Hong－Wei Guo，Jing Zhang，Rong－Qiang Liu，Zong－Quan Deng，Deng－Qing Cao

(1.School of Mechatronics Engineering，Harbin Institute of Technology，Harbin150001，China;2.School of Astronautics，Harbin Institute of Technology，Harbin150001，China)

1 Introduction

Space deployable truss structures(SDTS)play significant roles in earth observation，deep space exploration and satellite communication.Theyare widely used to deploy and support space instruments such as flexible solar array，antenna，synthetic aperture radar，and space telescope［1－2］.To achieve higher packaging efficiency，larger scale or longer length and light weight，numerous joints or hinges are used to connectlinkagesormodules.Both the 17-meter deployable truss antennas used in Japanese engineering testsatelliteⅧ andthe12.25-meteraperture deployable ring trussstructure used in American Thuraya satellite contains hundreds of joints in the structures.Due to the utilization of these joints，they can be compacted into much smaller volume［3－4］.The 60-meter long deployable articulated mast contains 87 bays was used to support the synthetic aperture radar in NASA’s Shuttle Radar Topography Mission，in each bays of the mast contains eight spherical joints，its compaction height was only 1.44 m［5］.Although the numbers of joints contained in the larger SDTS are considerable，these jointdominated structuresare primary choice forlarge scale orhigh precision deploying and supporting structures due to their high stiffness and accuracy［6］.

The main effect of the joints on the SDTS is their nonlinear behaviors.The nonlinear force-displacement of joints is the key recourse to the nonlinearities of the system，and joints introduce passive damping to the systems.Both joint damping and joint nonlinearity should be well understood and quantified in order to meet mission requirements.So nonlinear mechanical response within SDTS contributed by joints must be considered carefully，and dynamic model considering nonlinear joints must be accurately modeled.Because the joint dominated structures are widely used in space，these joint dominated structures have attracted numerous of scholars’attentions，especially focused on the jointsmechanic behaviorstudies.From t heir studies，finite element method，experiment method and analytical method have been used to study the joint nonlinear characteristics. Li［7］analyzed dynamic characteristics of deployable space structures considering joint clearance. D utson［8］developed a finite element analysis program to create a model of a single strut which accounted for friction，impacting，and damping in the joints，and then extended to model the entire truss to characterize the influence of joint on structural damping and dynamic behavior of pi n-jointed structures. Yoshida［9］proposed mathematical formulations to estimate the dynamic characteristics of free-vibrating structures that include joints.Crawley［10］proposed an experimental technique which called force state mapping to identify and quantify the potentially nonlinear dynamic properties of space truss joints.Its advantages are its ability to handle arbitrarily strong nonlinearities and direct graphical form of the data presentation.Later，Masters［11］extended force-state mapping to the characterization of realistic multiple degree-of-freedom(DOF)systems.Bowden［12］and Webster［13］used the describing function method to model nonlinear joints in multi-dof structures.

This paper investigates the nonlinearities of joints used in the SDTS based on describing function method，and the effects of the parameters which describe different joint characteristics on the dynamics response are analyzed systemically based on Bowden’s research.Then the overall dynamics of the SDTS with nonlinear joints are investigated，at last the dynamic responses of a physical deployable truss mast with spherical joints are tested under different exciting force level， and the mast nonlinearity and joints characteristics are identified.The layout of this article is as follows.First，the describing function method is introduced and the different nonlinear joint characteristicsare presented.Then the nonlinear characteristics of joints in a single degree of freedom system are analyzed. In the following section，dynamics of deployable truss mast with joints are investigated，and joint nonlinearitiesin thetruss structure are identified.At last concludes the paper.

2 Joint Nonlinearities Description

Joints are necessary components in deployable structures to realize fold and deployment.The joints used in the SDTS，either pin-joints or spherical joints present different nonlinear force-displacement relationship due to different construction of the joint，and the nonlinearities in the same kind of joint are also different because of different load paths and whether existence of latches or preload to restrict the joint.Four kinds of nonlinear force-displacement relationship are shown in Fig.1.

Fig.1(a)shows the nonlinear force-displacement of freeplay due to clearances between two components of the joint.The freeplay nonlinearity is characterized by two parameters，gap dimension δ and stiffness KFP.

Fig.1(b)shows the changes in stiffness due to the nonlinear contact between the two components of the joint.There are three parameters to characterize joint nonlinearity，κ，K1and K2respectively.

Fig.1(c)shows joint hysteresis nonlinearity due to friction between the joint components.The characterizing parameters are slipping force FSand stiffness KCF.Assuming AS=FS/KCFand 2AS＞ A，

Due to its symmetry，the force has inverse value when q＜0.

Fig.1 Force-displacement of four kinds of nonlinearities

Fig.1(d)shows joint cubic spring nonlinearity，and KCSis used to characterize its nonlinearity.

The describing function technique is a quasilinearization method，which can be far more powerful than most linearization techniques since it keeps the essential characteristic of nonlinear systems.So it is an effective tool for investigating the performance characteristics of nonlinear systems.The describing function formulation is used in describing joint nonlinear force by calculating the first harmonic in a Fourier series expansion of the nonlinear joint force and ignoring the higher harmonics［12］.

Based on describing function method，joint nonlinear force-displacement relationship can be expressed as where q=asinωt+bcosωt，，cpand cqare describing function coefficients，representing equivalent stiffness and damping of the joint used in the deployable truss structures respectively.cpand cqdepend on amplitude and frequency，which are expressed as

The nonlinearities of four kinds of nonlinear joints are described by describing function approach and summarized in Table 1. Describing function coefficients are the function of amplitude and corresponding characterizing parameters. Only hysteresis nonlinearitycontributes dampingdueto dissipation occurs when there is slipping.

Table 1 Describing function coefficients of different nonlinearities

3 Joint Characteristic Analysis

In order to further analyze the nonlinearities of these joints，the dynamics of one-DOF systems with these nonlinear joints are studied in this section.

The dynamic equation of one-DOF system with nonlinear joint can be written as

where M，C，K are linear mass，damping and stiffness respectively;FNLis the nonlinear force produced by nonlinear joint which is described by Eq.(5);F is harmonic exciting force，F(xiàn)=F0sin ωt.

It is assumed the dynamic response is

If insert the expression q and Eq.(5)into Eq.(6)，and divided by sine and cosine terms，two new equations are obtained

From above two Equations，the dynamic respond amplitude can be derived

So Eq.(9)is the forced response of one-DOF system with nonlinear joint.It is clearly found thatpandqare stiffness and damping contributed by joint to the system respectively.

If no damping is considered and the exciting force is set to zero，the motion equation becomes

The nonlinearities of the joints and how the characterizing parameters affectnonlinearities are analyzed as follows.Due to the physical parameters of the system is not easy to be determined，especially kFPand KCFmust be identified from experimental measures，so the following joint nonlinearity analysis mainly focuses on the change trend of nonlinear dynamic responses at different given parameters rather than magnitudes.

3.1 Freeplay

To keep the mobility of the components connected by joints，there is usually clearance remained between joint connect faces.As shown in Fig.2，the forced response of freeplay nonlinearity can be divided into two parts.The first part is that the amplitude is smaller than the gap in the joint(A＜δ)when the exciting force is at its low level，and the freeplay nonlinearity does not present in the response curves.The responses are the same as the linear system，and the responses are around the natural frequency ω0.However，the amplitude exceeds the gap in the joint(A ＞ δ)，and the responses change rapidly and around another frequency.The backbone curve(dash dotline)defined by Bowden［11］describes the responses change trend.Fig.2(a)shows the responses curves under different exciting force levels.Fig.2(b)is the responses curves with different δ，it will need larger exciting force to produce a larger amplitude to exceed the joint gap as δ increases.Fig.2(c)shows the responses of the system with different joint stiffness parameter KFP，and the nonlinearities become more distinct and the amplitudes at the same exciting force lever decrease as KFPincreases.

Fig.2 Freeplay nonlinearity

3.2 Changing Stiffness

Changing stiffness nonlinearity contains two parts of linear stiffness，and they are divided by κ;when the amplitude is smaller than κ，the stiffness is K1;while the amplitude is larger than κ，the stiffness changes into K2.The responses also will change with the stiffness switching.As shown in Fig.3，it is assumed that K1＞ K2，the response under different exciting force levels and the effect of K1，K2and κ on the nonlinearities are curved.As similar as the freeplay responses，when the amplitude is small(A ＜ κ)，the responses are around the first resonance frequency，while the amplitude is large(A ＞ κ)，the responses change quickly and go around the second resonance frequencyas shown in Fig.3(a).The larger exciting force is needed to excite larger amplitude and then convert to thesecond resonance frequency as κ increases as shown in Fig.3(b).It is obviously shown in Figs.3(c)and 3(d)that the larger the difference between K1and K2，the more nonlinear presents in responses.

Fig.3 Changing stiffness nonlinearity

The response curves when K1＜ K2are similar，the main difference is that the nonlinear responses present harden spring characteristics when K1＜K2.

3.3 Hysteresis

The responses of the hysteresis nonlinearity are similar with the responses of freeplay and changing stiffness，as shown in Fig.4.There are also two resonance frequencies，distinguished by whether there is slipping.When the amplitude is small(A ＜ AS)，slipping does not happen，and the resonance frequency.While the amplitude is larger(A ＞AS)，slipping happens and the stiffness of the joint decreases to zero rapidly，and the resonance frequency changes to ω0.The amplitude is very sensitive to the exciting force due to stiffness contributed by joint disappears when slipping happens，as shown in Fig.4(a).The amplitude is more sensitive to the exciting force，which increases significantly and rapidly with the increase of the exciting force at a small level.The larger KCFis，the more nonlinear presents in the response curves.To decrease the slipping force has the same effect as to increase the exciting force，as shown in Fig.4(c).

Fig.4 Hysteresis nonlinearity

3.4 Cubic Spring

Cubic spring nonlinearity is different from these three kinds of nonlinearities.The force-displacement is not a piecewise function，so there is no switch from one resonance frequency to another frequency;the responses are only around the curve which is calculated based on Eq.(10)

As shown in Fig.5(a)，the amplitude tends to change linearly with the increase of the exciting force frequency.Fig.5(b)shows that the resonance frequencies deviate far from ω0and the amplitude decreases correspondingly as KCSincreases.

Fig.5 Cubic spring nonlinearity

From above joint nonlinearities analysis，it can be seen that different kinds of joints present different nonlinearities，and the nonlinearities are depended on characterizing parameters of joints and the levels of the exciting force.There are two resonance frequencies except the system with cubic spring joint nonlinearity.When the exciting force at lower levels，the responses are around the firstresponse frequency，ifthe amplitudeincreasesand exceedsthe demarcation values(gap δ in the freeplay nonlinearity;κ dividing different stiffness in changing stiffness nonlinearity and ASin hysteresis nonlinearity)，the responses convert to the second resonance frequencies. However， the responses of the cubic spring nonlinearity are around a determined curve which relates with the characterizing parameter KCS.

4 Nonlinear Dynamic Experiment

4.1 Joint-dominated Deployable Truss Structure

To identify the nonlinear effect of the joints on the deployable truss mast，the dynamic responses of the jointdominated deployable truss structure under different exciting force level are measured.As shown in Fig.6，it is a deployable articulated truss mast construction，and it is a modular deployable structure which constructed by connecting basic bay repeatedly［14］.The longerons are connected to the battens by the spherical joints，and there are two spherical joints at two ends of each longeron，so the upper rigid face(constructed by four battens connected rigidly)and lower rigid face in the basic truss bay can rotate around the mast axial to realize deployment and compaction as shown in Fig.6(b).To keep the deployed truss bay as a stable structure and with enough shearing and torsionalstiffness，prestress diagonal cables are connected in the four side faces of the truss bay，and then the cables are locked by latch mechanisms，and the prestress also eliminates the clearances in the joints.There are eight joints in one basic bay，and the dynamic responses ofthe deployable truss mast contains five bays are tested in the experiment.There are forty joints in the approximate 1.8 meters long truss mast，so it is a typical joint-dominated deployable truss structure，and the nonlinear effect of the joints on the dynamic of the truss will be reflected distinctly in the dynamic test.

Fig.6 A typical joint-dominated deployable truss structure

4.2 Nonlinear Dynamic Test

The deployable truss mast is fixed on the working face of the vibration table at one end as a cantilever beam，as shown in Fig.7.The sine sweep vibration tests of the truss structure under different exciting force levels are tested.The dynamic responses of the truss are shown in Fig.8，and the plot spans the first three model frequencies.The response characteristics of the structure present a shift to lower resonant frequency and higher amplitude with the exciting force increases which is similar with hysteresis nonlinearity shown in Fig.4(a).The nonlinear dynamic behavior of the truss is mainly contributed by coulomb friction nonlinearity of the joints.The joint has fairly high stiffness because the friction mechanism is sticking due to prestress in the cables when at lower exciting force level，while the stiffness ofthe jointdecreases as soon as the mechanism slips athigherexciting level.Itis identified that the joints in the truss mast have softening spring characteristic，so the joint nonlinear dynamic behavior can be modeled as coulomb friction nonlinear hysteresis.

Fig.7 Truss vibration experiment

Fig.8 Dynamic responses under different exciting force level

It is found that the joints have significant nonlineareffecton dynamic ofdeployable truss structure from experiment test results.The joints can soften the stiffness of the structure although the contact surfaces of joint have been pre-pressed by large prestress in diagonal cables and the clearances in the joints are eliminated，and contact surfaces maintain to contact each other all the time.So nonlinear dynamic effects ofjointson the trussstructure mustbe considered.Nonlinear joint behaviors yield highly complex responses which are hard to model，but it is concluded from the analysis and experiment in this paper that joint nonlinearities can be represented conveniently and efficiently using describing function method.

5 Conclusions

1)Fournonlinearcharacteristicswhich can present in the nonlinearjointsare described by describing function method.The nonlinearities of these joints and the influence of the characterizing parameters on jointnonlinearitiesare illustrated by dynamic response curves，which are useful for estimating the nonlinear characteristics of the joints in the practical deployable joint-dominated truss structures and the change trends ofthe dynamic responses ofthe deployable truss structures under different exciting force levels.

2)The dynamic responses of a typical deployable joint-dominated truss mast are tested under different exciting levels，nonlinear behaviors are presented in the charts，and the joints nonlinearities are identified which agree with the hysteresis nonlinearities described by describing function method.So if the characterizing parameters of the nonlinear joints are identified from experimental data，and then the accurate nonlinear dynamic model of deployable joint-dominated structures can be developed by introducing the nonlinear model of the joints.

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