Successive Model-Updating of the dynamic behaviour of casing bodies on a practical example of an axial piston pump

Transcription

1 Successive Model-Updating of the dynaic behaviour of casing bodies on a practical exaple of an axial piston pup Ulrich Bittner Bosch Rexroth, Horb, Gerany Suary: These days, generally all new products are developed first with virtual odels rather than physical prototypes. This also involves the acoustic design, which deands valid siulation odels. Model Updating is a ore and ore widespread ethod to generate nuerical odels for structural dynaics based on experiental results. The paper describes the process of odel updating for a casing body of an axial piston pup, which is assebled by screwed connections. The prerequisites to atch experiental data with nuerical results and the process of updating based on the odal assurance criterion (MAC) are discussed. The odel is updated by solving an optiization proble in several steps to identify the unknown paraeters individually. The deviation of the corresponding eigenfrequencies between easureent and siulation could be reduced below 3 %. The paper concludes with the experiental identification of relevant boundary conditions, which have to be included in a valid FE odel besides the appropriate odelling of the structure itself. Keywords: Model Updating, Experiental Modal Analysis, Structural Dynaics, Screwed Joints 1

2 1 Introduction Siulation results of structural dynaics are highly dependent on the quality of finite-eleent (FE) odels. Although single parts in free-free boundary condition can be described fairly well by the standard aterial paraeters only, ore effort is needed if several coponents or boundary conditions are included in the FE-Model. The default values of standard FE contacts couple asseblies in a very stiff way, which does not satisfy the dynaic contact behaviour. Hence the dynaics of asseblies ust be set in contrast to experients. Model Updating defines the process of coparing data fro an experiental odal analysis (EMA) with the corresponding results fro a nuerical odal analysis (NUMA), finding the differences and odifying the input paraeters to adapt the odel to the easured results. Therefore the updated odel is adjusted to the real world odel. Before starting the updating process, soe aspects should be discussed in order to obtain a reasonable physical odel: - Usually, the initial odel already describes the reality quite well. This is also iportant in order to correlate results with experiental data, what would not be possible if the odel was too far off. - The deviation between virtual and real odels is caused by the uncertainty of one or ore paraeters as e.g. a deviation in geoetry, aterial, stiffness values or boundary conditions. The paraeters ay have different tolerances and sensitivities; soe are described by literature values and others are only roughly known. - Measureents always iply uncertainty, and different physical odels yield to different results, too. It ight be necessary to average a set of easureents. - It is iportant to copare the sae extent of the virtual with the physical odel. If physical boundary conditions have an influence on the results or additional parts are included in the easureent, both have to be considered in the virtual odel as well. - The nuber of input paraeter (unknown variables) should be as low as possible but as high as necessary. The updating process converges faster with a lower nuber of paraeters, therefore it is recoended to start with a lower nuber and increase the nuber of paraeters if necessary. As a practical exaple, the housing body of an axial piston pup is updated with experiental results. The pup represents a typical echanical structure, by which the driving echanis is covered with a casing body. 2 Axial Piston Pup 2.1 Functionality Axial piston pups have usually an odd nuber of pistons, which run in a cylinder block. The kineatics of the pistons is defined by the swash plate, whose angle can be changed to adjust the pup flow rate. Due to the pivoted swash plate, the rotation of the driving shaft and the cylinder causes an axial oveent of the pistons. The extension of the pistons sucks the oil into the cylinder, and the retraction discharges it. The control disk separates the suction port and the high pressure port. Air-borne sound Structure-borne sound Fluid-borne sound Structure-borne sound Structure-borne sound Fig. 1: Scheatic longitudinal section of an axial piston pup 2

3 As a result of the discrete nuber of pistons, the resulting force of the pistons is oscillating. Because of the fact that the casing body of the pup is located within the flow of forces, it is excited by the oscillating piston force. As seen in Fig. 1, the vibrating housing generates the air-borne sound directly, but also transits structure-borne sound to connected structures. The fluid-borne sound is caused by the discontinuous flow rate of the piston pup. 2.2 FE-Model As entioned before, the casing body is located within the flow of forces and is responsible for the radiated sound of the pup. Thus it is the ost iportant coponent in an acoustic FE-siulation and its dynaic behaviour has to be validated thoroughly. Fig. 2 shows the coponents of the casing body, which are assebled by screwed connections. Cover plate Port plate Housing Fig. 2: The three coponents of the casing body The FE-odel couples the coponents by bonded contacts, which have a linear-syetric behaviour. Ansys allows to paraeterize the noral and tangential stiffness separately. The stiffness is defined as a specific stiffness per area in units Force/Length³. Research on the paraeterization of the contact stiffness is given in [1-3]. It is generally known that the joint clearance of screwed plates depends on several paraeters as plate thickness, fastening torque or the distance to the screw. The interface areas ay even start to open, especially in case of thin plates. This was considered in the odel by separating the total contact area into a screw zone and a joint zone, as shown in Fig. 3. The paraeters of these zones ay be set individually. The diaeter of the screw zone was evaluated on the basis of the deforation cone of screwed connections [4]. a) Fig. 3: b) Separation of the contact area into a screw zone (a) and a joint zone (b). CAD geoetries often have a high level of detail which lead to fine FE eshs and high solution ties. For siulations in structural dynaics, geoetric discretization with all odel details is not as iportant as for other siulation disciplines, e.g. fatigue testing in which a very fine esh is needed. Therefore all inor details as rounded edges and iprintings were reoved before iporting the geoetries into the FE pre-processor (Fig. 4). The nuber of nodes could be reduced by 67 %, whereupon the siulation results hardly changed. 3

4 a) b) Fig. 4: Local reduction of details: esh before (a) and after (b) the reduction 3 Experiental Modal Analysis The EMA was proceeded by the roving haer ethod. All easureents were done in a free-free boundary condition, which can be realized in a FE odel, too. The coordinates of the easureent points were taken fro the CAD geoetry, as well as the origin of the global coordinate syste. This ensured the identification of corresponding nodes in the FE odel within a certain tolerance during the following correlation process. In general it is useful to easure with the help of ultiple reference sensors to resolve all odes, including repeated roots or local odes. There are nuerous papers describing the basics of EMA and the ajor points, such as [5-7]. For the later correlation process it is also iportant to easure equivalent configurations because additional coponents which are easured, though not included in the FE odel, will influence the vibrancy and lead to inaccurate results. 4 Model Correlation A ajor istake in the coparison of experiental and nuerical odal results is often ade by siply coparing the eigenfrequencies. This ight work for very siple structures such as plates, but for ost structures these results are incorrect because odes can be switched and thus have a different order. Not all siulated odes ay be resolved by the easureents, too or the FE odel ight iss soe odes because it is iproperly odelled or paraeters are set in the wrong way. For a proper odal correlation, siilar ode shapes have to be allocated, so that the differences of the corresponding eigenfrequencies give a conclusion about the correlation quality. 4.1 Modal Assurance Criterion Coparing ode shapes anually can be an elaborative process, but there are several criterions defined to assist finding siilarities of ode shapes [8]: - odal assurance criterion (MAC) - coordinate odal assurance criterion (COMAC) - frequency response assurance criterion (FRAC) - coordinate orthogonality check (CORTHOG) - frequency scaled odal assurance criterion (FMAC) - partial odal assurance criterion (PMAC) - scaled odal assurance criterion (SMAC) - odal assurance criterion using reciprocal odal vectors (MACRV) The ost coonly used criterion is the MAC, which is defined in the following way: MAC EMA, NUMA * 2 [{ φema} { φnuma} ] * * { φ } { φ } { φ } { φ } = (1) EMA EMA The MAC evaluates the orthogonality of odal shapes by applying the displaceents of the eigenvectors. The eigenvectors are allowed to have different scaling factors, so that residual vectors fro an EMA can be copared with ass-noralized vectors fro a NUMA, too. The MAC results in values between 0 and 1. The literature states that values between 0.9 and 1 ean identical shapes, values between 0.6 and 0.9 show a high siilarity and values less than 0.6 have no coincidence. Fig. 5 shows a saple MAC atrix with typically high MAC values in its diagonal and values close to zero next to the diagonal. The lower odes are often resolved better than higher odes and thus have higher MAC values. Fig. 5 also shows that ode 3 and 4 are switched. NUMA NUMA 4

5 The 5 th easured ode corresponds only with a low MAC to the 6 th FE ode. Even though the MAC is below the literature value of 0.6, the odes can still be the sae as no other odes correspond to these odes. Low MAC values ay be caused by a bad resolution of the easureent or by local odes to which other parts contribute with residual noise. Table 1: Corresponding Mode Pairs EMA FEM MAC % % % % % % % % Fig. 5: Saple MAC atrix To find all corresponding ode pairs, including local odes with lower MAC values, the following procedure has been adopted: - find the axiu MAC value for each row and each colun - if a MAC is the highest MAC in its row as well as in its colun, the MAC represents a successful correlation The corresponding ode pairs for the saple MAC atrix were shown in Table Prerequisites for atching FE results and easureent data It is not possible to copare results fro EMA and NUMA instantly in general. Node nubers and result coordinate systes have a different setup, which can be seen in Fig. 6. a) b) c) Fig. 6: Different views of the housing coponent: a) photo, b) FE esh, c) easureent structure The eigenvectors used for the MAC coparison ust have corresponding entries for each degree of freedo (DOF) in the sae order. Each DOF is defined by the position and the direction of the easureent. There are several ethods to expand the test odel or to reduce the analytical odel to the easured DOFs [9]. These ethods involve the transforation of syste atrices and thus are ore coplicated to handle. In this paper, the analytical results are truncated and apped to the easured DOFs but without reducing the analytical odel. Depending on the coplexity of the structure, several steps have to be applied to atch the data between the FE odel and the test odel: - check the origin of the global coordinate systes - find the corresponding nodes in the FE odel - read the FE results in the sae coordinate direction as the easured DOF - truncate uneasured DOF (e.g. tangential to the surface) 5

6 4.3 Forulation of an optiization proble Model updating can be understood as an optiization proble with input and output paraeters, an objective function and an optiization odel. It is possible to solve the optiization proble by any paraeter optiization software. In this paper, OptiSLang was used Input paraeters The input paraeters in the optiization cycle are used to update the analytical odel. Depending on the configuration of the odel the following paraeters were used as input paraeters: - young s odulus of each part - noral and tangential contact stiffness of the screw zone - noral and tangential contact stiffness of the joint zone The density of the odel s aterial was initially set by adjusting the weight of each part to the real coponents Output paraeters and objective function After applying the MAC criterion to the odal results as described in 4.1, a table according to Table 1 is created, including the deviations of the corresponding eigenfrequencies. Based on experience, the following objective function was forulated to consider the corresponding ode pairs in the objective function: f f, EMA MAC MAC 2 2 in The weighting factor MAC² ensures that ode pairs with a higher siilarity get a higher weight than ode pairs with a lower coincidence. (2) Optiization algorith Due to the procedure of atching the corresponding ode pairs, the objective function is nonlinear and discontinuous. By changing input paraeters, the nuber of corresponding ode pairs ight change and give a discontinuous objective function. In the beginning of the updating process, it is senseful to use an explorative algorith as an evolutionary optiization. Especially if the initial paraeters are only roughly known, the entire paraeter space is covered in order to find the global iniu. If the odel is already within the region around the global iniu, a direct or gradient based ethod is suggestive. 5 Successive odel updating The casing body has several unknown paraeters to identify during the odel updating process. As the aterial paraeters are already well described in literature, the contact paraeters are expected to underlie a greater variation. Therefore the odel updating is proceeded successively: - correlation of single parts to identify the young s odulus - correlation of partial asseblies with one screwed joint This botto-up process allows solving for one unknown during each odel updating step. After updating these steps independently, all configurations were included in one global optiization loop, together with the last configuration in order to allow a final fine-tuning: - correlation of the entire casing body Fig. 7 displays the MAC atrices before and after the odel updating. It can be seen that the MAC values have risen and ore odes correlate. 6

7 a) b) Fig. 7: MAC atrix of the casing body before (a) and after (b) odel updating A coplete overview over the optiization results is given in Fig. 8. The deviation of corresponding eigenfrequencies could be reduced fro partly ore than 20 % to less than 3 % for the single coponents as well as for the asseblies. Only soe few odes show a larger difference, which is caused by low MAC values and therefore a inor consideration in the global objective function. Fig. 8: Deviation of corresponding eigenfrequencies before (a, b) and after (c, d) odel updating The change of the input paraeters fro the initial to the optiized values can be seen in Table 2. As presued, the aterial paraeters were only slightly odified, whereas the joint paraeters had a ajor change. Another rearkable point is that the stiffness in the joint zone between port plate and cover plate was reduced to zero, but the other joint stiffness ade a ajor rise. This fact was further investigated by a static siulation in which the bolt forces were applied to the structure. The results can be seen in Fig. 9. The thinner cover plate is highly defored, so that the interfaces of the joint zone open and the contact stiffness drops to zero. The thicker port plate is hardly defored and the contact is still closed. Table 2: Change of input paraeters during odel update Material properties Density Young's odulus Housing -2 % +12 % Port plate -5 % -8 % Cover plate -3 % -3 % Joint stiffness Screw zone Joint zone Housing / port plate +110 % +110 % Port plate / cover plate +170 % -100 % 7

8 Fig. 9: Static displaceent siulation: bolt forces applied to the casing body 6 Boundary conditions Beside the appropriate paraeterization of aterials and joints, boundary conditions (BC) ay highly influence the structural dynaics of the casing body and therefore have to be considered in a FE odel for acoustic siulations. The following describes a ethod to identify the influence of BC and if they have to be considered in an FE odel or if certain BC ay by neglected. Applicable boundary conditions for axial piston pups are - support (fixed flange) - hydraulic hoses - hydraulic hoses (outlet hose pressurized) - support (daped flange) To copare the influence experientally, the level of structure-borne sound was easured reciprocally by applying the roving haer ethod with fixed sensors inside the pup [10]. The experient was perfored with an epty casing to isolate the effects fro operational noise. The results are displayed in Fig. 10. The influence of each boundary condition can be seen, including their effects on aplitude and frequency. The results conclude that all boundary conditions have an effect related to the reference easureent, except for the pressurized outlet hose which is identical with the unpressurized hose. Fig. 10: Influence of iscellaneous boundary conditions on the level of structure-borne sound 8

9 7 Conclusion The process of odel updating was described on a practical exaple of the casing body of an axial piston pup. The paper showed how the screwed connections were defined and correlated with experiental results in order to achieve a valid structural dynaics odel. The prerequisites to atch analytical results and experiental data were discussed, and an optiization proble has been forulated to update the odel iteratively. The deviation of the corresponding eigenfrequencies between easured and siulated odes could be reduced fro ore than 20 % below 3 %. In addition to the updating process, an experiental ethod has been proposed to identify whether applicable boundary conditions are relevant and thus have to be considered in a FE odel or not. The paper deonstrated the iportance to adapt dynaic odels to real easureents, which has to be carried out by a systeatic odel updating process rather than anual paraeter estiation. 8 References [1] Petuelli, G.: "Theoretische und experientelle Bestiung der Steifigkeits- und Däpfungseigenschaften noralbelasteter Fügestellen", Dissertation, RWTH Aachen, 1983 [2] Schober, U.: Gehäusedäpfung Untersuchung der Körperschalldäpfung durch Fügestellen i Motor, Forschungsvereinigung Verbrennungskraftaschinen, Frankfurt, 1989 [3] Mayer, M.: Zu Einfluss von Fügestellen auf das dynaische Verhalten zusaengesetzter Strukturen, Dissertation, Universität Stuttgart, 2007 [4] VDI 2230 Part 1, Systeatic calculation of high duty bolted joints Joints with one cylindrical bolt, Düsseldorf, 2003 [5] Avitabile, P.: Modal Space Back to Basics, Experiental Techniques, Society for Experiental Mechanics, [6] ME scope Application Notes #1-28, Vibrant Technologies, [7] Kollann, F. et.al.: Praktische Maschinenakustik, Springer Verlag, Berlin, 2006 [8] Alleang, R.: The odal assurance criterion - twenty years of use and abuse, Sound and Vibration, August 2003 [9] Dascotte, E.: Model updating for structural dynaics: past, present and future outlook, International conference on engineering dynaics, 2007 [10] Bittner, U.; Berneke, S.; Proppe, C.: Reziproke Messung des Körperschallaßes von Gehäusestrukturen, VDI-Tagung Maschinenakustik,