Response of a suspended cable to narrow-band random excitation with peaked P.S.D.

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SCIENCe ~D|RI=CTe ELSEVIER

MATHEMATICAL AND COMPUTER MODELLING

Mathematical and Computer Modelling 41 (2005) 1203-1212 www.elsevier.eom/locate/mcm

R e s p o n s e of a S u s p e n d e d C a b l e to N a r r o w - B a n d R a n d o m E x c i t a t i o n with Peaked P.S.D. M . H . KARGARNOVIN Mechanical Engineering D e p a r t m e n t , Sharif University of Technology Tehran, I.R. Iran B. MEHRI D e p a r t m e n t of M a t h e m a t i c a l Science, Sharif University of Technology Tehran, I.R. Iran D. YOUNESIAN Mechanical Engineering D e p a r t m e n t , Sharif University of Technology Tehran, I.R. Iran

(Received January 2003; revised and accepted November 2004) Abstract--The response of a suspended cables subjected to narrow-band random excitations with two types of peaked P.S.D. is formulated and analyzed. Banach fixed-point theorem is used for eigenfunction analysis of the differential-integral equations of motion for the first time in this paper. Fredholm approach also is used in the free vibration analysis of the suspended cable and then using Galerkin mode approximation method, power spectral density, and root mean square of the response are computed for two practical types of excitation. All of the calculated results converted to dimensionless quantities make their usage easier in vibration analysis of some practical cases such as vibration of moving track due to ground irregularity and vibration in power transmission lines due to vortex shedding. It is found that at the first crossover, at which repeated frequencies happen for the first two modes, the response of the cable is at lowest level. It is also found that the root mean square of the response of a suspended cable is lower than that of a linear cable. @ 2005 Elsevier Ltd. All rights reserved.

Keywords--Random vibration, Suspended cable, Power spectral density (P.S.D.), Galerkin's mode approximation method, Banach fixed-point theorem.

INTRODUCTION Because of its several applications, random vibration analysis of suspended cable has been an attractive subject in the engineering mechanics field in past couple of decades. Caughey [1] and Lyon [2] were among those researchers who tried to solve this problem for the first time. Both of them obtained the mean square response of a string subjected to a transverse planer Gaussian white noise excitation. T h e y found that the mean square deflection at each point was less than that of a linear string.

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et al.

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M.H. KARGARNOVIN

Later on, Anand and Richard [3,4] computed the mean square of large deflection of a nonlinear string subjected to a nonplaner Gaussion white excitation [3] and a planar narrow-band excitation [4] using perturbation and equivalent linearization techniques, respectively. They showed [3], transverse-transverse coupling reduces and longitudinal-transverse coupling increases the mean square deflection [4]. Furthermore, they demonstrated that in the case of a certain bandwidth, it is possible to have a triple-valued response exactly similar to the harmonic excitation. In 1983, Tagata calculated the amplitude probability density characteristic of a nonlinear stretched string under narrow-band random forcing by a polynomial approximation [5]. He also presented some results for the amplitude probability density function of the narrow-band random vibration of nonlinear string with three types of damping [6]. Chang and his coworkers [7] investigated the nonlinear interaction of in-plane and out-of-plane motions of a suspended cable in the neighborhood of 2:1 internal resonance. They used the Fokker-Planck equation together with Gaussian and non-Gaussian closures and Monte Carlo approach for numerical verification. Moreover, they found that away from the internal resonance condition, the response is governed by in-plane motion. In addition, the non-Gaussian closure solution is in good agreement with the numerical results and the bifurcation of the out-of-plane mode is predicted by Gaussion, non-Gaussian closures and also Monte Carlo simulation. Ibrahim and Chang [8] analyzed the response statistics such as R.M.S., P.S.D., and autocorrelation function of a suspended cable to random Gaussion process in the neighborhood of three simultaneous internal resonance conditions among four normal modes. They showed when the first in-plane mode is externally excited, its response can act as a parametric excitation to the other three modes. Martinelli and his coworkers [9] used finite-element method to analyze the dynamic response of suspended cables under stationary and nonstationary ]oadings. They showed the geometric nonlinearities and the aerodynamic forces produce a noticeable modal coupling characterized by both in-plane and out-of-plane components. In this paper, natural frequencies and mode shapes of a suspended cable are computed using two distinct methods, i.e., Banach fixed-point approach (BFPA) and the Fredholm approach (FA). The obtained results for natural frequencies and mode shapes are plotted versus a dimensionless parameter, known as static sag. By considering the principle of orthoganality of mode shapes [10], power spectral density of the response are computed using Galerkin mode summation approach and finally the root mean square of the response in two transverse and longitudinal directions are calculated at each point along the cable length. It has to be mentioned that the nature of excitations considered in this paper is in the form of peaked narrow-band power spectral densities, which are in good agreement with some practical excitations. Some demonstrable examples of this kind of excitation include vibration of transmission lines due to vortex shedding [11,12] and track vibration of crawlers due to ground corrugations [13]. 1. M A T H E M A T I C A L

MODELING

If U and V represent the respective longitudinal (x-direction) and transverse (y-direction) displacements of each point of the cable, the total kinetic and potential energies of the cable can be expressed as [10,13], IIK

IIp=

-

t ot / j

LL [T0~xx -V AEe2 ] dx, 2 ~x]

(1.2)

where L and A are the length and cross-sectional area of the cable, respectively, and T0, E, and p are the cable pretension, elasticity module, and mass per unit length, respectively, and Sxx is

Response of a Suspended Cable

1205

defined as [10],

OV

e=-

KV +-{1 0 U--d7 _

Ox

KV

(1.3)

with K being a constant parameter representing the static cable sag defined as K = ~.

(1.4)

Using Hamilton's principle and applying the technique of integration by-parts, we have

EA

-~z - K V

= p o t 2,

02V [OU ]02V f (x,t) + To-STx2 + E A K L -~x - K V = P-O-~ '

(1.6)

in which f ( x , t) represents the external force per unit length of the cable and can easily be replaced by -pa(t) when accelerating supports exist with acceleration of a(t) in the vertical (y) direction. By introducing the following dimensionless quantities,

U*

= _

UL,

V * = --VL,

t* = t

x

g,

S=-£, (1.7)

v~ = E A v2t _ To pgL ' pgL ' equations (1.5) and (1.6) reduce to

K*

K = -L '

f* (S,t*) - f (S,t*) p9

2 o [as* v~ ~ [ as

o2v,

2 f* (s, t*) + v~ ~

+

K*v~ [ o s

2. M E T H O D

OF

J =

- K'V*

]

-

a t *~ '

o2v. s t .~ '

(1.8) (1.9)

SOLUTION

In practice and over a technologically useful range of values of different parameters, the square of longitudinal wave speed (EA/pL) in the cable is much higher than the unity. Consequently, with an acceptable approximation, from equation (1.8), we have [13] o-g [ o s

- K'V*

= 0.

(2.1)

After integration and imposing boundary conditions (U*(0,t*) = U*(1, t*) = 0) on this equation, it yields

U*(S,t*) = - K * S

/0

V*Oht* ) d ~ + K *

/0

V*Oht*) &l.

(2.2)

By combining equations (1.9) and (2.2), in the case of free vibration, we have

202v*

vt OS 2

02v*

Ot,----~

v~K *~fo 1V* (7,t*) d~. -

(2.3)

-

Application of method of separation of variables, results in V* (S, t*) = g (S) h (t*),

(t*) + ~ h (t*) = 0,

(2.5)

g" (s) + a % (s) = ~ j~O1 g (~) d~, in which, ~t2

w2

= ~,

Vt

A2

-

(2.4)

.22 ( K ) vI V2

(2.6)

(2.7)

In order to solve the differential equation (2.6) and determine the resulting natural fi'equencies and mode shapes, the following two different approaches are utilized as described next.

1206

M.H. KARGARNOVINet al.

2.1. Banach Fixed-Point Approach (BFPA)

Banach fixed-point theorem is used for eigenfunction analysis of the differential-integral equation of motion (equation (2.6)) in this section. To do this, a Banach space is first defined as follows, B : {g (S) I 9 (S) E C [0, 1]},

(2.8)

in which 9(S) represents continuous (C) functions in interval of [0, 1]. Moreover, in defining a Banach space, a metric (d) needs to be defined as follows, d [91 (S), 92 (S)] = Max 19t (S) - 92 (S) I -= Supremum [91 (S), 92 (S)].

(2.9)

Now, an operator T is introduced in the defined Banach space to be the solution of the differential equation (2.6) as T [9 (S)]

=

C 1

)`2/01

Sin (f~S) + C2 Cos ([2S) + ~-[

9 (or) de,

(2.10)

where C1 and C2 are some constant coefficients. Now, it can be proved that the defined operator T is a contraction operator over the Banach space B. That is, d IT [91 (S)I, T [92 (S)]]--- Max ~22 f01 [gt (or)-92 (cr)] do" < Max~-ff A2 ~01]91 (or)-92 (a)[ da, Max~

fo I 191 (~) - 92 (o)l d~ < ~/k2 Max

191 (~) - 92 (~)1 = ~/~2 d [91 (S), 92 ( S ) ]

Finally,

A2 d [T [g, (S)], T [g2 (S)]] < ~--~d [gl (S), 92 (S)].

(2.11)

As a general rule, whenever the value of )` is less than [2, T will be a contraction operator over B [14], which determines the validity domain of this method. Therefore, on the domain of ),/[2 < 1, the operator T is a contraction operator over B and according to BFP theorem for a contraction operator, we have the following [14]. 1. Equation 9(S) -= T[9(s)] has one and only one solution. 2. The unique solution of equation 9(S) = T[9(s)] is computable from g(S)=

lim 9,~(S), n--+oo

9~ (s) = T [9~-1 (S)]. Now, using (2.10) and (2.12) and imposing the boundary conditions (9(0) = 9(1) = 0), we have

A2 1 - C o s ( f l ) + Sin ([2) + ~

ft

A2 [ S i ~ f~)

l+h~ +'"

Cos (a) + h~

+...

]

+...

C l l = O. c2

(2.13)

Consequently, numerically solving this eigenvalue equation, will result in the natural frequencies and mode shapes of the suspended cable.

Response of a Suspended Cable

1207

2.2. Fredholm Approach (FA) Recalling the definition of Fredholm integral equation, we can see that the equation (2.10) has the form of first-kind Fredholm integral equation, and it can be solved using the property of interchangeability of the kernel. That is, we can write

(2.14)

9 (S) = C1 Sin (flS) + C2 Cos (f~S) + a, in which, a

=

(2.15)

a2 Jo g (~) d(7.

Using these equations, we will have

g ( S ) = C1 [Sin (f~S) + - A2 _--f~2

A2 (Si~f~) A2Zf~2 )].

fl

(2.16)

Now, using equation (2.16) and imposing the boundary conditions (9(0) = 9(1) = 0) for the frequency equation, we will have tan

-- -f

(2.17)

p

Based on the previously described approaches, natural frequencies and mode shapes of a suspended cable can be computed. The first four natural frequencies versus dimensionless sag parameter A plotted in Figure 1. It can be seen that there are some special values for A in which repeated frequencies are occurred for two consecutive modes. This phenomenon, which is due to coupling between transverse and longitudinal displacements, is known as crossover phenomenon. For example, by looking at two first mode shapes we can find that prior to the first crossover A,

Validity regiot~ of BFPA

.j."" ...."':":

Q>X

......

Forth natural frequency

,~'"''Y Third natural frequency

~/~

Third natural frequency .~_~..,/.........-"~ ....--'""

'" ~ a t u r a l Second natural frequency , ~ " " ' - " - - First natural ~

.............

"

fre(luency

First natural frequency

~=~

O '£"

0

I

I

I

I

1

1

2

3

4

5

Figure 1. Variation of natural frequencies versus dimensionless sag parameter.

1208

M.H.

KAI~GARNOVIN et

al.

that is 2~r, one and two extremum in conjunction with two and three nodes are occurred in first transverse and longitudinal mode shapes, respectively. Beyond this point (A=2zr), two and one extremum in conjunction with three and two nodes appear for the first transverse and longitudinal mode shapes, respectively. Also, as shown in Figure 1, for the larger values of A, there is no variation in values of natural frequencies and it seems to be saturated relative to )~ variations. Also, line A = f~ is shown, indicating the validity domain boundary of the first approach, i.e., BFP theorem. Using integration by-parts and boundary conditions and equation (2.6) for two distinct mode shapes, one can show 1

f 0 g ~ ( s ) gj (s) ds

0,

(2.1S)

i # j.

This is known as theorem of orthogonality of mode shapes and it provides necessary conditions for the use of Galerkin mode approximation approach in the case of forced vibration. Therefore, in the case of forced vibration the solution of equation (1.9) can be written in the form of OO

V* (S, t*) = E

qi (t*) g~ (S),

(2.19)

i =1

where qi(t*) are time-dependent generalized functions to be determined in the course of solution procedure. After substituting equation (2.19) into equation (2.10) and multiplying both sides by gj(S) and integrating them from 0 to 1, one will get "

q~ (t*) + 2a~(~q; (t*) +

ft 2 "t*"

iq~ t J =

f: gi (S)p (S, t*) dS ,

f0'

(2.20)

(s) ds

in which p(S, t*) is equal to if(S, t*)/v~ and (i is the damping ratio of the ith mode. Using the Duhamel integral form, the solution of equation (2.20), i.e., the frequency response function for the ith generalized coordinate can be obtained in the form of e-Jut*

=

(2.21)

dOdt*.

Finally, for power spectral density of the response in y direction and its root mean square, the results are [15] OO

P.S.D.~

O4D

(S, w) = E E gi (s) gj (s) Hi (w) Hj (-w) P.S.D.in (w),

(2.22)

i=I j=l

R.M.S.,

=

E E g' (S) gj (S) Hi (w) Hj (-w) P.S.D.in (w) dw

,

i=1 j=l

(2.23)

in which P.S.D.m(aJ) stands for power spectral density of a zero mean stationary input random force. In a similar way and for the power spectral density of the response in x direction, we can write o~

P.S.D.~

OO

(S, w) : ~ ~ Q{ (S) Qy (S) Hi (w) Hy (-w) P.S.D.~, (w),

(2.24)

i:1 j=l

~ ' Q, (S) Q5 (S) H, (w) H3 (-w) P.S.D4~ (o~) dw

R.M.S.,, = --

'=

, (2.25)

j=l

in which according to equation (2.2), Qi can be computed as

Q,(S)

f

=

-K*S

1

I g,(,) d,+ do

r S

/

do

gi(,) d,.

(2.26)

Response of a Suspended Cable 3. N U M E R I C A L

1209

RESULTS

In this section, the root mean square of the response is computed for two types of narrow-band random uniform forces with peaked P.S.D. The first type of the power spectral density (P.S.D.) which corresponds to the vortex shedding induced force [15] is in the form of

~"

P.S.D. - J w , ~

exp

[ (1-I~Ilco.) '] -

(3.1)

in which/7 is the bandwidth of frequencies within which vortices are shed and cos is the vortex shedding frequency. The second type of P.S.D. corresponds to a eriticai state in which the dominant frequency of the road irregularities is exactly equal to the first natural frequency of a suspended cable. The mathematical representation of such condition is in the form of •'

-

P.S.D. co.v/_ q

exp -

.

(3.2)

In equations (3.2) and (3.3), the root mean square of the excitation force (2) is defined as

22 =///P.S.D.

da~.

(3.3)

It should be pointed out that in our analysis, the value of ~ is assumed to be 1.0 because the cable response is a linear function of it (equation (2.25)) and the values of/7 and aJs are assumed to be 0.1, 1.0, respectively, for a typical vortex shedding with small-scale turbulence [15]. According to the prescribed procedure, for each point along the cable length the root mean square of the transverse and longitudinal displacements are computed and plotted for different

12

,/"

IC

/ 8

/,/ / /

/f~--~-.

"N~k

/~=0.0

\

',,,,,.\. \ \.

\

;

Orj

/

......

i

~...

/'

\ ',,

4, i/'/.. "" .L /.."

f _

, '\ \ I

x=3.

/// } V ......... "-"-, ~/ C'¢'" X,=2.~ - - - . - ....... -._ ,/~,~___~__.-~--<.__._._~'

o

0

0.1

0.2

0.3

0.4

0.5 S

0.6

0.7

"\\. / \d/ ,~------__.;~ 0.8

0.9

1

Figure 2. Variation of root mean square of V* along the cable for different values of due to first type of excitation.

1210

M.H. KARGARNOVIN et al.

0.45

\

/ /

\

X,-~

/

0.4

'\

I'

'\

I

0.35

't

0.3 0.25 0.2

c/

X=3rc

0.15 0.1

'

- '

'"..

..:

K=2rc

""-,

o)

0.9

0.05 0 -0,05

r

i

-0.1

i

012

i

o.5

i

o5

$

Figure 3. Variation of root mean square of U*/K* along the cable for different values of ,k due to first type of excitation. 3.5

/

L--0.0 / 2.5

]

2

/

.+,

-.

\

\

i

"

\

x,

/

it

¢5 1.5

/1

I I

/ ./

]

/

)

.i

l

./

I

I

I

/

lI

0.5

/

//

/j

, .,

I

/,

I! J i

.... .

0.1

k=3g ..........................~x,=2'J'~

0.2

0.3

0.4

\\

\.

,...

0.5

-

.

0.6

, ...........

0.7

0.8

\

\,.\\

0.9

s Figure 4. Variation of root mean square of V* along the cable for different values of ,k due to second type of excitation.

values of • (see Figures 2-5). As seen, for )~ = 27r in which the first crossover p h e n o m e n o n occurs we have smallest root m e a n square along the cable• If we consider equation (2.21), it is obvious that in the case of uniform loading and for s y m m e t r i c m o d e shapes w i t h odd number of nodes, the value of integral in the n u m e r a t o r of this equation approaches to zero and therefore, we end up w i t h a small power in the frequency response of the s y s t e m under first natural frequency. If the first and second natural frequencies c o m e closer, above incident strengthens and at the crossover frequency, this effect is in its highest level and hence, the root m e a n square is m i n i m u m . For the first t y p e of r a n d o m force, this effect represents itself m u c h m o r e obvious because it has a broader band of spreading relative to the second type.

Response

0.14r--

,

,

of a Suspended

Cable

1211

,

/

/

I'

/ 0.1

I t

7" 0.08

\

/

I

¢,

\

2

0.12

"\

I

'~

i"

\

1

I

L

I'

'1

/

o.o8

I'

'1

L=3~ ,.....

• /

0.04

,

'~

. .... '

..

I

',

I

..

I

L=2~

.

,

,

:

',,

0.02

0 ~

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

S

Figure 5. Variation of root mean square of U*/K* along the cable for different values of A due to second type of excitation. Also, as seen in Figures 2 and 4 under crossover condition, the effects of the first two modes are faded away and the effect of third unsymmetrical mode shape becomes more noticeable in calculating the root mean square along the cable length. It is clear that for all cases, the root mean square of the response of a suspended cable is lower than that of a linear cable. In all cases, six modes are considered in the mode summation procedure and in order to calculate the damping ratio of the ith mode, the proportional damping ratio defined as following is used [16]. =

OL

+

(3.4)

in which

(3.5) The damping ratio 4, for the fundamental mode is assumed to be 0.001 and gt,, is the natural frequency of the first unsymmetric mode shape.

4. C O N C L U S I O N S The response of a suspended cable subjected to a narrow-band random excitation with two types of peaked P.S.D. is determined and analyzed using Galerkin mode approximation method. A novel approach using Banach fixed-point theorem for free vibration analysis was introduced in this paper. The root mean square of transverse and longitudinal components of displacement were calculated and plotted along the cable length. It was shown that at the first crossover in which repeated frequencies is taken place for the first two modes, the induced responses are minimum. This can be an important result applicable in the design of sag parameter of suspended cables. In other words, using the calculated optimum value of the sag parameter (A = 27r) can reduce vibration of the suspended cables subjected to the random excitations with peaked power spectral densities, It was also shown that the root mean square of the response of a suspended cable is lower than that of a linear cable. It means that using the linear theory of cables makes the response of the cable much higher than reality.

M . H . KARGAP~NOVINet al.

1212

REFERENCES 1. T.K. Caughey, Response of non-linear string to random loading, Journal of Applied Mechanics 26, 341-346,

(1959). 2. R.H. Lyon, Response of a non-linear string to random excitation, Journal of the Acoustical Society of America 32, 953-960, (1960). 3. G.V. Anand and K. Richard, Non-linear response of a string to random excitation, International Journal of Nonlinear Mechanics 91 251-260~ (1974). 4. K. Richard and G.V. Anand, Non-linear resonance in string narrow-band random excitation, Journal of Sound and Vibration 86 (1), 85-98, (1983). 5. G. Tagata, Analysis of a randomly excited non-linear stretched string, Journal of Sound and Vibration 58

(i), 95-107, (1983). 6. G. Tagata, Nonlinear string random vibration, Journal of Sound and Vibration 129 (3), 361-384, (1989). 7. W.K. Chang, R.A. Ibrahim and A.A. Afaneh, Planar and non-planar non-linear dynamics of suspended cables under random in-plane loading, International Journal of nonlinear Mechanics 31 (6), 837-859, (1996). 8. R.A. Ibrahim and W.K. Chang, Stochastic excitation of suspended cables involving three simultaneous internal resonance using Monte Carlo simulation, Computer Methods in Applied Mechanics and Engineering 168, 285-304, (1999). 9. L. Martinelli, V. Gattulli and F. Vestroni, Non-linear behavior of a suspended cable under stationary and non-stationary loading, In Proceeding of the Fourth International Conference on Structural Dynamics, pp. 893-898, Munich, (2002). 10. M.H. Kargarnovin, B. Mehri and D. Younesian, Free Vibration analysis of suspended cables using Banach fixed-point theorem, In Proceeding of AERO-2003 Conference, pp. 34-41, Tehran, (2003). 11. F.S. Hover, S.N. Miller and M.S. Triantafyllou, Vortex-induced oscillations in inclined cables, Journal of Wind Engineering and Industrial Aerodynamics 69, 203-211, (1997). 12. A.M. L-Souza and A.G. Davenport, The effects of high winds on transmission lines, Journal of Wind Engineering and Industrial Aerodynamics 74, 987-994, (1998). 13. C. Scholar, N.C. Perkins and Z.D. Ma, Low order vibration models for tracked vehicles, In Proceeding of the 1999 ASME Design Engineering Technical Conference, pp. 1-13, Las Vegas, NV, (1999). 14. B. Choudhary and S. Nanda, Functional Analysis With Applications, New Age International, New Delhi,

(2001). 15. C. Dyrbye and S.O. Hansen, Wind Loads on Structures, John Wiley &: Sons, New York, (1996). 16. R. Clough and J. Penzin, Dynamics of Structures, McGraw-Hill, New York, (1993).