vibrations of a taut string with linear damper

7/30/2019 Vibrations of a Taut String With Linear Damper

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CHAPTER 3

Vibrations of a Taut String with Linear Damper

To suppress problematic stay-cable vibrations like those discussed in the previous

chapter, viscous dampers are often attached to stay cables near the anchorages to

supplement the low levels of inherent damping. The potential for widespread application

of dampers for stay-cable vibration suppression necessitates a thorough understanding of

the dynamics of the cable-damper system to enable both an effective and economical

damper design. A simple model of the stay-cable with attached damper is a taut string

with a linear viscous dashpot. This simplified model neglects the bending stiffness of the

cable, the sag of the cable due to its static deflection under self-weight, and any

nonlinearity of the damper, whether unintentional or by design.

The effect of sag is not considered in the present study, as it has already been

studied in detail by several investigators, as will be discussed in the following section,

and it has been observed that for most stay cables, the sag is small enough that its

influence is not significant. The effect of bending stiffness has been shown to be

significant for many stay cables (Tabatabai and Mehrabi 2000), and a detailed study of its

effects is presented in the following chapter. The influence of damper nonlinearity is also

investigated in subsequent chapters. The simplifications introduced by the taut-string and

linear-damper approximations allow an analytical solution to be carried quite far,

providing insights into the complicated and unusual features of the cable-damper system.

These insights provide a basis for understanding the more complicated solution


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characteristics in the case of nonzero bending stiffness, and the formulation is extended in

subsequent chapters to develop an approximate solution for nonlinear dampers.

3.1 Previous Investigations

Carne (1981) and Kovacs (1982) were among the first to investigate the vibrations

of a taut cable with an attached damper, both focusing on determination of first-mode

damping ratios for damper locations near the end of the cable. Carne developed an

approximate analytical solution to the free-vibration problem, obtaining a transcendental

equation for the complex eigenvalues and an accurate approximation for the first-mode

damping ratio as a function of the damper coefficient and location, while Kovacs

developed approximations for the maximum attainable damping ratio (in agreement with

Carne) and the corresponding optimal damper coefficient (about 60% in excess of

Carnes accurate result) from consideration of the frequency response function for a cable

with attached damper under harmonic excitation.

Subsequent investigators formulated the free-vibration problem using Galerkins

method, with the sinusoidal mode shapes of an undamped cable as basis functions, and

several hundred terms were required for adequate convergence in the solution, creating a

computational burden (Yoneda and Maeda 1989; Pacheco et al. 1993). Several

investigators have worked to develop modal damping estimation curves of general

applicability (Yoneda and Maeda 1989; Pacheco et al. 1993), and Pacheco et al. (1993)

introduced nondimensional parameters to develop a universal estimation curve of

normalized modal damping ratio versus normalized damper coefficient, which is useful

and applicable in many practical design situations. Krenk (2000) developed an exact

analytical solution of the free-vibration problem for a taut cable and obtained an


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asymptotic approximation for the damping ratios in the first few modes for damper

locations near the end of the cable; Krenk also developed an efficient iterative method for

accurate determination of modal damping ratios outside the range of applicability of the

asymptotic approximation.

Several investigators have considered the influence of sag on the attainable

damping ratios. Pacheco et al. (1993) extended their formulation using Galerkins

method to investigate the influence of small sag on the modal damping ratios, and Xu et

al. (1997) developed an efficient transfer matrix approach using complex eigenfunctions

to consider the effect of cable sag and inclination. These studies revealed that moderate

amounts of sag can significantly reduce the first-mode damping ratio, while the damping

ratios in the higher modes are virtually unaffected (Pacheco et al. 1993; Xu et al. 1999).

However, using a database of stay cable properties from real bridges, Tabatabai and

Mehrabi (2000) found that for most actual stays, the influence of sag is insignificant,

even in the first mode.

Previous investigations have focused on vibrations in the first few modes for

damper locations near the end of the cable, and while practical constraints usually limit

the damper attachment location, it is important to understand both the range of

applicability of previous observations and the behavior that may be expected outside of

this range. Damper performance in the higher modes is of particular interest, as the full-

scale measurements discussed in the previous chapter revealed that vibrations of

moderate amplitude can occur over a wide range of cable modes, with vortex-induced

vibrations observed in quite high modes. In this chapter an analytical formulation of the

free-vibration problem is used to investigate the dynamics of the cable-damper system in


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higher modes and without restriction on the damper location. The basic problem

formulation in this chapter follows quite closely the approach recently used by Krenk

(2000), although it was developed independently as an extension of the transfer matrix

technique developed by Iwan (Sergev and Iwan, 1981; Iwan and Jones 1984) to calculate

the natural frequencies and mode shapes of a taut cable with attached springs and masses.

A similar approach was used by Rayleigh (1877) to consider the vibrations of a taut

string with an attached mass. Using this formulation, the important role of damper-

induced frequency shifts in characterizing the system is observed and emphasized.

Consideration of the nature of these frequency shifts affords additional insight into the

dynamics of the system, and when the shifts are large, complicated new regimes of

behavior are observed.

3.2 Problem Formulation

The problem under consideration is depicted in Figure 3.1. The damper is

attached to the cable at an intermediate point, dividing the cable into two segments,

where 2> 1. Assuming that the tension in the cable is large compared to its weight,

bending stiffness and internal damping are negligible, and deflections are small, the

following partial differential equation is satisfied over each segment of cable:

2 2

2 2

( , ) ( , )k k k k

k

y x t y x tm T

t x

=

(3.1)

whereyk(xk ,t) is the transverse deflection and xk is the coordinate along the cable chord

axis in the kth segment; m is the mass per unit length, and Tis the tension in the cable.


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L

1

1x

2

c

TT m2x

Figure 3.1: Taut Cable with Viscous Damper

This equation is valid everywhere except at the damper attachment point; at this location

continuity of displacement and equilibrium of forces must be satisfied. As in Pacheco et

al. (1993), a nondimensional time to1= is introduced, where mTLo )(1 = . To

solve (3.1) subject to the boundary, continuity, and equilibrium conditions, distinct

solutions over the two cable segments are assumed of the form:

exYxy kkkk )(),( = (3.2)

where is a dimensionless eigenvalue that is complex in general1. Substituting (3.2) into

(3.1) yields the following ordinary differential equation:

)()(

2

2

2

kk

k

kkxY

Ldx

xYd

=

(3.3)

Because

is complex, the solutions to (3.3), which are the mode shapes of the system, are

also complex. Continuity of displacement at the damper and the boundary conditions of

zero displacement at the cable ends can be enforced by expressing the solution as

1The assumed solution may be alternatively expressed as

jkkkk exYxy )(),( = , where 1=j

and is complex; this alternative formulation leads to trigonometric rather than hyperbolic expressions for

the eigenfunctions. Although the two forms are equivalent, when the damping is small, is mostly real,

and this alternative form leads to more direct expressions for the eigenfunctions. In cases of large damping,

to be discussed herein, is mostly real, and the form of solution in (3.2) is more direct.


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sinh( )

( )sinh( )

kk k

k

x LY x

L

=

(3.4)

where

is the amplitude at the damper. The equilibrium equation at the damper can be

written as

11

1122

1

1

1

2

2

===

=

x

xxt

yc

x

y

x

yT (3.5)

where c is the damper coefficient. Differentiating the assumed solution [(3.2) and (3.4)]

and substituting into (3.5) yields:

0)coth()coth(21

=++Tm

cLL (3.6)

Eq. (3.6) is equivalent to the frequency equation presented by Krenk (2000). Its roots

are the eigenvalues of the system, each corresponding to a distinct mode of vibration.

For specific values of Tmc/ and 1/L, (3.6) can be directly solved numerically to obtain

the damping ratios in as many modes as desired to an arbitrary degree of accuracy. Each

eigenvalue can be written explicitly in terms of real and imaginary parts as follows

( )21

1 iio

ii j

+= (3.7)

in which i is the damping ratio and i is the modulus of the dimensional eigenvalue,

which has been referred to as the pseudoundamped natural frequency (Pacheco et al.

1993). For convenience in subsequent manipulations, new symbols are introduced to

denote the real and imaginary parts of the ith

eigenvalue:

)Re( ii = ; )Im( ii = (3.8a,b)

It is noted that i is the nondimensional frequency of damped oscillation. The damping

ratio i can then be computed from i and i:


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1

2 2

21i ii

i i

= = +

(3.9)

Further insight into the solution characteristics can be obtained by expanding (3.6)

explicitly into real and imaginary parts. Taking the imaginary part of (3.6) and

expanding using the notation of (3.8a,b) yields the following equation, after some

simplification:

)2sin()2cosh()2sin()2cosh()2sin( 1221 =+ LLLL (3.10)

This equation is independent of Tmc/ , and its solution branches give permissible

values of

for a given 1/L, thus revealing the attainable values of modal damping with

their corresponding oscillation frequencies. Because it corresponds to enforcing equality

of phase in (3.5), (3.10) will be referred to herein as the phase equation. While solution

branches to (3.10) are symmetric about 0 = , only negative values of are considered,

as positive values of are associated with negative values of Tmc/ , which are not of

practical interest. Taking the real part of (3.6) yields the following equation, after some

simplification:

( )

( ) ( )

( )

( ) ( )0

2cos2cosh

2sinh

2cos2cosh

2sinh

22

2

11

1 =+

+ Tm

c

LL

L

LL

L

(3.11)

For a specific value of with real and imaginary parts and that satisfy (3.10), the

corresponding value of Tmc/ can be readily computed from (3.11). The expression

(3.4) for the complex mode shapes can be expanded explicitly in terms of the real and

imaginary parts using the notation of (3.8a,b), yielding the following expression

( ) ( ) ( ) ( )[ ]LxLxjLxLxAxY kkkkkkk sincoshcossinh)( += (3.12)


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in which the complex coefficients are given by )sinh( LA kk = . This expression

will be useful in considering the character of the mode shapes in the following sections,

in which three special types of solutions will be considered: a non-oscillatory decaying

solution (

=0), non-decaying oscillatory solutions (

=0), and oscillatory solutions

approaching critical damping ( and 1).

3.3 Special Limiting Solutions

3.3.1 The Case of Non-Oscillatory Decay

Solutions to (3.6) can exist for which is purely real, corresponding to non-

oscillatory, exponentially decaying solutions in time; in this case

=0, the phase equation

(3.10) is trivially satisfied, and (3.11) reduces to:

( )

( )

( )

( ) Tm

c

L

L

L

L=

+

12cosh

2sinh

12cosh

2sinh

2

2

1

1

(3.13)

in which it is emphasized that positive values of Tmc/ are of practical interest, leading

to negative values of

. Eq. (3.13) reveals that purely real solutions exist only when

Tmc/ exceeds a critical value; when , the two terms on the left-hand side of

(3.13) each approach unity, so that the critical value of c associated with this limit is

given by:

Tmccrit 2= (3.14)

For c2; for c>ccritthe magnitude of decreases monotonically with increasing c. In

the limit as

0, it can be shown that Tmc/ . Figure 3.2 shows a plot of vs.

Tmc/ from (3.13) with varying damper location 1/L .


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0.01

0.1

1

10

100

1 10 100

0.010.05

0.1

0.2

0.3

0.5

cTm

2

1/ :L

Figure 3.2: Nondimensional Decay Rate of Non-Oscillatory Solution vs. Nondimensional

Damper Coefficient with Varying Damper Location

This case of non-oscillatory decay may be compared to the case of supercritical

damping for a single-degree-of-freedom (SDOF) oscillator in free vibration. If an

exponential form of solution, x Ae= , is assumed for the SDOF oscillator, where x is

the displacement of the oscillator, /k m t= is a nondimensional time, kis the spring

constant, and m is the mass of the oscillator, then it is observed that there are two

solutions for the nondimensional decay rate, which can be expressed as

2

1,2 12 2

c c

km km

=

(3.15)


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If /(2 ) 1c km , then both values of the decay rate are purely real, and the

solution is non-oscillatory, a sum of decaying exponentials. The critical value of the

damper coefficient for the SDOF oscillator is then given by kmccrit 2= , which has a

remarkably similar form to (3.14) for the taut string. Using the notation of (3.8a,b),

= because is purely real, and Figure 3.3 shows a plot of the magnitude of the two

solutions for the decay rate from (3.15),1

and2

, as functions of the nondimensional

damper coefficient /c km . The magnitude of 2 increases with /c km (plotted in

gray), but the magnitude of1

decreases with /c km (plotted in black). Because the

overall decay of the solution is governed by the more slowly decaying exponential, the

net rate of decay decreases with increasing /c km , following the black curve in Figure

3.3, which has features similar to those observed for the taut string in Figure 3.2.

With

=0 in this case, the expression for the mode shapes (3.12) reduces to a

hyperbolic sinusoid with a purely real argument:

( )LxAxY kkkk sinh)( = (3.16)

Figure 3.4 illustrates the mode shape corresponding to a purely real eigenvalue,

plotted for several different values of Tmc/ , with an arrow indicating the trend with

increasing c. As c and 0 according to (3.13), the linear term in the series

expansion of )sinh( Lxk becomes dominant, and the mode shape tends to the static

deflected shape of the cable under a concentrated load.


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0.01

0.1

1

10

100

1 10 1002

ckm

1

2

1

2 2

c c

km km

= +

2

2

12 2

c c

km km

=

Figure 3.3: Nondimensional Decay Rate of Non-Oscillatory Solutions for SDOFOscillator vs. Nondimensional Damper Coefficient

increasing /c Tm

Figure 3.4: Mode Shape Associated with a Purely Real Eigenvalue


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3.3.2 Limiting Cases of Non-Decaying Oscillation

Solutions to (3.6) can exist for which

is purely imaginary (

= 0), corresponding

to non-decaying oscillation or zero damping. It can be shown that when = 0, the phase

equation (3.10) reduces to

0)sin()sin()sin( 21 = LL (3.17)

which indicates that there are three sets of frequencies associated with = 0,

corresponding to the zeros of each of the sinusoids in (3.17). One of these sets of

frequencies is associated with the limit ofc0 and the other two sets of frequencies are

associated with the limit of c (i.e., a perfectly rigid damper.) Many investigators

have made reference to these limiting cases in giving a physical explanation for the

existence of an optimum damper coefficient for a specific mode: in both of these limits,

physical intuition suggests that the damping ratios must be zero. These limiting cases can

be observed more clearly by rearranging the eigenvalue equation (3.6) into the following

form:

0)sinh()sinh()sinh( 21 =+ LLTm

c (3.18)

In the first limiting case, when c0, (3.18) reduces to 0)sinh( = , which can only be

satisfied if

is an integer multiple of 1=j . Consequently, must be purely

imaginary and

must take on integer values:

ioi = (3.19)

These frequencies simply correspond to vibrations in the undamped natural modes of the

cable.


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In the second limiting case, when c

, (3.18) becomes

0)sinh()sinh( 21 =LL , which yields two sets of roots, both of which are purely

imaginary. One set of roots, corresponding to the zeros of )sinh( 2 L , yields the

following frequencies:

)( 2Lici = (3.20)

These frequencies correspond to vibrations of the longer cable segment of length 2, with

zero displacement at the damper (the subscript c indicates a clamped frequency). The

second set of roots, corresponding to the zeros of )sinh( 1 L , yields the following

frequencies:

)( 1)1(

Lici = (3.21)

These frequencies correspond to vibrations of the shorter cable segment of length 1 (the

superscript indicates vibrations of the cable segment with index k=1), also with zero

displacement at the damper. As the three sets of frequencies (3.19)-(3.21) represent all

the solutions to (3.17), there no other frequencies associated with =0. Figure 3.5 shows

a plot of these three sets of frequencies against1/L ; this plot will prove useful in

explaining and categorizing the complicated solution characteristics that will be observed

under large frequency shifts, and its features will be discussed in more detail

subsequently.


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0

1

2

3

4

5

6

78

9

10

0 0.1 0.2 0.3 0.4 0.5

L1

)1(ci

ci

oi

Figure 3.5: Undamped Frequencies and Clamped Frequencies vs. Damper Location

When =0, the expression for the mode shapes (3.12) reduces to:

( )LxjAxY kkkk sin)( = (3.22)

which indicates that when =0, the mode shapes over each cable segment are sinusoids

with purely real arguments. Figure 3.6 depicts mode shapes associated with these three

special cases of non-decaying oscillation; the mode shape associated with o1 is depicted

in Figure 3.6(a), and the mode shapes associated with c1 and c1(1)

are depicted in Figure

3.6(b) and Figure 3.6(c), respectively.


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a)

b)

0=c

c)

c

c

Figure 3.6: Limiting Cases of Non-Decaying Oscillation

3.3.3 Oscillatory Solutions Approaching Critical Damping

Complex-valued solutions to (3.6) can exist for which as the

nondimensional frequency

tends to a constant value. When

with bounded,it

is evident from (3.9) that 1, so that while the real part of the eigenvalue increases

without limit, the damping ratio itself approaches a finite value (

=1), corresponding to

critical damping. The frequencies associated with these limiting cases of critical

damping can be determined from the phase equation (3.14) by taking the limit as ,

which yields the equation 0)2sin( 1 =L . The frequencies that satisfy this equation

can be divided into two distinct categories:

( )( )1, /2/1 Liicrit = (3.23)

( )(1) , 1/crit i i L = (3.24)

It is noted that the frequencies crit,i(1)

coincide with the clamped frequencies of the

shorter cable segment in (3.21). It can be readily shown using (3.11) that as

,

Tmc/ 2, or cccrit, regardless of the value of

. As

, the expression for the

mode shapes (3.12) tends to

( ) ( )LxjLxAxY kkkkk expsinh)( (3.25)


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Eq. (3.25) indicates that as

, the magnitude of the mode shape over each cable

segment takes the same hyperbolic form as (3.16) for the case of non-oscillatory decay.

3.4 Solution Characteristics for Small Frequency Shifts

By focusing their attention on vibrations in the first few modes for small values of

1/L, previous investigators restricted their attention to cases in which the damper-

induced frequency shifts are small. Consideration of the characteristics of solution

branches to the phase equation (3.10) in these cases provides additional insight into the

dynamics of the cable-damper system.

3.4.1 Asymptotic Approximate Relations

Figure 3.7 shows the first five branches of solutions to (3.10) for 1/L=0.05

plotted in the - plane; a distinct solution branch is associated with each mode. The

frequencies associated with the special cases of non-decaying oscillation discussed above

are plotted along the horizontal axis; the undamped frequencies, oi, are indicated by

circles and the clamped frequencies, ci, are indicated by crosses. Each solution branch

originates at oi with =0, terminates at ci with =0, and takes on nonzero values of for

intermediate frequencies.

0 1 2 3 4 50

0.01

0.02

0.03

0.04

05.01 =L

Figure 3.7: Phase Equation Solution Branches (

1/L = 0.05, First 5 Modes)


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The difference between these two limiting frequencies is given by

( )21/ioicii == (3.26)

The similar form of the solution branches for each mode suggests that it may be useful to

define a new parameter as a measure of the damper-induced frequency shift

10;


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generated by numerical solution of (3.10). A distinct curve is plotted for each mode, and

for this damper location, the numerically generated curves corresponding to each of the

five modes are indeed very nearly identical, agreeing quite well with (3.30). An

approximate relation between the clamping ratio and Tmc/ can be established by

substituting the expressions for i (3.28) and i (3.29) into (3.11) and using similar

asymptotic approximations, yielding the following relation:

ci

ci

1

12

(3.31)

where

is a nondimensional parameter grouping defined as

Li

mL

c

o

1

1

(3.32)

It is noted that )/(/ 1omLcTmc = , and the alternative normalization )/( 1omLc is

used in (3.32) to facilitate comparison with previous investigations;

is equivalent to the

nondimensional parameter grouping plotted on the abscissa in the universal curve of

Pacheco et al. (1993). Figure 3.8(b) shows a plot of (3.31) along with numerically

computed curves of

versus ci in the first five modes for 1/L=0.02, generated from

(3.11) after numerical solution of (3.10). Again, the numerically computed curves

corresponding to each of the five modes are nearly identical, agreeing well with the

approximation of (3.31). The clamping ratio increases monotonically with

, from ci =

0 when

= 0 to approach

ci = 1 as

. Substituting the optimal clamping ratio of

ci=1/2 into (3.31) reveals that the optimal value of modal damping is achieved when =

2 0.101, as determined by Krenk (2000). Eq. (3.31) can be inverted and solved for

ci:


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1)(

)(22

22

+

ci (3.33)

Substituting this expression into (3.30) yields the following approximate relation:

1)()/( 22

2

1 +

L

i

(3.34)

This expression was recently derived by Krenk (2000); it is an analytical expression for

the universal curve of Pacheco et al. (1993) and is a generalization of the first-mode

approximation obtained by Carne (1981). Figure 3.8(c) shows a plot of (3.34) along with

curves of i/( 1/L) versus for the first five modes for 1/L=0.02 generated numerically

from (3.10) and (3.11). It is evident that again the curves collapse very nearly onto a

single curve in good agreement with the approximation of (3.34).

It is important to note that the optimal damping ratio can only be achieved in one

mode for a linear damper. Noting that

is proportional to mode number, it is evident

from Figure 3.8(b) that ci increases monotonically with mode number, so that if the

damper is designed optimally for a particular mode, it will be effectively more rigid in the

higher modes and more compliant in the lower modes, resulting in suboptimal damping

ratios in other modes. As discussed in Chapter 5, recent investigations indicate that

nonlinear dampers may potentially overcome this limitation in performance, allowing the

optimal damping ratio to be achieved in more than one mode; however, in the case of a

nonlinear damper, the damping performance is amplitude-dependent, and optimal

performance can only be achieved in the neighborhood of a particular oscillation

amplitude.


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0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

L

i

1

ci

5-1modes

02.01 =L

asymptotic

a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ci

b)

asymptotic

5-1modes

02.01 =L

Figure 3.8: (a) Normalized Damping Ratio vs. Clamping Ratio; (b) Normalized Damper

Coefficient vs. Clamping Ratio; and (c) Normalized Damping Ratio vs. Normalized

Damper Coefficient.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

L

i

1

asymptotic

c)

5-1modes

02.01 =L


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3.4.2 Complex Mode Shapes

Asymptotic approximations can also be obtained for the complex mode shapes

when the damper-induced frequency shifts are small, affording additional insight into the

solution characteristics in this case. The complex mode shapes over the longer cable

segment can be expressed as follows, by setting2A jA= in (3.12):

( ) ( ) ( ) ( )2 2 2 2 2 2( ) cosh sin sinh cosY x A x L x L j x L x L = (3.35)

When is small, ( )cosh 1kx L and ( )sinh k kx L x L . Using these

approximations, expressing using (3.28), and using the approximation for in (3.29),

(3.35) can be approximated as:

2 1 2 22 2( ) sin ( ) (1 ) cos ( )

ci i ci ci ci i

x x xY x A i j i i

L L L L

+ + +

(3.36)

In obtaining an approximation for the mode shapes over the shorter cable segment, it is

noted that when1/L is small, ( )1cos 1x L and ( )1 1sin x L x L . Using

these approximations, in addition to the approximations used in obtaining (3.36), the

mode shape over the shorter cable segment can be expressed as

1 11 1 1( ) (1 ) ( )ci ci ci i

xY x A i j i

L L

+ +

(3.37)

Eq. (3.37) indicates that the mode shapes are approximately linear on the shorter cable

segment. The complex constant A1 is determined from the requirement of continuous

displacement:1 1 2 2( ) ( )Y Y= . Because the mode shapes are known to be approximately

linear over the shorter segment from (3.37), continuity of displacement can be enforced


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directly by expressing the mode shapes as1 1 1 1 2 2( ) ( / ) ( )Y x x Y = , yielding the following

expression:

1 2 1 2 2

1 11

( ) sin ( ) (1 ) cos ( )ci i ci ci ci ix

Y x A i j i iL L L L

+ + +

(3.38)

It will prove useful to shift the phase of the complex mode shapes so that the imaginary

part has zero amplitude at the damper attachment point. The mode shapes after shifting

the phase can be expressed as:

( ) ( )zj

k k k k Y x e Y x= (3.39)

where z is the phase of the complex amplitude at the damper, which is given by

[ ]

[ ]2 2

2 2

Im ( )tan

Re ( )z

Y

Y =

(3.40)

Substituting (3.36) with 2 2x = into (3.40) yields the following expression:

[ ]1 2

2

( / )( / ) (1 )tan

tan ( )( / )

ci ci

z

ci i

i L L

i L

+

(3.41)

With the substitution2 1/ 1 /L L= , the argument of the tangent function in the

denominator on the right-hand side of (3.41) can be expanded as follows:

1( )(1 / ) [ ( 1) ]

ci i ci ii L i + + (3.42)

in which the approximation1 1 2

( / ) ( / )ii L i has been introduced, and the term

1( / )i L has been neglected because i and 1( / )L are both assumed to be 1 . It

is then noted that tan[ ( ( 1) )] tan[ ( 1) ]ci i ci i

i + = , and because 1i

, the

tangent function can be approximated by its argument. Eq. (3.41) then reduces to

1 2( / )( / ) (1 )

tan( 1)

ci ci

z

ci i

i L L

(3.43)


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68

With the approximations2

( / ) 1L and1

( / )i

i L , (3.43) reduces to

tan1

ciz

ci

(3.44)

The following approximations can be obtained from (3.44):

sin z ci (3.45)

cos 1z ci (3.46)

Using the identity cos sinzjz z

e j = with the approximations in (3.45) and (3.46),

the phase-shifted mode shapes (3.39) can be expressed as

( ) ( 1 ) ( )k k ci ci k k Y x j Y x + (3.47)

Eq. (3.47) can be expanded into real and imaginary parts; the real part of the phase-

shifted mode shape is given by

Re ( ) 1 Re[ ( )] Im[ ( )]k k ci k k ci k k

Y x Y x Y x (3.48)

and the imaginary part is given by

Im ( ) Re[ ( )] 1 Im[ ( )]k k ci k k ci k k

Y x Y x Y x + (3.49)

Substituting (3.36) into (3.49) yields the following expression:

22 2

1 2 2

Re ( ) 1 sin ( )

1 cos ( )

ci ci i

ci ci ci i

xY x A i

L

x xA i i

L L L

+

+

(3.50)

The sine in (3.50) can be expanded as

2 2 2 2 2sin ( ) sin cos cos sinci i ci i ci ix x x x x

i i iL L L L L

+ = + (3.51)


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69

Because 1i

, ( )2cos / 1ci ix L and ( )2 2sin / / ci i ci ix L x L , and

(3.51) can be approximated by

2 2 2 2

sin ( ) sin cosci i ci ix x x x

i i iL L L L

+ + (3.52)

The cosine in (3.50) can be expanded as

2 2 2 2 2cos ( ) cos cos sin sinci i ci i ci ix x x x x

i i iL L L L L

+ = (3.53)

And using the same approximations as used in (3.52), (3.53) can be approximated by

2 2 2 2

cos ( ) cos sinci i ci ix x x x

i i iL L L L

+ (3.54)

Substituting (3.52) and (3.54) into (3.50) and introducing the approximation

1( / ) ii L allows some cancellation of terms, and an additional term can be neglected

because it has a factor of1

( / )i L , where i and 1( / )L are both assumed to be

1 . The real part of the phase-shifted mode shape can then be expressed as

( )2 2 2Re ( ) 1 sin / ciY x A ix L (3.55)

Eq. (3.55) is an expression for the undamped mode shapes of the cable, with a scaling

factor of 1 ci . It was previously shown in (3.37) that the mode shapes are

approximately linear over the shorter cable segment, and because1/ 1L , the real part

of the mode shape can be approximated as

( )Re ( ) 1 sin / ciY x A ix L (3.56)

in which x is a coordinate originating at the end of the cable nearer the damper and

spanning the entire length of the cable. Eq. (3.56) is approximately linear over the


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70

shorter segment, and is equivalent to (3.55) over the longer segment, but scaled by a

factor of 1( 1)i .

A similar asymptotic approximation can be developed for the imaginary part of

the phase-shifted mode shapes. Substituting (3.36) into (3.49) yields the following

expression:

22 2

1 2 2

Im ( ) sin ( )

(1 ) cos ( )

ci ci i

ci ci ci i

xY x A i

L

x xA i i

L L L

+

+ +

(3.57)

The sine in (3.57) can be alternatively expressed as

[ ]2 2sin ( ) sin ( (1 ) )ci i i ci ix x

i iL L

+ = + (3.58)

Eq. (3.58) can then be expanded as

2 2 2

2 2

sin ( ) sin ( ) cos (1 )

cos ( ) sin (1 )

ci i i ci i

i ci i

x x xi i

L L L

x x

i L L

+ = +

+

(3.59)

Because 1i

, the following approximations can be used: [ ]2cos (1 ) / 1ci ix L

and [ ]2 2sin (1 ) / (1 ) / ci i ci ix L x L . Eq. (3.59) can be approximated by

2 2 2 2sin ( ) sin ( ) (1 ) cos ( )ci i i ci i ix x x x

i i iL L L L

+ + + (3.60)

Substituting (3.60) into (3.57) and introducing the approximation 1( / ) ii L allows

cancellation of terms, and the expression for the imaginary part of the phase-shifted mode

shapes reduces to

[ ]2 2 2Im ( ) sin ( ) / ci iY x A i x L + (3.61)


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71

Noting that1 2

( / )i

i = , it can be shown that (3.61) is equivalent to

2 2 2 2Im ( ) sin( / )ciY x A ix (3.62)

Eq. (3.62) is an expression for the clamped mode shapes of the cable with zero

amplitude at the damper location, scaled by a factor of ci . Because (3.62) indicates a

zero amplitude at the damper location and (3.37) indicates that the mode shapes are

approximately linear over the shorter cable segment, the imaginary part of the phase-

shifted mode shape has approximately zero amplitude over the shorter cable segment, and

can be expressed as

[ ]1 1 2Im ( ) H( )sin ( ) / ciY x A x i x (3.63)

in which H( ) is the Heaviside function andx is a coordinate along the entire cable length

as in (3.56). Eq. (3.63) gives a zero amplitude over the shorter segment and is equivalent

to (3.62) over the longer segment, but scaled by a factor of 1( 1)i , which is the same

factor by which the real part in (3.56) was scaled, so that the relative phasing is

unchanged. Using (3.56) and (3.63) the asymptotic approximation for the complex mode

shapes can be written as

( ) [ ]{ }1 1 2( ) 1 sin / H( )sin ( ) / ci ciY x A ix L j x i x + (3.64)

The two sinusoids in (3.64) take on their maximum value of unity at approximately the

same location, so that the amplitude at this location of peak displacementpeak

x can be

approximated as

( ) 1peak ci ci

Y x A j + (3.65)


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72

Eq. (3.65) indicates that the magnitude of the peak displacement is asymptotically

equivalent to the amplitude coefficientA:

max[ ( )]Y x A (3.66)

Using (3.64) the amplitude at the damper location1

( )Y = can be approximated as

( )11 sin / ciA i L (3.67)

Eq. (3.67) indicates that 0 as 1ci

, as would be expected.

To understand the nature of the complex mode shapes it is helpful to consider the

evolution of the displaced profile with time; the cable displacement can be expressed as a

function of time using ( , ) Re[ ( ) ]y x Y x e = . Substituting (3.64) into this expression and

expanding e as (cos sin )e j + then yields the following expression:

( ) [ ]{ }1 1 2( , ) 1 sin / cos H( )sin ( ) / sinci ciy x Ae ix L x i x (3.68)

where = .

Figure 3.9 shows plots of the first complex mode for a damper location of

1/ 0.05L = and for three different values of the clamping ratio: 0.05

ci = , 0.5

ci = ,

and 0.95ci

= ; each column corresponds to a particular value of the clamping ratio. The

first plot in each column shows the real and imaginary parts of the complex mode shape,

from (3.56) and (3.63), with A = 1; as the clamping ratio increases, the amplitude of the

real part (which looks like the undamped mode shape) decreases and the amplitude of the

imaginary part (which looks like the clamped mode shape) increases. This decrease in

amplitude of the real part indicates that the amplitude at the damper location is tending to

zero as the damper becomes more rigid.


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73

Figure 3.9: Complex Mode Shapes for Three Different Clamping Ratios (mode 1,1/ 0.05L = ):

ci =

(second column), and 0.95ci

= (third column). First Row Shows Real (black) and Imaginary (gray)

Shapes; Subsequent Rows Show Displaced Profiles at Six Successive Instants in the First Half-

-1

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 =

-1

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

/5 =

-1

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2 /5 =

-1

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

/ 2 =

-1

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

3 /4 =

-1

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

=

0

0.2

0.4

0.6

0.8

1

0 0 .2 0 .4 0 .6 0 .8 1

Re[ ( )]

Im [ ( )]

Y x

Y x

0.5ci

=

0

0.2

0.4

0.6

0.8

1

0 0 .2 0 .

R e[ ( )]

Im [ ( )]

Y x

Y x

-1

0

1

0 0.1 0.2 0.3 0.4

-1

0

1

0 0.1 0.2 0.3 0.4

-1

0

1

0 0.1 0.2 0.3 0.4

-1

0

1

0 0.1 0.2 0.3 0.4

-1

0

1

0 0.1 0.2 0.3 0.4

-1

0

1

0 0.1 0.2 0.3 0.4

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0 .8 1

Re[ ( )]

Im [ ( )]

Y x

Y x

0.95ci

=

-1

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 =

-1

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

/5 =

-1

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2 /5 =

-1

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

/2 =

-1

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

3 /4 =

-1

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

=

/x L /x L

0.05ci

= 0.5ci

=


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74

Subsequent plots in each column of Figure 3.9 show the displaced profile at six

successive instants in the first half-cycle of oscillation, from (3.68) with A = 1, and

without including the decay due to the exponential factor e . Krenk (2000) presented

similar plots showing the evolution of the displaced profile for an optimal damper in

mode 1. When 0= , corresponding to the first instant at which the displaced profiles

are plotted, each profile is equivalent to the real part of the complex mode shape. At this

instant, the slope is continuous across the damper location and the velocity at this point is

zero, both of which imply that the damper force is zero at this instant. When / 2 = ,

corresponding to the fourth instant at which the displaced profiles are plotted, each

profile is equivalent to the imaginary part of the complex mode shape. At this instant, the

amplitude at the damper location is zero, while the velocity at the damper location takes

on its maximum value as do the slope discontinuity at the damper and the force in the

damper. For the case of the optimal clamping ratio, 0.5ci

= , the peak displacement

occurs when 3 / 4 = , at which instant the contributions from the real and imaginary

parts have the same amplitude. When = , the profile is the same as when 0= , but

with opposite sign.

3.5 Solution Characteristics under Large Frequency Shifts

When the frequency difference i in a given mode becomes large, as is the case in

higher modes and for larger values of 1/L, the approximate relations (3.30), (3.31), and

(3.34) become less accurate. Figure 3.10 shows the same plot as in Figure 3.8(c), but for

a damper located further from the end of the cable, at 1/L=0.05, and a distinct curve has

been plotted for each of the first eight modes, with an arrow indicating the trend with

increasing mode number. It is evident that the curves no longer collapse together. The


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75

solution curves for the first few modes agree quite well with the curves in Figure 3.8(c);

however, as the mode number increases, yielding larger values ofi, departures become

significant, and the optimal damping value i,opt/( 1/L) and the corresponding abscissa

value opt both increase noticeably with mode number. Numerical investigation in the

first five modes over a range of 1/L revealed that i,opt/( 1/L) increases monotonically

with i in a given mode, to take on a value of 1.2 - 1.3 as i0.5, and opt also

increases monotonically with i, taking on a value of about 0.18 as i0.5. When

i reaches 0.5 in a given mode, new regimes of behavior are observed which are

discussed in the following sections.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

L

i

1

increasing mode number

Figure 3.10: Normalized Damping Ratio versus Normalized Damper Coefficient; ( 1/L =

0.05, First 8 Modes)

3.5.1 Special Cases of Vertical Solution Branches

In investigating and categorizing the complicated characteristics of solution

branches to the phase equation (3.10) for large frequency shifts, it useful to begin by


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76

considering some special cases of vertical solution branches, which serve as boundaries

between different regimes of behavior. These branches extend from

=0 to

=

[from

=0 to =1 according to (3.13)] at a constant frequency . Inspection of (3.10) reveals

that if the following conditions are satisfied:

0)2sin( 1 =L (3.69)

2sin(2 ) 0L = (3.70)

sin(2 ) 0 = (3.71)

then (3.10) is trivially satisfied for all values of , yielding a vertical solution branch.

The values of that satisfy (3.69) are the two sets of critically damped frequencies

previously introduced in (3.23) and (3.24). The condition (3.71) is satisfied ifcan be

expressed as i or as i+1/2, where i is an integer. It is noted that the additional requirement

(3.70) follows directly from (3.69) and (3.71), and consequently, (3.69)-(3.71) are all

satisfied in four distinct cases, corresponding to four types of vertical solution branches.

To aid in interpreting these special cases, Figure 3.11 presents a plot of the different sets

of nondimensional frequencies, oi (3.19), ci (3.20), ci(1) (3.21) which is equal to

crit,i(1) (3.24), and

crit,i (3.23), against 1/L. The combinations of 1/L and

corresponding to the four distinct types of vertical solution branches are indicated with

different symbols. These four types are discussed in the following paragraphs, and the

contexts in which they arise are discussed in subsequent sections.


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77

0

1

2

3

4

5

6

7

8

9

10

0 0.1 0.2 0.3 0.4 0.5

Frequency Curves

Vertical Branches

Type 1

Type 2

Type 3

Type 4

L1

)1(

,

)1(

,

icritci

icrit

ci

oi

=

Figure 3.11: Undamped Frequencies, Clamped Frequencies, and Critically Damped

Frequencies versus Damper Location with Vertical Solution Branches Indicated

Along a vertical solution branch, (3.11) can be expressed as

( )

( )

( )

( )0

2cosh

2sinh

2cosh

2sinh

2

2

1

1 =++

++ Tm

c

bL

L

aL

L

(3.72)

where ( )La 12cos = and ( )Lb 22cos = , and because is constant along

each branch, a and b are also constant.

Type 1: crit,n = i. In these cases, indicated with hollow circles in Figure 3.11, the

nth

critically damped frequency crit,n (3.23), where n is an integer, coincides with an

undamped frequency, oi=i. Equating these expressions and solving for 1/L yields

1/L=(n-1/2)/i, where n

(i+1)/2, which corresponds to a damper located at the nth


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78

antinode of mode i. In this case, a=1 and b=1 in (3.72), from which it can be shown that

for this case, Tmc/ 2 monotonically from zero as from zero.

Type 2: crit,n = i+1/2. In these cases, indicated with solid squares in Figure 3.11,

the nth critically damped frequency,

crit,n (3.23), equals i+1/2. Equating these

expressions and solving for 1/L yields 1/L=(n-1/2)/(i+1/2), where n

(i+1)/2. It is also

noted that in these cases crit,n coincides with a clamped frequency cp (3.20), wherep is

also an integer. Using these expressions, the intersection frequency can be expressed in

terms of n and p as (i+1/2)=n+p-1/2, and the corresponding damper location can be

expressed as 1/L = (n-1/2)/(n+p-1/2), where n

p. In this case, a=1 and b=1 in (3.72),

and using the resulting expression it can be shown that Tmc/ takes on a minimum

value of

11 2

1min

sinh( 2 )(1 )

cosh( 2 ) 1

o

o

Lc

LTm

= +

+

(3.73)

which is less than 2, at a value o which satisfies the equation

1)2cosh()()2cosh()( 1221 = LLLL . As decreases from o to

approach zero, Tmc/ increases monotonically to approach infinity, and as increases

from o to approach infinity, Tmc/ increases monotonically to approach 2. It will be

observed subsequently that this type of vertical solution branch actually corresponds to

two solution branches that intersect at the intermediate value = oand diverge; along one

branch as Tmc/ 2, and along the other branch 0 as Tmc/ .

Type 3: crit,n(1) = i. In these cases, indicated with solid circles in Figure 3.11, the

nth

critically damped frequency associated with the shorter cable segment, crit,n(1)

(3.24),


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79

coincides with an undamped frequency, oi=i. Equating these expressions and solving for

1/L yields 1/L=n/i, where n

i/2, which corresponds to a damper located at the nth node

of mode i. In this case, a=1 and b=1 in (3.72), and the resulting expression is the same

as (3.13) for the case of a purely real eigenvalue, for which it was observed that Tmc/

increases monotonically from 2 to approach infinity as 0 from .

Type 4: crit,n(1) = i+1/2. In these cases, indicated with hollow squares in Figure

3.11, the nth critically damped frequency associated with the smaller cable segment,

crit,n(1)

(3.24), equals i+1/2. Equating these expressions and solving for 1/L yields

1/L=n/(i+1/2), where n

i/2. In this case, a=1 and b=1 in (3.72), and it can be shown

that Tmc/ increases monotonically from 2 to approach infinity as 0 from .

3.5.2 Behavior in a Single Mode

To further explore the dynamics of the cable-damper system, is it helpful to

consider the evolution of a single solution branch of the phase equation (3.10) as the

damper is moved from the end of the cable past the first antinode to the first node of the

corresponding undamped mode; over this range, three distinct regimes of behavior are

observed for a given mode. The evolution of the solution branch associated with mode 2

will be considered, and Figure 3.12 shows a surface plot of

versus

and 1/L for mode

2; the surface was generated by incrementing 1/L from 0 to 0.5 and repeatedly solving

(3.10). Figure 3.13 is a companion to Figure 3.12 with a separate plot for each of the

three regimes; each curve in Figure 3.13 is a slice through the surface of Figure 3.12 at a

specific value of 1/L.


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80

Figure 3.12: Attainable Damping Ratios in Mode 2 vs. Frequency and Damper Location


36/45

81

0

0.05

0.1

0.15

0.2

0.25

0.3

2 2.1 2.2 2.3 2.4 2.5

05.01 =L

15.01 =L

19.01 =L

199.01 =L

a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

25.01 =L

3.01 =L

33.01 =L

22.01 =L

205.01 =L

b)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2

34.01 =L

4.01 =L

45.01 =L

49.01 =L

c)

Figure 3.13: Phase Equation Solution Branches for Mode 2 in Three Regimes:(a) Regime 1; (b) Regime 2; (c) Regime 3


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82

Regime 1: Characteristics of regime 1 are exhibited in a given mode i when

0

1/L


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83

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3

2.01 =L

Figure 3.14: Phase Equation Solution Branches for an Intermediate Case

Regime 2: Characteristics of regime 2 are exhibited in mode i when

1/(2i+1)


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84

from the familiar sinusoid to a shape resembling a hyperbolic sinusoid, as indicated by

(23). Interestingly, when c>ccrit, then no solution exists along a solution branch in regime

2; this indicates that ifc>ccritvibrations are completely suppressed for modes in regime 2.

However, other solutions emerge when c>ccrit; it was previously observed that solutions

with purely real eigenvalues exist only when c>ccrit, and it will be observed subsequently

that solutions associated with significant vibration of the shorter cable segment also

emerge when c>ccrit.

0

0.5

1

0 0.25 0.5 0.75 1

increasing c

)(xY

Lx

Figure 3.15: Evolution with Damper Coefficient of a Mode Shape in Regime 2(

1/L = 0.25, mode 2)

When 1/L=1/(2i1), corresponding to the upper limit of regime 2, an intersection

of adjacent solution branches occurs, similar to that depicted in Figure 3.14. In this case,

the (i1)th clamped frequency and the first critically damped frequency

crit,1 are both

equal to i1/2, and a vertical solution branch of type 2 occurs at =i1/2.

Regime 3: Characteristics of regime 3 are exhibited in mode i when 1/(2i

1)


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85

approaches oi, and the solution branch for the i

th mode shrinks to a single point at

=i

with

=0. This indicates, as expected, that no damping can be added to a given mode

when the damper is located at a node of that mode. Interestingly, when the damper is

located at the node of mode i, a vertical solution branch of type 3 occurs at =i. Vertical

solution branches of type 3 and 4 are associated with emergent vibrations of the shorter

cable segment, and the characteristics of these solution branches are considered in the

following section.

3.5.3 Modes with Significant Vibration in the Shorter Cable Segment

Figure 3.16 depicts several solution branches associated with the first clamped

mode of the shorter cable segment for several different values of 1/L, ranging from 1/3 to

1/2. Each curve originates at the clamped frequency of the shorter cable segment c1

(1)

(3.21) with

=0 (indicated with squares along the

-axis in Figure 3.16), and tends to the

critically damped frequency crit,1(1), (3.24) as 1 ( ) (indicated with diamonds

along the top of Figure 3.16). Because ci(1)

= crit,i(1)

, in each case the solution branches

originate and terminate at the same frequency. When 1/L=1/3, the damper is located at

the node of mode 3, and a vertical solution branch of type 3 occurs at

=3, plotted at the

right-hand edge of Figure 3.16. A circle and a cross are also plotted at this frequency

with =0 to indicate that the third undamped frequency, o3, and the second clamped

frequency, c2, also coincide with crit,1

(1) and c1

(1) for this damper location, as can be

seen in Figure 3.11. Similarly, when 1/L=1/2, the damper is located at the node of mode

2, and a vertical solution branch of type 3 occurs at

=3. When 1/L=0.4,

crit,1(1)=

c1(1)=2.5, and a vertical solution branch of type 4 occurs at

=2.5, plotted at the

center of Figure 3.16. Vertical solution branches of type 3 and 4 have similar


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86

characteristics, as discussed previously, and for each of these solution branches, the

origin of the solution branch at

= 0 is associated with c , and c decreases

monotonically to ccrit as 1 ( ). Consequently, ifcccrit. Fig. 13

shows a plot of the magnitude of the mode shape associated with the first clamped mode

of the shorter cable segment for 1/L=0.4 and for several different values of Tmc/ . For

slightly supercritical values of c, the damping is very large, and the magnitude of the

mode shape resembles a hyperbolic sinusoid. As c and 0, the mode shape takes

on the expected form of a half sinusoid to the left of the damper.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3

34.01 =

L

37.01 =L

4.01 =L

43.01 =L

47.01 =L

3

11 =L

2

11 =L

Figure 3.16: Phase Equation Solution Branches Associated with the First Clamped Modeof the Shorter Cable Segment

0

0.5

1

0 0.2 0.4 0.6 0.8 1

increasingc

Lx

)(xY

Figure 3.17: Magnitude of the Mode Shape Associated with the First Clamped Mode of

the Shorter Cable Segment ( 1/L = 0.4)


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87

3.5.4 Global Modal Behavior

The previous discussion of the three regimes of behavior considered damper

locations only up to the location of the first node, 1/L

1/i. When 1/L is increased past

the first node, the branches of solutions to the phase equation (3.10) initially exhibit the

same characteristics as those observed in regime 1. As the damper is moved still further

along the cable to approach the second antinode, a vertical solution branch of type 2 is

encountered at a frequency of i+1/2, and for damper locations past this point,

characteristics of regime 2 are exhibited. As the damper is moved past the second

antinode to approach the second node, a vertical solution branch of type 2 is encountered

at a frequency ofi1/2, and for damper locations past this point, characteristics of regime

3 are exhibited. The cycle continues past subsequent nodes and antinodes until 1/L=1/2,

at which point the subsequent behavior is dictated by symmetry. This periodic cycling

through the three regimes of behavior is conveyed graphically in Fig. 14. The curves of

limiting frequencies versus 1/L depicted in Figure 3.11 are used as the basis for this

graphical depiction, and the regions corresponding to each of the three regimes have been

indicated using different shadings. The shaded areas indicate ranges of and 1/L for

which solutions to (3.10) exist, and the type of shading indicates the characteristics of

solution branches in that region. Situations in which solution branches corresponding to

two adjacent modes intersect at a vertical solution branch of type 2 (as depicted in Figure

3.14) are indicated by a solid vertical line connecting the undamped frequencies of the

two adjacent modes.


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Figure 3.18: Regime Diagram of Frequency vs. Damper Location

In the interesting case of 1/L=1/2, the damper is located at an antinode of all the

odd-numbered modes and at a node of all the even-numbered modes. Vertical solution

branches of type 1 then occur at the undamped frequencies of each odd-numbered mode,

and 1 ( ) in these modes as cccrit, while =0 in the even-numbered modes.

When c>ccrit, vibrations in the odd-numbered modes are completely suppressed, and new

modes of vibration emerge, including a non-oscillatory decaying mode and a set of

modes with the same frequencies as the even-numbered modes, associated with vertical

solution branches of type 3. As c , 0 in this set of emergent modes, and these

modes take on an appearance similar to that of the undamped even-numbered mode

shapes, except that they are symmetric about 1/L=1/2, with a discontinuity in slope at the

damper.

3.6 Summary

Free vibrations of a taut cable with attached linear viscous damper have been

investigated in detail, a problem that is of considerable practical interest in the context of


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stay-cable vibration suppression. An analytical formulation of the complex eigenvalue

problem has been used to derive an equation for the eigenvalues that is independent of

the damper coefficient. This phase equation (3.10) reveals the attainable modal

damping ratios i and corresponding oscillation frequencies for a given damper location

1/L, affording an improved understanding of the solution characteristics and revealing

the important role of damper-induced frequency shifts in characterizing the response of

the system. The evolution of solution branches of (3.10) with varying 1/L reveals three

distinct regimes of behavior.

Within the first regime, when the damper-induced frequency shifts are small, an

asymptotic approximate solution to (3.10) has been developed, relating the

damping ratio i in each mode to the clamping ratio ci, a normalized measure

of damper-induced frequency shift. Asymptotic approximations have also been

obtained relating ci and i to the damper coefficient, and it is observed that these

approximations grow less accurate when the damper-induced frequency shifts

become large.

Characteristics of the second regime are exhibited when the damper is located

sufficiently near the antinode of a specific mode. For modes in this regime the

damping ratio increases monotonically with the damper coefficient, approaching

critical damping as the damper coefficient approaches a critical value

Tmccrit 2= . When c>ccrit, vibrations are completely suppressed for modes in

this regime; however, other modes of vibration emerge, including a non-

oscillatory decaying mode of vibration and a set of modes with significant

vibration in the shorter cable segment.


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Characteristics of the solution branches in regime 3 are similar to those in regime

1 except that the oscillation frequencies are reduced, rather than increased, by the

damper, and the attainable damping ratios approach zero as the damper

approaches the node.

A regime diagram has been presented to indicate the type of behavior that may be

expected in each of the first 10 modes for any given damper location. The complexities

of the solution characteristics in different regimes are potentially significant in practical

applications, and the approach and analysis presented herein facilitate an improved

understanding of the dynamics of the cable-damper system.

vibrations of a taut string with linear damper

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