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Estimating the energy dissipated in a bolted spacecraft at resonance
Computers and Structures 84 (2006) 340–350 www.elsevier.com/locate/compstruc
Estimating the energy dissipated in a bolted spacecraft at resonance A.D. Crocombe a
a,*
, R. Wang a, G. Richardson b, C.I. Underwood
c
School of Engineering, University of Surrey, Guildford, Surrey GU2 7XH, UK b Surrey Space Technology Ltd., Guildford, Surrey GU2 7XH, UK c Surrey Space Centre, Surrey GU2 7XH, UK Received 7 October 2004; accepted 29 September 2005 Available online 15 November 2005
Abstract In this paper a method is established to estimate the energy dissipated in the bolted joints of a satellite structure. The technique is equally applicable to any other bolted structure. A 3D FE model of a detailed bolted joint was created and the relationship between energy dissipated in the joint and the transverse excitation force established using non-linear FE analyses. Experimental tests were carried out on bolted joint specimens to provide data for the FE modelling techniques used. Local joint forces in the satellite were obtained from a FE model of the satellite using a frequency response analysis. These forces were used in conjunction with the joint force–energy relationship to obtain the energy dissipated in the joints. Analysis revealed that, without including joint macro-slip, the energy dissipated in the joints of the current satellite was relatively small. 2005 Elsevier Ltd. All rights reserved. Keywords: Bolted joints; Spacecraft structure; FE model; Contact elements; Non-linear analysis; Energy dissipation
1. Introduction Surrey Space Technology Ltd. (SSTL) is developing a new 400 kg spacecraft which is constructed from bolted honeycomb panels. Modelling this kind of spacecraft accurately and optimising its structure is important for its success. An important aspect of this modelling is a clear understanding of the damping effect of the bolted joints. The first stage of this research was focused on determining how much of the energy dissipated during launch can be attributed to the bolted joints. A subsequent objective was to seek ways to maximise this energy dissipation. The Finite Element Method (FEM) is the most commonly used tool in structural design and analysis. Many researchers have investigated bolted structures using FEM. This work can be traced back to 1976 when Krishnamurthy (reported by Bursi and Jaspart [1]) established a 3D model of a bolted connection. He used 8-node sub*
Corresponding author. Tel.: +44 (0)1483 689194; fax: +44 (0)1483 306039. E-mail address: [email protected] (A.D. Crocombe). 0045-7949/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2005.09.024
parametric bricks in order to reproduce the behaviour of a bolted end plate connection. The contact was included artificially by attaching and releasing nodes during each load step on the basis of stress distribution. The analysis was only linear elastic but was still very time intensive [1]. Others researchers have written specific code to solve the problem. Krzyzanowski et al. [2] undertook research on static and vibration energy dissipation in flange bolted joints using a load–displacement hysteresis loop. Analysis was performed using the FE method with their own contact macro-element. The relationship between vibration damping of the system, the interface surface finish and the applied load was investigated. Also considered was the relationship between vibration damping of the system, flange thickness and the applied bolt preload. Kukreti et al. [3] developed a three-dimensional finite element programme to analyse the moment-rotation hysteretic behaviour of end-plate connections subjected to seismic loading. The results were compared with experimental data. With the development of advanced commercial FE software, researchers have used the contact elements within the
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software to investigate this problem. Much modelling and analysis have been undertaken. Sun [4], Sherbourne and Bahaari [5] and Ramadan et al. [6] have used ANSYS in their modelling. Sun [4] developed a 3D finite element model. Superelements, gap elements, constraint equations and submodelling have been used to model a bolted flange interface in jet engines and undertake thermal/stress analysis. Substructure techniques were a key feature of this modelling, being used to reduce both the model size and complexity. These techniques have been used very successfully in life management problems, field investigations and design changes in aircraft engines. Sherbourne and Bahaari [5] established a 3D finite element model to study the stiffness and strength of the T-stub to unstiffened column flange bolted connection subjected to a moment. The prying action and gradual plasticity of the components were investigated. Sherbourne and Bahaari [5] developed a 3D model to study the stiffness and strength of the T-stub connections. Ramadan et al. [6] studied the static behaviour of bolted steel single shear lap connections. The load–displacement relationship of the joint was studied and the yield behaviour of the connection and stress around the hole were analysed. The analysis results were validated against experimental data. Bursi and Jaspart [1] and De Matteis et al. [7] have undertaken analyses using ABAQUS. Bursi and Jaspart [1] investigated the effect of constitutive relationships, step size, integration points, kinetic description, element types and discretisation. They used a 3D finite element model based on solid and contact elements in ABAQUS and LAGAMINE to investigate the behaviour of rotation, force and stress with the displacement in extended end plate connections. The experimental results validated the numerical models. De Matteis et al. [7] carried out FEA on aluminium alloy T-stub joints. The procedure was calibrated against existing experimental results and failure mechanisms were investigated. The bolt preloading, the effect of the heat affected zone due to welding and the effect of material strain hardening were taken into account in a sensitivity analysis. Mistakidis et al. [8], Sawa and Matsumoto [9] and Gantes and Lemonis [10] have used MARC and NASTRAN. Mistakidis et al. [8] established a numerical 2D finite element model to study the behaviour of a steel T-stub connection subjected to tensile loading. The development of zones of plasticity and unilateral contact effects on the interfaces between connection members and bolts were taken into account. The results were compared with those obtained from laboratory tests. Sawa and Matsumoto [9] studied the stress and sealing performance in pipe flange connections. The finite element analyses were undertaken in MARC where contact conditions including friction could be considered. A flange gasket was modelled using an elastoplastic material. Gantes and Lemonis [10] developed a finite element model for simple T-stub steel connections subjected to tensile loading. Gap elements were used to model the contact behaviour. The bolt length was varied
341
to take into account the actual flexibility provided by the bolt shank and the nut. The results were compared with experimental data found in the literature. Pratt and his co-workers [11] investigated single- and dual-conical head, bolted lap joints. The relationship between the load and elongation was obtained by finite element modelling as well as from experiments. A non-linear finite element code named NIKE3D was used. Slidelines were used in the code to model the contact condition. The yield strength and the deformation energy were also found from testing. The aim of the work presented in this paper was to determine the energy absorbed during launch by the bolted joints in a satellite. Initially a preliminary study of bolted joints was undertaken, which included both modelling and testing. Then, a detailed FE model of the actual satellite joint was constructed for stress analysis. From this model the relationship between the energy dissipated by the joint and the excitation force was found. Then the forces in the actual joints on the satellite were calculated using a dynamic frequency response analysis of the satellite model. This was repeated for excitations in all three directions for a number of natural frequencies and assumed global damping levels. The energy dissipated by each joint was estimated from the relationship determined from the detailed satellite joint analysis. Finally, the energy dissipated from all the joints was compared with the excitation energy input to the satellite model to see the influence of the joints at each natural frequency and mode of excitation. 2. Bolted joint modelling and testing It is very difficult to incorporate detailed bolted joint models in a FE model of the whole satellite because of the small dimension and large number of the joints. The behaviour of bolted joints should be known clearly before it can be incorporated into the whole satellite model. Moreover, the energy dissipated in a joint, determined from a detailed joint model, can be used directly to estimate the energy dissipated in all joints in the satellite. In order to investigate the behaviour of a bolted joint, a simple specimen was designed, see Fig. 1. Numerical modelling and experimental testing were carried out. The joint used was a double lap joint composed of two 80 · 25 · 6 mm aluminium main plates with a 34 · 24 · 2 mm aluminium clamping plate on either side. Two M6 steel bolts were used to join the assembly together. A 2 mm space was maintained between the two main plates. The diameter of the holes in all plates was 7 mm to give enough space to accommodate bolt movement due to slipping.
Fig. 1. Preliminary bolted joint specimen assembly.
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Making use of symmetry, only one-eighth of the joint was modelled. The friction between the main plates and clamping plates was of primary concern, so the bolts and nuts were not included in the model explicitly. The preload was modelled as a uniform pressure applied over a ring of 3.5 mm inner radius and 6 mm outer radius around the hole. This ring represents the action of the washer. Shell elements were used, as these reduced the complexity of the mesh compared with solid elements, and were appropriate for this structure. The boundary conditions were established according to the rules of symmetry. The FE model is shown in Fig. 2. Contact between the clamping plates and main plates was modelled using Nastran gap elements. Two points lying on adjacent contact surfaces were connected by a gap element. When the surfaces were in contact with each other, the gap elements were closed. There were contact forces along the gap axis as well as shear forces (friction forces) perpendicular to gap axis. When closed, there are three different conditions: sliding (no friction), sticking (static friction) and slipping (kinetic friction) under different applied forces. When the contact surfaces were not in contact at the points, the gap elements were open and no axial or tangential load transfer could occur. Non-linear static analyses were carried out on this model. The relationship between the applied force and displacement at the free end, for different coefficients of friction, is shown in Fig. 3. The same response, but for different preloads, is shown in Fig. 4. The coefficient of friction and the preload directly affected the force at which the
Fig. 2. FE model of the preliminary joint specimen.
Forces (N) 1500
µ = 0.8
1200
µ = 0.6 900
µ = 0.4 600
2
300
3
1
0 0
0.01
0.02
0.03
0.04
0.05
Displacement (mm) Fig. 3. The variation of joint force with displacement for different coefficients of friction (preload = 1.5 kN).
Forces (N) 1000
P = 2 kN
800
P = 1.5 kN
600
P = 1 kN
400 200 0 0
0.01
0.02
0.03
0.04
0.05
Displacement (mm) Fig. 4. The variation of joint force with displacement for different preloads (coefficient of friction = 0.4).
bolt started to slip. It can be seen that a good approximation for the slipping force (F) is F = lP, where l is the coefficient of friction, and P is the preload. From Figs. 3 and 4 it can be seen that the curves had two clearly different parts. Before the force reached a certain value, it increased steadily with small but discernable non-linearity, representing extension of the constituent parts and micro-slip in the gap elements. After a certain value, the movement increased with no significant increase in force. In the former stage only part of the contact surface slipped and this has been called the micro-slip stage. In the latter stage the entire contact surface slipped and this has been called the macro-slip stage. The work presented in this paper considers only the micro-slip behaviour of bolted joints as it was initially felt that macro-slip should be avoided under service loading conditions. A simple check on the model stiffness can be made. Considering the curve in Fig. 3 with a coefficient of friction (l) of 0.4, the displacement at which macro-slip began was about 0.022 mm. Based on simple tensile loading, an estimate of the displacement from the aluminium main plate outside the contact region was about 0.0139 mm. Thus the remaining extension was taken up by micro-slip of the gap elements and extension within the contact region and totalled less than 0.008 mm. This seems entirely reasonable. In this kind of joint configuration, the gap elements outside the washer had only a small preload. Thus, only a small applied displacement was required to cause some of these gap elements to slip The contact states were checked from the results to the curve in Fig. 3 with a coefficient of friction (l) of 0.4. It was found that, even at the beginning (point 1), some gap elements had begun to slip. At point 3, all gap elements had slipped. Point 2 lies between point 1 and point 3 and there were a significant number of gap elements in the ‘‘slip’’ state. The force and slipping behaviour of the gap elements at these three points are shown in Figs. 5 and 6 in which (1), (2) and (3) correspond to points 1, 2 and 3 in Fig. 3. It was found that only the gap elements under, and immediately adjacent to, the pressure area were closed. This is shown in Fig. 5 where the zero shear force corresponds to gap
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Fig. 5. Shear forces in gap elements (N).
Fig. 6. Relative displacements in gap elements (mm).
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elements that were not closed. It can be seen that the shear force increased with increasing applied load (points 1–3) as expected. Plots of gap element displacement are shown in Fig. 6. It can be seen from Fig. 6 that the displacements increased with increasing distance from the bolt centre. Further, an element on the right hand side had a larger displacement than the corresponding element on the left hand side. It was found from the results file that the closed gap elements farthest from the bolt centre began to slip first. As the applied load increased the number of elements moving from the stick to the slip state increased as the band of slipping elements moved in towards the centre of the bolt. From point 3 onwards, no gap elements were in the ‘‘stick’’ state, all had slipped. In order to obtain the mechanical parameters for the bolted joints, a series of experiments were undertaken. Preload is so important in the response of a bolted joint that it was necessary to measure it with considerable accuracy. A special strain gauge, Kyowa KFG-1.5-120-C20-11 internal bolt gauge, was used. A hole of 2 mm diameter and 8 mm depth was drilled in the head of a bolt. The gauge was put into the hole and bonded using adhesive EP-18. The strain, which was proportional to the preload, was measured using a digital strain gauge indicator, making use of Wheatstone bridge theory. Although the strain–preload relationship can be calculated theoretically it is better to measure both parameters and hence calibrate the bolt gauge experimentally. This was done and the resulting calibration factor was found to be 0.0052 kN/le. Many static tests were undertaken to observe the behaviour of the joint. An experimental result is compared with the corresponding numerical result in Fig. 7. It can be seen that, after an initial region in the experimental data which was probably attributable to taking up slack in the loading train, the loading stiffness of the numerical model was in good agreement with the experimental result. At higher loads, differences became a little more apparent, with the numerical model exhibiting a little more compliance. This was possibly because the geometry of the numerical model was not exactly the same as the real one (the washer was
Force (N) 3500 3000 2500 Experimental result Numerical result Numerical result with washer
2000 1500 1000 500
Displacement (mm)
0 -0.01
0
0.01
0.02
0.03
Fig. 7. Experimental and numerical results in the bolted joint specimen.
not taken into consideration etc.). A second numerical result is shown in Fig. 7 where the thickness of the shell elements in the region corresponding to the washer (i.e. within a 6 mm radius from the bolt centreline) was increased by a factor of 3 to incorporate the stiffness of the washer. It can be seen that the result gave excellent correlation with the experimental data, thus establishing confidence in the results of the FE joint modelling work. 3. Analysis of the satellite joint Fig. 8 shows the detailed model for the actual satellite joint. It included the aluminium skin, the honeycomb core, the aluminium connecting plate, the foam and the aluminium bobbin. The geometry and the material properties were derived from the actual satellite panel. The energy dissipation of the joint is important and this dissipation is mainly from the friction between the connecting plate and the panel when they experience relative movement. Again, the bolt did not appear explicitly in the model, the preload was applied as a pressure. The skin and the plate were modelled using 3-node shell elements and other parts were modelled using 4-node tetrahedral elements. The contact condition was again modelled using Nastran gap elements. Because of the symmetry of the joint connection it was only necessary to model a quarter of one bolted connection with appropriate symmetric boundary conditions applied. The force on a complete bolt structure was four times the force on the FE model while the displacement was the same. Non-linear static analyses were carried out on the detailed joint model shown in Fig. 8. A preload of 8 kN was applied and the coefficient of friction was set at 0.35, the former matched the intended assembly preload values and the latter was determined from the preliminary joint testing work. The resulting force–displacement response exhibited the same characteristics of micro-slip (where only part of the contact surfaces have relative movement) and macro-slip (where the whole contact surfaces move relative to each other) that were seen in the preliminary bolted joint configuration. The macro-slip force in this model was about 1380 N. In practice it is required that the bolted joints should operate within the micro-slip level. So the model excitation force was confined to less than 1380 N for this configuration of joint. Different magnitudes of excitation force were applied to the left end of the model in Fig. 8. A typical saw tooth loading profile is shown in Fig. 9. The force–displacement response of the joint generated a hysteresis loop, as shown in Fig. 10. The area of the loop provided the energy dissipated per cycle under the given excitation force by a quarter of the joint. This analysis was repeated with different excitation forces and different energy values were obtained. These data were used to determine the relationship between the applied force and dissipated energy for both in-plane and corner joints on the satellite. As only a quarter of the joint was modelled (see Fig. 8) the total energy and load
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Fig. 8. Detailed model of the actual satellite bolted joint.
Fig. 9. A typical loading profile for the actual satellite joint model.
Shear force (N) 1000
Fig. 12. Energy dissipated in a corner joint.
500
-0.05
-0.03
0 -0.01 -500
0.01
0.03
0.05
Displacement (mm)
-1000
Fig. 10. A typical satellite joint hysteresis loop.
The configuration of the corner joints on the satellite only used a single joining plate and thus had only one friction surface (rather than two as the in-plane joints). Thus at a given state of slip these corner joints only carried half the load and dissipated half the energy of the in-plane joints. The resulting force–energy response for the corner joint is shown in Fig. 12. A fourth order polynomial was used to fit these data for processing of the satellite results. For the plane joint (Fig. 11), the polynomial was E ¼ 0:0005F 4 þ 0:0009F 3 þ 0:0013F 2 þ 0:0031F þ 0:0023
ð1Þ
For the corner joint (Fig. 12), it was E ¼ 0:0002F 4 þ 0:0004F 3 þ 0:0007F 2 þ 0:0016F þ 0:0012
ð2Þ
These two formulae were used to calculate the energy dissipated in the bolted joints in the satellite model. 4. Satellite model analysis Fig. 11. Energy dissipated in an in-plane joint.
carried for each bolt by the in-plane joint were determined by multiplying both the force and the energy found from the FE modelling by a factor of 4. The resulting force– energy response for the in-plane joint is shown in Fig. 11.
The satellite consisted of honeycomb panels connected by bolted joints forming the bus to hold all the equipment and two propulsion tanks. The model is shown in Fig. 13. The majority of the structure was modelled using 4-node shell elements. Most of the equipment was considered as non-structural mass. However some tall subsystems, such
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Fig. 13. Satellite model and grouping of the joints.
as the power system and the laser reflectors etc., were modelled using 8-node brick elements skinned by 4-node shell elements. The propulsion tank shell was modelled using 4-node shell elements and the propellant was modelled using low modulus solid 6-node wedge elements. The joints were located on the lines indicated by the circles in Fig. 13. The (joint) nodes on connected panels were linked together using multipoint constraints (MPC), causing the nodes to have the same translational displacements. The forces in the bolted joints were obtained by extracting the MPC forces. For all analyses the four separation system attachment points were connected to a single external control point using translation MPCs. Base loads and base constraints were applied to this point. These four points were connected to the nodes above them on the bottom of the satellite model using MPCs to enforce the same x and y displacements. In the z direction four high stiffness springs were used to connect them. This modelled the launch release springs, used in practice. Because of the symmetry of the satellite, only the energy dissipated in the joints of a quarter of satellite need to be estimated. The total energy dissipated in the satellite will be four times this value. The joints were divided into the eight groups shown in Fig. 13, according to locations and their connecting configurations. This grouping made the energy calculation easier. No non-linear elements were used in this model. Damping was modelled using a structural damping coefficient (G). When using structural damping, the damping force of an element (f) was found using: fs = i * G * kp*ffiffiffiffiffiffi u,ffi where k is the stiffness, u is the displacement, i ¼ 1. This parameter was used for calculation convenience as only one parameter is needed to model the damping of the whole structure. A value for G was found by comparing the vibration testing displacements with the modelling results. An approximation of this parameter for the satellite is 0.04 based on this experimental testing.
Normal mode analysis was undertaken on the model first, to get the natural frequencies of the satellite. For this analysis, all six degrees of freedom of the external control point were constrained. The modal effective mass fractions of the model were also found. This parameter indicated how much of the whole satellite mass was involved in a given modal motion. Only those frequencies with large modal mass fraction values were important. Six frequencies whose modal effective mass fractions were over 10% were chosen, they ranged from 27 Hz to 109 Hz. In this paper, f1, f2, . . . , f6 are used to indicate these six frequencies. Frequency response analyses were then undertaken on the satellite model. These analyses were of a base driven type, using a large mass approach. A large mass was attached to the control node and a force applied, resulting in a known acceleration. Additional translation constraints were applied to the control node to prevent unwanted rotation. These analyses were run over a frequency range of 20– 150 Hz. The frequency recovery points were chosen as the six natural frequencies mentioned above. When the influence of the load excitation level was investigated, the response was, as anticipated, simply scaled by the appropriate load factor. The response analyses were repeated for different excitation directions and with different structural damping coefficients. The forces at each node of the connecting parts were extracted from result files. Using a structural damping coefficient of 0.04 and an excitation acceleration of 2g, the MPC forces at each node in all groups were extracted and the resultant forces were calculated. It was found that the resultant forces at f1 and f2 under x and y excitation and at f3 and f4 under z excitation were much bigger than those under other conditions. They are shown in Fig. 14. The axis labelled distance refers to the x, y or z location along the group of joints. The origin of the coordinate system is at the bottom centre of the satellite model shown in Fig. 13. The von Mises stress distribution for an x excitation at the frequency f1 is given in Fig. 15. This corresponded to
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Fig. 14. The variation of the resultant MPC forces along the joint lines.
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frequencies and different excitation directions as shown in Table 1. 5. Estimating the energy dissipated in the satellite model The forces in the joints after optimisation are shown in Fig. 16. For those with two contact surfaces the joints were optimised so that, ideally, all the joints transmitted a force of 5000 N. For those with one contact surface the joints were optimised so that all the joints transmitted a force of 2500 N. Although it was impossible to control all the forces at 5000 N or 2500 N (some of them even exceeded these limits in other modes of excitation), the distribution was much better after this single pass optimisation strategy and the energy dissipated by the joints was then investigated. The joint forces were put into Eqs. (1) and (2) as appropriate, and then summed to obtain the energy dissipated in the joints. The total energy input to the satellite was calculated from the force–displacement hysteresis loop at the excitation points. The effects of structural damping coefficient and excitation levels on the satellite energy were investigated. Figs. 17 and 18 show the energy from the x excitation at the first natural frequency. The variation of the energy with structural damping coefficient is shown in Fig. 17 and the energy variation with the excitation level is shown in Fig. 18. It can be seen that the energy dissipated from the bolted joints and the energy input to the satellite decreased with an increase of structural damping coefficient and increased with an increase in excitation level. The ratio of the energy dissipated from all the joints to the excitation input energy also varied with structural damping coefficient and excitation level. A smaller G and a smaller excitation level caused the ratio to increase. The ratio at a G value of 0.04 and an excitation level of 2g (typical launch conditions) is shown in Table 2. It should be noted that the above results were obtained with the following assumptions:
Fig. 15. Von Mises stress distribution for x excitation at f1.
one of the maximum force curves in Fig. 14. By comparing this figure with curves in Fig. 14, it can be seen that the MPC forces were reasonable. For example, the MPC forces of the nodes near the middle of the satellite in group 3 and near the bottom of the satellite in group 8 (near zero on the distance axes) were quite high in Fig. 14. The stresses at the corresponding points in Fig. 15 were also quite high. This is because these two locations are near the excitation points. From Fig. 14, it can be seen that there were some peaks in the MPC force curve for group 7 joints. This was because there were local instruments mounted at these locations. They changed the local stress as can be seen from Fig. 15. It was found that the forces vary a lot within each group. Ideally the forces in the bolted joints should be almost the same to avoid excess weight and to maximise the damping. So the positions of the bolts have been optimised, increasing the pitch in areas of high load transfer and reducing it in areas of low load transfer. However, in reality, the joint distribution can be optimised only for one excitation and one frequency. Thus it was necessary to develop a rule to choose the conditions for which the joints were to be optimised. In this work the maximum summed force of all nodes in a group was used as a selection criterion. They appeared in different planes at different
(a) The forces derived from the edges of the satellite model were the forces at the joints. (b) The summed force at one edge of the satellite model was independent of how many bolts were used. (c) The amount of energy dissipated by all joints with the same number of contact surfaces was the same. (d) The coefficient of friction in the joints was constant and the preload was distributed uniformly on the washer area.
Table 1 Critical forces for optimisation
Excitation direction Frequency number Resultant force
Group 1
Group 2
Group 3
Group 4
Group 5
Group 6
Group 7
Group 8
x 1 xz
y 2 xz
y 1 xz
y 1 yz
z 3 yz
x 1 yz
x 1 yz
x 1 yz
A.D. Crocombe et al. / Computers and Structures 84 (2006) 340–350
Fig. 16. The variation of the resultant MPC forces along the joint lines after joint location optimisation.
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Satellite analyses produced input for determining energy dissipated by joints during satellite launch. This two step approach proved to be a cost effective way to determine energy dissipation by the bolted joints of a complex structure for a wide range of parameters. Indeed, to assess a totally different joint design it would only be necessary to determine the force–energy relationship for the new joint. From the above work, it can be seen that for this kind of satellite structure, if the bolted joints are only working in the micro-slip range, the energy dissipated in the joints will be small. For more control of the vibration level, additional damping mechanisms at the joints should be considered. Two approaches could be investigate: a sub-set of the joints operating in macro-slip or using a viscoelastic layer in the joints. From the data obtained already, it is easy to estimate that if 10% of the joints are allowed to have a modest macro-slip, 20% percent of the total energy will be dissipated from the joints. Further research will be focussed on using viscoelastic material in the bolted joints to improve the damping levels.
Fig. 17. Energy changes with G at frequency f1.
References
Fig. 18. Energy changes with excitation level at frequency f1.
Table 2 Percentage of energy dissipated from joints to the energy input to the satellite
x excitation y excitation z excitation
f1
f2
f3
f4
f5
f6
3.1 2.9 2.0
1.5 1.6 2.1
0 0.37 1.6
0 0.30 1.6
0.45 1.0 0.83
1.4 1.6 0.39
6. Conclusions and future tasks Experimental joint testing was used to successfully validate the results of an FE model of energy dissipative joints. The relationship between the energy dissipated and the applied harmonic joint force was successfully established using FEA.
[1] Bursi OS, Jaspart JP. Basic issues in the finite element simulation of extended end plate connections. Comput Struct 1998;69:361–82. [2] Krzyzanowski J, Skoczynski W, Augul K. Mechanism of vibration energy dissipation in flange bolted joints. In: Proceedings of the international seminar on modal analysis, CONF 23 1999;2:1071–6. [3] Kukreti AR, Biswas P. Finite element analysis to predict the cyclic hysteretic behavior and failure of end-plate connections. Comput Struct 1997;V65(October):127–47. [4] Sun W. Bolted joint analysis using ansys superelements and gap elements. In: 1989 Ansys conference proceedings, vol. 2—Engineering Design Options: From Pcs To Parallels. F66-F75, 1989. [5] Sherbourne AN, Bahaari MR. 3D simulation of bolted connections to unstiffened columns—I. T-stub connections. J Construct Steel Res 1996;40:169–88. [6] Ramadan HM, Mourad SA, Rashed AA, Bode H. Finite element modelling and experimental testing of single shear bolted joints. Adv Civil Struct Eng Comput Pract 1998:117–24. Editors: Topping BHV. [7] De Matteis G, Mandara A, Mazzolani FM. T-stub aluminium joints: influence of behavioural parameters. Comput Struct 2000;78:311–27. [8] Mistakidis ES, Baniotopoulos CC, Bisbos CD, Panagiotopolous PD. Steel T-stub connections under static loading: an effective 2-D numerical model. J Construct Steel Res 1997;44:51–68. [9] T. Sawa, M. Matsumoto. FEM stress analysis and sealing performance in pipe flange connections with gaskets subjected to internal pressure and external bending moment. PVP2002-1086, ASME2002. [10] Gantes CJ, Lemonis ME. Influence of equivalent bolt length in finite element modeling of T-stub steel connections. Comput Struct 2003;81:595–604. [11] Pratt JD, Pardoen G. Comparative behavior of single-bolted and dual-bolted lap joints. J Aerospace Eng 2002(April):55–63.
Estimating the energy dissipated in a bolted spacecraft at resonance A.D. Crocombe a
a,*
, R. Wang a, G. Richardson b, C.I. Underwood
c
School of Engineering, University of Surrey, Guildford, Surrey GU2 7XH, UK b Surrey Space Technology Ltd., Guildford, Surrey GU2 7XH, UK c Surrey Space Centre, Surrey GU2 7XH, UK Received 7 October 2004; accepted 29 September 2005 Available online 15 November 2005
Abstract In this paper a method is established to estimate the energy dissipated in the bolted joints of a satellite structure. The technique is equally applicable to any other bolted structure. A 3D FE model of a detailed bolted joint was created and the relationship between energy dissipated in the joint and the transverse excitation force established using non-linear FE analyses. Experimental tests were carried out on bolted joint specimens to provide data for the FE modelling techniques used. Local joint forces in the satellite were obtained from a FE model of the satellite using a frequency response analysis. These forces were used in conjunction with the joint force–energy relationship to obtain the energy dissipated in the joints. Analysis revealed that, without including joint macro-slip, the energy dissipated in the joints of the current satellite was relatively small. 2005 Elsevier Ltd. All rights reserved. Keywords: Bolted joints; Spacecraft structure; FE model; Contact elements; Non-linear analysis; Energy dissipation
1. Introduction Surrey Space Technology Ltd. (SSTL) is developing a new 400 kg spacecraft which is constructed from bolted honeycomb panels. Modelling this kind of spacecraft accurately and optimising its structure is important for its success. An important aspect of this modelling is a clear understanding of the damping effect of the bolted joints. The first stage of this research was focused on determining how much of the energy dissipated during launch can be attributed to the bolted joints. A subsequent objective was to seek ways to maximise this energy dissipation. The Finite Element Method (FEM) is the most commonly used tool in structural design and analysis. Many researchers have investigated bolted structures using FEM. This work can be traced back to 1976 when Krishnamurthy (reported by Bursi and Jaspart [1]) established a 3D model of a bolted connection. He used 8-node sub*
Corresponding author. Tel.: +44 (0)1483 689194; fax: +44 (0)1483 306039. E-mail address: [email protected] (A.D. Crocombe). 0045-7949/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2005.09.024
parametric bricks in order to reproduce the behaviour of a bolted end plate connection. The contact was included artificially by attaching and releasing nodes during each load step on the basis of stress distribution. The analysis was only linear elastic but was still very time intensive [1]. Others researchers have written specific code to solve the problem. Krzyzanowski et al. [2] undertook research on static and vibration energy dissipation in flange bolted joints using a load–displacement hysteresis loop. Analysis was performed using the FE method with their own contact macro-element. The relationship between vibration damping of the system, the interface surface finish and the applied load was investigated. Also considered was the relationship between vibration damping of the system, flange thickness and the applied bolt preload. Kukreti et al. [3] developed a three-dimensional finite element programme to analyse the moment-rotation hysteretic behaviour of end-plate connections subjected to seismic loading. The results were compared with experimental data. With the development of advanced commercial FE software, researchers have used the contact elements within the
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software to investigate this problem. Much modelling and analysis have been undertaken. Sun [4], Sherbourne and Bahaari [5] and Ramadan et al. [6] have used ANSYS in their modelling. Sun [4] developed a 3D finite element model. Superelements, gap elements, constraint equations and submodelling have been used to model a bolted flange interface in jet engines and undertake thermal/stress analysis. Substructure techniques were a key feature of this modelling, being used to reduce both the model size and complexity. These techniques have been used very successfully in life management problems, field investigations and design changes in aircraft engines. Sherbourne and Bahaari [5] established a 3D finite element model to study the stiffness and strength of the T-stub to unstiffened column flange bolted connection subjected to a moment. The prying action and gradual plasticity of the components were investigated. Sherbourne and Bahaari [5] developed a 3D model to study the stiffness and strength of the T-stub connections. Ramadan et al. [6] studied the static behaviour of bolted steel single shear lap connections. The load–displacement relationship of the joint was studied and the yield behaviour of the connection and stress around the hole were analysed. The analysis results were validated against experimental data. Bursi and Jaspart [1] and De Matteis et al. [7] have undertaken analyses using ABAQUS. Bursi and Jaspart [1] investigated the effect of constitutive relationships, step size, integration points, kinetic description, element types and discretisation. They used a 3D finite element model based on solid and contact elements in ABAQUS and LAGAMINE to investigate the behaviour of rotation, force and stress with the displacement in extended end plate connections. The experimental results validated the numerical models. De Matteis et al. [7] carried out FEA on aluminium alloy T-stub joints. The procedure was calibrated against existing experimental results and failure mechanisms were investigated. The bolt preloading, the effect of the heat affected zone due to welding and the effect of material strain hardening were taken into account in a sensitivity analysis. Mistakidis et al. [8], Sawa and Matsumoto [9] and Gantes and Lemonis [10] have used MARC and NASTRAN. Mistakidis et al. [8] established a numerical 2D finite element model to study the behaviour of a steel T-stub connection subjected to tensile loading. The development of zones of plasticity and unilateral contact effects on the interfaces between connection members and bolts were taken into account. The results were compared with those obtained from laboratory tests. Sawa and Matsumoto [9] studied the stress and sealing performance in pipe flange connections. The finite element analyses were undertaken in MARC where contact conditions including friction could be considered. A flange gasket was modelled using an elastoplastic material. Gantes and Lemonis [10] developed a finite element model for simple T-stub steel connections subjected to tensile loading. Gap elements were used to model the contact behaviour. The bolt length was varied
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to take into account the actual flexibility provided by the bolt shank and the nut. The results were compared with experimental data found in the literature. Pratt and his co-workers [11] investigated single- and dual-conical head, bolted lap joints. The relationship between the load and elongation was obtained by finite element modelling as well as from experiments. A non-linear finite element code named NIKE3D was used. Slidelines were used in the code to model the contact condition. The yield strength and the deformation energy were also found from testing. The aim of the work presented in this paper was to determine the energy absorbed during launch by the bolted joints in a satellite. Initially a preliminary study of bolted joints was undertaken, which included both modelling and testing. Then, a detailed FE model of the actual satellite joint was constructed for stress analysis. From this model the relationship between the energy dissipated by the joint and the excitation force was found. Then the forces in the actual joints on the satellite were calculated using a dynamic frequency response analysis of the satellite model. This was repeated for excitations in all three directions for a number of natural frequencies and assumed global damping levels. The energy dissipated by each joint was estimated from the relationship determined from the detailed satellite joint analysis. Finally, the energy dissipated from all the joints was compared with the excitation energy input to the satellite model to see the influence of the joints at each natural frequency and mode of excitation. 2. Bolted joint modelling and testing It is very difficult to incorporate detailed bolted joint models in a FE model of the whole satellite because of the small dimension and large number of the joints. The behaviour of bolted joints should be known clearly before it can be incorporated into the whole satellite model. Moreover, the energy dissipated in a joint, determined from a detailed joint model, can be used directly to estimate the energy dissipated in all joints in the satellite. In order to investigate the behaviour of a bolted joint, a simple specimen was designed, see Fig. 1. Numerical modelling and experimental testing were carried out. The joint used was a double lap joint composed of two 80 · 25 · 6 mm aluminium main plates with a 34 · 24 · 2 mm aluminium clamping plate on either side. Two M6 steel bolts were used to join the assembly together. A 2 mm space was maintained between the two main plates. The diameter of the holes in all plates was 7 mm to give enough space to accommodate bolt movement due to slipping.
Fig. 1. Preliminary bolted joint specimen assembly.
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Making use of symmetry, only one-eighth of the joint was modelled. The friction between the main plates and clamping plates was of primary concern, so the bolts and nuts were not included in the model explicitly. The preload was modelled as a uniform pressure applied over a ring of 3.5 mm inner radius and 6 mm outer radius around the hole. This ring represents the action of the washer. Shell elements were used, as these reduced the complexity of the mesh compared with solid elements, and were appropriate for this structure. The boundary conditions were established according to the rules of symmetry. The FE model is shown in Fig. 2. Contact between the clamping plates and main plates was modelled using Nastran gap elements. Two points lying on adjacent contact surfaces were connected by a gap element. When the surfaces were in contact with each other, the gap elements were closed. There were contact forces along the gap axis as well as shear forces (friction forces) perpendicular to gap axis. When closed, there are three different conditions: sliding (no friction), sticking (static friction) and slipping (kinetic friction) under different applied forces. When the contact surfaces were not in contact at the points, the gap elements were open and no axial or tangential load transfer could occur. Non-linear static analyses were carried out on this model. The relationship between the applied force and displacement at the free end, for different coefficients of friction, is shown in Fig. 3. The same response, but for different preloads, is shown in Fig. 4. The coefficient of friction and the preload directly affected the force at which the
Fig. 2. FE model of the preliminary joint specimen.
Forces (N) 1500
µ = 0.8
1200
µ = 0.6 900
µ = 0.4 600
2
300
3
1
0 0
0.01
0.02
0.03
0.04
0.05
Displacement (mm) Fig. 3. The variation of joint force with displacement for different coefficients of friction (preload = 1.5 kN).
Forces (N) 1000
P = 2 kN
800
P = 1.5 kN
600
P = 1 kN
400 200 0 0
0.01
0.02
0.03
0.04
0.05
Displacement (mm) Fig. 4. The variation of joint force with displacement for different preloads (coefficient of friction = 0.4).
bolt started to slip. It can be seen that a good approximation for the slipping force (F) is F = lP, where l is the coefficient of friction, and P is the preload. From Figs. 3 and 4 it can be seen that the curves had two clearly different parts. Before the force reached a certain value, it increased steadily with small but discernable non-linearity, representing extension of the constituent parts and micro-slip in the gap elements. After a certain value, the movement increased with no significant increase in force. In the former stage only part of the contact surface slipped and this has been called the micro-slip stage. In the latter stage the entire contact surface slipped and this has been called the macro-slip stage. The work presented in this paper considers only the micro-slip behaviour of bolted joints as it was initially felt that macro-slip should be avoided under service loading conditions. A simple check on the model stiffness can be made. Considering the curve in Fig. 3 with a coefficient of friction (l) of 0.4, the displacement at which macro-slip began was about 0.022 mm. Based on simple tensile loading, an estimate of the displacement from the aluminium main plate outside the contact region was about 0.0139 mm. Thus the remaining extension was taken up by micro-slip of the gap elements and extension within the contact region and totalled less than 0.008 mm. This seems entirely reasonable. In this kind of joint configuration, the gap elements outside the washer had only a small preload. Thus, only a small applied displacement was required to cause some of these gap elements to slip The contact states were checked from the results to the curve in Fig. 3 with a coefficient of friction (l) of 0.4. It was found that, even at the beginning (point 1), some gap elements had begun to slip. At point 3, all gap elements had slipped. Point 2 lies between point 1 and point 3 and there were a significant number of gap elements in the ‘‘slip’’ state. The force and slipping behaviour of the gap elements at these three points are shown in Figs. 5 and 6 in which (1), (2) and (3) correspond to points 1, 2 and 3 in Fig. 3. It was found that only the gap elements under, and immediately adjacent to, the pressure area were closed. This is shown in Fig. 5 where the zero shear force corresponds to gap
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Fig. 5. Shear forces in gap elements (N).
Fig. 6. Relative displacements in gap elements (mm).
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elements that were not closed. It can be seen that the shear force increased with increasing applied load (points 1–3) as expected. Plots of gap element displacement are shown in Fig. 6. It can be seen from Fig. 6 that the displacements increased with increasing distance from the bolt centre. Further, an element on the right hand side had a larger displacement than the corresponding element on the left hand side. It was found from the results file that the closed gap elements farthest from the bolt centre began to slip first. As the applied load increased the number of elements moving from the stick to the slip state increased as the band of slipping elements moved in towards the centre of the bolt. From point 3 onwards, no gap elements were in the ‘‘stick’’ state, all had slipped. In order to obtain the mechanical parameters for the bolted joints, a series of experiments were undertaken. Preload is so important in the response of a bolted joint that it was necessary to measure it with considerable accuracy. A special strain gauge, Kyowa KFG-1.5-120-C20-11 internal bolt gauge, was used. A hole of 2 mm diameter and 8 mm depth was drilled in the head of a bolt. The gauge was put into the hole and bonded using adhesive EP-18. The strain, which was proportional to the preload, was measured using a digital strain gauge indicator, making use of Wheatstone bridge theory. Although the strain–preload relationship can be calculated theoretically it is better to measure both parameters and hence calibrate the bolt gauge experimentally. This was done and the resulting calibration factor was found to be 0.0052 kN/le. Many static tests were undertaken to observe the behaviour of the joint. An experimental result is compared with the corresponding numerical result in Fig. 7. It can be seen that, after an initial region in the experimental data which was probably attributable to taking up slack in the loading train, the loading stiffness of the numerical model was in good agreement with the experimental result. At higher loads, differences became a little more apparent, with the numerical model exhibiting a little more compliance. This was possibly because the geometry of the numerical model was not exactly the same as the real one (the washer was
Force (N) 3500 3000 2500 Experimental result Numerical result Numerical result with washer
2000 1500 1000 500
Displacement (mm)
0 -0.01
0
0.01
0.02
0.03
Fig. 7. Experimental and numerical results in the bolted joint specimen.
not taken into consideration etc.). A second numerical result is shown in Fig. 7 where the thickness of the shell elements in the region corresponding to the washer (i.e. within a 6 mm radius from the bolt centreline) was increased by a factor of 3 to incorporate the stiffness of the washer. It can be seen that the result gave excellent correlation with the experimental data, thus establishing confidence in the results of the FE joint modelling work. 3. Analysis of the satellite joint Fig. 8 shows the detailed model for the actual satellite joint. It included the aluminium skin, the honeycomb core, the aluminium connecting plate, the foam and the aluminium bobbin. The geometry and the material properties were derived from the actual satellite panel. The energy dissipation of the joint is important and this dissipation is mainly from the friction between the connecting plate and the panel when they experience relative movement. Again, the bolt did not appear explicitly in the model, the preload was applied as a pressure. The skin and the plate were modelled using 3-node shell elements and other parts were modelled using 4-node tetrahedral elements. The contact condition was again modelled using Nastran gap elements. Because of the symmetry of the joint connection it was only necessary to model a quarter of one bolted connection with appropriate symmetric boundary conditions applied. The force on a complete bolt structure was four times the force on the FE model while the displacement was the same. Non-linear static analyses were carried out on the detailed joint model shown in Fig. 8. A preload of 8 kN was applied and the coefficient of friction was set at 0.35, the former matched the intended assembly preload values and the latter was determined from the preliminary joint testing work. The resulting force–displacement response exhibited the same characteristics of micro-slip (where only part of the contact surfaces have relative movement) and macro-slip (where the whole contact surfaces move relative to each other) that were seen in the preliminary bolted joint configuration. The macro-slip force in this model was about 1380 N. In practice it is required that the bolted joints should operate within the micro-slip level. So the model excitation force was confined to less than 1380 N for this configuration of joint. Different magnitudes of excitation force were applied to the left end of the model in Fig. 8. A typical saw tooth loading profile is shown in Fig. 9. The force–displacement response of the joint generated a hysteresis loop, as shown in Fig. 10. The area of the loop provided the energy dissipated per cycle under the given excitation force by a quarter of the joint. This analysis was repeated with different excitation forces and different energy values were obtained. These data were used to determine the relationship between the applied force and dissipated energy for both in-plane and corner joints on the satellite. As only a quarter of the joint was modelled (see Fig. 8) the total energy and load
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Fig. 8. Detailed model of the actual satellite bolted joint.
Fig. 9. A typical loading profile for the actual satellite joint model.
Shear force (N) 1000
Fig. 12. Energy dissipated in a corner joint.
500
-0.05
-0.03
0 -0.01 -500
0.01
0.03
0.05
Displacement (mm)
-1000
Fig. 10. A typical satellite joint hysteresis loop.
The configuration of the corner joints on the satellite only used a single joining plate and thus had only one friction surface (rather than two as the in-plane joints). Thus at a given state of slip these corner joints only carried half the load and dissipated half the energy of the in-plane joints. The resulting force–energy response for the corner joint is shown in Fig. 12. A fourth order polynomial was used to fit these data for processing of the satellite results. For the plane joint (Fig. 11), the polynomial was E ¼ 0:0005F 4 þ 0:0009F 3 þ 0:0013F 2 þ 0:0031F þ 0:0023
ð1Þ
For the corner joint (Fig. 12), it was E ¼ 0:0002F 4 þ 0:0004F 3 þ 0:0007F 2 þ 0:0016F þ 0:0012
ð2Þ
These two formulae were used to calculate the energy dissipated in the bolted joints in the satellite model. 4. Satellite model analysis Fig. 11. Energy dissipated in an in-plane joint.
carried for each bolt by the in-plane joint were determined by multiplying both the force and the energy found from the FE modelling by a factor of 4. The resulting force– energy response for the in-plane joint is shown in Fig. 11.
The satellite consisted of honeycomb panels connected by bolted joints forming the bus to hold all the equipment and two propulsion tanks. The model is shown in Fig. 13. The majority of the structure was modelled using 4-node shell elements. Most of the equipment was considered as non-structural mass. However some tall subsystems, such
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Fig. 13. Satellite model and grouping of the joints.
as the power system and the laser reflectors etc., were modelled using 8-node brick elements skinned by 4-node shell elements. The propulsion tank shell was modelled using 4-node shell elements and the propellant was modelled using low modulus solid 6-node wedge elements. The joints were located on the lines indicated by the circles in Fig. 13. The (joint) nodes on connected panels were linked together using multipoint constraints (MPC), causing the nodes to have the same translational displacements. The forces in the bolted joints were obtained by extracting the MPC forces. For all analyses the four separation system attachment points were connected to a single external control point using translation MPCs. Base loads and base constraints were applied to this point. These four points were connected to the nodes above them on the bottom of the satellite model using MPCs to enforce the same x and y displacements. In the z direction four high stiffness springs were used to connect them. This modelled the launch release springs, used in practice. Because of the symmetry of the satellite, only the energy dissipated in the joints of a quarter of satellite need to be estimated. The total energy dissipated in the satellite will be four times this value. The joints were divided into the eight groups shown in Fig. 13, according to locations and their connecting configurations. This grouping made the energy calculation easier. No non-linear elements were used in this model. Damping was modelled using a structural damping coefficient (G). When using structural damping, the damping force of an element (f) was found using: fs = i * G * kp*ffiffiffiffiffiffi u,ffi where k is the stiffness, u is the displacement, i ¼ 1. This parameter was used for calculation convenience as only one parameter is needed to model the damping of the whole structure. A value for G was found by comparing the vibration testing displacements with the modelling results. An approximation of this parameter for the satellite is 0.04 based on this experimental testing.
Normal mode analysis was undertaken on the model first, to get the natural frequencies of the satellite. For this analysis, all six degrees of freedom of the external control point were constrained. The modal effective mass fractions of the model were also found. This parameter indicated how much of the whole satellite mass was involved in a given modal motion. Only those frequencies with large modal mass fraction values were important. Six frequencies whose modal effective mass fractions were over 10% were chosen, they ranged from 27 Hz to 109 Hz. In this paper, f1, f2, . . . , f6 are used to indicate these six frequencies. Frequency response analyses were then undertaken on the satellite model. These analyses were of a base driven type, using a large mass approach. A large mass was attached to the control node and a force applied, resulting in a known acceleration. Additional translation constraints were applied to the control node to prevent unwanted rotation. These analyses were run over a frequency range of 20– 150 Hz. The frequency recovery points were chosen as the six natural frequencies mentioned above. When the influence of the load excitation level was investigated, the response was, as anticipated, simply scaled by the appropriate load factor. The response analyses were repeated for different excitation directions and with different structural damping coefficients. The forces at each node of the connecting parts were extracted from result files. Using a structural damping coefficient of 0.04 and an excitation acceleration of 2g, the MPC forces at each node in all groups were extracted and the resultant forces were calculated. It was found that the resultant forces at f1 and f2 under x and y excitation and at f3 and f4 under z excitation were much bigger than those under other conditions. They are shown in Fig. 14. The axis labelled distance refers to the x, y or z location along the group of joints. The origin of the coordinate system is at the bottom centre of the satellite model shown in Fig. 13. The von Mises stress distribution for an x excitation at the frequency f1 is given in Fig. 15. This corresponded to
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Fig. 14. The variation of the resultant MPC forces along the joint lines.
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frequencies and different excitation directions as shown in Table 1. 5. Estimating the energy dissipated in the satellite model The forces in the joints after optimisation are shown in Fig. 16. For those with two contact surfaces the joints were optimised so that, ideally, all the joints transmitted a force of 5000 N. For those with one contact surface the joints were optimised so that all the joints transmitted a force of 2500 N. Although it was impossible to control all the forces at 5000 N or 2500 N (some of them even exceeded these limits in other modes of excitation), the distribution was much better after this single pass optimisation strategy and the energy dissipated by the joints was then investigated. The joint forces were put into Eqs. (1) and (2) as appropriate, and then summed to obtain the energy dissipated in the joints. The total energy input to the satellite was calculated from the force–displacement hysteresis loop at the excitation points. The effects of structural damping coefficient and excitation levels on the satellite energy were investigated. Figs. 17 and 18 show the energy from the x excitation at the first natural frequency. The variation of the energy with structural damping coefficient is shown in Fig. 17 and the energy variation with the excitation level is shown in Fig. 18. It can be seen that the energy dissipated from the bolted joints and the energy input to the satellite decreased with an increase of structural damping coefficient and increased with an increase in excitation level. The ratio of the energy dissipated from all the joints to the excitation input energy also varied with structural damping coefficient and excitation level. A smaller G and a smaller excitation level caused the ratio to increase. The ratio at a G value of 0.04 and an excitation level of 2g (typical launch conditions) is shown in Table 2. It should be noted that the above results were obtained with the following assumptions:
Fig. 15. Von Mises stress distribution for x excitation at f1.
one of the maximum force curves in Fig. 14. By comparing this figure with curves in Fig. 14, it can be seen that the MPC forces were reasonable. For example, the MPC forces of the nodes near the middle of the satellite in group 3 and near the bottom of the satellite in group 8 (near zero on the distance axes) were quite high in Fig. 14. The stresses at the corresponding points in Fig. 15 were also quite high. This is because these two locations are near the excitation points. From Fig. 14, it can be seen that there were some peaks in the MPC force curve for group 7 joints. This was because there were local instruments mounted at these locations. They changed the local stress as can be seen from Fig. 15. It was found that the forces vary a lot within each group. Ideally the forces in the bolted joints should be almost the same to avoid excess weight and to maximise the damping. So the positions of the bolts have been optimised, increasing the pitch in areas of high load transfer and reducing it in areas of low load transfer. However, in reality, the joint distribution can be optimised only for one excitation and one frequency. Thus it was necessary to develop a rule to choose the conditions for which the joints were to be optimised. In this work the maximum summed force of all nodes in a group was used as a selection criterion. They appeared in different planes at different
(a) The forces derived from the edges of the satellite model were the forces at the joints. (b) The summed force at one edge of the satellite model was independent of how many bolts were used. (c) The amount of energy dissipated by all joints with the same number of contact surfaces was the same. (d) The coefficient of friction in the joints was constant and the preload was distributed uniformly on the washer area.
Table 1 Critical forces for optimisation
Excitation direction Frequency number Resultant force
Group 1
Group 2
Group 3
Group 4
Group 5
Group 6
Group 7
Group 8
x 1 xz
y 2 xz
y 1 xz
y 1 yz
z 3 yz
x 1 yz
x 1 yz
x 1 yz
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Fig. 16. The variation of the resultant MPC forces along the joint lines after joint location optimisation.
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Satellite analyses produced input for determining energy dissipated by joints during satellite launch. This two step approach proved to be a cost effective way to determine energy dissipation by the bolted joints of a complex structure for a wide range of parameters. Indeed, to assess a totally different joint design it would only be necessary to determine the force–energy relationship for the new joint. From the above work, it can be seen that for this kind of satellite structure, if the bolted joints are only working in the micro-slip range, the energy dissipated in the joints will be small. For more control of the vibration level, additional damping mechanisms at the joints should be considered. Two approaches could be investigate: a sub-set of the joints operating in macro-slip or using a viscoelastic layer in the joints. From the data obtained already, it is easy to estimate that if 10% of the joints are allowed to have a modest macro-slip, 20% percent of the total energy will be dissipated from the joints. Further research will be focussed on using viscoelastic material in the bolted joints to improve the damping levels.
Fig. 17. Energy changes with G at frequency f1.
References
Fig. 18. Energy changes with excitation level at frequency f1.
Table 2 Percentage of energy dissipated from joints to the energy input to the satellite
x excitation y excitation z excitation
f1
f2
f3
f4
f5
f6
3.1 2.9 2.0
1.5 1.6 2.1
0 0.37 1.6
0 0.30 1.6
0.45 1.0 0.83
1.4 1.6 0.39
6. Conclusions and future tasks Experimental joint testing was used to successfully validate the results of an FE model of energy dissipative joints. The relationship between the energy dissipated and the applied harmonic joint force was successfully established using FEA.
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