Influence of Bending Stiffness

7/30/2019 Influence of Bending Stiffness

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CHAPTER 4

Influence of Bending Stiffness

In the problem formulation for the taut string in the previous chapter, the

equilibrium of forces at the damper attachment point required a discontinuity in slope at

that point. Evidently, a real stay cable has a nonzero value of bending stiffness and

cannot kink in this way at the damper, but must have some finite curvature at the

damper attachment point. Tabatabai and Mehrabi (2000) considered the combined

effects of bending stiffness and sag on first mode damping ratios using a complex

eigenvalue analysis of a cable discretized into multiple elements. They also used a

database of stay cable properties from actual bridges to evaluate the range of relevant

parameters, and their investigation indicated that for a significant number of stay cables

the influence of bending stiffness could be significant, especially in the case of a damper

located near the end of the cable. This chapter presents a detailed investigation of the

influence of bending stiffness on the vibrations of a cable with attached damper by

modeling the cable as a tensioned beam and developing an analytical solution to the free-

vibration problem.

4.1 Dynamic Stiffness Formulation

The formulation in this chapter is developed using the dynamic stiffness method,

which has the following advantages for this problem:

For the important case of fixed-fixed boundary conditions, the global

dynamic stiffness matrix takes on a very simple form, allowing much of


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the solution to be carried out analytically and enabling important insights

into the solution characteristics.

The formulation in terms of end displacements and slopes allows more

complex boundary conditions to be readily incorporated (e.g., torsional

springs at the end supports).

It can be readily extended to more complex systems (e.g., a tensioned

beam with dampers and springs attached at multiple locations, or a system

of parallel tensioned beams interconnected with springs and/or dampers)

by simply summing the contributions from each cable segment into the

global stiffness matrix.

The formulation is numerically well-conditioned and generally applicable,

avoiding some difficulties that would be encountered in the higher modes

by a transfer matrix formulation [To avoid such difficulties in the transfer

matrix formulation Franklin (1989) used different solution technique for

higher modes and lower modes].

Clough and Penzien (1975) presented the derivation of the dynamic flexural-

stiffness matrix for a beam, and discussed the extension of this formulation to include the

effects of axial force on transverse-bending stiffness. The formulation was developed for

harmonic (non-decaying) oscillations, and hyperbolic functions are used to express the

spatial variation of the solution. Leung (1985) extended the dynamic stiffness

formulation to exponentially varying harmonic oscillations for both damped and

undamped beam structures (without axial force), also using hyperbolic functions for the

spatial solution. Gradin and Chen (1995) used a substructuring approach, combining the


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dynamic stiffness formulation of Leung (1985) with a transfer matrix formulation to

obtain a reduced-order global dynamic stiffness matrix for beam structures (without axial

force) with attached springs, masses, and dashpots.

The dynamic stiffness formulation developed herein includes the effect of axial

force and is developed for exponentially decaying harmonic solutions, as are expected in

the case of attached dampers. Rather than using hyperbolic functions for the spatial

variation of the solution, which have been observed to result in numerically ill-

conditioned problems, the present formulation expresses the solution in terms of

exponentials decaying from each end of the cable segment.

The dynamic stiffness matrix will be developed for a segment of a tensioned beam

with length , mass per unit length m, bending stiffnessEI, and an axial tension ofTas

depicted in Figure 4.1.

TT

m, EI

x

a2a1

12

y(x)

Figure 4.1: Tensioned Beam Segment

Assuming small deflections in a single plane and neglecting internal damping, the partial

differential equation of motion for transverse bending vibrations is given by:

4 2 2

4 2 20

y y yEI T m

x x t

+ =

(4.1)


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where y(x ,t) is the transverse deflection and x is the coordinate along the cable chord

axis. To solve this governing differential equation subject to the boundary conditions and

the continuity and equilibrium conditions, a separable solution is assumed of the form:

( , ) ( ) i ty x t Y x e = (4.2)

in which 1i = and the frequency is complex in general. Substituting the assumed

form of solution (4.2) into the equation of motion (4.1) yields the following ordinary

differential equation in the spatial coordinate:

4 22

4 20

d Y d Y EI T mY

dx dx + = (4.3)

Assuming a solution of the form

( ) xo

Y x Y e= (4.4)

yields the following equation

4 2 2 0EI T m + = (4.5)

Eq. (4.5) is a quadratic equation in 2 and its solutions are given by

2

2 2

2 2

T T m

EI EI EI

= +

(4.6)

Because is complex in general, the argument of the square root in (4.6) is also

complex, and [ ] represents either of the two values of the multi-valued function

1/ 2( ) . Eq. (4.6) yields four distinct values of :1

p = ,2

p = ,3

iq = , and

4iq = , where

2

2

2 2

T T mp

EI EI EI

= + +

(4.7)


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and

2

2

2 2

T T mq

EI EI EI

= + +

(4.8)

The following identity is readily verified from (4.7) and (4.8):

2 2 Tp qEI

= (4.9)

The general solution to (4.3) can then be expressed as

( ) px px iqx iqxY x A e B e C e D e = + + + (4.10)

Previous investigations formulated the dynamic stiffness matrix by expressing the spatial

variation of the solution (4.10) using hyperbolic and trigonometric functions. However,

as has been demonstrated in a previous study (Franklin 1989) and confirmed in the

context of the present investigation, expressing the solution in terms of hyperbolic

functions leads to a numerically ill-conditioned problem. For moderate values of their

arguments, the hyperbolic sine and cosine terms take on very large values, while their

difference is quite small, which results in numerical difficulties for many practical

problems. It is also observed qualitatively that for large values of axial tension, the

hyperbolic terms contribute primarily in a small boundary layer region near the ends of

the cable, where they allow the solution to satisfy the boundary conditions on

displacement, slope, moment, and shear. For this reason, it is preferable to express the

solution in the following equivalent form, with an exponential term decaying from each

end of the cable segment.

( )( ) sin( ) cos( )px p xY x Ae Be C qx D qx = + + + (4.11)


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In formulating the dynamic stiffness matrix, the spatial variation of the solution can be

expressed in terms of displacements and slopes at the ends of the member using

displacement shape functions:

1 1 2 21 1 2 2( ) ( ) ( ) ( ) ( )Y x y x y x y x y x = + + + (4.12)

The displacements and slopes at the ends of the beam segment can be related to the

solution coefficients in (4.11) as follows:

1

1

2

2

(0) 1 1 0

(0) 0

( ) 1 cos sin

( ) sin cos

p

p

p

p

Y Ae

Y Bp pe q

Y Ce ql ql

Y Dpe p q ql q ql

= =

(4.13)

This relation, which can be expressed as WA

= , can then be inverted to solve for the

solution coefficients in terms of the end displacements, WA 1= :

1 2 3 4 1

1 2 3 4 1

1 2 3 4 2

1 2 3 4 2

a a a aA

b b b bB

c c c cCd d d d D

=

(4.14)

Explicit expressions for the terms in the matrix W-1 are given in the Appendix, and using

these terms, the displacement shape functions can then be expressed as

1

1

2

2

1 1 1 1

( )2 2 2 2

3 3 3 3

4 4 4 4

( )

( )

( )cossin( )

px

p x

y x a b c d e

y x a b c d e

a b c d y xqx

a b c d qxy x

=

(4.15)

Using these shape functions, the dynamic stiffness matrix can then be formulated by

enforcing equilibrium of shear force and bending moment at the ends of the beam


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segment. Using the moment-curvature relation for a tensioned beam with the assumed

separable form of solution (4.2), moment equilibrium at the two ends can be expressed

as:

2

1 2

0

( 0, ) i t i t

x

d YM x t M e EI e

dx

=

= = = (4.16)

2

2 2( , ) i t i t

x

d YM x t M e EI e

dx

=

= = =

(4.17)

Similarly, shear equilibrium at the two ends can be expressed as:

3

1 3

0( 0, )

i t i t

x

d Y

V x t V e EI edx

=

= = = (4.18)

3

2 3( , ) i t i t

x

d YV x t V e EI e

dx

=

= = =

(4.19)

Substituting into (4.16) (4.19) the expression (4.12) for Y(x) in terms of displacement

shape functions and end displacements, writing the result in matrix form, and canceling

the exponential terms from both sides yields the following equation:

1 1 2 2

1 1 2 2

1 1 2 2

1 1 2 2

1 1

1 1

2 2

2 2

(0) (0) (0) (0)

(0) (0) (0) (0)

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

y y y yV

y y y yMEI

V y y y y

M y y y y

=

(4.20)

This dynamic stiffness relation can be expressed as KF local= , and using (4.15), the

local dynamic stiffness matrix can be expressed as:

3 3 31 2 3 4

2 2 21 2 3 4

3 3 3 31 2 3 4

2 2 2 21 2 3 4

0

0

sin cos

cos sin

p

p

local p

p

a a a ap p e q

b b b bp p e qEI

c c c cp e p q q q q

d d d d p e p q q q q

=

K

(4.21)


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The following symmetry properties of the displacement shape functions are observed:

1 2(0) ( )y y = (4.22)

1 2( ) (0)y y = (4.23)

1 2(0) ( )y y = (4.24)

1 2( ) (0)y y = (4.25)

As a consequence of these properties, the terms in the third and fourth rows of the local

stiffness matrix can be expressed using the terms in the first two rows:

11 12 13 14

21 22 23 24

13 14 11 12

23 24 21 22

local

K K K K

K K K K

K K K K

K K K K

=

K (4.26)

Explicit expressions for these terms are given in the Appendix.

4.2 Fixed-Fixed Tensioned Beam with Damper

The dynamic stiffness formulation is now applied to the particular problem

depicted in Figure 4.2: an axially loaded beam with fixed supports at both ends and a

damper attached at an intermediate point, dividing the beam into two segments, where

2 1> . This problem is of particular interest in the context of stay cable vibration

suppression in bridges.


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L

1

1x

2

c TTm, EI

2x

A

Figure 4.2: Fixed-Fixed Tensioned Beam with Damper

A formulation of the problem with more general support conditions is presented in

Section 4.7, but the fixed-fixed case is investigated in detail here because the problem is

of smaller order, allowing a more concise presentation of the problem, and revealing

many of the important features. The dynamic stiffness method is formulated in terms of

displacements and slopes at the ends of each segment, and because the both the

displacement and slope are constrained to be zero at the fixed supports at each end, the

problem depicted in Figure 4.2 can be formulated in terms of only two unknowns: the

amplitude and slope at the damper, denoted A and , respectively. The force in the

damper is linearly proportional to the velocity of the beam at the damper attachment

point, and can be expressed as a function of the amplitude at the damper:

1

( ) i tdx

yF t c ci Ae

t

=

= =

(4.27)

Assembling the contributions from the two beam segments into a global stiffness matrix

then yields the following equation:

(1) (2) (1) (2)

33 11 34 12

(1) (2) (1) (2)

43 21 44 22

( ) ( )

0( ) ( )

A AK K K K ci

K K K K

+ + =

+ + (4.28)


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in which the superscript indicates the number of the beam segment corresponding to each

term in the global stiffness matrix. Noting the symmetry properties given in (4.26), the

contributions from beam segment 1 can be expressed using the terms in the first two rows

of the local stiffness matrix. Because the damper force is proportional to the amplitude at

the damper, it can be moved to the left hand side of the equation, and the complex

eigenvalue problem for free vibrations of the beam-damper system can be written as:

(1) (2) (2) (1)

11 11 12 12

(2) (1) (1) (2)

21 21 22 22

( ) ( )

( ) ( )

AK K ci K K

K K K K

+ + =

+ 0 (4.29)

The complex eigenvalues then correspond to values of for which the determinant of

the 2-by-2 matrix in (4.29) equals zero. Setting this determinant equal to zero and

solving for the damper coefficient c yields the following equation:

(2) (1) (2) (1)(1) (2) 21 21 12 12

11 11 (1) (2)

22 22

( )( )( )

( )

K K K K iK K c

K K

+ =

+ (4.30)

Noting that the damper coefficient c is purely real, the left-hand side of (4.30) must also

be purely real, which yields the following equation, independent ofc.

(2) (1) (2) (1)(1) (2) 21 21 12 12

11 11 (1) (2)

22 22

( )( )1Re ( ) 0

( )

K K K K K K

K K

+ =

+ (4.31)

Eq. (4.31) will be referred to as the phase equation, and is analogous to the equation

referred to by the same name in the treatment of the vibrations of a taut string with

attached damper in Chapter 3. Solution branches to (4.31) give permissible values of the

complex frequency for a given 1/L , thus revealing the attainable values of modal

damping with their corresponding oscillation frequencies, and the evolution of these

solution branches under varying parameters is helpful in characterizing the response of

the system. It is important to note that the contributions to (4.30) and (4.31) from the


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local stiffness matrix for each beam segment (e.g., (1)11

K and (2 )11

K ) are functions ofp

and q, and consequently, they also depend on the complex frequency . The real and

imaginary parts of the complex frequency will be denoted as follows

Re( ) = (4.32)

Im( ) = (4.33)

These quantities are analogous to the previous definitions for the taut string in (3.8a,b):

is the oscillation frequency and is the rate of exponential decay. However, positive

values of correspond to decaying oscillation with the presently assumed form of

solution in (4.2), whereas negative values of corresponded to decaying oscillation for

the taut string. Also, (4.32) and (4.33) are dimensional, in contrast with the

nondimensional definitions in (3.8a,b). Alternative nondimensional versions of (4.32)

and (4.33) will be introduced in subsequent sections.

4.2.1 Nondimensionalization of Stiffness Matrix

To achieve results of general applicability and to facilitate presentation, it is

useful to nondimensionalize the problem formulation. The amplitude at the damperA is

nondimensionalized by the cable length:

( / )A A L= (4.34)

The terms in (4.29) from the stiffness matrix for each segment are normalized by the

bending stiffness and the length:

( ) 3 ( )

11 11( / )k kk L EI K = (4.35)

( ) 2 ( )

12 12( / )k kk L EI K = (4.36)


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( ) 2 ( )21 21

( / )k kk L EI K = (4.37)

( ) ( )

22 22( / )k kk L EI K = (4.38)

With these normalizations, the complex eigenvalue problem (4.29) can then be rewritten

in nondimensional form as:

(1) (2) 3 (2) (1)

11 11 12 12

(2) (1) (1) (2)

21 21 22 22

[ ( / ) ] ( )

( ) ( )

k k i cL EI k k A

k k k k

+ + =

+ 0

(4.39)

Explicit expressions for the nondimensional terms in the stiffness matrices (4.35) (4.38)

are given in the Appendix; these terms are functions of nondimensional versions of the

mode shape parametersp (4.7) and q (4.8):

p pL= (4.40)

q qL= (4.41)

Alternative explicit expressions for p and q are given in the following sections,

depending on the choice of nondimensionalization for the complex frequency .

Nondimensionalization of the identity in (4.9) yields the following relation:

2 2 2p q = (4.42)

where is a nondimensional bending stiffness parameter, as used by Tabatabai and

Mehrabi (2000):

2TL

EI= (4.43)

When becomes large, bending effects are less significant, and the tensioned beam

behaves more like a taut string; when is small, bending effects predominate. Using a

database of stay-cable properties, Tabatabai and Mehrabi (2000) report that nearly all


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bending stiffness parameters () are within the range of 10-600, with 82% of the cables

having values of larger than 100.

The appropriate choice of nondimensionalization for the complex frequency

depends on the magnitude of. In most cases a taut-string nondimensionalization will

be used, to facilitate comparison with the taut-string results, but in cases of zero tension,

the taut-string nondimensionalization cannot be used, and an alternative beam

nondimensionalization will be employed.

4.2.2 Beam Nondimensionalization of Frequency

When is small, indicating that the axial tension T is small relative to the

bending stiffness 2/EI L , the undamped natural frequencies are close to the natural

frequencies of a fixed-fixed beam, and it is helpful to introduce the following beam

nondimensionalization of the frequency:

12

2

EI

L m

= (4.44)

Substituting (4.44) into the definitions forp (4.7) and q (4.8), the following expressions

for the nondimensional values p (4.40) and q (4.41) can be obtained:

2 2 2 2 2/ 2 ( / 2) ( )p = + + (4.45)

2 2 2 2 2/ 2 ( / 2) ( )q = + + (4.46)

Using this nondimensionalization of frequency, the following expression for the

nondimensional damper coefficient, analogous to (4.30), can be obtained:


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(2) (1) (2) (1)

(1) (2) 21 21 12 1211 112 (1) (2)

22 22

( )( ) ( ) ( )

k k k k i cLk k

EI mk k

+ =

+ (4.47)

From (4.47) the following beam nondimensional version of the phase equation (4.31)

can be obtained:

(2) (1) (2) (1)(1) (2) 21 21 12 12

11 11 (1) (2)

22 22

( )( )1Re ( ) 0

( )

k k k k k k

k k

+ =

+

(4.48)

Under the beam nondimensionalization, the real and imaginary parts of the complex

frequency will be denoted and , respectively, as in (4.32) and (4.33). When

0

, corresponding to a beam without axial tension, p

and q

are equivalent and can

be expressed as:

p q = = (4.49)

4.2.3 Taut-String Nondimensionalization of Frequency

When is large, indicating that the tension T is large relative to the bending

stiffness 2/EI L , the natural frequencies in the absence of the damper are close to the

natural frequencies of a taut string, and it is convenient to introduce the following

nondimensionalization of frequency:

1

T

L m

=

(4.50)

From (4.50) and (4.44) it can be shown that the alternative nondimensional frequencies

are related by

( / ) = (4.51)


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It is also noted that ( / ) is equal to the ratio of the fundamental frequency of a taut

string ( / ) / L T m to the fundamental frequency of a pin-supported beam

Using (4.50), p (4.40) and q (4.41) can be expressed as:

2 2 2 2/ 2 ( / 2) ( )p pL = = + + (4.52)

2 2 2 2/ 2 ( / 2) ( )q qL = = + + (4.53)

Using this taut-string nondimensionalization of frequency, the following expression for

the nondimensional damper coefficient, analogous to (4.30), can be obtained:

(2) (1) (2) (1)(1) (2) 21 21 12 12

11 112 (1) (2)

22 22

( )( )( )

( )

k k k k i ck k

k k Tm + =

+

(4.54)

From (4.54) the following taut-string nondimensional version of the phase equation

(4.31) can be obtained:

(2) (1) (2) (1)(1) (2) 21 21 12 12

11 11 (1) (2)

22 22

( )( )1Re ( ) 0

( )

k k k k k k

k k

+ =

+

(4.55)

Under the taut-string nondimensionalization, the real and imaginary parts of the

complex frequency will be denoted and , respectively, as in (4.32) and (4.33).

4.2.4 Nondimensional Mode Shapes

For the fixed-fixed beam, the mode shapes can be expressed in terms of the

nondimensional amplitude A and the slope at the damper location. The

nondimensional mode shape over the shorter cable segment can be expressed as

2 2

(1) (1) (1)

1 1 1( ) ( ) ( )Y x A y x y x = + (4.56)

And the nondimensional mode shape over the longer segment can be expressed as


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1 1

(2) (2) (2)

2 2 2( ) ( ) ( )Y x A y x y x = + (4.57)

The shape functions in (4.56) and (4.57) are nondimensional versions of those in (4.15)

and are given by

1

1

2

2

( ) ( ) ( ) ( ) ( ) /1 1 1 1

( ) ( ) ( ) ( ) ( ) ( ) /

2 2 2 2

( ) ( ) ( ) ( ) ( )

3 3 3 3

( ) ( ) ( ) ( ) ( )

4 4 4 4

( )

( )

cos / ( )

s( )

k

k k

k k k k k px Lk

k k k k k p x Lk

k k k k k kk

k k k k k k

y x a b c d e

y x a b c d e

qx Ly x a b c d

y x a b c d

=

in /kqx L

(4.58)

Explicit expressions for each term of the coefficient matrix in (4.58) are given in the

Appendix. For a given value of the complex frequency , the amplitude and slope at the

damper of the corresponding mode shape at the damper can be related by the following

equation, obtained from the second row of (4.39):

(1) (2) (1) (2)

21 21 22 22( ) ( )k k A k k = + (4.59)

4.3 Non-Oscillatory Decaying Solutions

In the case of a taut cable without bending stiffness, it was previously observed in

Section 3.3.1 that solutions exists for which the cable decays without oscillation. Such

solutions also exist for the tensioned beam, and for these solutions, the frequency is zero

and is purely imaginary: i = . Using the taut-string nondimensionalization, p

(4.52) and q (4.53) can then be rewritten as:

2 2 2 2/ 2 ( / 2) ( )p = + (4.60)

2 2 2 2/ 2 ( / 2) ( )q = + (4.61)

Similarly, (4.54) can be rewritten as


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(2) (1) (2) (1)

(1) (2) 21 21 12 1211 112 (1) (2)

22 22

( )( )1( )

( )

k k k k ck k

k k Tm

+ =

+

(4.62)

For given values of and1/L , the nondimensional damper coefficient /c Tm can be

computed from (4.62) over a range of values of ; Figure 4.3 shows a resulting plot of

versus /c Tm for 1/ 0.05L = and for several different values of the bending

stiffness parameter . The curve corresponding to a taut string with zero bending

stiffness plotted previously in Figure 3.2 is also plotted with these curves for

reference. The curves corresponding to 1000= and to the taut-string result terminate

on the plot because numerical difficulties were encountered in evaluating the solution for

large values of, not because the solution actually ceases to exist.

Similar to the taut-string result, it is evident in Figure 4.3 that zero-frequency

solutions only exist when /c Tm is greater than some critical value. In the taut-string

case that critical value was / 2c Tm = , and Figure 4.3 shows that this critical value is

increased by the influence of bending stiffness. The critical value of /c Tm for the case

of 10= is indicated in Figure 4.3 by a vertical dotted line at the lowest value of

/c Tm for which a solution exists. In contrast with the zero-frequency solution for the

taut string, for which only one solution for the decay rate existed for any given

supercritical value of /c Tm , two solutions for exist for each supercritical value of

/c Tm when the bending stiffness is nonzero. A vertical dotted line is plotted in Figure

4.3 at a supercritical value of /c Tm for the case of 10= , and the two solutions are

indicated with circles. The larger value of , which corresponds to a more quickly

decaying solution, is denoted the fast solution, and the smaller value of is denoted


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the slow solution. This behavior can be compared with that of the zero-frequency

solution for the SDOF oscillator, plotted in Figure 3.3, for which two solutions for the

decay rate exist for supercritical values of the damper coefficient. Figure 4.4 shows the

evolution with increasing /c Tm of the mode shapes associated with the fast and

slow solution branches. Both branches begin at the critical value of /c Tm with the

same value of , so initial mode shapes, plotted with a heavier line in Figure 4.4, are

identical. The evolution of the slow solution is similar to that previously observed for

the taut string in Figure 3.4, approaching the static deflected shape of the beam under a

concentrated load at the damper location as /c Tm .

0.01

0.1

1

10

100

1000

1 10 100 1000

10

50

100

1000

Taut String

cTm

2

:

1/ 0.05L =

"critical" damping

"slow"

solution

"fast"

solution

Figure 4.3: Nondimensional Decay Rate vs. Nondimensional Damper Coefficient with

Varying Bending Stiffness (1/ 0.05L = , Taut-String Nondimensionalization)


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0

1

0 0.2 0.4 0.6 0.8 1

increasing /c Tm"slow"

solution

a)

( )Y x

1

100

/ 0.3L

==

0

1

0 0.2 0.4 0.6 0.8 1

b)

increasing /c Tm

"fast"

solution

/x L

( )Y x

1

100

/ 0.3L

=

=

Figure 4.4: Evolution of Slow and Fast Non-Oscillatory Mode Shapes with

Increasing Nondimensional Damper Coefficient ( 100= ,1/ 0.3L = )

Figure 4.5 is similar to Figure 4.3, but corresponds to a damper located further

from the end of the cable:1/ 0.3L = . The critical values of /c Tm are closer to the

taut-string result in this case than in Figure 4.3. Unlike the taut-string case, for which the

critical value /crit

c Tm is independent of damper location, for a given value of the

bending stiffness parameter , /critc Tm varies with 1/L .


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0.01

0.1

1

10

100

1000

1 10 100 1000

10

50

1001000

Taut String

cTm

2

:

1/ 0.3L =

Figure 4.5: Nondimensional Decay Rate vs. Nondimensional Damper Coefficient with

Varying Bending Stiffness (1/ 0.3L = , Taut-String Nondimensionalization)

The critical value of the nondimensional damper coefficient /critc Tm can be

computed for given values of1/L and the bending stiffness parameter by computing

/c Tm over a range of values of , and determining the minimum value of /c Tm , as

indicated schematically in Figure 4.3 for the curve corresponding to 10= . Figure 4.6

shows a contour plot of /crit

c Tm , generated in this manner over a range in values of the

damper location (from 1/ 0.005L = to 1/ 0.1L = ) and bending stiffness parameter

(from 10= to 1000= ). For a given value of , it can be seen that /crit

c Tm

decreases toward the taut-string value of 2 as1/L increases. This indicates that the

influence of bending stiffness is most significant when the damper is located near a fixed


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support, as may be expected, because, as has been previously noted (e.g. Franklin 1989),

the bending effects for a tensioned beam are most significant in the region near the

supports.

Figure 4.6: Contour Plot of the Critical Value of the Nondimensional Damper Coefficient

/crit

c Tm vs. Bending Stiffness Parameter and Damper Location

In the case of a beam without axial tension ( 0= ), using (4.49) with i =

(beam nondimensionalization) p and q can written as

( 1)

2

ip q

+= = (4.63)

For a given damper location, the nondimensional damper coefficient /cL EI m can

then be computed from (4.47) over a range of values of (evidently, the

nondimensionalization /c Tm is not appropriate when 0T ), and as previously, the


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critical value of the nondimensional damper coefficient is given by its minimum value.

Figure 4.7 shows a plot of the critical value of the nondimensional damper coefficient

/crit

c L EI m against the damper location1/L for the fixed-fixed beam with zero

tension. The critical value /crit

c L EI m takes on very large values as the damper

approaches a fixed support and decreases to a value of 16.619 when the damper is near

midspan. Figure 4.8 shows a plot of the critical value of the nondimensional damper

coefficient normalized by the damper location ( ) 1/ ( / )critc L EI m L against 1/L , and

this normalized critical value is virtually constant for1/ 0.2L


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6.8

7

7.2

7.4

7.6

7.8

8

8.2

8.4

0 0.1 0.2 0.3 0.4 0.5

LmEI

Lccrit 1

L/1

6.934

Figure 4.8: Normalized Critical Value of the Nondimensional Damper Coefficient

( ) 1/ ( / )critc L EI m L vs. Damper Location for a Beam with Zero Tension ( 0= )

4.4 Limiting Cases of Non-Decaying Oscillation

In the case of a taut cable without bending stiffness, it was previously observed that

there are solutions for which the cable oscillates without decay; these solutions were

associated with the limits of 0c and c . For such solutions, the frequency is

purely real, = .

4.4.1 Undamped Modes

When 0c , the problem reduces to computing the natural frequencies and mode

shapes of a fixed-fixed tensioned beam. This problem has been previously solved and is

discussed by Wittrick (1986), who presents an iterative solution technique for the

Influence of Bending Stiffness

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