Energy Dissipation in Spacecraft Structures Incorporating Bolted Joints with Viscoelastic Layers

Transcription

1 Energy Dissipation in Spaeraft Strutures Inorporating Bolted Joints with Visoelasti Layers R. Wang and A. D. Croobe University of Surrey, Guildford, Surrey, GU 7XH, UK G. Rihardson Surrey Spae Tehnology Ltd., Guildford, Surrey, GU 7XH, UK and C. I. Underwood University of Surrey, Guildford, Surrey, GU 7XH, UK The energy dissipation apaity of bolted joints with visoelasti layers in a spaeraft struture was investigated. Initially a linear spring dashpot odel was used to represent the bolts in a satellite struture. A relationship was developed between the odel paraeters (stiffness and daping oeffiient) and the visoelasti aterial and geoetri properties (shear odulus, loss fator, operating area, and thiness) of the atual bolted joint. This odel was then developed into the non-linear doain. Experients on bolted joints with visoelasti layers were arried out to provide inforation for the non-linear joint odel. These odels were inorporated into a siple spaeraft odel to investigate the effet on the spaeraft response. Based on these nuerial analysis, it was found the joints an dissipate uh energy and the response of the spaeraft struture to vibrations during launh an be dereased signifiantly. I. Introdution Surrey Satellite Tehnology Liited (SSTL) developed a new spaeraft design, oposed of honeyob panels onneted by any bolted joints. The energy dissipation apaity of those joints was of interest to the designers. Forer researh [] showed that the daping apaity was quite low for plain joints operating in the iro-slip range. By using Finite Eleent analaysis, it was found that the perentage of the energy dissipated in the satellite deriving fro the joints was not over 5% when the strutural daping oeffiient was.4 and the exitation aeleration was g, an aeleration used to haraterise the launh of this satellite. Ways of inreasing the energy dissipation apaity of bolted joints in the spaeraft, thus dereasing the strutural response, have been investigated and an approah inorporating visoelasti layers in the bolted joints is presented here. Visoelasti dapers have long been used in the ontrol of vibration and noise in aerospae strutures, engine ounts and industrial ahines as well as large ivil engineering strutures. It is well nown that

2 the behaviour of visoelasti aterials is both strain and strain rate dependent whilst also depending on teperature. A widely used odel for visoelasti aterials involves speifying a oplex odulus []. Both the real and the iaginary parts hange with frequeny and teperature. This odel fors a good approxiation in any situations. However, when the strain is large and ore auray is required, ore opliated odels are needed to deal with the non-linearity. The sipler type of odels, whih do not onsider strain dependeny have been tered linear odels in this paper. Those odels that inlude both strain and strain rate dependeny, have been tered non-linear odels. In the linear doain, uh researh has been done. For exaple, Johnson [3] presented three different ways of solving the equations of otion for strutures with a visoelasti aterial: a) the oplex eigenvalue ethod; b) the odal strain energy ethod (SEM); and ) the diret frequeny response ethod. These ethods are quite oonly used in Finite Eleent software. Method ) has been used in this paper for a detailed Finite Eleent analysis of an individual bolted joint with linear visoelasti layer in Nastran. The visoelasti aterial was represented by frequeny dependent storage and loss shear oduli. When analyzing a struture with any bolted joints it is not feasible to odel the joints in suh detail and a sipler representation for their tie dependent response is required. There are any siple analytial odels that an be used to odel a linear viso-elasti response. The ost oon ones are spring dashpot systes, for exaple, Maxwell and Kelvin odels. In the Maxwell odel a spring and a dashpot are used in series. In the Kelvin odel they are used in parallel. These two odels are the siplest analytial fors for a visoelasti aterial. Many other analytial odels are different obinations of these two odels. These inds of odels have been alled standard ehanial odels (SMM) in Par s paper [5] and have been found to be quite effiient. Other oon linear odels inlude the Zener odel [6] and Havrilia-Negai odel [7] provide a better frequeny representation but at the ost of additional odel paraeters. In the strutural odelling reported in this paper the sipler odels have been shown to be entirely appropriate and the sipler forulation fors a good foundation for the non-linear developent undertaen. Considering the non-linear doain a oon approah [eg 8, 9] is based on a for of the onvolution integral with non-linear aterial desriptors. Kitagawa et al. [] gave a review of literature that foused on defining a

3 onstitutive law in non-linear stress and strain spae using the onept of overstress, where the rate dependeny was dependent of the deviation for an equilibriu non-linear aterial response. Bergstro et al. [] extends this overstress onept to networs, an equilibriu networ orresponding to the state that is approahed in long ter stress relaxation tests and a seond networ apturing the non-linear rate-dependent deviation fro the equilibriu state. Many experients have been arried out to validate these non-linear viso-elasti odels. Suh wor provides a good understanding of the non-linear response of viso-elasti aterials. However, although they have been found to provide a good representation of seleted aterial data, extensive alibration is required to deterine the odel paraeters. Furtherore, when odelling a struture with any bolted joints a uh sipler, non-linear representation of the joints is required. In the wor reported here this has been ahieved by a novel non-linear extension of the linear spring-dashpot odels disussed in the previous paragraph. In the next two setions, ethods of inorporating a linear and non-linear visoelasti layered bolted joint odel into a spaeraft strutural odel will be disussed. The effets of the joints on the response of the spaeraft struture will be shown. II. Energy dissipated in a spaeraft struture inorporating linear visoelasti layered bolted joints The wor reported in this setion fouses on linear viso-elasti odels. Initially a siplified analytial odel governing the behaviour of a bolted joint unit are developed and opared with ore aurate FE odels. This validated siplified analytial odel is then used as the basis for developing spring-dashpot representations of the bolted joint unit. Finally these spring dashpot odels are used to represent bolted joints in a satellite odel and estiates of inreased daping levels obtained. A. Modeling of bolted joints with a visoelasti layer. Visoelasti aterial properties

4 A visoelasti aterial has both elasti and visous properties. So, under dynai loads there is a phase differene between the exitation load and the displaeent response. Norally this is represented by a oplex odulus G * =(+iη)g. It has been found that the shear odulus G and the loss fator η depend on frequeny as well as teperature. Muh researh has been undertaen on different aterials by other researhers. Here Stahle s [] experient data for SMRD F9, shown in Fig., will be used for the visoelasti layers in bolted joints beause this aterial has been used in spaeraft appliations already. The values in Table were obtained fro Fig. at a teperature 5 C. This teperature is hosen beause the fous of this researh is on the launh situation.. Estiation of the energy dissipated in a visoelasti layer in shear In order to estiate the energy dissipation apability of a visoelasti aterial in shear, a siple alulation has been introdued. The haroni shear fore F and the orresponding displaeent x ating on the eleent of aterial with shear area of A and transverse thiness of h are shown in Fig.. This fore and displaeent are lined, via the shear stress and strain through the oplex shear odulus as F A x ( iη )G () h For haroni loading, where F = F oost, Eqn an be solved to give the haroni displaeent x as x x where x o o os( t ) Fo h ; AG sin The energy dissipated E an be found as the area defined by the fore-displaeent hysteresis loop. This is given below and the haroni displaeent x an be substituted to give πf hη πfxsin () AG( η ) E where θ is the phase angle between the fore and the displaeent. 3. Detailed odel of a bolted joint with a visoelasti layer

5 A detailed bolted joint with a visoelasti layer is shown in Fig. 3a. This represents the real joint shown in Fig. 4. The effetive area of the visoelasti layer in the whole joint is the area of the overlap less the area of the bolt hole. The thiness of the visoelasti layer is.5. Ten node tetrahedral solid eleents were used in the FE odel shown in Fig. 3b. A.5 N preload was applied over the washer area, whih an be seen in Fig. 3a. The left end of solid 3 was onstrained by MPCs (ultipoint onstraints) so that the whole surfae will have the sae displaeent. A fore was applied on the independent node of the MPCs.. A quarter of a bolted joint was odeled aording to syetry onditions. The 3 planes of syetry are i) the right hand edge of solid, ii) the botto of solid 3, and iii) the front surfaes of solids,, and 3. A siilar esh has been used in previous studies and has been found to give good results. A linear visoelati analysis was then undertaen. The following stiffness atrix was used for the visoelasti aterial: where [ Kdd ] v {[ g REFTR( f )] i[g g REFTI( f )]}[ K dd ] v TR( f ) g TI( f ) g (3) REF REF G ( f ) [ ] G REF G( f ) [ g ] G REF and g is the global strutural daping oeffiient, whih equals to twie the ritial daping. The paraeter g REF is the referene strutural daping oeffiient of the visoelasti aterial. It an be seen fro (3) and (4) that g REF is finally anelled. It is inluded for onveniene and the value it taes is not iportant. The paraeter G REF is the referene odulus of visoelasti aterial and is also used only for the initialisation of atrix. G'(f) is the frequeny dependent storage odulus and G"(f) is the frequeny dependent loss odulus of the visoelati aterial. These two paraeters are used in the alulations. The atrix [K dd] v is the stiffness atrix of the visoelasti aterial oputed on the basis of G REF. Futher details of equations (3) and (4) an be found in [3]. This stiffness atrix was inorporated into the dynai equations of otion for the eleents and then the equations were solved diretly by nuerial alulation. (4) The aterial of solid and 3 is aluiniu with an elasti odulus of 7 MPa, a Poisson s ratio of.33, and a density of 8 g/ 3. Many frequeny response analyses have been undertaen. The results fro a single exaple are disussed below for illustrative purposes: At Hz and an exitation fore of N it was found that

6 x E F x sin( ). 863 J where x o is the aplitude of the displaeent at the free (left) end of the joint and E is energy dissipated in one oplete visoelasti layer, This is why the fator appears in both the finite eleent and the estiation forulae. By using the following paraeters F N, h. 5,G Pa,η.683 A 6 5 π 3 Fro Eq. () it an be found that Fro Eq. () it an be found that x E.898 J When the results fro the FE odel and the estiation forula are opared it is seen that the displaeent error of the estiation forula was.%, the phase angle error was.9%, and the error of dissipated energy was.9%. It an be seen that using the estiation forula in the preliinary analyses is quite reasonable. A siilar level of auray was found fro other analyses with different loading paraeters. B. Analytial odels of bolted joints with visoelasti layers In order to represent the bolted joints as spring-dashpots in the strutural satellite odel it is neessary to find the spring and dashpot paraeters that represent the bolted joint response. The following equations an be obtained aording to spring and dashpot ehanis in the Maxwell odel. F x F x F F F x x x (5) Fro these equations, it an be shown that:

7 F dt df dt dx (6) Assuing a haroni exitation fore, F F = F osωt (7) Eq. (6) an be solved to give ) os( ) sin os ( sin os t x t t F t F t F x (8) where sin os F x (9) Fro above equations it an be shown that: sin os x F x F () By substituting the results fro the detailed bolted joint odel analysis (at Hz) into Eq. () it an be shown that N.s/.7 N/ () A spring dashpot FE odel was reated in Nastran. The paraeters in Eq. () were used. The result was:

8 x E.863 J (fro equation a) It an be seen that the spring dashpot odel gave the sae displaeent and the sae phase angle under the sae exitation fore as those given by the detailed joint odel. Sine the error between the estiation odel and the detailed joint odel was found to be sall in setion II.A.3, the forer an be used to obtain the relationship between A, G, η, h, and,. This approah aes the odelling proedure sipler by reoving the need for ultiple bolted joint analyses. Using the estiation forula (Eq. ()) to obtain and through Eq. () gives: A( )G h A( )G h () In the sae way (also nuerially tested in Nastran), the estiation forula of and for Kelvin odel an be obtained: AG h AG h (3) Equations () and (3) give appropriate equations to evaluate the equivalent Maxwell and Kelvin spring dashpot paraeters fro a speifiation of the basi visoelasti aterial paraeters. Either of these equations an be hosen for use in a spaeraft odel. In the following setion the Kelvin odel is used. C. Developing a siple satellite odel inorporating bolted joints with a visoelsti layer. Satellite odel The siple satellite odel used is shown in Fig. 5. This satellite was oposed of 7 honeyob panels whih were odelled with 4 shell eleents. On eah botto edge there were 8 joints and on eah top edge there were 9 joints. So there were 74 joints and they were divided into 9 groups, nubers in eah group being syetrially loated. For exaple the four node pairs at the top verties of the satellite belong to group. The group nubers

9 are shown in Fig. 5. The planes of syetry were at the entre of the satellite and perpendiular to x and y axes. All groups had 4 node pairs exept group 9 whih was in the iddle of the top edges and had only node pairs. Eah joint (or node pair) represented a bolted onnetion with a bolt on eah of the two perpendiular faes Thus eah pair of oinident nodes along the four edges was onneted by three springs and three dashpots. These transitted loads in the x, y, and z diretions respetively. The oordinate syste used is shown in the figure. The rotational degrees of freedo were onstrained by MPCs.. Modeling and results The shear odulus G and the loss fator η of the visoelasti aterial in Table were used in Eq. (3) along with geoetry of the visoelasti layer in the bolted joint (ie A and h in Eq. (3)). Several thinesses of the visoelasti layer were used to investigate the effet on the satellite response. The width of the bolted joints was assued to be.. The length of the satellite edge was.9 and there were 9 bolts along eah edge and the hole in the joint for the bolts was 5. Thus the shear area of eah joint in Eq. (3) was alulated as A =.9. 9-π.5-6 = The frequeny dependent spring dashpot paraeters of the bolted joint with a layer thiness of.6 are shown in Table. Due to the orientation of the lines of joints the opliane in the global Y diretion is twie the opliane in the other two diretions. Therefore, the stiffness and the daping oeffiient of spring dashpot in the Y diretion were half those in X and Z diretions. Frequeny response analyses were arried out with an exitation aeleration of g and a global strutural daping oeffiient of.4. These values are quite typial for the ind of satellite entioned in this paper. It was found that the natural frequeny was Hz. At this frequeny the fore-displaeent response at eah joint was obtained. The energy dissipated in all these joints was about 7.3 J whih is 5% of the input energy, 7. J. The daping ratio ς for this visoelasti ase was found by using half power bandwidth approah at point N to be about.4. Joints with.4 visoelasti layers were also onsidered. The displaeent of node N in the satellite odel with (none),.4, and.6 layers is shown in Fig. 6. The von-mises stress in eleent M is shown in Fig. 7. The displaeent dereased by about 9% and the stress dereased by about % fro to.6 ases. Due to this iproved response, inorporating suh daped bolted joints ould result in a derease in the satellite

10 ost, for exaple, less aterial an be used due to the derease of stress in strutures. The thiness of the visoelasti layers an be used to ontrol the response of the satellite. III. Energy dissipated in a spaeraft struture inorporating non-linear visoelasti layered bolted joints The fous of this setion shifts to ore oplex, non-linear viso-elasti representation. Experiental wor was undertaen to haraterise bolted joints with daping layers operated in the non-linear doain. A siplified way of representing suh non-linear behaviour of bolted joints was developed based on non-linear spring dashpots. Finally these were used to odel the bolted joints in a satellite struture and a ethod of deterining the daping of this syste was developed and applied to the satellite struture. A. Experiental wor In order to obtain a better understanding of real visoelasti behaviour and to generate data to define the aterial property for odelling, a series of yli tests were arried out. The joint tested was a double lap joint oposed of two 5x8x6 aluiniu bars with two 5x34x3 aluiniu laping plates. The learane hole for the bolts had a diaeter of 8 and the bolts were of 6 diaeter. The learane was suffiient to allow the visoelasti layer to develop signifiant levels of strain. Thin layers of various visoelasti aterials were used between the laping plates and the laped bars. An assebled speien is shown in Fig. 4. An Instron 85 test ahine was used to undertae the experients. When teperature is fixed, the ain fators that ould affet the non-linear behaviour of visoelasti aterials in the joints are frequeny, strain level and preload. Sinusoidal dynai tests were arried out to investigate these paraeters. They inluded tests at various frequenies (. Hz, Hz, and Hz), with various yli displaeent aplitudes (ranging between. and.8) and with preload torques varying between 6 N. and 4 N.. The experiental aterial was a ind of silione rubber naed Versasil anufatured by Nusil. It annot arry a signifiant load, but an be used to investigate typial visoelasi behavior and validate the non-linear spring dashpot odel that was developed. Before uring, the rubber is very flexible. It was rolled into a thin sheet and then

11 ured in the oven at C for about.5 hours. Following this it is lie a noral rubber and an be ut into the shape needed for insertion into the joints. A VersaSil 45 gaset having a thiness of about.8 was used in the yli tests. At this thiness the axiu shear strain ould be 5% due to the liitation of the joint learane of.. This did not ae full use of the dutility of the aterial. Different frequenies and different aplitudes of displaeent were applied. Fig. 8 shows two hysteresis loops. One is at Hz and the other is at Hz. The aplitude of the displaeent was. in both ases. The stiffness of the rubber inreased with frequeny as disussed in the literature. More experients were arried out at Hz with different displaeent aplitudes in extensoeter ontrol. The results are shown in Fig. 9. Fro Figs. 8 and 9, the dependene of properties on both strain and strain rate an be observed, but the dependene was doinated by the strain aplitude effets. In pratie the joints would be required to sustain a higher shear fore in order to be used in load arrying strutures. However, the deforation of the joint is liited for two reasons, a) the learane in the joint is liited and b) the deforation has to eet design requireents. For a fixed deforation when the rubber is thinner, the strain is higher and hene sustains a higher level of stress. As the elongation of the VersaSil aterial was quite large, the thiness of the rubber ould be redued to enable it to bear ore fore for a given displaeent. It was also assued that if a prier was used, the rubber should bond to the substrates and thus sustain ore shear loading. Thus a thin layer of VersaSil 45 was used with a prier to see if the shear fore ould be inreased. For the results in Fig. the thiness of the rubber used was about.5. The test was ontrolled by extensoeter at an aplitude.4 and a frequeny Hz. The Mullins effet is learly apparent in the data. It an be seen fro Fig. that the shear fore did inrease opared with the data in Fig. 9, hanging fro about N to 5 N in the stabilized state as the strain aplitude was inreased. The stiffness of the rubber is still too sall to be used for signifiant strutural loading. However, the results an be used to orrelate the nuerial odel that was developed Soe experients were also undertaen to investigate the effet of the preload of the bolted joints on the resulting hysteresis loop. The results are shown in Fig.. Prier was used and the thiness of the rubber was about.5. A variety of preloading was applied through the use of 8 N, N, N, and 4 N tightening torques. It appears that the effet of the preload was not very signifiant. The stiffness of the rubber inreased slightly with an inrease in the preload.

12 B. Modeling of bolted joints with visoelasti layers It was neessary to inorporate the non-linear visoelasti bolted joints into a satellite struture to investigate the global response of the satellite. The question is, how to ahieve this? It an be seen fro the literature review that uh researh has been arried out on the non-linear behavior of visoelasti aterials. Soe good onstitutive odels have been developed. However, these odels need to be inluded in a detailed joint odel and it was not possible to inlude this level of detail in the global satellite odel. Thus a sipler representation of the non-linear bolted joint response has been developed in this paper. For a linear spring-dashpot odel lie the Kelvin odel, whih is oposed of a spring and a dashpot in parallel, the fore-displaeent hysteresis loop is ellipse. Based on this, it ay be onjetured that if the spring beae nonlinear, the hysteresis loop ay hange and beoe ore lie the experiental data shown in Figs. 8-. Further, if one an define the non-linear values of stiffness and daping at any point, it ay be possible to derive an exat ath. The spring-dashpot odel used in this setion is shown in Fig.. It was a non-linear spring (whose stiffness depends on the relative oveent of the two nodes) and a linear dashpot (whose daping oeffiient is a onstant). It ay be easier to ontrol the shape using stiffness instead of daping (or both) beause the stiffness is ore diretly related to the stiffness of the syste. The value of the stiffness (Δ) and the onstant daping oeffiient an be deterined fro the experiental results so that the syste an repliate the observed behaviour of the joints. Different paraeters for the spring and dashpot were used to assess the effet of the paraeters on the resulting loop shape. The fore-deforation relationship was taen to be of the for F = aδ b = (Δ)Δ Different paraeters (as shown in Table 3) were used for the systes. The results are shown in Fig. 3. It an be see fro Fig. 3 that if a proper for of non-linear spring was hosen, it ight be possible to repliate the data seen in the experients. By observing the loops in the figures showing the experiental data it an be seen that at both ends of the loops the stiffness is bigger and in the iddle the stiffness is saller. A stiffness lie that shown in Eq. (4) was used to fit the urve. = a Δ +a Δ+a 3 (4)

13 One set of the experiental data (the last loop fro Fig. ) was seleted and used to find the orresponding stiffness of the spring so that the orrelation between the spring-dashpot syste is as lose as possible to the experiental data. An optiization funtion in Matlab was used to find the best oeffiients a, a, a 3 in Eq. (4) and the best daping oeffiient onstant. The objetive funtion used was (fi-fi) where f i is the experiental fore, F i is the nuerially predited fore at the sae displaeent. The results are shown in Fig. 4. The sae optiization proedure was used to find the best stiffness and the best daping oeffiient of a linear spring-dashpot syste. This is also shown in Fig. 4. It an be seen that the urve fro the non-linear spring dashpot syste is uh loser to the experiental data. Interestingly it was found the energy error (4.5%) of the linear syste was saller than the nonlinear syste (5.7%). The energy error is alulated fro: ( e E) / e where e and E are experiental energy and nuerial energy separately. This proble was resolved by defining another objetive funtion inluding energy dissipated as well as fore: (fi-fi) +w e-e where w is siply a weighting paraeter used to balane fore and energy errors. When w was large enough, the energy error an be quite sall (%) for both linear and non-linear systes. To give a oprehensive evaluation of the systes the fore error was also alulated, whih is given as f F / i Soe results are shown in Table 4. It an be seen that if only the energy is of onern, both linear and non-linear systes an provide a good representation of the aterial. If the fore and the energy are both of onern, the nonlinear syste is a uh better representation. i f i C. Inorporating non-linear joints into the satellite odel Suh non-linear spring-dashpot joints need to be inluded in the global satellite odel. In order to see if they an be used in an FE odel of the satellite where there are any bolted joints and to investigate their effet on the

14 behavior of the whole struture, the sae siple satellite odel shown in Fig. 4 was used. The non-linear joints were on the four edges at the top and botto of the satellite, in the sae positions as in the linear odeling, Fig. 4. They were also divided into the sae 9 groups as the linear ones. A non-linear aterial was defined, the behavior of whih is shown in Fig. 5. The paraeters were hosen by oparing with a linear odel, to ensure that the stiffnesses and the energies dissipated will be siilar. A stiffness of 7 6 N/ and a daping oeffiient of 4 4 N.s/ were hosen for the linear spring-dashpot odel, whih are lose to the values used in siple satellite odel in Setion II when the thiness of the visoelasti layer was.6 and the frequeny was around 3 Hz. After several nuerial trials the following paraeters were used: 5 4 (N.s/) 6 (N/) (5) These paraeters were used in the y diretion at eah joint. In the x and z diretions linear spring-dashpots with stiffness of.4 7 N/ and daping oeffiient of 8 4 were used. This is beause in these two diretions the stiffnesses were twie as large (see Table ) and the energy dissipated in the was quite sall opared with the energy dissipated in the Y diretion. It is diffiult to find the natural frequeny when using the non-linear odel beause the tehniques available (for exaple frequeny response analysis) do not onsider the deforation dependeny of the spring stiffness. Fro Eq. (5) and the deforation in Setion II it an be seen that the stiffness of the springs lay between 6 N/ and 7 6 N/. Frequeny response analyses with a strutural daping oeffiient of.4 were arried out with spring stiffness of 6 N/, 4 6 N/, and 6 6 N/. It was found the natural frequeny was 34.8 Hz, Hz, and 34.5 Hz respetively. The ai of this setion was to see the effet of the non-linearity on the strutural response, so sall differenes in the natural frequeny are not signifiant. In fat there are ways to find the atual natural frequeny. For exaple a fairly good result an be obtained by use of tie onsuing non-linear transient dynai iterative alulations. It would also be appropriate to arry out analyses at several frequenies around the natural frequeny to investigate the response of the odel near the natural frequeny. However, this was not the priary fous of onern of this wor. In the following analyses a frequeny of Hz was hosen to see the effet of the non-linear spring dashpot odel. Non-linear transient analyses were arried out at Hz on the siple satellite odel. The strutural daping oeffiient was.4 and the exitation aeleration was g. The displaeent of node N (Fig. 4) was obtained. It

15 was.94 when the exitation aeleration was /s and was.975 when the exitation aeleration was g. If the odel was linear the displaeent of node N should be.89 at an exitation aeleration of g. The relative differene is 8.5%. Assue that the non-linear odel is an aurate odel, then linear odel will have an error of 8.5%. IV. Conlusions This paper has outlined a possible ethodology for representing bolted joints with viso-elasti layers in large strutural odels by using spring-dashpot odels of varying degrees of oplexity. Relationships have been derived whih an be used to deterine linear spring-dashpot properties in ters of the viso-elasti aterial properties and the geoetry of the bolted joint. By undertaing dynai FE analyses of a satellite struture that inorporates these siplified bolted joints there is evidene that it ay be possible to dissipate a not insiginifiant portion of the exitation energy in a typial bolted joint onnetions that inlude a viso-elasti layer. Experiental testing has shown that andidate aterials for the viso-elasti layer ay exhibit a signifiantly non-linear viso-elasti response. A non-linear spring dashpot odel was developed that was a better fit to the experiental data than the linear odel. By inorporating these odels in the global satellite FE odel it was shown that, for the onfiguration onsidered, the differene in predited response between equivalent linear and non-linear odels, for the sae input, were of the order of % and thus the use of non-linear odels are reoended. Appendix: List of notation g = aeleration of gravity, global strutural daping oeffiient G * = oplex odulus of visoelasti aterials G = shear odulus of visoelasti aterials η = loss fator of visoelasti aterials F = exitation fore A = shear area of a aterial eleent x = displaeent h = thiness of a aterial eleent e, E = energy dissipated θ = phase angle between fore and displaeent [ = stiffness atrix of visoelasti aterials K dd ] v g = referene strutural daping oeffiient of visoelasti aterials REF

16 G = REF referene odulus of visoelasti aterials G ( f ) = frequeny dependent storage odulus of visoelasti aterials G ( f ) = frequeny dependent loss odulus of visoelasti aterials = stiffness of spring = daping oeffiient of dashpot ω = radian frequeny Referenes Croobe A. D., Wang R., Rihardson G., and Underwood C. I., Estiating the Energy Dissipated in a Bolted Spaeraft at Resonane, Coputers and Strutures, Vol. 48, No. 5-6, 6, pp Legai R., Johnson D. W., Waler W. J., and Be C. J., the Appliation of Visoelasti Passive Daping to Satellite Equipent Support Strutures, Journal of Vibration, Aoustis, Stress and Reliability in Design, Vol. 7, 985, pp Johnson C. D., and Kienholz D. A., Predition of Daping in Strutures with Visoelasti Materials, CSA Engineering In., Moeyev V., a Generalized Coplex Eigenvetor Method for Dynai Analysis of Heterogeneous Visoelasti Strutures, International Journal for Nuerial Methods in Engineering, Vol. 5,, pp Par S. W., Analytial odelling of visoelasti dapers for strutural and vibration ontrol, International Journal of Solids and Strutures, Vol. 38,, pp Ouis D., Charaterization of Polyers by Means of a Standard Visoelasti Model and Frational Derivate Calulus, International Journal of Polyeri Material, Vol. 53, 4, pp Kalgaonar R. A., Nandi S., Tabe S. S., and Jog J. P., Analysis of Visoelasti Behaviour and Dynai Mehanial Relaxation of Copolyester Based Layered Siliate Nanooposites Using Havrilia-Negai Model, Journal of Polyer Siene: Part B: Polyer Physis, Vol. 4, 4, pp Lee L.-H., Adhesive Bonding, Plenu Press, Bans H. T., Pinter G A, Potter L. K., Gaitens M. J., and Yanyo L C, Modeling of Nonlinear Hysteresis in Elastoers under Uniaxial Tension, Journal of International Material Systes and Strutures, Vol., 999, pp Kitagawa M., Tatsuya M., and Toohio M., Rate-dependent Nonlinear Constitutive Equation of Polypropylene, Journal of Polyer Siene Part B-Polyer Physis, Vol. 7, No., 989, pp Bergstro J. S., and Boye M. C., Constitutive Modeling of the Large Strain Tie-dependent Behavior of Elastoers, J. Meh. Phys. Solids, Vol. 46, No. 5, 998, pp Stahle C. V., and Staley J. A., Appliation of Daping to Spaeraft Strutures, National SAMPE Syposiu and Exhibition (Proeedings), 984, pp

17 3 MSC.Nastran/Patran douentation, Maheal-Shwendler Corporation, 5

18 Table Stiffness and daping properties of SMRD F9 (at 5 C) Frequeny Storage odulus Daping odulus (Hz) ( 6 Pa) ( 6 Pa) Loss fator Table Frequeny dependent properties of spring dashpot (for a joint with.6 layer) f(hz) K x K z Cx Cz ( 6 N/) ( 4 N.s/) K y ( 6 N/) C y ( 4 N.s/) Table 3 Paraeters for different spring-dashpot syste Paraeters of urves in Fig. 3 a Paraeters of urves in Fig. 3 b urve urve urve 3 urve urve urve 3 exitation fore (N) a (N/ b ) b 3.6 (Ns/) Table 4 Coparison of linear and non-linear syste w linear non-linear energy error fore error energy error fore error.6% 6.% 4.7% 7.7%.4% 6.3%.35% 7.6%

19 List of figures Fig. Visoelasti properties for SMRD F9 Fig. Siple odel of visoelasti aterial Fig. 3 Detailed bolted joint with visoelasi layer a) diension of the joint and b) FE Model Fig. 4 Bolted joint with visoelasti layers Fig. 5 Siple satellite odel Fig. 6 The variation of displaeent of node N with thiness of visoelasti layer Fig. 7 The von Mises stress of eleent M with thiness of visoelasti layer Fig. 8 Hysteresis loops of a bolted joint with VersaSil 45 at Hz and Hz Fig. 9 Hysteresis loops of a bolted joint with VersaSil 45 (Hz) Fig. Hysteresis loops of a bolted joint with thin VersaSil 45 Fig. The effet of pre-load on the hysteresis loops Fig. Non-linear spring dashpot odel Fig. 3 Fore displaeent loops for spring-dashpot syste (a) F=Δ 3 and (b)f=δ.6 Fig. 4 The variation of experiental data with data fro optiised spring dashpot syste for VersaSil Fig. 5 Coparison of the non-linear and the linear spring dashpot odels

20 L O S S F A C T O R E T A S H E A R M O D U L U S G N / M TEMPERATURE T DEG C 6 75 G F R E Q U E N C Y F H Z Fig. Visoelasti properties for SMRD F9 Fig. Siple odel of visoelasti aterial Solid Solid x z y Exitation fore Visoelasti layer Solid 3 a) b)

21 Fig. 3 Detailed bolted joint with visoelasi layer a) diension of the joint and b) FE Model Fig. 4 Bolted joint with visoelasti layers Node N 9 Eleent M 9

22 Fig. 5 Siple satellite odel. Displaeent () Thiness of visoelasti layer () Fig. 6 The variation of displaeent of node N with thiness of visoelasti layer Stress (MPa) Thiness of visoelasti layer () Fig. 7 The von Mises stress of eleent M with thiness of visoelasti layer

23 fore(n) Hz -.3 Hz -.35 displaeent() Fig. 8 Hysteresis loops of a bolted joint with VersaSil 45 at Hz and Hz.3 Fore (N) Displaeent () Fig. 9 Hysteresis loops of a bolted joint with VersaSil 45 (Hz)

24 Fore (N) Displaeent () all yle yle Fig. Hysteresis loops of a bolted joint with thin VersaSil 45.4 fore(n) displaeent() 8N- N- N- 4N- Fig. The effet of pre-load on the hysteresis loops Node j () Node i

25 Fig. Non-linear spring dashpot odel (N) (N) () a) b) () Fig. 3 Fore displaeent loops for spring-dashpot syste (a)f=δ 3 and (b)f=δ.6 Fig. 4 The variation of experiental data with data fro optiised spring dashpot syste for VersaSil

26 Applied load (N) Linear_5N Linear_N Linear_5N NonLinear_5N NonLinear_N NonLinear_5N Displaeent () Fig. 5 Coparison of the non-linear and the linear spring dashpot odels