dynamic response of an overhead crane system, 1998

7/23/2019 Dynamic Response of an Overhead Crane System, 1998

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Journal of Sound and Vibration(1998) 213(5), 889906

DYN AMIC R ESPONSE O F AN OVER HEAD CR ANE

SY STEM

D. C. D. O J. S. H

Institute for Aerospace Studies, University of Toronto, 4925 Dufferin Street, Downsview,Ontario, Canada M3H5T6

G. R. H

Systems Design Engineering,University of Waterloo ,Waterloo,Ontario,Canada N2L3G1

(Received24 February 1997, and in final form 3 February 1998)

A simply supported uniform EulerBernoulli beam carrying a crane (carriage andpayload) is modelled. The crane carriage is modelled as a particle as is the payload whichis assumed to be suspended from the carriage on a massless rigid rod and is restricted tomotion in the plane defined by the beam axis and the gravity vector. The two coupledintegro-differential equations of motion are derived using Hamiltons principle andoperational calculus is used to determine the vibration of the beam which is, in turn, usedto obtain the dynamics of the suspended payload. The natural frequencies of vibration ofthe beamcrane system for a stationary crane are investigated and the explicit frequencyequation is derived for that set of cases. Numerical examples are presented which covera range of carriage speeds, carriage masses, pendulum lengths and payload masses. It isobserved that the location and the value of the maximum beam deflection for a given setof carriage and payload masses is dependent upon the carriage speed. At very fast carriagespeeds, the maximum beam deflection occurs close to the end of the beam where thecarriage stops as a result of inertial effects and at very slow speeds occurs near the middleof the beam because the system reduces to a quasi-static situation.

1998 Academic Press Limited

1. INTRODUCTION

The primary use of overhead cranes is in the transfer of heavy payloads from onelocation to another, thus they are found in areas such as airports, shipyards, andautomobile plants. While the swing of the payload link poses an interesting controlproblem, a proper representation of the system dynamics is, in our view, an essentialcomponent in designing an effective controller.

The system reduces to an elastic beam with a moving load when the attachment to thepayloadhenceforth called a pendulumis removed. This is a common feature in thedesign of railroads, highways, and bridges. While these are traditional civil engineeringapplications, the model is now used in ballistic machining problems, high-speed precision

drilling, and problems involving two-phase flows [13].One of the earliest studies on beams with moving load was done by Stokes [4] and themonograph by Fryba [5] is an excellent reference, with many analytical solution methodsfor simple cases. Lee [6] investigated the dynamic response of a beam with intermediatepoint constraints subjected to a moving load via the method of assumed modes. The pointconstraints were modelled as linear springs with stiffnesses of enough magnitude toguarantee a numerically zero deflection at these points. It was observed that the point

0022460X/98/250889+ 18 $25.00/0/sv981564 1998 Academic Press Limited


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... .890

constraints resulted in a significant reduction in the deflection of the beam for slow movingloads. The method of assumed modes was also used by Esmailzadeh and Ghorashi [7] tostudy the dynamic response of a single-span beam with a moving load that was causedby a uniform partially distributed moving mass. A similar analysis was implemented byMichaltsoset al. [8] with a concentrated moving load.

The application of the finite element method to the moving load problem can be foundin references [3] and [911]. The study by Stanisic [12] is a departure from theaforementioned studies because the position of the moving load is included in the modeshapes derivation thus ensuring satisfaction of both the boundary and transient conditions.This technique is based on the use of operational calculus and was shown to exhibit fastconvergence.

Khalilyet al. [13] have used the technique by Stanisic [12] in modelling a cantilever beamwith a moving mass. They showed that the inclusion of the position of the mass in theeigenfunctions provides superior results when compared with the conventional and moreprevalent method in which the eigenfunctions are independent of the position of the mass.

This paper is an attempt to increase our understanding of the dynamics of an overheadcrane system. We assume that the beam is simply supported and that it can be adequatelymodelled using EulerBernoulli beam theory. Further, we assume that the crane carriagetransverses the beam at a known uniform speed and that the pendulum may be adequatelymodelled as a rigid massless bar. The motion of the pendulum is assumed to be planarwith small angular displacements and displacement rates from the vertical.

A set of coupled, non-linear equations of motion is derived via Hamiltons principle.The complexity of these equations is increased by the presence of Dirac delta functionterms. We obtain the transverse vibration of the beam using the method outlined inreference [12]. A polynomial is fitted to this result and then used to determine the dynamicsof the pendulum. The effects of varying the various parameters are investigated.

2. DESCRIPTION OF THE SYSTEM

A schematic of the problem is depicted in Figure 1. An overhead crane carriage of massmc moves along the beam with a prescribed velocity xc . This implies that the position ofthe carriage is known at all times. A rigid rod of length L has one end attached to thecarriage while the other end carries a payload of massmL . The rod is displaced from thevertical by angle .

The beam is simply supported and is assumed to be adequately modelled usingEulerBernoulli beam theory. The properties of the beam are Youngs modulus E, volumedensity , cross-sectional area A, length Lb , and second moment of area I.

3. EQUATIONS OF MOTION

A dextral inertial frame Fa with basis vectors a1, a2, and a3 and co-ordinates xyz isattached at the left-hand end of the beam such that the a 1basis vector passes through theright-hand end of the beam and the a2 basis vector points in the direction of the staticdeflection as illustrated in Figure 1.

The position vector of an elemental mass of the beam rb , the position vector of thecarriage rc , and the position vector of the load rL , can be expressed as

rb = xa1 +y(x,t)a2, rc = xc (t)a1 +y(xc ,t)a2 (1, 2)


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, , , ,

a1

a2

xc

mc

mL

y

L

x

,A,Lb,E I }{

891

and

rL = (xc (t) + Lpsin )a1 + (y(xc ,t) + Lpcos )a2. (3)

The corresponding velocities are

rb =y(x,t)

t a2, rc =

dxc (t)dt

a1 +dy(xc ,t)

dt a2 (4, 5)

and

rL =dxc (t)dt + Lpcos ()d(t)dta1 +dy(xc ,t)dt Lpsin d(t)dta2. (6)The kinetic energy of the system T is

T= Tb + Tc + TL , (7)

where

Tb = 12A Lb

0y(x,t)t 2

dx, Tc = 12mcdxc (t)dt2

+dy(xc ,t)dt 2

, (8, 9)

TL = 12mLdxc (t)dt2

+ 2Lpcos dxc (t)

dtd(t)

dt 2Lpsin

dy(xc ,t)dt

d(t)dt

+dy(xc ,t)dt 2

+ L2pd(t)dt2

. (10)

The potential energy of the system U is

U= 12EILb

0y(x,t)x 2

dx (mc + mL )gy(xc ,t) mLgLpcos . (11)

Figure 1. Schematic of the system.


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... .892

UsingTand U in Hamiltons principle and taking variations over (t) andy (x,t) whileobserving their independence, we find the equations of motion, after some algebraicmanipulations, to be

mLL2pd2(t)

dt2 + mLLpcos

d2xc (t)

dt2 mLLpsin

d2y(xc ,t)

dt2 + mLgLpsin = 0, (12)

Lb

0A2y(x,t)t2

+ EI4y(x,t)

x4 + (mc + mL )d2y(x,t)dt2 g(x xc )

mLLpsin d2(t)dt2 +cos ()d(t)dt2

(x xc )dx = 0, (13)with the boundary conditions

y(0,t) =2y(0,t)x2

=y(Lb ,t) =2y(Lb ,t)

x2 = 0 (14)

and initial conditions

y(x, 0) =y(x, 0)

t =

d(0)dt

= xc (0) = 0 and (0)=0. (15)

In the simple case where one has a simply supported beam with a mass fixed somewherein the span one can find that the governing frequency equation is

mA

(sin (Lb )(cosh ((L1 L2))cosh (Lb ))

+sinh (Lb )(cos ((L1 L2)) cos (Lb )))

4 sin (Lb ) sinh (Lb ) = 0, (16)

whereL1is the position of the mass as measured from the left end of the beam wherex = 0.

The other parameters in equation (16) are defined as

4 =2A

EI , Lb = L1 + L2 and m = mc + mL .

This relation ignores the motion of the payload and treats the carriage and payload asa single point mass. It may be observed that in the event that m = 0, L1 = 0 or L2 = 0,equation (16) reduces to the frequency equation for a uniform simply supported beam,

sin (Lb )=0.

3.1. -

To obtain the non-dimensionalized forms of the equations of motion, equations (12) and(13), we follow Stanisic [12] and define

v(,t) =y(x,t)Lb, = xLb

, c (t) = xc (t)Lb, 1 = mcALb

,

2 = mLALb

, t =, 2 =AL4b

EI , = 1 + 2, (17)

p1 =mcgL2b

EI , p2 =

mLgL2bEI

, p =p1 +p2, L =LpLb

.


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893

Using the above definitions, the non-dimensional form of equation (12) can be written as

2Ld2()

d2 + 2cos

d2c ()d2

2sind2v(c ,)

d2 +p2sin= 0 (18)

and that of equation (13) as

1

02v(,)2

+4v(,)

4 +

d2v(c ,)d2

( c )

2sin d2()d2 cos ()d()d2

+p( c )d= 0. (19)The corresponding non-dimensionalized boundary conditions are

v(0,) =2v(0,)2

= v(1,) =2v(1,)2

= 0 (20)

and the non-dimensionalized initial conditions are

v(, 0) =v(, 0)

=

d(0)d

= 0 and (0)=0. (21)

The non-dimensional form of the frequency equation given in equation (16) is

(sin ()(cosh ((2c 1)) cosh ())

+sinh ()(cos ((2c 1))cos ()))

4 sin () sinh () = 0, (22)

with sin () = 0 corresponding to the frequency equation for a uniform simply supportedbeam.

4. SOLUTION METHOD

If we assume a crane carriage speedc = constant and also assume that() and d()/dare small and that their products are negligible, then the non-dimensionalized equationsof motion can be written as

d2()d2

1L d2v(c ,)d2 2 gLp() = 0 (23)and

1

02v(,)2

+4v(,)

4 +

d2v(,)d2

(c ) p( c )d= 0. (24)The use of the small angle and small angular rate assumption allows the uncoupling ofthe beam motion from the payload motion but not conversely. Except for thesimplifications enjoyed as a result of this assumption the governing equations remainnon-linear coupled integro-differential equations. Equation (23) could result in aHillMathieu equation under some conditions and this would imply the possibility of anunstable motion. Note that equation (24) is identical to equation (5) in reference [12]


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... .894

without the spring term. Thus, one can follow reference [12] and derive an approximatesolution for the vibration of the beam. The actual derivation process is very lengthy andtedious and will not be repeated here, instead, the reader is urged to consult Stanisic [12]for more details.

Assume a separable solution to equation (24) in the form

v(,) = V() ej, (25)

where is a non-dimensional frequency. Using this expression, the following frequencyequation results [12]:

sin () sinh () +

2(sin ((c 1)) sin (c ) sinh ()

sinh ((c 1)) sinh (c ) sin ()) = 0, (26)

where 2 =. The orthonormalized eigenfunction is expressed as

Vn (,c ) = 1

Bn (c )

nL (,c 1),nR ( 1,c ),

0 c ,c 1,

(27)

where

nL (,c 1)= sin (n) sin (n (c 1)) sinh (n )

sinh (n) sinh (n (c 1)) sin (n ), (28)

nR ( 1,c ) = sin (n ( 1) sin (nc ) sinh (n )

sinh (n ( 1)) sinh (nc ) sin (n ) (29)

and

Bn (c ) = 2 sin (n ) sinh (n (3n (c ) c'nL (c ) +'nR (c )))

+ 2n (sin2 (nc ) sinh2 (n )sinh2 (nc ) sin2 (n )) + 2n(c ). (30)

The solution to equation (24) can be expressed as a summation of the product of theorthonormal eigenfunctions Vm (,c ) and undetermined coefficients qm (,c ) such that

v(,) =

m = 1

Vm (,c )qm (,c ). (31)

It was assumed earlier that the carriage travels with a constant velocity. Now, it is alsoassumed that the expressions Vm (,c )/candqm (c ,)/cand their second derivativesare small enough that their products can be ignored when compared with(2qm (c ,)/2)Vm (,c ) [12]. Thus,

2

2(Vm (,c )qm (c ,))

2

qm (c ,)2 Vm (,c ). (32)

Substituting equation (31) into equation (24) and using equation (32), the followingequation results after some algebraic manipulations:

2qm (c ,)2

+ 4m(c )qm (c ,) =pVm (c ,c ), (33)


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895

T1

Material properties

Parameter Value

80103 kgm3

E 21171011 PaLb 100 mA 160104 m2

I 213107 m4

g 981 ms2

ALb 128 kg 1685

with initial conditions

qm (0,c ) = qm (0,c ) = 0. (34)

The following is the solution to the above equation which is obtained from operational

calculus [14]:

qm (,c ) =p

0

()Vm (c (),c ()) d

0

()4m(c ())qm (,c ()) d. (35)

Equation (35) is substituted into equation (31) to determine the beam deflection under thecarriage v(c ,). A Chybeshev polynomial series is then fitted to the resulting beamdeflections and its second derivative is used in equation (23) to numerically obtain thedynamics of the payload.

5. NUMERICAL EXAMPLES

The beam is assumed to be square in cross-section and to be made of steel with thematerial properties outlined in Table 1. The crane carriage travels from one end of thebeam to the other in all simulations. Before investigating the dynamics of the pendulumour analysis is validated by simulating the dynamics of the beam. In particular we seekto reproduce Figure 4 in reference [12].

The ratio of the mass of the carriage to the mass of beam 1is varied over the values00, 005, and 010, the ratio of the mass of the payload to the mass of the beam is fixedat2 = 015, and the ratio of the length of the pendulum to the length of the beam is fixedatL = 015. The simulation is carried out for three constant carriage speedsc =05, 10and 20 s1.

Before considering the cases where the carriage and payload are moving, it is instructiveto first consider the natural frequencies of the beam when the carriage and payload arestationary. The purpose of this preliminary investigation into the frequency behaviour ofthe system for stationary masses is to demonstrate the complexity of the behaviour evenin this simple scenario. Careful examination of equation (22) reveals that when the massis positioned at = k/n, k = 0, 1, 2, . . . ,n of the span, (i.e., at the modal nodes) thenatural frequency of moden is identical to the natural frequency of the same mode in the


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2

4

6

8

0 2 4 6 8 10

Distance from the left end support (m)

Naturalfrequency

(Hz)

... .896

Figure 2. Natural frequencies of a beam with a fixed mass in the span for =015 ( ), 020 (), 025;(); , 1st mode; , 2nd mode; , 3rd mode.

case where there is no mass on the beam. This is illustrated in Figure 2 for the first threenatural frequencies. The material and geometric properties used to produce this figure aregiven in Table 1.

As expected, the addition of the mass in the span of the beam decreases the naturalfrequency of the system, except as noted above, and the larger the mass the greater thedecrease. The location of the minima in Figure 2 is dependent not only on the size of the

T2

Stationary mass, 1Hzc =015 =020 =025

0000 09323203 09323203 093232030500 08173761 07874778 076060991000 09323203 09323203 09323203

T3

Stationary mass, 2Hz

c =015 =020 =025

0000 37292810 37292810 37292810

0225 33111085 32133922 31296904

0230 33103962 32133021 313034280235 33103629 32139979 313185510500 37292810 37292810 372928100765 33103629 32139979 313185510770 33103962 32133021 313034280775 33111085 32133922 312969041000 37292810 37292810 37292810


9/18

0.005

0.000

0.010

0.015

0.020

0.025

0.00 0.25 0.50 0.75 1.00

Carriage position,c

v(x

,t)

897

T4

Stationary mass, 3Hz

c =015 =020 =025

0000 83908824 83908824 83908824

0135 75320338 73458314 719274020140 75263722 73431626 719350660145 75240848 73442846 719827380333 83908824 83908824 839088240500 75381336 73657407 722459100666 83908824 83908824 839088240855 75240848 73442846 719827380860 75263722 73431626 719350660865 75320338 73458314 719274021000 83908824 83908824 83908824

mass but also on the position of the mass as may be seen in the data presented in Tables24 where the local minimum values are in bold. Of interest here is that, in the third modedata, the global minimum does not occur when the mass is at midspan.

Figure 3 depicts the beam deflection under the carriage as the carriage travels atc = 05 s1. It is observed that the magnitude of the maximum deflection increases withthe mass of the carriage. This is to be expected given the higher gravitational loadsassociated with the greater masses. It may further be observed in Figure 3 that for the lowervalues of 1 (i.e., 00 and 005) the maximum deflection occurs at or very near to themidspan of the beam but for the larger carriage mass (1 = 01) the peak deflection underthe carriage definitely occurs past the middle of the beam.

This effect is accentuated in Figures 4 and 5 where results for the same system, but withthe carriage speed increased from c =05 to 10 and 20 s1, respectively, are illustratedHere it may be observed that the location of the maximum deflection (under the carriage)has moved forward, away from the middle of the beam to the right; the higher the carriagespeed, the further the maximum is moved to the right. This is caused by a coupling of the

Figure 3. Beam deflection under the carriage forc =05 s1 and2 =015; ,1 =0; , 005; , 010.


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0.005

0.000

0.010

0.015

0.020

0.025

0.00 0.25 0.50 0.75 1.00

Carriage position,c

v(x

,t)

... .898


beam motion and the motion of the carriage. Because the beam motion is dynamic it isvibrating as the crane carriage advances and when the speed of the carriage is sufficientlylarge it can reach a point on the beam that is more than half way across before aconstructive superposition of the gravitational loading and the beam motion combine toyield a maximum deflection under the carriage. Figures 3 and 4 demonstrate that this effectis more sensitive to changes in carriage speed than it is to changes in the carriage mass.It may also be noted in Figure 4 that, just as in the previous case, the magnitude of themaximum deflection increases with the mass of the carriage.

Further, upon close examination of Figures 4 and 5 it may be noted that the magnitude

of the various maxima is lower than their corresponding values for a carriage speed of05 s1. This is due to the increased speed of the carriage and can be appreciated byconsidering the two limiting cases ofc0 and c. In the former limit we approacha sequence of quasi-static loadings where at each instant the maximum deflection of thebeam is under the carriage and dynamic effects are negligible. In the later limit the beamwould not deflect at all due to inertial effects and the negligible transit time.

With reference to Figure 4 the (1 =005, 2 = 015) combination is the nearest to, andis in close agreement with, the scenario presented in Stanisic [12].

5.1.

The vibration of the pendulum carrying the payload is examined under three differentscenarios to determine the effect of the length of the pendulum, the effect of the mass ofthe carriage and the mass of the payload, and finally, the effect of the carriage speed.

5.1.1. Case 1: effect of the pendulum length

To examine the effect of the length of the pendulum a carriage speed c of 10 s1, acarriage mass to beam mass ratio 1 of 005 and a payload mass to beam mass ratio2 = 015 are used. The results are depicted in Figure 6 where a definite change in frequencyis evident. While the payload is suspended from a moving carriage it is, by virtue of theconstant carriage velocity, in an inertial frame (ignoring the transverse acceleration of the


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0.002

0.000

0.004

0.006

0.008

0.010

0.00 0.25 0.50 0.75 1.00

Carriage position,c

v(x

,t)

0.005

0.010

0.000

0.005

0.00 0.20 0.40 0.60

Non-dimensional time,

Angle,

()

899


beam) and should behave just as a stationary pendulum would. Figure 6 shows that, asexpected, the swing frequency is inversely proportional to the square root of the lengthof the pendulum and, for this example, the beam vibration has no discernible effect onthe payload motion as compared to changes in the pendulum length.

5.1.2. Case 2: effect of carriage mass

In this example the ratio of the length of the pendulum to the length of the beam L ,the ratio of the payload mass to the beam mass2, and the carriage speed care each fixedat 010, 015, and 10 s1, respectively, while the ratio of the mass of the carriage to the

Figure 6. Effect of the pendulum length: , L = 005; , 010; , 015.


12/18

0.005

0.010

0.000

0.005

0.00 0.20 0.40 0.60


Angle,

()

... .900

Figure 7. Effect of carriage mass: , 1 = 005; , 010; , 015.

mass of the beam 1 is varied as 005, 010 and 015. The swing angle versusnon-dimensional time is plotted in Figure 7 for each 1.

The beam motion and carriage mass contribute an effective stiffness coupling term tothe dynamics of the pendulum as can be seen by considering the leading factor of thepenultimate term in equation (23), i.e., the factor

d2v(c ,)d2

(36)

which also appears in the beam equation (24) with as a leading factor. The plots do notreveal any apparent change in the swinging frequency of the pendulum but, the magnitude

of the swing of the pendulum increases slightly with decreasing carriage mass.From the equations of motion, equations (23) and (24), it is expected that for fixed values

of a variation in the mass of the payload would give the same effect as a variation inthe mass of the carriage. Thus, that situation will not be considered explicitly.

5.1.3. Case 3: effect of carriage speed

Here, the ratio of the carriage mass to the beam mass, the ratio of the payload massto the beam mass, and the ratio of the pendulum length to the beam length are fixed at1 =005, 2 =015 and L = 010, respectively, while the carriage speed is varied asc =02, 05 and 10 s1. The results are illustrated in Figures 810.

The non-dimensional terminating time for each speed is determined according to

= t = Lb

xc = 1xcLbEIA .These plots are not presented in the same figure because the time window that is

available for the pendulum to swing is different for each carriage speedit increases withdecreasing carriage speed. Thus, a full cycle is observed at the lowest speed because theperiod is smaller than the total travel time of the carriage.


13/18

0.008

0.004

0.000

0.004

0.008

0.0120.0 0.5 1.0 1.5 2.0 2.5 3.0


Angle,

()

0.005

0.000

0.005

0.010

0.0100.0 0.4 0.6 0.8 1.00.2


Angle,

()

901

Figure 8. Effect of carriage speed for 1 =005, 2 =015, L = 010 and c =02 s1.

The results at the slowest speed (Figure 8) is of particular interest in that the amplitudeof vibration starts to slowly grow after about = 20. Figure 9 shows that the pendulumcompletes one-half period by the time the carriage reaches the end of the beam when thecarriage speed is 05 s1 while a quarter period is observed at a carriage speed of 10 s1

(Figure 10).To further improve our understanding of the effects of the carriage speed on the

pendulum motion, Figure 11 has been included which depicts the beam transversedeflection under the carriage. These plots are similar to those of Figures 35 but for fixedcarriage and payload masses and varied carriage speeds. Taking the results depicted in



14/18

0.005

0.000

0.005

0.010

0.010

0.0 0.4 0.60.2


Angle,

()

0.005

0.010

0.015

0.0000.00 0.25 0.50 0.75 1.00

Carriage position,c

v(x

,t)

... .902


Figures 35 into consideration, one can conclude that there is a threshold carriage speedthat will generate beam deflections under the carriage that are symmetric about the middleof the beam.

What the above suggests is that the location and the value of the maximum beamdeflection for a given set of carriage and payload masses is dependent upon the carriagespeed. At very fast carriage speeds, the maximum beam deflection is expected to occur closeto the end of the beam where the carriage stops because in those cases the deflections are

Figure 11. Beam deflection under the carriage for 1 =005, 2 =015 andL = 010; ,c =02 s1; ,05 s1; , 10 s1.


15/18

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.0141.0

0.80.6

0.40.2

0.0 0.00.2

0.40.6

0.81.0

v(x,t

)(*

1.0

m)

903

Figure 12. Beam deflection for a given carriage location for 1 =005, 2 =015, L =010 and c =02 s1

(time = 50 s, beam length= 100 m).

dominated by inertial effects. At very slow speeds, the maximum beam deflection isexpected to occur at the middle of the beam because the system reduces to a quasi-staticsituation. With regard to the slow speeds, it is observed that the maximum beam deflectionoccurs before the middle of the beam within a particular range of carriage speeds. Thismay be explained by noting that the beam vibration frequency is varying because of thevariation in mass distribution. An analytical deduction of this range of carriage speeds fora given configuration is difficult, if not impossible.

Plots of the beam deflection history for a carriage speed of 2 m/s over a 10-m long beamare presented in Figure 12. This figure is presented to provide a qualitative understandingof the interaction between the carriage speed and beam vibration frequency. The negativeof the deflections has been plotted for aesthetic reasons because the reference frame of the

beam was positioned in such a manner that downward deflections of the beam werepositive.

Each curve in the figure is a beam deflection profile traced when the carriage hastravelled along the beam for a particular time. The dots represent the carriage locationat time intervals of 05 s as it traverses the beam. The diagonal line at the bottom of thefigure is the projection of the carriage trajectory in the timeposition plane. In a likemanner the plot for a carriage speed of 02 s1 in Figure 11 is the projection of the carriagetrajectory onto the beam deflectiontime plane for this same example.

The beam deflection profile traced when the carriage has travelled 4 m along the beamprovides the maximum beam deflection for all the profiles. Note, however, that thismaximum deflection does not occur under the carriage but rather at a point forward ofthe carriage. If one considered the beam as simply supported without any concentratedload, the period for the ith mode is

Ti= 2i2AL4EI

1/2

=2i2

.

From Figure 2 it can be seen that, for the first mode, this will yield a reasonably closeestimate for the dominant period.


16/18

... .904

The period of the first mode of the beam is 10726 s and the carriage will travel 429 mafter 2T1periods which correlates very well with the maximum deflection under the carriageat the slowest carriage speed ( = 02 s1) in Figure 11. The other local peaks inthat figure occur at 1T1, 3T1and 4T1periods indicating that this carriage speed is sufficientlyslow to allow the beam vibrations to be observed in the response. At the next higher carriage

speed, i.e., = 05 s1 the peak beam deflection under the carriage occurs at a location (time)that corresponds very closely to 1T1period. The peak displacement under the carriage forthe highest carriage speed reported in Figure 11 does not correlate with an integer multipleofT1or, apparently, any other system characteristic period.

6. SUMMARY

The dynamics of an overhead crane system with the carriage moving at a specifiedconstant speed has been examined. The beam portion of the crane was modelled as anEulerBernoulli beam, the carriage as a particle and the pendulum as a rigid massless rodwith a particle tip mass. The coupled integro-differential equations of motion were derivedusing Hamiltons principle.

Our assumption of a small angle of rotation (or swing) of the pendulum has led to asystem of equations in which the dynamics of the pendulum are dependent on the dynamicsof the beam but not the converse. Allowing a large angular displacement of the pendulumwould introduce a bidirectional coupling between the beam and the pendulum andintroduce the possibility of pendulum instability due to the flexible support offered by thebeam; this would be a worthwhile area for future inquiry. The acceleration of the beamunder the carriage acts as an effective stiffness contribution in the governing equation ofmotion of the pendulum but this effect is not unidirectional since the term can take eitherpositive or negative values and hence the pendulum acts as if it has a time varying stiffnesscoefficient.

The mode shapes of the beam were determined as functions of the instantaneous locationof the carriage on the beam. The vibration of the beam was determined via operational

calculus and fitted to a Chybeshev polynomial series. The resulting expression was thenused to simulate the dynamics of the pendulum.The results obtained from the numerical calculations that employed the non-dimensional

formulation of the problem indicate that the deflection of the beam is dependent on boththe carriage speed and the mass of the combined carriage and payload mass. In particular,the magnitude of the deflection increases with increasing crane mass at all carriage speeds.For a given crane mass, there is a threshold carriage speed at which the beam deflectionunder the carriage is symmetric about the middle of the beam. The position of themaximum deflection tends to drift to the right or the left of the beam centre dependingon whether the operating speed is above or below this threshold. Further, the magnitudeof the maximum deflection tends to decrease with increasing drift from symmetry.

The threshold speed and the location of the maximum beam deflection for a givencarriage speed are observed to be functions of the beam vibration frequency due to what

may be considered as constructive interference between the gravitational deflection of thebeam and the vibrational behaviour of the beam. The value of the maximum beamdeflection for a given set of carriage and payload masses is dependent upon the carriagespeed. For high carriage speeds, the maximum beam deflection occurs close to the end ofthe beam where the carriage stops because the deflections are dominated by inertial effectsof the carriage. At very slow speeds the beam behaves in a quasi-static manner and themaximum beam deflection occurs close to the middle of the beam.


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The dynamics of the pendulum are dependent on the carriage mass, the payload mass,the length of the pendulum, and the carriage speed. The carriage mass or payload masseffect is reflected in the magnitude of the vibration while changes in the pendulum lengthaffect both the frequency and magnitude of the swing.

Finally, an unstable pendulum swing is possible at some carriage speeds and

configurations because the governing equation will reduce to a HillMathieu equation.

REFERENCES

1. F. H1976 Transactions of the JSME 42, 24002411. Two-phase flow induced vibration ina horizontal piping system.

2. R. K, C. W. L, A. G. Uand R. A. S1987ASME Journal of Vibration,Acoustics,Stress and Reliability in Design 109, 361365. Dynamic stability and response of a beam subjectto a deflection dependent moving load.

3. Y. H. L and M. W. T1990 Journal of Sound and Vibration 136, 323342. Finiteelement analysis of elastic beams subjected to moving dynamic loads.

4. G. G. S 1849 Transactions of the Cambridge Philosophical Society , Part 5, 707735.Discussion of a differential equation relating to the breaking of railway bridges.

5. L. F 1972Vibration of Solids and Structures under Moving Loads . Groningen: Noordhoff.

6. H. P. L1994Journal of Sound and Vibration 171, 369395. Dynamic response of a beam withintermediate point constraints subject to a moving load.7. E. Eand M. G1995 Journal of Sound and Vibration 184, 917. Vibration

analysis of beams traversed by uniform partially distributed moving masses.8. G. M, D. Sand A. N. K1996Journal of Sound and Vibration

191, 357362. The effect of a moving mass and other parameters on the dynamic response ofa simply supported beam.

9. J. H, T. Yand N. A1985Journal of Sound and Vibration 100,477491. Vibration analysis of non-linear beams subjected to a moving load using the finiteelement method.

10. T. Y, J. Hand N. A1985Journal of Sound and Vibration 104,179186. Vibration analysis of non-linear beams subjected to a moving load by using theGalerkin method.

11. M. O1991Journal of Sound and Vibration 145, 299307. On the fundamental moving loadproblem.

12. M. M. S 1985Ingenieur-Archiv 55, 176185. On a new theory of the dynamic behaviorof the structures carrying moving masses.

13. F. K, M. F. Gand G. R. H1994Nonlinear Dynamics 5, 493513. Onthe dynamic behavior of a flexible beam carrying a moving mass.

14. B. F1956 Principles and Techniques of Applied Mathematics. New York: John Wileyand Sons.

APPENDIX: NOMENCLATURE

A beam cross-sectional areaE Youngs modulusI second moment of the areaL ratio of the length of the pendulum to the length of the beamLb length of the beamLp length of the pendulum

mc mass of the carriagemL mass of the payload

p summation of non-dimensional carriage and payload weightsp1 non-dimensional weight of the carriagep2 non-dimensional weight of the payloadqm undetermined coefficientsrb position vector of an elemental massrc position vector of the carriage


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rL position vector of the payloadt timeT system kinetic energyVm orthonormal eigenfunctionsU system potential energyv(,) non-dimensional transverse displacement

x co-ordinate of the beamxc position of the carriage on the beam

y(x,t) transverse displacement natural frequency constant of the beamW system virtual work summation of the non-dimensional carriage and payload masses1 non-dimensional mass of the carriage2 non-dimensional mass of the payload pendulum angle from the vertical square of the non-dimensional frequency non-dimensional co-ordinate of the beamc non-dimensional position of the carriage volume mass density non-dimensional time non-dimensional frequency

( ) time derivative( )' spatial derivative

dynamic response of an overhead crane system, 1998

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