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American Institute of Aeronautics and Astronautics

1

A NONLINEAR FINITE ELEMENT MODEL OF A

TRUSS USING PINNED JOINTS

Joseph D. Dutson

and Steven L. Folkman

Mechanical and Aerospace Engineering Department

Utah State University

Logan, UT 84322-4130

Graduate Research Assistant Associate Professor, Senior Member AIAA, Member ASME

Copyright 1996 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Abstract

Researchers at Utah State University have

been studying the dynamic behavior of a three bay

cantilevered truss. The truss includes eight

tang/clevis-type joints with clearance fit pins. The

clearance in these joints significantly changes the

dynamic behavior of the truss. When strut preloads are

minimized the structural damping in the truss

increases dramatically. Struts have been subjected to

cyclic loading to characterize the dynamic behavior ofan individual strut containing a pinned joint. The

three bay truss was tested in different gravity

environments which alter the dynamic behavior of the

truss by changing strut preloads. This paper reports on

efforts made to develop a nonlinear finite element

model of the truss which simulates the observed truss

behavior and provides insights into the damping

mechanisms occurring in the joints. The finite element

analysis program LS-DYNA3D was used to create a

simple model of a single strut which accounts for

friction, impacting, and equivalent viscous damping in

the joints and gives reasonable agreement with

measured data. The strut model was extended to

model the entire truss. The predicted results correlate

well with measured data, particularly when strut

preloads are minimized.

Introduction

As researchers explore the possibility of

constructing space structures, they are confronted with

the task of determining a cost effective, safe method for

assembling these structures in space. One feasible

approach utilizes a deployable structure assembled with

light-weight flexible trusses connected with revolute

joints. However, vibration modes are easily induced in

these light-weight trusses. Therefore, structural

damping is an important consideration in the design of

deployable space structures.

If the revolute joints in deployable structures

are designed such that large preloads are present across

the mating surfaces, the behavior of the joint

approaches that of a tightly clamped or welded joint.In this configuration, the joints contribute little to the

damping of the structure, and the structure could be

accurately modeled using linear finite element models.

On the other hand, the dynamic characteristics of the

structure will be altered dramatically if the joints are

designed with a small amount of slop or deadband

between the mating surfaces and if joint preloads are

small. The deadband and friction characteristics in

this type of joint can introduce nonlinearities into the

joint behavior. The energy dissipation associated with

friction and impacting in the joint results in increased

structural damping. Predicting the dynamic behavior

of a structure with friction and impacting in the joints

is difficult.

Large-scale testing of space structures is

expensive and often impractical. Testing of a

prototype usually occurs far into the design process and

may result in expensive redesigns. Also, testing

complete structures on the ground may not reveal how

the structure will behave dynamically in the micro-

gravity environment of space. However, individual

components of the structure are easily tested on the

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ground early in the design process. If the dynamic

behavior of individual struts is understood, the overall

system might be more accurately modeled. This paper

will document the creation of nonlinear finite element

models of individual struts of a truss. These strut

models allow selection of properties for equivalent

viscous damping elements and friction coefficients.The strut models were then extended to model a three

bay truss and the ability of the model to predict the

dynamic behavior of the truss is reported.

Models of Joint Damping

The structural damping associated with

connections or joints has been studied extensively.

Models of joint or interface damping are usually

derived from two mechanisms, friction and impacting.

Impacting implies that two surfaces, initially separated

by some finite distance, come into contact and a

momentum exchange occurs. Modeling the energydissipation associated with impacting is complicated.

Crawley, Sigler, and van Schoor1 suggested that a

measure of energy dissipation due to impacting is the

coefficient of restitution. The coefficient of restitution

is a material property, but it is also a function of the

geometry of the contacting surfaces. Also, the

compliance of the structure attached to the joint can

influence the dynamic response. Therefore, the

coefficient of restitution by itself is felt to be too

simplistic to model damping in a joint.

Friction is attributed to either extensional

motion or rotation. Models of friction damping

generally fall into two categories, microslip and

macroslip. In microslip models there is no relative

motion between the two mating surfaces; however,

small surface imperfections allow localized,

microscopic slippage. Microslip damping levels are

generally very low and Plunkett2 said that we are still

far from being able to predict the damping associated

with microslip. Macroslip models assume that no

damping occurs until there is relative motion between

the two mating surfaces. Relative motion occurs when

the forces parallel to the surface interface exceed the

Coulomb friction force. The Coulomb friction force isproportional to the normal force between the two

interfaces. Den Hartog3 analyzed this classical friction

model and determined that for small loads the energy

dissipated increases linearly with displacement.

Although there are no general models

available for predicting joint damping, many

mathematical models have been proposed. Ferri4

created a model of a nonlinear sleeve joint and

concluded that the three major sources of energy

dissipation were damping due to Coulomb friction,

damping due to impact, and material damping. He

also showed that the overall damping was similar to

viscous damping. Lankarani and Nikravesh5

created amodel for impacting of two elastic bodies. They

concluded that energy dissipation during impact was a

function of initial velocity, coefficient of restitution,

and material properties. Tzou and Rong6 presented a

mathematical model of a three-dimensional spherical

joint which included the effects of friction and

impacting. Onoda and others7

showed that impacting

causes energy to be transferred from lower to higher

modes. The excitation of higher modes results in

greater structural damping.

Although numerous mathematical models of

pinned joints have been developed, the literature doesnot describe attempts to model the dynamic response of

pinned joints using commercial FEA programs.

Folkman and others8 described a combination of finite

elements that could be used to model a pinned joint;

however, they did not attempt to construct the model

and correlate the results with measured data.

Experiment Description

Researchers at Utah State University have

developed an experiment titled the Joint Damping

Experiment (JDX). This project is funded by IN-

STEP, NASAs In-Space Technology Experiment

Program. The objectives of JDX include development

of a small-scale pin-jointed truss which was flown as a

Get Away Special payload on the Space Shuttle. This

truss has allowed researchers to characterize the

influence of gravity and joint gaps on the structural

damping and dynamic behavior of pin-jointed

structures. The objectives also include development of

a nonlinear finite element model of the truss to

simulate the measured behavior. This model will

increase understanding of the structural dynamics

occurring in a pin-jointed truss and will facilitate

future modeling of such structures.

The flight model of the JDX truss is shown in

the photograph in Fig. 1. This three-bay truss is

mounted to an aluminum base plate to provide a

cantilevered boundary condition. A stainless steel tip

mass is attached to the other end of the truss to lower

the natural frequency of the structure. The truss is

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excited by electromagnets deflecting and then releasing

the tip mass. The resulting free vibrational decay for

the twang test is recorded. Only the eight joints at

the top of the bottom truss bay are free to act like

pinned joints. These eight joints exhibit nonlinear

dynamic behavior and are referred to as unlocked

joints. Figure 1 illustrates the location of the unlockedjoints. The pins in the remaining joints are press fit

into place. These joints with press fit pins behave

linearly as if they were welded connections and are

referred to as locked joints

Figure 1. Photograph of the JDX flight model truss.

The design of an unlocked joint in the flight

model truss is shown in Fig. 2. This is a tang/clevis-

type joint with a clearance fit pin. The two holes in the

clevis and the hole in the tang are each press fit withhardened steel sleeves to reduce wear. The pin used is

a hardened steel shoulder bolt. Therefore, the pin-

sleeve interface where impacting occurs is very hard.

The diametral gap in the unlocked joints of the truss

averages about 0.00063 in.

The truss excitation system preferentially

excites three modes in the truss; two bending modes

and a torsional mode. The two bending modes are the

two lowest frequency modes of the truss and are

described as the bend 1 and bend 2 modes. The

torsional mode consists of a rotational motion about the

long axis of the truss. Figure 3 is a top view of thetruss tip mass which illustrates the direction the

magnets move to excite the bend 1, bend 2, and torsion

modes. Ground based truss twang tests were conducted

with the truss in two orientations with respect to the

gravity vector. Figure 4 illustrates the 0- and 90-deg

truss orientations. In the 0-deg orientation, gravity

induced preloads in the struts are minimized, while

preloads are maximized in the 90-deg orientation. The

influence of gravity on joint damping was observed by

comparing the results from the two orientations.

Figure 2. Illustration of the JDX joint design.

Figure 3. Illustration of the tip mass and the magnets

used to excite truss by deflecting the tip mass.

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As part of the JDX research, a force-state

mapping (FSM) technique was used to characterize

individual struts from the truss. The FSM technique

involves applying a sinusoidal load to a strut and

measuring the resulting axial displacements,

accelerations, and applied forces. A strut with bothjoints locked as well as a strut with one joint locked

and the other unlocked were characterized. The FSM

tests were conducted on both long (i.e. the diagonal

struts in each bay) and short struts. The FSM

technique resulted in parameters such as strut stiffness,

deadband, and equivalent viscous and friction damping

for use in the analytical computer model of the truss.

Ferney9 documents the FSM results.

90-DEG TRUSSORIENTATION

0-DEG TRUSS

ORIENTATION

DIRECTION OF

DIRECTION

OF

EXCITATION

GRAVITY EXCITATION

Figure 4. Illustration of the 0- and 90-deg truss

orientations for ground tests.

The JDX flight model truss was flown on

NASAs KC-135 low-G aircraft on October 25-28,

1994 and on the Space Shuttle Endeavor on September

7-18, 1995 in order to test the truss in a micro gravity

environment. Comparing these tests with ground tests

demonstrates the influence of gravity on the structural

damping and dynamic behavior of the truss. These

tests also provided experimental data for comparison

with finite element model of the truss.

Finite Element Model of a Strut

It was desired to construct a finite element

model of a strut with an unlocked joint which would be

reasonably simple while capturing the most important

features of the struts behavior. A model was madethat would account for the deadband, impacting,

extensional friction, rotational friction, and equivalent

viscous damping in the joint. LS-DYNA3D10, a

commercial finite element analysis program, was

chosen to model the strut. Impacting and friction can

be modeled in LS-DYNA3D by a point contacting or

sliding along a surface. The point and surface used to

model impacting and friction form a sliding interface.

No stiffness is assigned to a sliding interface until

contact is made, at which time a very high stiffness is

assigned in the direction perpendicular to the surface.

After contact, stiffness between the node and surface in

the lateral direction is based on the Coulomb friction

force.

Figure 5 illustrates a strut and the beam

elements used to model the strut in LS-DYNA3D. An

unlocked joint is located between nodes 2 and 3. The

upper half of Fig. 5 shows the elements used to model

the unlocked joint. The coordinate system was defined

such that the x axis is aligned with the strut and the y

and z axes are orthogonal to the x axis. Nodes 1, 2, 3,

4, 8, and 9 all lie on the x axis but are offset in the

upper part of Fig. 5 for clarity. Elements 1 and 2 are

beam elements used to model the clevis and tang,

respectively. Element 10 is the large square in Fig. 5.

Element 10 is a rigid element which is actually formedfrom several solid elements to define contact surfaces

in the joint. The width of element 10 is the pin

diameter used in the joint. Nodes 3, 8, and 9 were

rigidly connected by elements 8 and 9 which penetrate

element 10. Under a tensile strut load, node 9 impacts

the surface of rigid element 10, while node 3 impacts

element 10 when the load is compressive. Nodes 3 and

9 are initially located a distance equal to half the joint

deadband away from the surface of element 10. Node

8 is located inside a narrow slot in element 10 and has

two functions. First, it is the hinge point for joint

rotations. The slot that contains node 8, although only

shown in two dimensions, is three dimensional and

prevents relative displacement between node 8 and

element 10 in the y and z directions. The slot is very

narrow (2x10-6 inch) and is a sliding interface for node

8. Second, it provides extensional friction as the joint

traverses the deadband. A force FN, applied to both

element 10 and node 8, maintains a constant

compressive force at the friction interface (assuming

lateral shearing forces are not present). Element 7 is a

viscous damper which damps oscillations that occur at

the friction interface when normal force FN is initially

applied. Element 6 provides equivalent viscous

damping as the joint traverses the deadband. Nodes 2and 10 are rigidly connected to element 10 to form a

single rigid part. Rotational friction may be present

when either node 3 or node 9 is in contact with

element 10 and there is relative rotation between

element 10 and the rigid line of nodes 3, 8, and 9.

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Element 10 is actually a composite of several

elements formed by combining six solid elements into

a single part. These six solid elements are combined in

such a way that a slot is left in the center of the part.

Figure 6 is a three-dimensional cutaway view of the

unlocked joint model which shows five of the blocks

used to construct element 10. Care must be takenwhen constructing element 10 so that node 8 cannot

slip out of the slot. Note that node 8 can penetrate

slightly into the sliding interface of the slot. If the

solid blocks used to construct element 10 do not

overlap each other, this penetration of node 8 could

allow the node to slip out of the slot at the interface

between two blocks. Thus in Fig. 6, block B is shaded

to show how it overlaps A, C, D, and E.

y

x

z

1 1 2

9

67

FN

10

8

8 9

FN

3 42

10

UNLOCKED JOINT

ASSEMBLY

ELEMENT NUMBERS

NODE NUMBERS

1 2 3 4 5

1 (2 & 3) 4 5 6 7

ELEMENT

NUMBER

1, 5

2, 4

3

6, 7

8, 9

10

DESCRIPTION

HUB/CLEVIS BEAM ELEMENTS

TANG BEAM ELEMENTS

TUBE BEAM ELEMENT

VISCOUS DAMPING ELEMENTS

RIGID BAR ELEMENTS

RIGID SOLID ELEMENT

Figure 5. Finite element model of a strut with an

unlocked joint.

Although the model shown in Fig. 5 is

simplistic, it captures many of the desired features of

an unlocked joint. Extensional and rotational friction,

equivalent viscous damping, deadband, and impacting

are all included in the LS-DYNA3D joint model.

Figure 7 shows the expected quasi static force-

displacement relationship for the unlocked joint. KC

represents the strut stiffness when the gap is closed in

compression while KT represents the stiffness in

tension. DB is the width of the deadband. For a

x

y

z

PART

A

B, C, D, E

DESCRIPTION

END BLOCK

OVERLAPPING BLOCKS USED TO

FORM FRICTION SLOT

C

DE

A

ELEMENT 10

NODE 8

NODE 10

B

Figure 6. Three-dimensional view of the unlocked

joint model.

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APPLIEDFORCE

DISPLACEMENT

KT

KC

DB

SYMBOL DEFINITION

KT

KC

DB

W

STRUT STIFFNESS IN TENSION

STRUT STIFFNESS IN COMPRESSION

LENGTH OF THE DEADBAND

WIDTH OF THE DEADBAND (1/2 FRICTION

FORCE)

W

Figure 7. Expected quasi static force-displacement

curve for unlocked joint.

perfectly aligned strut with identical hole diameters,

the deadband width is equal to twice the difference of

the hole diameter and the pin diameter. Finally, W

represents the width of the hysteresis loop and is twice

the friction force for the quasi static case. At highervelocities the hysteresis loop is wider than the quasi

static loop shown in Fig. 7 due to the viscous damping

losses.

Model Parameters from Force-State Mapping

Force-state mapping (FSM) tests reported in

reference 9 were used to characterize individual struts

from the JDX truss. During FSM tests a sinusoidal

load was applied to the right end (node 7 in Fig. 5) of

the strut while the left end (node 1 in Fig. 5) was

constrained. The applied force as well as the resulting

displacements and accelerations (of node 7) weremeasured. These data were used to produce a map of

the force-displacement-velocity domain. The FSM

tests provided parameters for the finite element model

of a single strut.

By plotting the measured force-displacement

results from a quasi-static pull test of a strut, the

stiffness of the strut can be obtained from the slope.

The force-displacement curve for a strut with both

joints locked is nearly linear. Therefore, the struts

with both joints locked can be modeled using simple

beam elements. The force-displacement curves forstruts with one joint locked and one joint unlocked

(locked-unlocked) are nonlinear and demonstrate

hysteresis similar to that shown in Fig. 7 when the

loading is quasi static.

Figure 8 is an example of the force-

displacement curve for a short locked-unlocked strut

subjected to a quasi static (0.1 Hz) applied load. The

figure demonstrates how parameters can be obtained

for use in a finite element model. Again, KC and KT

represent the strut stiffness in compression and tension,

respectively. DB is the width of the deadband. Due to

strut misalignment the observed deadband is less thanthe expected value. The FSM tests showed that the

deadband predicted from the force-displacement curve

was generally 0.0004 to 0.0007 inches less than the

expected deadband. The joint deadband in the model

was set equal to the measured deadband rather than the

deadband predicted by measuring the hole and pin

diameters. W, the width of the quasi static hysteresis

loop, represents two times the extensional friction force

as the joint moves through the deadband.

Stiffness values were chosen for the five beam

elements shown in Fig. 5 such that the overall strut

stiffness would be equal to the average of KC and KT.

The stiffness of the tubing was easily calculated

because it has a constant, known cross section. The

tang and clevis stiffness values could not be estimated

by hand calculations. Therefore, stiffness values were

selected such that the overall model strut stiffness

would be approximately the same as the measured strut

stiffness values for long and short struts as well as for

struts with both joints locked and struts with one joint

unlocked.

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KC = 72,700 LB/IN

W

KT = 71,700 LB/IN

DB = 0.001 IN

0.

FORCE

(LBS)

-20.

-40.

-60.

-80.

-100.

20.

40.

60.

80.

100.

-.003 -.002 -.001 0.0 .001 .002 .003

DISPLACEMENT (INCH)

Figure 8. Measured quasi static force-displacement

plot illustrating determination of parameters.

The width of the force-displacement hysteresis

loop is related to the energy dissipated per cycle in the

strut. This width can be modeled by either friction orviscous damping. The equivalent viscous damping in

the unlocked joints was chosen such that the width of

the model hysteresis curve was approximately equal to

the width of the measured hysteresis curve from a FSM

test with dynamic loading.

Figure 9 illustrates force-displacement curves

which compare results from a single strut finite

element model with measured data. The comparison

in Fig. 9 is for a 35 Hz sinusoidal load applied to a

short locked-unlocked strut. The force shown is the

force applied to node 7 (see Fig. 5) while the

displacement is the axial displacement of node 7.

Although there are differences between the two curves,

the areas (and the energy dissipated per cycle of the

strut) are nearly the same. Figure 10 shows a

comparison between the measured and predicted

displacement of node 7 as a function of time. The

lowest frequency oscillations are a result of the 35 Hz

applied force. The natural frequency of the strut causes

the higher frequency oscillations. This higher

frequency strut mode causes the irregular hysteresis

loops in Fig. 9. Figure 10 shows that the model

predicts higher amplitude high frequency oscillations.

We were unable to achieve better agreement betweenthe measured data and the model by changing the

properties of the model. It is suspected that a more

detailed model of the joint may be needed to get better

agreement.

Figure 9. Measured and predicted hysteresis curves.

Figure 10. Measured and predicted time-displacement

curves.

Finite Element Model of the Truss

The single strut finite element models were

extended to model the entire JDX truss. Other than the

sliding interfaces at the unlocked joints, the truss was

modeled using beam and plate elements. The expected

deadband in a joint can be computed as twice the

average clevis and tang hole diameters minus the pin

diameters. The actual deadband at each unlocked joint

is influenced by strut misalignment. It was not possibleto measure the effective deadband of each unlocked

joint after the truss was assembled. An estimate of

effective deadband for each joint was obtained by

using two times the smallest of the two clevis hole

diameters and the tang hole diameter minus two times

the diameter of the pin.

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A parameter called global damping is used in

LS-DYNA3D to provide a small amount of damping

for each node in a deformable structure. Global

damping was used to represent low level material

damping. In essence, global damping defines a viscous

damper between each node of the structure and ground.

The equivalent viscous damping for each node isproportional to the mass assigned to the node.

In order to find the initial deflected position of

the truss in either the bend 1 or bend 2 directions, a 40

lb ramped force was applied to the tip mass for 0.2

seconds, then the force was held constant for 0.3

seconds to allow the structure to come to rest. A large

value of global damping was used while the truss was

being deflected so that all truss vibrations would damp

out quickly. At 0.5 seconds the global damping was

decreased and the force was removed from the tip mass

to allow the truss to vibrate freely. The displacements,

velocities, and accelerations for each node were storedat 3000 samples per second which was the same

sampling rate used in measured data.

A truss model with all of the joints locked was

used to determine an appropriate value for global

damping. The global damping was adjusted in the

truss model until the results matched the measured

data for the truss with all joints locked. When the truss

was excited in the bend 1 direction, a global damping

parameter of 3.0 lb-sec/in modeled the measured data

well. The global damping was set to 20.0 lb-sec/in

during the 0.5 second truss deflection period. Figure

11 illustrates the deflection of the center of the tip mass

in the bend 1 direction. Figure 12 shows the

acceleration of the center of the tip mass for both the

measured data and the LS-DYNA3D model. The

release of the tip mass for the model was shifted to 0

seconds in Fig. 12 to match the measured data. It can

also be seen from Fig. 12 that the locked truss natural

frequency predicted by the model matches the

measured results for the bend 1 direction.

The truss model was modified to include eight

joints unlocked. Figure 13 compares the model and

measured results for a test in the bend 1 direction in amicro gravity environment. When power to the

magnets in the flight model truss is turned off, the

magnetic force decays in an exponential fashion. The

time constant for the decay is not known, but it is

approximately 0.01 seconds. This decay occurs during

the first, short peak in the acceleration data. The

analytical model uses an instantaneous release of the

tip mass. The release of the truss in the LS-DYNA3D

model was shifted to occur at about 0.1 seconds so as to

coincide with the second acceleration peak in the

measured data. The model predicts well the natural

frequency in the bend 1 direction (which is a function

of amplitude), the high frequency hash in the

acceleration data, and the structural damping of thetruss. The results for the bend 2 direction are similar

to the bend 1 results and thus are not included in this

paper.

Figure 11. Bend 1 displacement of tip mass in locked

truss model.

Figure 12. Bend 1 acceleration for a locked truss in 0-

deg orientation.

Figure 14 shows the measured and predicted

results for the bend 1 direction in the 0-deg orientation

(see Fig. 4) in a 1 G environment. The measured data

shows a significant decrease in damping while the

predicted decay is very similar to the micro gravity

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environment. Figure 15 compares the bend 1 results in

the 90-deg orientation when gravity induced strut

preloads are maximized. In this case the model does a

good job of simulating most of the effects of high

gravity preloads. However, structural damping in the

model is too low. The cause of the discrepancies in the

1 G environment tests is currently unknown.

Figure 13. Bend 1 acceleration for an unlocked truss

in micro gravity.


in 0-deg orientation.


in 90-deg orientation.

Onoda and others7 showed that impacting in

joints can excite higher frequency modes in a structure.Accelerations predicted for the truss model were

obtained at 25,000 samples per second in the bend 1,

micro gravity test to examine higher frequency mode

content. Figure 16 shows a plot of the Fourier

transform of a 0.01 second segment of the predicted

decay. The truss model predicts that higher frequency

modes are being excited in the truss with unlocked

joints. An attempt is currently being made to use high

frequency accelerometers to confirm the presence of

higher frequency modes in the JDX truss.

Figure 16. Bend 1 frequencies for an unlocked truss inmicro gravity.

The truss model torsion mode was excited and

compared with the measured data. Two 20 lb forces

were applied to the arms of the tip mass in order to

twist the truss. Figures 17 and 18 compare the

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measured and predicted torsion tests for a locked truss

and an unlocked truss, respectively. The results were

shifted in time in order to have a similar amplitude

peak in the predicted output occur at the same time as

a measured peak. It is seen from Fig. 17 that the

global damping chosen for the bend 1 mode is too large

for the torsion mode. An accurate model of the torsionmode requires a new global damping parameter.

Figure 17. Torsion test acceleration for a locked truss.

Figure 18. Torsion test acceleration for an unlocked

truss in micro gravity.

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It is informative to look at the frequencies

being excited in the truss during the torsion tests.

Figures 19 and 20 show the locked truss frequencies

for measured and predicted torsion tests, respectively.

Both figures were generated with about 0.2 seconds of

data after the release of the tip mass. The torsion mode

is seen at approximately 110 Hz. Although not at thesame frequencies, both figures show that higher

frequency modes are being excited in the locked truss

torsion test. A variety of modes in the tip mass and

torsion arms could produce the observed response.

Figures 21 and 22 illustrate the frequencies

excited in the measured and predicted torsion tests for

a truss with eight unlocked joints and a truss

orientation of 0-deg.. In both cases the 110 Hz torsion

mode disappears and only the higher frequency modes

can be seen. It is significant that a mode that would be

predicted by a linear model of the truss can disappear

when a few clearance fit pinned joints are included inthe structure. The cause of this response is unknown.

It is, however, interesting that the model and measured

data agree in the disappearance of the torsion mode

when the truss uses a few unlocked joints.

Figure 19. Measured torsion test frequencies for

locked truss in 0-deg orientation.

Figure 20. Predicted torsion test frequencies for

locked truss in 0-deg orientation.

Figure 21. Measured torsion test frequencies for

unlocked truss in micro gravity.

Figure 22. Predicted torsion test frequencies forunlocked truss in micro gravity.

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Conclusions

A truss structure was described which has

been used to characterize the influence of gravity and

joint gaps on structural damping and dynamic behavior

of pin-jointed structures. A finite element model of a

single strut with a pinned joint was constructed in LS-DYNA3D. The model included extensional and

rotational friction, equivalent viscous damping, and

impacting in the joint. The results of force-state

mapping tests were used to determine appropriate

parameters for the finite element model. The single

strut model was extended to model the pin-jointed truss

structure. The truss model results correlated well with

measured data from tests conducted in a micro gravity

environment; however, the model did not predict as

well the truss behavior when gravity caused strut

preloads. The finite element model predicts that

impacting in the pinned-joints excites higher frequency

modes in the truss, thereby increasing structuraldamping. Much work remains to be done to determine

the effect of each joint parameter on the overall

structural damping of pin-jointed structures.

Nevertheless, the ability to predict many of the

observed behaviors has been demonstrated.

Additionally, a procedure for estimating model

parameters such as joint deadband, friction, and

equivalent viscous damping from tests characterizing

individual joints has been demonstrated.

Acknowledgments

This research was performed under the NASA

INSTEP program, funded through NASA Langley

Research Center (LaRC) under Contract NAS1-19418.

The support of Mark Lake at LaRC as technical

monitor is gratefully acknowledged.

References

1Crawley, E. F., Sigler, J. L., and van Schoor, M. C.,

Prediction and Measurement of Damping in Hybrid

Scaled Space Structure Models, Space Systems

Laboratory, Dept. of Aeronautics and Astronautics,

MIT, Report SSL 7-88, Cambridge, July 1988.

2Plunkett, R., Friction Damping,Damping

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3Den Hartog, J. P.,Mechanical Vibrations, 4th ed.,

McGraw-Hill, New York, 1956.

4Ferri, A. A., Modeling and Analysis of Nonlinear

Sleeve Joints of Large Space Structures,AIAA Journal

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354-365.

5

Lankarani, H. M., and Nikravesh P. E., A ContactForce Model With Hysteresis Damping for Impact

Analysis of Multibody Systems,Journal of

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6Tzou, H. S., and Rong, Y., Contact Dynamics of a

Spherical Joint and a Jointed Truss-Cell System,

AIAA Journal, Vol. 29, No. 1, Jan. 1991, pp. 81-88.

7Onoda, J., Sano, T., and Minesugi, K., Passive

Vibration Suppression of Truss by Using Backlash,

Proceedings of the 34th AIAA/ASME/ASCE/AHS/ASC

Structures, Structural Dynamics, and Materials

Conference (LaJolla, CA); also AIAA Paper 93-1549.

8Folkman, S. L., Rowsell, E. A., and Ferney, G. D.,

Influence of Pinned Joints on Damping and Dynamic

Behavior of a Truss,Journal of Guidance, Control,

and Dynamics, Vol. 18, No. 6, Nov-Dec., 1995, pp.

1398-1403.

9Ferney, B. D., and Folkman, S. L., Results of

Force-state Mapping Tests to Characterize Struts Using

Pinned Joints,Proceedings of the 36th

AIAA/ASME/ASCE/AHS/ASC Structures, Structural

Dynamics, and Materials Conference (New Orleans,

LA); also AIAA Paper 95-1150.

10LS-DYNA3D Users Manual, Livermore Software

Technology Corporation, Livermore, California, 1995.

sdm96_joe

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