216

CHAPTER 6

Vibrations of a Taut Str ing with Nonlinear Damper.

I I : Numer ical Solution

In order to verify the equivalent linear solutions developed in the previous chapter

for the power-law damper and the viscous damper with a friction threshold, a numerical

formulation of the free vibration problem of a taut string with a nonlinear damper

attached is developed using Galerkin’s method. Pacheco et al. (1993) used Galerkin’s

method in their numerical formulation of the complex eigenvalue problem for a taut cable

with a linear damper. The sinusoidal mode shapes of an undamped cable were used as

shape functions, and several hundred terms were required for adequate convergence in

the solution, creating a computational burden. To develop a numerical solution for the

case of a nonlinear damper, a more efficient formulation is necessary.

In the context of a study on semi-active damping of stay cables, Johnson et al.

(1999) found that the convergence could be improved by introducing a shape function

corresponding to the static deflected shape of the cable with a concentrated load at the

damper location (i.e., a piecewise linear function). It is noted that the equilibrium of

forces at the damper location requires a discontinuity in slope of the cable to balance the

force in the damper; because sinusoids are smooth functions, a large number of sine

terms is required to adequately approximate this discontinuity. It is also noted that the

shape functions used in the Galerkin’s method formulation must be able to capture the

two limiting cases depicted in Figure 3.6(a,b), corresponding to the limits of a zero

217

damper coefficient (the undamped mode shape) and an infinite damper coefficient (the

“clamped” mode shape). In the case of a linear damper it was observed that when the

damper-induced frequency shifts are small, if a given complex mode shape is normalized

so that the amplitude at the damper is purely real, then the real part of the mode shape is

asymptotically equivalent to the undamped mode shape, and the imaginary part is

asymptotically equivalent to the “clamped” mode shape. These observations all suggest

that the “clamped” and undamped mode shapes may be appropriate choices to use as

shape functions in developing an approximate numerical formulation of the problem for

application to nonlinear dampers

6.1 Formulation

Motivated by these observations, an approximate numerical solution will be

developed in which a solution is assumed of the form:

( , ) ( ) ( ) ( ) ( ) ( ) ( )T T Ty x L x L x xτ τ τ τ = = +� o o c c

�b

�b

�b (6.1)

where ( ) [ ( ) ( ) ]T T Tx x x= o c

� � � is a vector of shape functions and

( ) [ ( ) ( ) ]T T Tτ τ τ= +o cb b b is a vector of nondimensional time-varying coefficients. The

elements of ( )xo

� are undamped mode shapes of the cable

��

���

�=L

ixx

�

oiπ

sin)( (6.2)

and the elements of ( )xc

� are “clamped” mode shapes of the right-hand cable segment

11

1

( )( ) H( )sinci

i x�x x

L

π −= − � �−�

��

� (6.3)

where H( ) is the Heaviside function. These shape functions are illustrated in Figure 6.1.

218

)(1 xoΦ

)(2 xcΦ

)(1 xcΦ

)(2 xoΦ

•••

•••

Figure 6.1: Shape Functions for Numerical Formulation

The governing partial differential equation of motion for the cable (3.1) can be expressed

in terms of nondimensional time to1ωτ = , where mTLo )(1 πω = as follows:

2( , ) ( / ) ( , )y x L y xτ π τ′′=�� (6.4)

in which the overdot represents differentiation with respect to � , and the prime represents

differentiation with respect to x. A smooth residual over the two cable segments can then

be defined by substituting the assumed form of solution into the governing partial

differential equation (6.4) and defining the residual as the error in satisfying that

equation:

2

( , ) ( ) ( ) ( ) ( )T TS

Lr x x xτ τ τ

π� � ′′= − � �� �

�b

�b�� (6.5)

The equilibrium of forces at the damper location can be written as

� �1

( , ) dxT y x Fτ

=′ =

� (6.6)

in which � �1

( , )x

y x τ=

′�

denotes the change in slope of the cable across the damper and dF

is the force in the damper, which in the present study can be expressed as a function of

the velocity at the damper attachment point 1( , )v y τ= � � . The tension in the cable can be

expressed as 2 21( ) /oT m Lω π= ; substituting this expression into (6.6) and rearranging

219

yields the following form of the equilibrium equation, normalized commensurately with

(6.4):

� �1

2

21

( , ) dx

o

FLy x

mτ

π ω=� � ′ =� �� � �

(6.7)

For each of the elements of ( )xo

�, the slope is continuous across the damper, while for

each of the elements of ( )xc

�, the slope is zero to the left of the damper and nonzero to

the right. Consequently, the change in slope across the damper can be expressed in terms

of only the “clamped” shape functions as:

� �1

1( , ) ( ) ( )T

xy x Lτ τ

=′ ′= c c

�b

�� (6.8)

For each of the elements of ( )xc

�, the amplitude at the damper location is zero, and

consequently, the velocity is also zero. The velocity v at the damper can then be

expressed in terms of only the undamped shape functions as:

1 1( ) ( ) ( )Tov Lω τ= o o

�b�� (6.9)

An interior node residual at the damper location can then be defined by substituting (6.8)

and (6.9) into the equilibrium equation (6.7), and defining the residual as the error in

satisfying that equation:

2

1 1

121

( ) ( ) ( )( ) ( ) ( )

( )

Td o T

Do

F L Lr L

m L

ω ττ τ

ω π � �� ′= − � �

� �

o o

c c

�b �

b��

� (6.10)

Galerkin’s method is then applied, setting the inner product of the residual with each of

the comparison functions to zero to make the residual orthogonal to each of the

comparison functions:

1

110

, ( , ) ( ) ( , ) ( ) ( ) ( ) 0L

s s Dr r x x dx r x x dx rτ τ τ< >= + + =� �� � � ��

�� (6.11)

220

Integrating by parts allows some cancellation of terms, leading to the following equation:

1( )

0 DT T f′ ′� � � � � �

+ = −� � � � � �� � � �′ ′ � �� � � �� �

o oc o oc o oo

oc c oc c cc

� � � �b �b

� � � �bb

�� �

�� (6.12)

where the terms in each sub-matrix of (6.12) are defined as follows:

[ ] ij

L

ojoiij dxx�

x�

Lδ== �0 )()(

2o

� (6.13)

[ ] ij

L

cjciij Ldxx

�x

�L

�

���

� −== � 1

01)()(

2 �c

� (6.14)

[ ] ( ) ( )( )

1 1

20 2 21

2 1 / sin /2( ) ( )

1 /

L

oi cjij

j L i L�x

�x dx

L j i L

ππ

−= =

− −�

�oc

� � �

� (6.15)

[ ] 22 0

2( )

L

oi oj ijij

L �(x)

�x dx i δ

π′ ′ ′= =�o

� (6.16)

[ ] ( ) 1212 0

2( ) 1 /

L

ci cj ijij

L �(x)

�x dx i L δ

π−′ ′ ′= = −�c

�� (6.17)

[ ] ( ) ( )( )

21 1

2 20 2 21

2 1 / sin /2( )

1 /

L

oi cjij

i j L i LL (x)

x dx

j i L

ππ π

−′ ′ ′= =

− −�

�oc

� �

� (6.18)

and the normalized damper force Df on the right-hand side of (6.12) is given by:

2

1

2

( )d

Do

Ff

m Lω= (6.19)

The formulation represented by (6.12) becomes poorly conditioned numerically if a large

number of shape functions is used, because the undamped mode shapes ( )xo

�

themselves form a complete basis for the solution, so that the set of shape functions

( ) [ ( ) ( ) ]T T Tx x x= o c

� � � becomes linearly dependent as its size becomes large.

However, the advantage of the present formulation is that a good approximate solution

can be obtained using a small number of shape functions. In fact, for vibrations in a

221

particular mode i, an accurate approximation can be obtained using only two shape

functions: the i th undamped mode shape ( )oi xΦ and the i th clamped mode shape ( )ci xΦ .

Using only these two shape functions, the equations of motion (6.12) reduce to

121

1 1

1 1 sin( / )

(1 / ) (1 / ) 0oc oc oioi

Doc oc cici

b i Lbi f

L L bb

π−

Γ � � Γ � � � �� �+ = −� � � � � �� � � �Γ − Γ −� � � �� � � �� �

�� �

��� � (6.20)

in which

( ) ( )( )( )

1 1

1 1

2 1 / sin /

2 / /oc

L i L

L i L

ππ

−Γ =

−� �

� � (6.21)

and the normalized damper force Df (6.19) is a function of the velocity at the damper

attachment point v, which in this case can be expressed simply as:

1 1( )sin( / )o oiv L i L bω π= �� (6.22)

The small order of this “single-mode” formulation (6.20) offers considerable advantage

when performing numerical integration with a nonlinear damper force.

6.2 Estimation of Effective Damping Ratios

Eq. (6.12) can be integrated numerically for a given type of nonlinear damper to

yield a set of time histories of decaying oscillation. For comparison with the

performance of a linear damper and for evaluation of the equivalent linear

approximations developed in the previous chapter for nonlinear dampers, it is desirable to

compute from these time histories effective damping ratios, which should be expected to

depend on the amplitude of oscillation. The logarithmic-decrement method is often used

to estimate damping ratios from time histories of decaying free-vibration; in this method,

an exponential form of decay is assumed, and the damping ratio is estimated from the

decay in peak displacement amplitudes. However, for certain types of nonlinear

222

dampers, the decay of the resulting time histories may be far from exponential, and in

these cases, the logarithmic-decrement method may not give good results. Consequently,

an alternate method for estimating the damping ratio on the basis of energy dissipation

will be used herein. The method is first presented for a viscously damped single-degree-

of-freedom (SDOF) oscillator, and then expressions for the energy of cable vibration are

developed so that the method can be extended to estimate the amplitude-dependent

effective modal damping ratios for the cable with nonlinear damper.

6.2.1 Single-Degree-of-Freedom Oscillator

For a damped SDOF oscillator with harmonic forcing at the natural frequency, the

damping ratio can be estimated from the energy loss per cycle (Clough and Penzien

1975), and a derivation of this result is presented in the following, with a view to

extending the method to the vibrations of a taut string with attached damper. If the

damped oscillator is vibrating with a frequency ω and amplitude A under harmonic

excitation, the displacement can be expressed as:

sinx A tω= (6.23)

The potential energy can then be expressed as

2 2 21 1sin

2 2potE kx kA tω= = (6.24)

The velocity is given by

cosx A tω ω=� (6.25)

And the kinetic energy can be expressed as

2 2 2 21 1cos

2 2kinE mx m A tω ω= =� (6.26)

223

The total energy is then given by

2 2 2 21sin cos

2totE A k t m tω ω ω = +� (6.27)

If the forcing frequency is equal to the natural frequency, /k mω = , then the total

energy is invariant with time:

21

2totE A k= (6.28)

The work done by a viscous damper in one cycle of oscillation was computed previously

in (5.49), and is given by

2dW cAπ ω∆ = (6.29)

If the work done by the damper in one cycle of oscillation is normalized by the total

energy, the following expression results:

2d

tot

W c

E kmπ∆ = (6.30)

Noting that the damping ratio for an SDOF oscillator is defined as /(2 )c kmζ = , the

following expression for the damping ratio can be obtained from (6.30):

4

d

tot

W

Eζ

π∆= (6.31)

While this expression has been derived for forced vibration of an SDOF oscillator at the

natural frequency, it can be extended to obtain quite accurate approximations of the

modal damping ratios for free vibrations a cable with an attached damper.

6.3 Energy of Cable Vibration

To extend the approximation in (6.31) to estimate the damping ratios for a

damped cable, it is first necessary to develop expressions for the energy of cable

224

vibration. The total energy of vibration of a taut string can be expressed as the sum of its

potential energy and its kinetic energy. The kinetic energy of vibration of a taut string

can be expressed as

2

0 2

L

kin

m yE dx

t

∂� �= � �∂� �� (6.32)

Substituting the assumed form of solution (6.1) into (6.32) yields the following

expression for the kinetic energy:

2

21 0

1( ) ( ) ( ) ( ) ( )

2

L T Tkin oE m L x x dxω τ τ = +� � o o c c

�b

�b� � (6.33)

The orthogonality of the shape functions, as indicated by (6.13) and (6.14), allows some

simplification, and the kinetic energy can be expressed as

2 2 2 2 21 0 0 0

( ) 22

L L L

kin o oi oi ci ci oi cj oi cji j

mE L dx b dx b dx b bω

� �= Φ ⋅ + Φ ⋅ + Φ Φ ⋅� �

� �� �� � �� � � � (6.34)

Expressions for the integrals in (6.34) are given in (6.13), (6.14), and (6.15). Recalling

that the i th mode of the damped cable is well approximated by the two shape functions

( )oi xΦ and ( )ci xΦ , (6.34) can be rewritten as

3 2 ( ) ( )1

ˆ ˆ4

i xkin o kin kin

i

mE L E Eω = +� �

� � (6.35)

where ( )ˆ ikinE is the nondimensional kinetic energy in the i th damped mode:

( ) 2 21 1 1

1 1

2(1 / ) sin( / )ˆ 1 2(2 / ) ( / )

ikin oi ci oi ci

L i LE b b b b

L L i L

ππ

−� �= + − +� � −� �

� � �� � � �

� � (6.36)

and ( )ˆ xkinE is the kinetic energy coupled among the damped modes:

( ) 1 12 2 2

1

2 (1 / )sin( / )ˆ 2(1 / )

xkin oi cj

i j i

j L i LE b b

j i L

ππ≠

−= − −�

��� � � �

�

(6.37)

225

The potential energy can be treated in a similar manner. The potential energy of

vibration of a taut string can be expressed as

2

0 2

L

pot

T yE dx

x

∂� �= � �∂� �� (6.38)

Substituting the assumed form of solution (6.1) into (6.38) yields the following

expression for the potential energy:

2 2

0( ) ( ) ( ) ( )

2

L T Tpot

TLE x x dxτ τ′ ′ = +� � o o c c

�b

�b (6.39)

The orthogonality of the derivatives of the shape functions, as indicated by (6.16) and

(6.17), allows some simplification, and the potential energy can be expressed as

2

2 2 2 2

0 0 02

2

L L L

pot oi oi ci ci oi cj oi cji j

TLE dx b dx b dx b b

� �′ ′ ′ ′= Φ ⋅ + Φ ⋅ + Φ Φ ⋅� �

� �� �� � � (6.40)

Expressions for the integrals in (6.40) are given in (6.16), (6.17), and (6.18). In a similar

manner as in (6.35), (6.40) can be rewritten as

2

( ) ( )ˆ ˆ4

i xpot pot pot

i

E TL E Eπ = +� �

� � (6.41)

where ( )ˆ ipotE is the nondimensional potential energy in the i th damped mode:

1

( ) 2 2 21 1 1

1 1

2(1 / ) sin( / )ˆ 1 2(2 / ) ( / )

ipot oi ci oi ci

L i LE i b b b b

L L i L

ππ

− −� �= + − +� �� � −� �� ��

� � �

� � (6.42)

and ( )ˆ xpotE is the potential energy coupled among the damped modes:

( ) 2 1 12 2

1

2 (1 / )sin( / )ˆ 2(1 / )

xpot oi cj

i j i

j L i LE i b b

j i L

ππ≠

−= − −�

��� �

�

(6.43)

The total energy is given by the sum of the kinetic and potential energy:

tot kin potE E E= + (6.44)

226

Substituting mTLo )(1 πω = into (6.35) yields the following alternate expression for

the kinetic energy, in which it is evident that the dimensional coefficient is the same as

that for the potential energy in (6.41)

2

( ) ( )ˆ ˆ4

i xkin kin kin

i

E TL E Eπ = +� �

� � (6.45)

The total energy can then be expressed in nondimensional form as

( ) ( )2

4ˆ ˆ ˆi xtottot

i

EE E E

TLπ= = +� (6.46)

where ( ) ( ) ( )ˆ ˆ ˆi i ikin potE E E= + is the total nondimensional energy in the i th damped mode, which

can be expressed as

( ) ( ) ( ) ( )ˆ ˆ ˆ ˆi i i io c ocE E E E= + + (6.47)

where ( )ˆ ioE is the energy of vibration in the undamped mode shape:

( ) 2 2 2ˆ io oi oiE b i b= +� (6.48)

( )ˆ icE is the energy of vibration in the “clamped” mode shape:

1

( ) 2 2 21 1ˆ 1 1ic ci ciE b i b

L L

−� � � �= − + −� � � �� � � �

� �� (6.49)

and ( )ˆ iocE is the energy of coupling between the “clamped” and undamped mode shapes:

( )( ) 21 1

1 1

2(1 / ) sin( / )ˆ 2(2 / ) ( / )

ioc oi ci oi ci

L i LE b b i b b

L i L

ππ

−= +−� � � �

� � (6.50)

The expression ( ) ( ) ( )ˆ ˆ ˆx x xkin potE E E= + in (6.46) is the total nondimensional energy of coupling

among the damped modes:

( )( ) 21 12 2 2

1

2 (1 / )sin( / )ˆ 2(1 / )

xoi cj oi cj

i j i

j L i LE b b i b b

j i L

ππ≠

−= + − −�

��� � � �

�

(6.51)

227

When 1 / 1L� � and only the first few modes are included, the coupling term ( )ˆ xE is

quite small, because even if several modes have comparable amplitudes so that the terms

2( )oi cj oi cjb b i b b+� � are moderately sized for i j≠ , the coefficients of these terms are very

small, as plotted in Figure 6.2 for 1i = , for 1, 2, 3, and 4j = , and for three different

damper locations. The small magnitude of the coupling term ( )ˆ xE indicates that the total

energy of cable vibration is well approximated by the sum of the energy in each of the

damped modes, where the i th damped mode is represented by the two shape functions

( )oi xΦ and ( )ci xΦ .

0%

1%

2%

3%

4%

5%

6%

7%

8%

2 3 4 5

0.010.020.05

1 12 2 2

1

2 (1 / )sin( / )

(1 / )

j L i L

j i L

ππ

� �� ��

−− −� �

�1/ :L�

1i =

j

Figure 6.2: Magnitude of the Coefficients in the Energy Coupling Term ( )ˆ xE

If the “single-mode” formulation of (6.20) is used, then there is no potential for coupling

among damped modes, and the total energy is simply given by (6.47).

228

6.3.1 Cable with Attached Damper

As the cable oscillates with an attached damper, the damper performs dissipative

work on the cable, reducing its energy of vibration; the work done by the damper in one

half-cycle of oscillation can be expressed as:

1k

kd dW F v d

τ

ττ+∆ = ⋅� (6.52)

where kτ and 1kτ + are two successive instants at which the damper force passes through

zero, and v is the velocity of the cable at the damper location. The work done by the

damper can be nondimensionalized in the same way as the nondimensional expression for

the total energy in (6.46):

1

2

4ˆ k

kd dW F v d

TL

τ

ττ

π+∆ = ⋅� (6.53)

The work done by the damper is equivalent to the decrease in the total energy of cable

vibration over the same interval: ˆ ˆd totW E∆ = −∆ , where ˆ

totE∆ is given by

1ˆ ˆ ˆ( ) ( )tot tot k tot kE E Eτ τ+∆ = − (6.54)

Using (6.46), the change in total energy can be expressed as the sum of the change in

energy in each mode plus the change in the energy of coupling:

( ) ( )ˆ ˆ ˆi xtot

i

E E E∆ = ∆ + ∆� (6.55)

Where the change in nondimensional energy in mode i can be written as

( ) ( ) ( )1

ˆ ˆ ˆ( ) ( )i i ik kE E Eτ τ+∆ = − (6.56)

and the change in the energy of coupling ( )ˆ xE∆ is analogously defined. As discussed

previously in the context of Figure 6.2, the energy of coupling ( )ˆ xE is expected to be

229

small, and if it can be neglected, the work done by the damper is approximately

equivalent to the sum of the energy loss in each of the damped modes:

( )ˆ ˆ id

i

W E∆ ≅ − ∆� (6.57)

The expression (6.31) for the damping ratio of an SDOF oscillator can then be

generalized to estimate the damping ratios in each mode of the damped cable by

normalizing the drop in energy in each mode by the total energy in that mode

( )

( )

ˆ

ˆ2

i

i iave

E

Eζ

π∆≅ − (6.58)

where ( )ˆ iaveE is the average energy in mode i over the half-cycle of oscillation:

( ) ( ) ( )1

ˆ ˆ ˆ[ ( ) ( )] / 2i i iave k kE E Eτ τ+= + (6.59)

It is noted that (6.58) has a factor of 2π in the denominator, whereas the approximation

for the SDOF oscillator (6.31) has a factor of 4π ; this difference results because (6.58)

was developed for a half-cycle of oscillation, whereas (6.31) is for a full cycle of

oscillation. Eq. (6.58) was formulated using a shorter interval of time in order to obtain

increased resolution in the estimation of amplitude-dependent damping ratios and so that

the decaying value of the energy in the i th mode ( )ˆ iE would deviate less from its average

value ( )ˆ iaveE over the interval.

6.4 Ver ification for L inear Damper

The preceding formulation is first verified by applying it to the vibrations of a taut

string with linear viscous damper, for which the exact solution has previously been

obtained. For the linear damper, the damping ratios can be computed with this

formulation either directly from an eigenvalue analysis or by numerical integrating the

230

equations of motion and estimating the damping ratios on the basis of energy dissipation

using (6.58). The eigenvalue analysis is much more efficient, and the numerical

integration is only performed to verify the approximate expression (6.58) for the modal

damping ratios, which will be used for nonlinear dampers. Both methods are

implemented in the following and both show excellent agreement with each other and

with the exact solution.

6.4.1 Eigenvalue Analysis

In the case of a linear damper, the damper force (6.19) can be expressed as:

11

2 ( ) ( )TD

o

cf

mLτ

ω= o o

�b�� (6.60)

Because the force is directly proportional to the cable velocity in this case, the damper

force can be conveniently expressed as a damping matrix, and (6.12) can be rewritten as:

0

0 0T T

′ ′� � � � � � + + =� � � � � �� � � �� � ′ ′� � � � �� � � �

o oc o oc ooo o

oc c oc c cc c

� � bCb b

0 � �bb b

�� �

�� � (6.61)

or

=Mb +Cb + Kb 0�� � (6.62)

where the elements in the damping matrix are given by

[ ] ( ) ( )1 11

2 sin / sin /ij

o

ci L j L

mLπ π

ω=oC � � (6.63)

Eq. (6.62) can be recast in state-space form as

1 1− −

� � � �=� � � �� �− −� � �� �

0 I bbM K M C bb

�

��� (6.64)

or

231

=q Aq� (6.65)

Assuming an exponential form of solution in time, eλτ=q � , and substituting into (6.65)

then yields the following eigenvalue problem:

( )λ− =A I � 0 (6.66)

Eq. (6.66) can then be solved for the complex eigenvalues λ and eigenvectors � , and the

modal damping ratios can be computed from the real and imaginary parts of the

eigenvalue as in (3.9). Figure 6.3 shows a comparison of the first-mode damping ratio

computed from the approximate formulation (6.66) using the “single-mode” formulation

(6.20) with the exact solution computed from (3.6) for a damper location of 1 / 0.025L =�

– the normalized damping ratio 1/( / )i Lζ � is plotted against the normalized damper

coefficient κ , which was defined as

1

1o

ci

mL Lκ

ω≡ �

(6.67)

The agreement is remarkably good: the two curves are indistinguishable, and error

calculation for selected data points near the optimal portion of the curve indicated that the

approximate value for the damping ratio in mode 1 is obtained within 10-4 percent of the

exact value. This excellent agreement confirms the appropriateness of the selected shape

functions for capturing the features of this problem.

232

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

Galerkin

Exact

��

���

�

L

i

1�

ζ

κ

1 / 0.025

mode 1

L =�

Figure 6.3: Comparison of Damping Ratio Computed from Galerkin’s Method with the Exact Solution for a Linear Damper

6.4.2 Numer ical Integration

Alternatively, the damping ratios can be estimated using (6.58) from numerical

integration of the equations of motion (6.12), yielding an estimate of the damping ratio

from each half-cycle of oscillation over which the integration is performed. Figure 6.4

shows a resulting plot of the estimated first-mode damping ratio for 1 / 0.01L =� obtained

from numerical integration of the “single-mode” equations (6.20) against the normalized

peak modal amplitude of oscillation /A Lτ (the damper coefficient was selected for

optimality in mode 1). The damping ratio is very nearly independent of the oscillation

amplitude, as it should be for a linear damper, and the estimated damping ratio agrees

233

very well with the value obtained from an eigenvalue analysis using (6.66), which was

previously observed to be virtually indistinguishable the exact solution.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.002 0.004 0.006 0.008 0.01

Eigenvalue AnalysisNumerical Integration

��

���

�

L

i

1�

ζ

0.5050

0.5051

0.5052

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

/A Lτ

1

2

/ 0.01

opt

L

κ κ π −

=

= =

�

Figure 6.4: Amplitude-Dependence of Damping Ratio Computed from Numerical Integration (Inset Shows Highly Magnified Vertical Axis)

By varying the normalized damper coefficient 1/( )oc mLω and performing a series

of numerical integrations, the normalized damping ratio 1/( / )i Lζ � can be estimated over

a range of values of the normalized damper coefficient κ . Figure 6.5 shows a resulting

plot of 1/( / )i Lζ � against κ in which values obtained from numerical integration are

compared with values obtained from the eigenvalue analysis, and the two results are

virtually indistinguishable. This excellent agreement between the values obtained from

the eigenvalue analysis and from the numerical integration confirms the appropriateness

234

of the approximation (6.58) used in estimating damping ratios from numerical integration

on the basis of energy dissipation.

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

Eigenvalue AnalysisNumerical Integration

0.49

0.5

0.51

0.08 0.09 0.1 0.11 0.12 0.13

1 / 0.01

mode 1

L =�

��

���

�

L

i

1�

ζ

κ

Figure 6.5: Comparison of Damping Ratio Obtained from Eigenvalue Analysis and from Numerical Integration using Galerkin’s Method (Inset Shows Magnified Optimal Portion

of the Curve)

6.5 Power-Law Damper

In the case of a nonlinear power-law damper, with the force-velocity relationship

given in (5.4), the damper force (6.19) can be expressed as:

1 121

2 ( ) ( ) sgn ( ) ( )( )

T TD

o

cf

m L

β

β τ τω −

= � o o o o

�b

�b� �� � (6.68)

The parameter 21/ ( )oc m L βω − appeared previously in the equivalent linear solutions for

the power-law damper, for which the normalized damper coefficient κ was defined as

235

1

(2 1) 12

1( )o

Aci

m L L L

β ββτ

βκω

−−

−� � � �≡ � �� �

� �� �

� (6.69)

For specific values of 1 / L� , 21/ ( )oc m L βω − , and β , free vibrations of a taut cable with a

power-law damper can be simulated by solving (6.61) with the damper force given by

(6.68). Initial conditions should be specified that satisfy equilibrium of forces at the

damper location. Using the definition of the interior-node residual in (6.10), equilibrium

of forces for the power-law damper can be written as

21 1 12

1

( ) ( ) sgn ( ) ( ) [ ( )] ( )( )

T T T

o

cL

m L

β

βπ τ τ τω −

′ =� o o o o c c

�b

�b

�b� �� � � (6.70)

where it is noted that the elements in the vector 1[ ( )]L ′c�� are given by

11

( )(1 / )ci

iL

L

π′Φ =−

��

(6.71)

To initialize the cable to produce vibrations in a given mode i, an arbitrary initial velocity

( )oi ob τ� in the undamped mode shape can be specified (with ( ) 0oi ob τ = ), and the

corresponding displacement ( )ci ob τ in the clamped mode shape (with ( ) 0ci ob τ =� ) can be

computed from equilibrium (6.70):

11 12

1

(1 / )( ) sin( / ) ( ) sgn sin( / ) ( )

( )ci o oi o oi oo

L cb i L b i L b

i m L

β

βπτ π τ π τω −

− = �

� � �� � (6.72)

With these initial conditions, the equations of motion (6.12) can be integrated

numerically with the damper force given by (6.68), and the amplitude-dependent

damping ratios can be estimated over each half-cycle of oscillation using (6.58).

Figure 6.6 shows segments of time histories obtained from numerical integration

of the “single-mode” equations (6.20) for a square-root damper ( 0.5β = ) in the first