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to random loads. Assuming that vehicles travel at the same velocity, are of equal weight, and

that the bridge response is a Poisson process, he obtained, based on numerical procedures,

the density function of the response and its excursion rate, and he estimated the fatigue

life of highway bridges. Bryja and Sniady [12] studied the dynamic response of a beam to

the passage of a train of concentrated forces with random amplitudes. Based on the

introduction of two in#uence functions, one of which satis"es a non-homogeneous, theother a homogeneous di!erential equation for beam response, they obtained, explicitexpressions for the expected value and variance of the beam de#ection. Chatterjee et al. [13]presented a linear dynamic analysis for determining the coupled #exural and torsionalvibration of multispan suspension bridges. The analysis duly considers the non-linear

bridges}vehicle interactive force, eccentricity of vehicle path, surface irregularity of thebridge pavement, cable-tower connection and end conditions for the sti!ening grider. Thedynamic analysis duly considers the non-linear bridge}vehicle interactive force, eccentricityof vehicle path, surface irregularity of the bridge pavement, cable-tower connection and end

conditions for the sti!ening girder. The random vibration of a simply supported elasticbeam subjected to random loads moving with constant and time-varying velocity and axial

forces was considered by Zibdeh [14]. Accelerating, decelerating and constant velocity

motions were assumed for the stream of loading. In a recent paper, Abu-Hilal and Zibdeh

[15] considered the transverse vibrations of homogeneous isotropic beams with general

boundary conditions subjected to a constant force travelling with accelerating, decelerating

and constant velocity motion.

In this paper, the dynamic response of elastic homogeneous isotropic beams with

di!erent boundary conditions subjected to a harmonic force travelling with accelerating,decelerating, and constant velocity types of motion is treated. The four classical boundary

conditions considered are pinned}pinned, "xed}"xed, pinned}"xed, and "xed}free. Closed-

form solutions of the dynamic response of the studied beams are obtained. Also thesesolutions are presented graphically for di!erent values of speed and frequency of the movingharmonic force and discussed.

2. ANALYTICAL ANALYSIS

The transverse vibration of a uniform elastic Bernoulli beam is described by the equation

EIv#vK#r?vR#r

GvR"p (x, t), (1)

where EI, , r?

, and rG

are the #exural rigidity of the beam, the mass per unit length of thebeam, the coe$cient of external damping of the beam, and the coe$cient of internaldamping of the beam respectively. The external and internal damping of the beam are

assumed to be proportional to the mass and sti!ness of the beam respectively, i.e., r?"

,

rG"

EI, where

and

are proportionality constants.

In modal form, the beam de#ection v (x, t) at point x and time t is written as:

v (x, t)"I

XI

(x) yI

(t), (2)

where yI

(t) is the kth generalized de#ection of the beam and XI

(x) is the kth normal mode of

the beam and is given as

XI

(x)"sin I

x#AI

cos I

x#BI

sinh I

x#CI

cosh I

x, (3)

704 M. ABU-HILAL AND M. MOHSEN

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where I

, AI

, BI

, CI

are unknown constants and can be determined from the boundary

conditions of the beam. Substituting equation (2) into equation (1) and then multiplying by

XH(x), and integrating with respect to x between 0 and L yields

I

yI

*

EIXI

XH

dx#I

yKI

*

XI

XH

dx#

I

yRI

*

XIX

Hdx#

I

yRI

*

EIXI

XH

d

x

"*

XH

p(x, t) dx (4)

Considering the orthogonality conditions

*

XI

XH

dx"0, kOj, (5)

and the relations

kI"

*

EIXI

XI

dx and mI"

*

XI

(x) dx

yields the di!erential equation of the kth mode of the generalized de#ection:

yKI

(t)#2II

yRI

(t)#I

yI

(t)"QI

(t), (6)

where

I"(k

I/m

I"

I(EI/, (7)

I"

(#

I)

I

, (8)

QI

(t)"1

mI

*

XI

(x) p(x, t) dx, (9)

kI"

*

EIXI

XI

dx (10)

and

mI"

*

XI

(x) dx

"

2I

[I

(1#AI!B

I#C

I)#2C

I!2A

IBI!B

IC

I!

(1!A

I) sin 2

I

#2AI

sin I#(B

I#C

I) sinh

Icosh

I#2(B

I#A

IC

I) cosh

Isin

I

#2(AI

CI!B

I) sinh

Icos

I#2(C

I#A

IBI

) sinh I

sin I

#2(AI

BI!C

I) cosh

Icos

I#B

IC

Icosh 2

I] (11)

VIBRATION OF BEAMS 705

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are respectively, the natural circular frequency of the kth mode, the damping ratio of the kth

mode, the generalized force associated with the kth mode, the generalized sti!ness of the kthmode, and the generalized mass of the beam associated with the kth mode. The constant

Iin equation (11) is de"ned as

I"

I. (12)

The load p(x, t), which moves on the beam from left to the right is written as

p (x, t)"[x!f(t)] P (t), (13)

where

P(t)"P

sin t (14)

is the moving harmonic force with the constant amplitude P

and the circular frequency ,

and

f(t)"x#ct#

at (15)

is a function describing the motion of the force at time t, where x

, c, a are the initial

position of application of force P at instant t"0, the initial speed, and the constant

acceleration of motion respectively.

Substituting equations (13) and (14) into equation (9) yields

QI

(t)"Pm

I

XI

[ f(t)] sint. (16)

Assuming the beam is originally at rest (i.e., v(x, 0)"0, *v(x, 0)/*t"0), the solution ofequation (6) is then written as

yI

(t)"R

hI

(t!)QI

() d, (17)

where hI

(t) is the impulse response function de"ned as

yI

(t)"1

BI

e\DISIR sin BI

t,

0

t*0,

t'0,(18)

where

BI"

I(1!

I(19)

is the damped circular frequency of the kth mode of the beam. Substituting equations (17)and (18) into equation (17) yields

yI

(t)"P

e\DISIRm

I

BI

R

e\DISIO sin BIX

I[ f()] sin d. (20)


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Substituting equations (3) and (15) into equation (20), carrying out the integration and

substituting the result into equation (2) yields the de#ection v(x, t) of the beam by theaccelerated (a'0) and decelerated (a(0) motion of the force:

v(x, t)"

I

XI

(x) yI

(t),

where

yI

(t)"F

Re+!r

r

eX>X[erf(r

t#r

z

)!erf(r

z

)]

#r

r

eX>X[erf(r

t#r

z

)!erf(r

z

)]

!r

r

eX>X[erf(r

t#r

z

)!erf(r

z

)]

#r

r

eX>X[erf(rt#r

z

)!erf(r

z

)]

!i(2r

r

e\X>X>X[erf(!ir

t#r

z

)!erf(r

z

)]

#i(2r

r

e\X>X>X[erf(!ir

t#r

z

)!erf(r

z

)]

#(2r

r

eX\X>X[erf(r

t#r

zN

)!erf(r

zN

)]

!

(2rreX\X

>X

[erf(rt#

rzN

)!

erf(rzN

)], (21)

with F

, r

to r

, and z

to z

given in the Appendix A.

The de#ection v (x, t) due to a moving load with constant velocity, does not followautomatically from equation (21) because of the nature of the error function. Also setting

a"0 in equation (21) to obtain the dynamic response for the case of constant velocity

(a"0) leads to in"nite values ofyI

because of the de"nition ofr

, r

, r

, r

, and r

. In the

case of constant velocity, equation (15) becomes

f(t)"ct. (22)

Substituting this equation into equation (20) and carrying out the integration yields

v(x, t) due to a moving load with constant velocity:

v(x, t)"I

XI

(x)yI

(t),

where

yI

(t)"F!

q#AIII

q

#q!AIIIq

cos(I

c#) t

#q#A

II

Iq

#q!A

II

Iq

cos(Ic!)t


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TABLE 1

Maximum static de-ection and its location of the studied beams

Pinned}pinned Fixed}"xed Pinned}"xed Fixed}pinned Fixed}free Free}"xed

v

P48El

P192El

P

48(5ElP

48(5ElP3El

P3El

xK?V

2

2

(51!

1

(5 0

!I

I!A

Iq

q

!I

I#A

Iq

q

sin(Ic#) t

#I

I!A

Iq

q

!I

I#A

Iq

q

sin(Ic!) t

!q

q

q

!q

q

q

eGAR cos(t)#q

q

q

!q

q

q

e\GAR cos(t)

#q

q

q

#q

q

q

eGAR sin(t)!q

q

q

#q

q

q

e\GAR sin(t)

#q#A

II

Iq

!q#A

II

Iq

!q!A

II

Iq

#q!AIIIq

#qqq

!qqq

!qqq

#qqq

e\DISIR cos BIt

#I

I!A

Iq

q

!I

I!A

Iq

q

!I

I#A

Iq

q

#I

I#A

Iq

q

!q

q

q

#q

q

q

#q

q

q

!q

q

q

e\DISIR sin BIt (23)with F

, and q

to q

given in Appendix.

3. RESULTS AND DISCUSSION

To clarify the analysis, the dimensionless de#ection

vN"v (x

K?V, t)

v

(24)

versus the dimensionless time s is given for beams with di!erent boundary conditions, where

v and xK?V are the maximum static de#ection and the position at which v occurrespectively. x

K?Vand v

are given in Table 1 for the considered beams which are

pinned}pinned, "xed}"xed, pinned}"xed, "xed}pinned, "xed}free, and free}"xed. The


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Figure 1. Dynamic response of a pinned}pinned beam, varying the excitation frequency : "0)25, "0)05.

de#ection v(xK?V

, t) is obtained either from equation (21) or equation (23), where only the

"rst term of the summation is considered (i.e., k"1).The studied beams are homogeneous, isotropic and originally at rest. They are subjected

to concentrated harmonic forces with constant amplitudes. The forces enter the beams from

the left-hand side at position x"0 and move to the right with the following three types ofmotion.

Accelerated motion: Force P starts to act on a beam at rest at position x"0. Its motion

is uniformly accelerated so that it reaches the velocity c at position x". The time t

needed to cross the beam and the corresponding acceleration are given as [1]

t"

2

c, a"

c

2. (25)

Decelerated motion. A force P moving with constant velocity enters a beam at rest from

the left at position x"

0. Its motion along the beam is uniformly decelerated so that itstops at the end of the beam, i.e., x". The time t

needed to cross the beam and the

corresponding deceleration are given as

t"

2

c, a"

!c

2. (26)


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Figure 2. Dynamic response of a pinned}pinned beam, varying the speed c. "1, "0)05.

;niform motion: A force P moving with constant velocity enters a beam at rest from left

at position x"0. During its travel along the beam its velocity remains constant. The time

t

needed to cross the beam is given as

t"

c. (27)

The dimensionless time s is de"ned by the accelerated/decelerated motion as

s"t

tG

"ct

2, i"1, 2 (28)

and by the uniform motion as

s"t

t

"ct

. (29)

Thus when s"0 (t"0) the force is at the left-hand side of the beam, i.e., x"0, and when

s"1 (t"tG, i"1, 2,3) the force is at the right-hand side of the beam, i.e., x".


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Figure 3. Dynamic response of a "xed}"xed beam, varying the speed c. "1, "0)05.

In Figures 1}7, the e!ect of speed c and excitation frequency are presented. The e!ect ofspeed is represented by the dimensionless speed parameter , where

"

c

cAP , (30)

with cAP

the critical speed, de"ned as [1]

cAP"

. (31)

The e!ect of excitation frequency is represented by the frequency ratio where

"

. (32)

The damping ratio is assumed to be "0)05.Figure 1 shows the e!ect of the excitation frequency represented by the frequency ratio

for a simply supported beam, where the speed parameter is held constant ("0)25).


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Figure 4. Dynamic response of a pinned}"xed beam, varying the speed c. "1, "0)05.

From the "gure it is clear that by all three types of motion, the maximum response

vK?V"

Max+v (xK?V

, t),

v

(33)

is increased by increasing the values of , reaches a maximum value at "1, andthen decreases. The other beams considered behave similarly by varying the excitation

frequency .

Figure 2 shows the dynamic response of a simply supported beam for di!erent values of and motions at resonance, i.e., "1. It is noticed that in the accelerated and deceleratedmotions the beam has a much higher maximum dynamic response v

K?Vthan in the uniform

motion. The maximum response vK?V

is reached in the accelerated and uniform motions at

a later time than in the decelerated motion. The di!erences in the dynamic response to thedi!erent types of motion are due to the kinematics involved. Independent of the type ofmotion, the maximum response v

K?Vbecomes smaller by increasing the values of since the

acting time tG

of the load on the beam becomes shorter.

Figure 3 shows the dynamic response of a "xed}"xed beam for di!erent values of andmotions at resonance ("1). This beam behaves similarly to a simply supported beam;however, v

K?Vof the "xed}"xed beam is relatively smaller than v

K?Vof the simply supported


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Figure 5. Dynamic response of a "xed}pinned beam, varying the speed c. "1, "0)05.

beam. On the other hand, the absolute dynamic response v?@Q"v

vK?V

of the "xed}"xedbeam is essentially smaller than v

?@Qof the simply supported beam, since the maximum static

de#ection v

of the "xed}"xed beam (vDD

) is much smaller than v

of the simply supported

beam (vQQ

) as shown in Table 1 (vDD

(/2)"0)25vQQ

(/2)).

Figure 4 shows the dynamic response of a pinned}"xed beam for di!erent values of andmotions at resonance. Independent of the type of motion, the maximum response v

K?Vis

decreased by increasing the values of , since the acting time of the load on the beambecomes shorter. The "gure shows that in the accelerated and decelerated motions the beamhas a higher maximum response v

K?Vthan in the uniform motion. The maximum response

vK?V

is reached in the decelerated motion at an earlier time than in the accelerated and

uniform motions.

Figure 5 shows the dynamic response of a "xed}pinned beam for di!erent values of andmotions at resonance. This beam behaves, in general, similarly to a pinned}"xed beam;however, due to the direction of the motion, there are di!erences between the behaviour ofthese two beams. The dynamic response of the pinned}"xed beam becomes higher at earliertime than the dynamic response of the "xed}pinned beam, since the pin support at theleft-hand side of the pinned}"xed beam permits rotational motion. Also the pinned}"xedbeam shows a higher maximum dynamic response v

K?Vin the accelerated motion but

a lower maximum response in the decelerated motion than the "xed}pinned beam.


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Figure 6. Dynamic response of a "xed}free beam, varying the speed c. "1, "0)05.

Figure 6 shows the dynamic response of a "xed}free beam for di!erent values of andmotions at resonance. It is noticed that the beam needs some time to show a noticeable

response since the "xed support on the left-hand side prevents rotational motion.Independent on the type of motion, the maximum response v

K?Voccurs at the end of the

beam. A higher maximum response occurs in the case of decelerated motion than in theother two types of motion.

Figure 7 shows the dynamic response of a free}"xed beam for di!erent values of andmotions at resonance. Independent of the type of motion, v

K?Voccurs by smaller values of

before the load reaches the midspan of the beam; by increasing the values of, vK?V

occurs

at a later time. The maximum response vK?V

is higher in the accelerated motion than in the

other two types of motion. The di!erences in the behaviour of the free}"xed and "xed}freebeams are due to the direction of the load motion.

4. CONCLUSIONS

The dynamic response for homogeneous isotropic elastic beams with general boundary

conditions due to a moving harmonic force was discussed in detail for di!erent cases. Thee!ect of support type, variation of speed, direction of load motion, and type of load motion


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Figure 7. Dynamic response of a free}"xed beam, varying the speed c. "1, "0)05.

were studied. Also the dynamic response for the simply supported beam was calculated

at di!erent values of the frequency ratio ; the highest response was obtained for theresonance case "1. Due to the kinematics of accelerated and decelerated motion, itwas found that their e!ect on beams is greater than the case when motion is uniform.

The direction of motion a!ects the beam response and the location of the maximumdynamic response.

REFERENCES

1. L. FRYDBA 1972

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6. R. IWANKIEWICZ and P. SNIADY 1984 Journal of Structural Mechanics 12, 13}26. Vibration ofa beam under a random stream of moving forces.

7. R. SIENIAWSKA and P. SNIADY 1990 Journal of Sound and

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z"!r

(

BI!i(

I

I#

Ic) (

BI!)),

z"r

(

BI#i(

I

I#

Ic) (

BI#)),

z"r

(

BI#i(

I

I!

Ic) (

BI!)),

z"!r

(

BI!i(

I

I!

Ic) (

BI#)),

z"

BI!#i(

I

I#

Ic),

z"

BI##i(

I

I#

Ic),

z"

I

I!

Ic#i(

BI!),

z"

I

I!

Ic#i(

BI#),

z"0)5r

[

I

I#

Ic!

BI!],

q"II#Ic, q"II!Ic,

q"B

I#C

I, q

"B

I!C

I,

q"

BI!

Ic!, q

"(

I

I)#(

BI!

Ic!),

q"

BI!

Ic#, q

"(

I

I)#(

BI!

Ic#),

q"

BI#

Ic!, q

"(

I

I)#(

BI#

Ic!),

q"

BI#

Ic#, q

"(

I

I)#(

BI#

Ic#),

q"

BI! , q

"

BI#,

q"(

I

I#

Ic)#(

BI!), q

"(

I

I#

Ic)#(

BI#),

q"(

I

I!

Ic)#(

BI!), q

"(

I

I!

Ic)#(

BI#).


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