an entirely different point of view towards the

cause of (and hence control of) resonances. To understand this point of view, consider the following

thought study:

First, consider an system which extends to infinity such as a large pond or an infinite string. Such a

system would have no resonances since any initial conditions would disappear as the energy radiates

out to infinity similar to ripples in a pond.

Next, suppose that you wanted to create a resonance. To do this, you must prevent the energy from

radiating out to infinity - i.e. the energy must be confined somehow. To confine the energy, some

method of reflecting it back into the region when it gets too far away is required. Since reflections are

caused by discontinuities in the medium, this means that you must surround a region with a

discontinuity. This is analogous to replacing the pond with a glass of water - the waves now reflect

off the edges back into the glass, containing the energy and creating resonances.

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Finally, suppose that you had a resonance that you wanted to eliminate. The prior line of reasoning

suggests that the resonances exist due to discontinuities surrounding confining the energy. This implies

that: 1) The controller that tries to minimize the resonances should attack the cause of the problem:

the discontinuities, and 2) The controller should be located at or near the point where the

discontinuities are created. Ideally, the controller should make one discontinuity disappear so that,

relative to the system, you have an system that extends to infinity and has no resonances.

This paradigm was first hinted at by Vaughan (1968), who proposed an infinite-order controller for a

beam in bending that acts as an infinite beam, thus eliminating all reflections. von Flotow (1983) took

this idea and developed wave analysis technique for large truss networks, similar to space platforms.

Von Flotow (1986a) then used the wave technique to define joints so that waves could be channeled

to certain members of a truss. Energy absorbing sections could then be added as an energy "kill

zone"

Further extension of wave analysis techniques to beams with shear and bending moments was

presented in (1986b) and (1986c). In (1986b), von Flotow found the endpoint contrtaints necessary

to absorb all energy at the end of a hanging Timeshinko beam. Since medium is dispersive, control

was frequency dependent. The "best" viscous damper then found using least squares ofer a specified

frequency range. Further, the scattering matrix had to be approximated sine it was 4x4 including both

incident and reflected bending and shear waves.

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Results very similar to existing controllers based upon modal analysis such as proposed by Balas

(1979). Point is that better understanding of how and why controllers work is obtained since the

control is attacking the cause of resonances, not just the effect. New insight into different controllers

now possible, such as ones which shunt energy away (von flotow 1986c) or absorb energy in certain

frequencies possible.

Complexity of wave analysis for trusses and beams due to bending and sheer creating 4th-order

dynamics. Further, beams and trusses are dispersive in nature, which results in

frequency-dempendent controllers.

Such in NOT the cast for strings under tension, however: 2nd-order dynamics lending themselves

well to 2x2 scattering matricies and are non-dispersive in nature. Likewise, the wave analysis point

of view should be easier to apply to strings and have simpler results.

This paper looks at applying von-Flotow's approach to problems with damping vibrations in strings.

Two different cases are investigated:

First, the controller is assumed to be attached to the earth and apply a force to the string. This

is similar to the design of winglets for high-voltage transmission lines or terminating a

high-tension cable.

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Second, a mass-damper is assumed for the controller where no earth reference is permitted.

This is similar to the design of Stockbridge dampers for high-voltage transmission lines to

dampen vibrations.

By using a paradigm based upon traveling wave analysis, potential and problems with each approach

ahould be more clear than with other approaches. The merits of using an analysis and design

paradigm as proposed by von Flotow for such problems will then be more obvious.

Notation

Term: Definition

m The mass of the mass damper

w The frequency of the oscillations in the incident wave

c The wave velocity of the string

T The tension in the string

p The density of the string

M the mass damper's mass

B The damping in the mass damper

K The spring stiffness in the mass damper

Introduction

Consider first the problem of designing a mass damper type of controller for a vibrating string when the

controller is attached to a fixed reference as per Figure 1. Such a problem is encountered when

designing "winglets" for damping vibrations in high-voltage transmission lines. These winglets are pieces

of metal or plastic which are placed along the span of a line. The mass of the winglet would then

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correspond to m while the wind resistance generates some resistance to motion, B. In practice, K is

zero since attaching a spring to a fixed reference (such as a tower) is not done. Conceptually, however,

such a spring could be added. For analysis purposes, therefore, the general case of K not equal to zero

will be analyzed.

The purpose of the control design is then to choose m, B, and K to minimize the oscillations in the

string. Using the paradigm of von Flotow, this is equivalent to choosing these terms to minimize the

reflection/transmission terms, which is equivalent to maximizing the energy dissipation.

Since vibrating strings are linear systems and the objective is to maximize energy dissipation, the system

presented in Figure 1 can be similified somewhat. Beacuse the system is linear, the "optimal" controller

for a general wave will also be "optimal" when two identical waves are traveling to the right for x0. This symmetry allows a controller to be designed for only 1/2 of the string as presented in

Figure 2.

The advantage of analyzing the 1/2 string is that the string's displacement for x


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First, an infinite string is considered. A sinusoidal wave traveling left to right is assumed for the

exciting signal. The amplitude and phase of the reflected wave will then be determined using a

force balance at x=0.

Second, the "optimal" values for M, B, and K will be determined by minimizing the reflection with

respect to each.

Last, the implications of this "optimal" choice with respect to the design of "winglets" will be

assessed by looking at the controller's sensitivity to changes in frequency and in the paremeters M,

B, and K.

Derivation

Assume without loss of generality that a sinusoidal waveform excites the string of the form

(1)g(x, t) = cos (cx t) + cos(cx + t

or, using phasor notation,

(2)g(x, t) = ej( cxt) + ej( cx+t

Here, it is assumed that a wave is traveling right with an amplitude of 1.00 and a frequency of w. Upon

encountering the discontinuity at x=0, it is reflected back tothe left with a reflection coefficient of .The amplitude of relates to the amplitude of the reflection while the angle of determines the phase

lag of the reflection.

In order to determine , a force balance is used at x=0 is used. From Figure 3, the net force at x=0 is

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(3) F= 0 = Tdgdx

+ md y

dt2

+Bdy

dt+ K

where the subscript gxand y

tdenotes differentiation with respect to x and t respectively. Substituting for

g, evaluating at x=0, and simplifying results in

(4)0 = K m2 +j(T/c +B) ejt+ K m2 +j(T/c B)

An implicit assumption behind using phasors is that one will take the real part eveltually. Beacuse of

this, the e-jwt term can be converted to a ejwt term using the relationship

(5)Re((a +jb)ejt) =Re((a jb)ejt).

With this, (4) can be rewritten as

(6)0 = K m2 +j(T/c +B) + K m2 j(T/c B)

Solving for results in

(7) =Km 2j(T/cB)

Km

2+k(T/c+B)

.

Next, in order to minimize the reflections, the amplitude of needs to be minimized:

(8) 2 =Km

2

2

+(T/cB)2

(Km) 2+(T/c+B)2

To find the optimal choice of M, B, and K, the partials of with respect to each are set to zero:

(9) 2

m = 0 = (4BT/c)2

2

K m

2

(10) 2

K= 0 = (4BT

2/cK m

2

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(11) 2

B= 0 =

K m

2

2

+ (T/c B)2

(2B)(T/c +B)

From (9) and (10), several solutions exist. Most of these solutions are not of interest, however, for

practical considerations. The solution of T=0 implies no tension on the string. This clearly will not work

for high-voltage transmission lines since some tension is necessary to keep the lines off the ground. The

solution w=0 implies a stationary string. This paper assumes motion since it is analyzing vibrating

strings. The solution of 1/c=0 implies infinite wave velcity, onlly possible with zero mass or infinite

tension - both of which are not possible.

The only solution which makes sense is to select . This solution is encouraging since it is them2 =

well-known practice of tuning the resonance of the mass-spring system to the frequency of oscillation.

Assuming that , (11) can be simplified toK m2

=

(12)0 = 3

(B)(T/c)(T/c B)(T/c +B)

The solution B=0 is actually a maximum since it results in =-1. At the frequecny of w, the impedanceof the mass-spring system is infinite. It likewise prevents any motion of the string at x=0 at a frequency

of w. The solution B=-T/c results in an unstable system since the controller adds energy to the system.

The desired solution, therefore, is B=T/c.

Since this analysis was for 1/2 of the string, the optimal choice for selecting M, B, and K is twice the

values previously calculated:

(13)m2 =

B = 2T/c

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Analysis

Aparently, the selection of M and K are arbitrary so long as they are tuned for eachother. Note,

however, that is a function of frequency as well as M, B, and K. If the frequency of the excitation is

unknown or can vary, then the choice of M and K does affect the effectiveness of the damper. In

Figure 4, the amplitude of is plotted vs. frequency as M varies from 0.01 to 100. Here, the terms M,B, and K are tuned to minimize at w=1 using normalized values for M, B, and K:

(14)mc

T=M0

K= TCM0

B = Tc

so that

(15) 2 =M0

2(1/) 2

M02(1/) 2+

Note from Figure 4 that the range where the mass damper reduces the reflections is drastically reduced

as M increases. The "optimal" choice of a controller, therefore, would be to make the mass as small as

possible (prefereably zero) and tune B to 2T/c.

This is similar to the design of "winglets" for high-voltage transmission lines where small "wings" are

placed on the lines. The wings add damping due to wind resistance as the line moves. The mass of the

winglets are kept small by making them out of sheet metal or plastic. Further, as stated previously, no

springs are attached from the transmission line to the earth or to a tower, resulting in K=0.

It is interesting to note that a competing design for damping vibrations is to place hollow spheres along

the transmission line. These spheres serve to make the line more visable and to dampen vibrations.

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While the spheres certainly do make the line more visable, they add more mass than the winglets and

provide less wind resistance (due to their aerodynamic shape), and likewise should be less effective at

damping vibrations.

Mass Damper Design

While the addition of winglets appears to be one way of effectively damping vibrations in a transmission

line, other methods exist and are used. One such method is to attach a mass damper to the transmission

line as per figure 5, also termed a stockbridge damper. Here, instead of the controller "pushing" off of a

fixed reference, a spring and damper connect the string to a small mass.

Examples of this form of controller are widespread. In civil engineering, such controllers are termed

harmonic oscillators are are found in skyscrapers to dampen wind-induced oscillations and to mitigate

earthquake damage (Snowdon 1968, Iwuchukwu 1985) as well as support beams for floors where the

objective is to reduce the amplitude of the resonance (Hunt, 1972). In control systems, such a

controller is termed a servo-compensator and is used to force a signal (usually the tracking error) to

zero at the frequency of the disturbance. In high-voltage transmission lines, such controllers are often

called Stockbridge Dampers where they are used to reduce wind-induced oscillations in transmission

lines.

In each case, the objective is the same: minimize the resonance and/or maximize the energy dissipation

by the controller. While the objective may be the same, the analysis techniques vary considerably

ranging from input-impedance methods to pole placement to time responses via finite-element

simulations.

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Here, the method proposed by von-Flotow for analyzing such a controller from a wave-standpoint will

be used instead. Using this approach, a more intuitive feel for how and why the mass damper works as

well as how to optimize the design is sought.

The method used to optimize M, B, and K will be as before:

First, assuming sinusoidal excitation, the reflection coefficient, , is determined using a force

balance at the end of the string

Second, M, B, and K are "optimized" by setting the partial of reflection with respect to each to

zero

Last, the results are analyzed and interprited to determine why the given solution is "optimal" and to

assess what the effect of varying the control parameters or wave frequency will be.

Derivation

Similar to the previous section, consider only the left-half og the string as presented in Figure 6.

Assume also that a sinusoidal signal excites the string traveling from left-to-right with a reflection

traveling right to left with an amplitude of :

(16)g(x, t) = ej(cxt) + ej(

cx+t)

Referring to Figure 6, the net force at point A is

(17) F= 0 = Tdgdx

B ddt

(g y) K(g y)

while at point B it is

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(18)md2y

dt2=B d

dt(g y) + K(g y)

Using the "s" operator to denote differentiation with respect to time, the transfer function from g to y can

be found:

(19)y = Bs+K

ms2+Bs+K

g

which can be rewritten as

(20)

(g

y

) =

ms2

ms2+Bs+K

g

Substituting (20) into (17) and simplifying results in

(21)Tmd3g

dxdt2

+ TBd2g

dxdt+ TK

dg

dx+ mB

d3g

dt3

+ mKd2g

dt2

= 0

Substituting in (16) for g and analyzing at x=0 gives

(22)0 = TB

2/c mK2 +jTm

3/c + TK/c + mB3 e

j

+ TB2/c mK2 +j Tm3/c + TK/c mB3

Using (5), the e-jwt term can be converted to a ejwt term allowing (22) to be solved for as

(23) =(TB/cmK)+jTm

2+TK/c+mB 2

(TB/c+mK)+jTm

2

TK/c+mB2

The amplitude of the reflections is likewise

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(24) 2 =(TB/cmK) 2+Tm

2+TK/c+mB2 2

(TB/c+mK) 2+Tm2TK/c+mB2

2

Setting the partial of with respect to M, B, and K to zero will again allow one to find the "optimal"2

choice of M, B, and K. From the previous solution, it is aparent that the "optinal" choice will resultng

=0. This allows M, B, and K to be determined by inspection.

In order for to be zero, both terms in the numerator of (24) must be zero, resulting in

(25)B = mKcT

(26)K= T2m 2

T2+m 2c

Once again, the mass appears to be arbitrary since Gamma can be set to zero for any positive mass.

One interesting implication of this equation is that quite different designs can be obtained, depending

upon whteher large or small masses are used. Assume first that the mass is arbitrarilly large such as

would be obtained when the spring and damper are tied to the earth. In this case, the numerator of (24)

becomes

(27)0 = (K)2 + T2/c +B2

This gives the prior solutoin of K=0, B=T/c. (Note that this still corresponds to 1/2 of the controller so

the total damping would be twice this.) Here, a spring should not be used. Instead, a relative large

damper tuned to T/c should be used. A design practice could then be to start with K=0, B=0 and

increase B until the reflections are minimized.

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Assume next that an arbitrarilly small mass is used. This would be exemplified by adding harmonic

oscillators to skyscrapers where the mass of the damper relative to the mass of the structure would be

quite small. In this case, numerator of (24) becomes

(28)0 = TB/c mK

and

(29)0 = Tm2/c + TK/c + mB2

A first order solution for m small is

(30)K m2

B (c/T)(mK) = (c/T)(m2 2 )

This corresponds to the typical design for mass dampers for structures. Here, B is often considered

equal to zero and the mass and spring are tuned to the resonance of the structure. To keep the mass'

displacement finite at resonance, B is then increased from zero. [Snowdon]

In order to determine which solution is superior from a theoretical standpoint, consider first the

sensitivity of to changes in B and K. To determine this, a numerical example corresponding to a

string long made out of 10mm diameter Aluminum with a 5m long traveling wave.

string density = p = 212 g/mTension = T = 500N

wave velocity = c = 48.56 m/s

wave frequency = = 0.5Hz. = rad/sec.

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In Figure 7, the contour plots for constant Gamma vs B and K are presented for masses between 0.01g

and 1kg. Consider first the contour plot corresponding to a mass of 0.01g. This situation is similar to

the dessign of mass dampers for buildings and other large structures where the mass of the controller is

relatively small compared to that of the structure. In this case, the spring is tuned so that the

mass-spring system has a resonance near the system's resonance

K= 0.09868 0.09870 = m2

A reasonable design procedure for this case would likewise be to set B=0 and then tune K to the

resonance of the structure (equal to frequency of the incident wave in this case). B could then be

increased until the reflected energy is minimized - moving the operating point from the left axis to the

center of the contour plots where Gamma=0. This is in fact the design procedure mentioned by

Iwuchuwu .

In addition to the method for tuning a mass damper, this plot reveals some potential problems with mass

dampers with relatively small masses. Note that the region where Gamma is small is fairly tight: if either

B or K vary from their nominal value by more 10%, Gamma increases from 0 to 0.1. From similar

contour plots, this tollerance becomes even tighter when amaller mass are used. It likewise appears that

tuning a mass damper may be a difficult procedure if small masses are used. As reported in Hunt this is

one of the drawbacks behing using such controllers for large structures: they are notoriously difficult to

tune.

Contrast the first graph to the extreme case of using a Stockbridge damper-like controller with a mass

of 1kg - a not very realistic mass but one that serves to demonstrate the effect of increasing the mass.

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Note that remains less than 0.1 for K between 0 and 0.2 N/m. This means that if K is tuned

incorrectly by up to 2x high or 0x small, the reflected wave will only increase from 0% of the incident

wave to less than 10%. Note also that B can vary from 0.18 to 0.25 Ns/m with increaseing Gamma

from 0% to 10%.

In figure 8, the effect of varying the frequency of the incident wave on the reflected wave is presented

for various masses. Note from this figure that the reflected wave at 0.5Hz is always zero, implying that

the mass is arbitrary if the exact frequency of the incident wave is known. If the incident wave drifts or

has some bandwidth, however, mass has a pronounced effect. Mass dampers with relatively small

mass have a very narrow frequency range where they are effective. The larger the mass on the mass

damper, however, the wider the range where the mass damper does some good.

A common problem reported with mass dampers is that they are notoriously difficult to tune and are

very sensitive to changes in the structures resonance. With structures, the mass must remain relatively

small due to structural concerns. From the prior analysis based upon von Flotow's wave paradigm, that

this is the case is not at all surprising. Similarly, if a Stockbridge damper is designed with a relatively

small mass, similar problems with tuning the damper of sensitivity to frequency drift should be

encountered. The benefit of increasing the mass on the damper can likewise be observed.

Summary and Conclusions

In this paper, a design approach proposed by von Flotow for controlling beams and trussses was

used to dampen the vibrations of a flexible string: an approach that tries to minimize reflections.

Using this point of view, the objective of the controller was obvious: to minimize the reflections. The

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location of the controller was clear: given a choice, the controller should be placed where the

reflections are created i.e. at the endpoint. The form of the controller was also clear: some form of

collocated feedback is required since reflections are determined by the constraints at a particular

point on the string.

To illustrate the implications of this design technique, a controller was designed for a vibrating string

under two conditions: first that the controller was attached to the earth and second that the controller

was not (i.e. external forces were zero). It was shown that a very simple and highly effective

controller resulted from this design approach for the first two cases. It was also shown that the third

was more complicated and resulted in a tradeoff between the bandwidth of the controller and the

mass of the damper.

Since this design approach gives simple and effective controllers along with good insight as to why the

controllers work along with their limitations, it is should be considered for any system capable of

being modeled as a string under tension.

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[19] Vaughan, D.R., "Application of Distributed Parameter Concepts to Dynamic Analysis and

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[20] von Flotow, A. H., "A Traveling Wave Approach to the Dynamic Analysis of Large Structures,"

Proceedings of the AIAA Strictures, Dynamics, and Materials Conference, South Lake Tahoe,

CA, May 2-4, 1983.

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[21] von Flotow, A. H., Disturbance Propogation in Structural Networks," Journal of Sound and

Vibrations, 1986a.

[22] von Flotow, A. H., and Schafer, B., "Wave Absorbing Controllers for a Flexible Beam,"

Journal of Guidance, Control, and Dynamics, 1986b.

[23] von Flotow, A. H., "Traveling Wave Control for Large Space Structures," Journal of Guidance,

Control, and Dynamics, Vol 9, No. 4, July-Aug 1986c, pp. 462-468.

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Figure 1: A Mass Damper is Added which Is Able to Generate External Forces to Dampen Vibrations

Figure 2: Using Symmetry, A Controller for 1/2 of the String is Designed

Figure 3: The Forces at the Endpoint of the String

Figure 4: The Effect of Changing Mass on the Reflection, . Note that at the point where the mass

damper is tuned (w=1), the mass is arbitrary. Away from this frequency, smaller masses work better.

Figure 5: Damping Vibrations by Using a Mass Damper Attached to the String

Figure 6: Assuming Symmetry in the Excitating Waves, 1/2 of the String is Analyzed

Figure 7: Contour Plots Showing Constant Reflection Coefficients vs. B and K

Figure 8: The Amplitude of the Reflection vs. Frequency as the Mass Increases

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94 analysis of vibrational dampers

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