ON THE DYNAMIC PROPERTIES OF A CONVEYOR BELT AND ITS DRIVE

Dr. Neville F. RiegerChief Scientist

STI Technologies, Inc.1800 Brighton-Henrietta Townline Road

Rochester, NY 14623nrieger@sti-tech.com

Abstract:The vibrations and dynamic properties of a conveyor belt and drive system have been investigated. Naturalfrequencies and modes of the drive-conveyor system were determined from tests on a mining conveyor andits geared drive. This data was used to prepare several computer models which were used to study thetransient response properties of the drive and the conveyor during startup. Results of these studies arepresented, and the usefulness of the simple models is evaluated. An estimate of the size of the transientdynamic magnifier occurring at each start of the conveyor was obtained.

Keywords:

ConveyorEigenvectorsEigenvaluesGeardriveTransientTorsional

Notation:

A cross-sectional areaA, B integration constantsa {gE/w}1/2

[D] dynamical matrixE modulus of elasticityE line voltageF forcef(t) function of timeG shear modulus of elasticityg gravitational accelerationI inertia of componentI current[I] inertia matrixIp polar second moment of area[K] stiffness matrixk spring stiffnesskij individual stiffness within [K]L length of beltM mass of componentm {�/a}mij individual mass within [M]N rotational speed rev/minR radius of pulleyS s(x) f(t), time varying displacements(x) displacementT torque

t timeU u(x) f(t), time varying displacementu(x) {A cos mx + B sin mx}w specific weight, lbf/in3

x length coordinate�,� Rayleigh damping coefficients� 3.14159265…� angular displacement

�2βω

2ωα

� , Rayleigh damping value

� circular frequency, rad/sec

Introduction:

In a recent paper Rieger [1] examined the interaction between a conveyor and its drive system, and studiedthe dynamic stresses experienced by the drive gear teeth during start-up and operation. A schematic of thedrive and conveyor involved is shown in Figure 1. It was found that the pitting damage experienced by thegear teeth of this drive system was not due to the few peak transient start-up load cycles, but was morelikely due to some aspect of tooth loading experienced during gear operation.

The purpose of the present paper is to discuss the dynamic properties of the conveyor system in greaterdetail, and in particular to show that although the drive and conveyor are mechanically coupled, thesecomponents actually function as de-coupled systems during operation. This is due to the chosen layout ofthe system, and to the separation of the natural modes on the frequency scale. Recognition of this de-coupling effect allows simplifications to be made in the modeling of the system. This in turn simplifies thecalculation of natural frequencies of the drive and of the conveyor.

The response of the conveyor belt to start-up and shutdown transients from the drive was also studied, andthe results are compared with results obtained in Ref. [1]. It was found that if the modes of the twosubsystems can be represented with sufficient accuracy by simplified models, then the transient responseproperties of both components can also be obtained more simply.

Figure 1. Layout of Drive Showing Positions of Strain Gages and Accelerometers. Ref. [1]

Details of Conveyor Drive System:

The conveyor drive system is shown in Figure 1. The drive consists of two 650 Hp wound rotor inductionmotors, one at each end of the geared drive system. These motors drive the gearboxes and the mainconveyor pulley, and hence the conveyor. Both gearboxes contained opposed double reduction helical gears.The motor drive shaft operates at 960 rpm and the 63 inch diameter conveyor drive pulley rotates at 30 rpm.

The conveyor belt is 42 inches wide by 4800 feet in length. It delivers coal from a seam 1400 feet beneaththe surface of the earth. Up to 600 tons of coal per hour are carried on the conveyor when it is fully loaded.The vertical rise from the lower end of the conveyor to the drive pulley is 1400 feet.

The conveyor belt is 1.25 inches thick. It is made from neoprene synthetic rubber reinforced with 12 wovensteel cables of diameter 0.375 inch. The weight density of the neoprene is 0.10 lb/in3, and the density of thecoal is taken as 75 lb/ft3 on the upper surface of the conveyor. At 600 tons per hour the average depth of thecoal across the 36 inch active width of the conveyor is 6.0 inches.

The conveyor dynamic properties of interest in the present case are those in the axial, i.e., lengthwisedirection of the upper (coal-carrying) surface of the conveyor. The upper section of the conveyor belt is in astate of tension during operation, due to the flow of drive power. The lower portion of the conveyor belt isassumed to exert no effect on the conveyor dynamics due to its slackness. The actual conveyor has a take-upmechanism to absorb start-up and shut-down transients at its lower end, and a recoil mechanism to absorbshock loads from the continuous addition of coal lumps to the belt. The effect of factors such as (a) therelative flexibility of the belt in the axial direction, (b) the distributed mass of the belt and coal, (c) theinertia of the take-up pulleys, and (d) the flexibility of the recoil mechanism were considered the presentcalculations, and the extent to which they appear to influence the dynamic properties of the conveyor wasinvestigated.

Conveyor Modes and Natural Frequencies:

Results for the modes and natural frequencies of the geared drive and conveyor system have been studiedand presented in [1]. These results have been verified against test data from the system, running with andwithout coal. The model used for calculation of the complete system is shown in Figure 2. The calculatedresults for natural frequencies of the system are listed in Table I, and compared with test results. Modeshapes for the conveyor are shown in Figure 3 and for the geared drive in Figure 4.

Several simplified models of the conveyor alone were developed and tested against available data, in orderto study the oscillations of the conveyor belt during operation. The theory for these models is given in theAppendix. Equation 14 can be used to obtain natural frequencies for each of these models. This simplifiedtheory allowed a number of aspects of the conveyor system to be studied in greater detail. Six simplifiedmodels of the conveyor system were considered for calculation purposes. These models were:

1. Complete multi inertia model of the conveyor plus drive system. From Ref. [1]. Shown in Figure 2. Forreference purposes.

2. Conveyor belt alone with coal, and without end pulleys. Distributed system. Shown in Figure 5(a).

3. Conveyor belt, drive pulleys, with coal. Distributed system with end inertias and mass. Shown in Figure5(b).

4. Conveyor belt, geared system inertia at drive end, idler pulley, inertia and mass, with coal. Shown inFigure 5(c) with M3 = 0.

5. Conveyor belt, small end inertias, with tensioning mass and stiffness, with coal. Shown in Figure 5(c.)

6. Conveyor belt plus drive system inertia. Idler end pulley and tensioning mass, with coal. Distributedbelt system plus 3 discrete d.o.f. Shown in Figure 5(c).

Figure 2. Lumped Inertia Model of Conveyor and Drive System. Ref. [1]

Figure 3. Conveyor Axial Modes. Ref. [1]

Figure 4. Drive System Modes. Ref. [1]

Table I. Measured Test Frequencies and Correlation with Calculated Frequencies, Hz. Ref. [1]

Source Belt Loaded Belt Unloaded Calculated Comments

T1 2.0 2.2 2.153 Mode 6T1 6.3 7.5 5.902 Mode 11

RVA 16.5 HSS 1/revT2 28.75 28.75 28.99 Mode 12

30.0 29.02 Mode 13T2 33.0 HSS 2/rev

RVA 35.0VA1, VA3, T1 50.0 50.0 Line Freq.VA1, RVA 56.25 L.S. Tooth Pass

85.81 Mode 14T1 112.0 107.45 Mode 17

149.5 Mode 18149.5 Mode 19

VA1 150.0 3 x Line Freq.216.2 Mode 20216.2 Mode 21

VA1, RVA 230.0 230.0 H.S. Beam n.f.260.3 Mode 22260.3 Mode 23

VA1, RVA 365.0 H.S. Tooth PassLegend: T1, T2 Torsional Strain Gages 1, 2RVA Armature Current Meter

VA1, VA3 Accelerometers 1, 3HSS High Speed ShaftLSS Low Speed Shaft

These results show that apart from some rigid-body contributions the first four modes of the conveyor beltlie below 2.0 Hz, and are separated from the torsional modes of the geared drive which all lie above 2.0 Hz.This observation was used to simplify the system model into a conveyor-pulley system (for the lowermodes), and into a geared drive system (for the higher modes). The distributed and discrete-mass modelsshown in Figures 5a and 5b were used to make natural frequency calculations for the conveyor system.Model (a) consists of the belt alone modeled as a distributed system, with added mass from coal load alongits upper surface. Using Equation 18 the eigenvalue equation becomes sin ml = 0. Model (b) contains thetwo drive pulleys at either end of the belt and the distributed belt itself plus coal. Eigenvalues may be foundfrom Equation 18. Two end conditions were studied; (a) free-free ends with the pulleys free to rotatetorsionally, and (b) clamped-free ends with the drive pulley restrained from angular movement. The naturalfrequencies for this case are given in Table II.

Figure 5. Simplified Models of Conveyor

Table II. Natural Frequencies from Conveyor Models

Model Description Figure f1 f2 f31 Complete System 2 0.428 0.909 1.3982 Distributed Belt 5(a) 0.494 0.988 1.4823 Belt & Small Identical End Inertias 5(b) 0.439 0.881 1.3294 Drive System Inertia, Belt, Small Pulley 5(c) 0.351 0.764 1.2095 Belt, Small Pulley Inertias Tensioning Mass 5(c) 0.309 0.767 1.2516 Drive Inertia, Belt Tensioning Mass 5(c) (0.192)

(0.240)0.309

0.726 1.241

Model (b) consists of the conveyor belt alone, but including an allowance for the idler pulleys, with andwithout coal. Frequency results are listed in Table II.

Model (c) consists of Model (b) plus the tensioning mass and spring. This model also allowed translationalmovement as well as angular movement of the lower belt pulley, and included the effect of the tensioningmass spring system. This is the four d.o.f. model for which the matrix frequency equation is derived in theAppendix. Natural frequency results for this model are listed in Table II.

From the correlation between the results from these models and the data obtained in Ref. [1] for thecomplete system, it is evident that the lower four modes of the conveyor can be modeled with acceptableaccuracy for frequency calculations by treating the conveyor as a separate component, i.e., without includingthe geared drive system. The accuracy of this simple model decreases for the higher modes where effectsfrom the geared drive begin to enter the conveyor modes. The simple belt plus pulley inertia model givesacceptable frequency results for design calculations using the formula obtained in the Appendix, Equation18. Inclusion of the tensioning mass and the effect of pulley mass translation does not appear to improve theaccuracy of the frequency results or mode shapes. The value of such additions to the computer model isdifficult to assess without having available test data on the belt frequencies for comparison purposes.

Figure 6(a). Start-up Sequence – Current I1 and Torque T1 vs. Time – Belt Loaded.

Figure 6(b). Simulation of Start-up Torque at Motors

Transient Response Analysis of Drive and Conveyor:

Transient response properties of the geared-conveyor system were studied using the measured curves forstart-up current given in Figure 6(a), and by calibrating these curves for use as start-up torques applied toboth the motor armatures. Representation of these curves by the straight line curve segments shown inFigure 6(b) was used. Response torque values throughout the system were obtained from the computermodel plus the start-up torque transients. Examples of these transient torque responses at the drive pulley areshown in Figure 7. The transmitted shaft torque on both sides of the drive pulley, and the sum of these

torques are shown. The details in these curves agree closely with the input data and with measured straingage data from tests on the conveyor drive (not shown). The initial start-up transient effect is seen to decayin accordance with the level of damping chosen for the computer model (� = 0.08). The applied torquedecrease with time is in accordance with the simulated input torque curve, and the torque step increases inthe input curve are apparent. These include the transient decay oscillations from the lowest drive systemmode at 2.2 Hz. The significance of impulsively applied loads on the drive is evident from Figure 7. Aninput torque transient of 66,000 + 52,000 = 118,000 lb.in. has an oscillatory peak value of 166,000 lb.in. atthe conveyor, representing a maximum torque magnifier of 1.406.

Figure 7. Drive Torques Acting on Conveyor Drive Pulley

The variation in dynamic drive force in the conveyor belt is shown in Figure 8. In accordance with thelowest mode of the conveyor at 0.428 Hz, an initial period is needed to apply drive torque to belt, whichaccounts for the relatively long build-up of belt force. Transient oscillations then occur at the lowest beltfrequency of 0.428 Hz, as expected. The force values shown are negative because of the form of theoscillations (negative wave at drive end).

Discussion:

Comparing the calculated results for natural frequencies with the measured results from strain gages andaccelerometers indicates that the general computer model of the drive system was adequate, because thefrequency correlation obtained between calculate and test data for the drive system was consistently closethroughout the range of frequencies observed. Experimental correlation for the belt modes could not beobtained because no frequencies corresponding to the calculated belt modes were observed in the data fromthe test: see Table I. Such data was not sought at the time because the emphasis of the study was on thenature of the gear tooth loads. However, the computed belt results appear to be valid because (a) they are inaccordance with classical results, e.g., the wave equation results from Model 1, and (b) they agree well withverified theoretical results, i.e., Equation 18 from Ref. [2]. This result is considered to validate the beltmodes with sufficient accuracy for the present study.

Figure 8. Dynamic Force in Belt at Main Pulleys

For validation of torques, belt forces, and vibration amplitudes it was necessary to consider the nature of theinputs used for the drive torque, and for the damping values. Test data obtained for steady values of thearmature currents I1 and I2 was used to determine the applied motor torque. Current data was converted topower by multiplying these current values by the line voltage E. The product EI = Watts, Watts/746 = H.P.,and H.P. = TN/63025. This gives the torque T lb.in. where N is in rpm. This torque allowed the transientmagnifiers for the currents and torques to be determined from the current curves under start-up conditions.Simulated start-up torque curves developed and calibrated by this procedure were applied to the computermodel. Computed results from these calibrated models gave data on the transient loads developed in thesystem under a variety of conditions; start-up, shut-down, with coal, without coal, and gave dynamicmagnifiers corresponding to the same transient conditions. As a result it can be said that as long as thesystem remains linear, it appears that the transient loads observed will never reach the theoretical maximumvalue of 2.0 associated with a simple sustained step transient. This is because the motors were designed toimpose transient loads having a decaying exponential form. The size of the dynamic magnifiers found weretypically between 1.3 and 1.5 for both the drive system and conveyor. A theoretical damping value of � =0.08 gave the best correlation with the observed test data for transient decay. In the calculations thisdamping was expressed as a Rayleigh quotient with values of � = 0.054 and � = 0.089.

Conclusions:

1. Th dynamic properties of a conveyor system have been obtained from test data and have been simulatedby several calculation methods.

2. For the symmetrical two motor drive system involved, the lower conveyor modes are sensibly de-coupled from the higher modes of the drive. This is due to the relative mass-elastic properties of the twosystems, although modal de-coupling is also facilitated by the symmetry of the drive system layoutabout the conveyor.

3. The simple conveyor modes were found to occur at frequencies between 0.4 Hz and 2.0 Hz. Modesinvolving the geared drive system were found above 2.0 Hz.

4. The first four natural frequencies and mode shapes of the conveyor were predicted with good accuracyusing simple models.

5. Higher modes of the conveyor drive system show an increase in the amount of coupling which occurswith the geared drive system.

6. Torque transient response of the system showed good correlation between results measuredexperimentally and calculated transient results.

7. The transient response of the conveyor appeared to involve only the lowest axial mode of the conveyorat 0.428 Hz.

8. Test data shows that while the transient response gives rise to the highest conveyor (and gear) loads, thetransient conditions die out quickly, leaving only steady drive load effects under operating conditions.

9. A range of torque magnifiers between 1.3 and 1.5 were obtained from the results.

References:

1. Rieger, N. F., “Experiments and Calculations on the Torsional Dynamic Properties of Coal ConveyorDrive,” Proceedings, IFToMM 6th International Conference on Rotor Dynamics, Sydney, Australia,September 30-October 3, 2002.

2. Timoshenko, S. P., “Vibration Problems in Engineering,” D.Van Nostrand & Company, Inc.,Publishers, Princeton, NJ, Third Edition, Art 50, p.p. 318-320, 1955.

Appendix. Modes and Natural Frequencies:

The modal properties of the conveyor model shown in Figure 5(c) may be determined as follows. Assumingfor dynamical purposes that the main features of the conveyor are the massive elastic belt (with or withoutcoal) the two end pulley inertias, and the shock load absorber shown, the following equations of motion maybe written for harmonic motions of the system:

Belt 2

2

2

2

xt)(x,uEA

tt)(x,u

gwA

�

��

�

�(1)

Drive Pulley Intertia TRFtθI 112

12

1 ��

�

�(2)

Tail Pulley Intertia 2222

2

2 RFtθI ��

�

�(3)

Belt Tension Spring Mass 3222

2

2 FFtSM ���

�

�(4)

)Sk(SFtSM 2332

32

3 �����

�

�(5)

and writing tcos(x)u(t)f(x)ut)(x,U ���� (6)

2,3nωt,cossf(t)s(x)t)(x,S nn ��� (7)

where mxsinBmxcosA(x)u �� (8)

andwgEaand,

aωm 2

�� (9)

For the boundary conditions we write:

1xx(x)uEAF1 �

�

�� (10)

2xx(x)uEAF2 ��

�� (11)

111 θRu � (12)

2222 Rθsu �� (13)

Substituting and reducing leads to the following 4�4 matrix equation in the unknown displacements {u1, u2,s2, s3}T as a function of the harmonic excitation vector {T, 0, 0, 0}T:

{u},[D]{u}[I]]ω[[K]{T} 2��� (14)

where T0}0,0,{T,{T} � (15)

����

�

�

����

�

�

�

���

�

��

kk00kkHcosmLHcosmLH0HcosmLHcosmLH0HHHcosmL

[k] (16)

�����

�

�

�����

�

�

�

3

2

222

211

M0000M00

00/RI0

000/RI

[I] (17)

The determinant of [D] in Equation 14 contains eigenvalues corresponding to the natural frequencies of themodes of vibration of the conveyor system in Figure 5(c). If the tensioning mass is effectively detuned fromthe system, i.e., either extremely massive or negligible compared to the conveyor inertia, the remainingsystem may be considered either by making the spring displacement s3 = 0, or without the spring, i.e., S2 = S3= 0. Both cases reduce the size of the determinant. For the case S2 = S3 = 0 the resulting 2x2 determinant maybe further reduced to the frequency equation:

222

21

214

222

221

1

(EAm)RRIIω

ω)RI

RI(EAm

mLtan�

�

� (18)

This transcendental equation is readily solved numerically for given values of the parameters. It can befurther reduced to that given in Ref. [2] on page 320, by setting R1 = R2 and GIp = EA. This has the effect ofmaking the two systems identical. Equation 18 was used in this study to verify the computer results shownin Table II. Equation 18 further reduces to the well-known eigenvalue expression sin mL = 0 when I1 and I2are set to zero, corresponding to the case for a uniform belt with free ends.