Analysis of ground vibration transmission in high precision equipment by Frequency Based Substructuring

Transcription

1 Analysis of ground vibration transission in high precision equipent by Frequency Based Substructuring G. van Schothorst 1, M.A. Boogaard 2, G.W. van der Poel 1, D.J. Rixen 2 1 Philips Innovation Services, Departent Mechatronics Technologies High Tech Capus 7, 5656 AE Eindhoven, The Netherlands e-ail: Gert.van.Schothorst@philips.co 2 University of Technology, Faculty 3E, Dep. Precision and Microsystes Engineering Mekelweg 2, 2628 CD Delft, The Netherlands Abstract Machines with high accuracy that are sensitive to ground vibrations are generally designed using crude assuptions on the dynaic properties of the floor where they are placed. The effect of dynaic coupling between floor dynaics and achine dynaics is generally oitted. This also holds for the prediction of achine accuracy in the design phase, based on expected or easured ground vibration spectra. In this paper, experiental dynaic substructuring ethods are exploited to predict achine vibrations in situations where the achine is placed on a non-rigid, dynaic floor. More specifically, a new Transfer Path Analysis is deonstrated based on the Frequency Based Substructuring technique for the case that ground vibration levels are easured for free interface conditions. The ethod can be seen as the dual counterpart to earlier presented approaches [1], where the disturbance vibrations have been easured in fixed interface conditions (so-called blocked forces or equivalent forces). After proper coupling of the achine odel with the experiental characteristics of the floor dynaics, these ground vibrations are translated into achine vibrations. The ethod is deonstrated on a practical ipleentation. A siplified experiental odel, siilar to the dynaics present in high-precision achines, has been built and easured. Fro experiental work perfored, soe lessons regarding the applicability of this ethod will be presented. In conclusion, using the Frequency Based Substructuring ethod, ore accurate perforance prediction of (high precision) equipent on factory floors is ade possible, potentially saving costly and conservative design choices in the achine design. 1 Introduction It is well known that floor vibrations are one of the ain disturbance sources for high precision equipent [2, 3]. Especially for equipent in the seiconductor industry, like photolithographic achines, but also in the field of etrology (e.g. electron icroscopes), the ipact of floor vibrations on the achine accuracy is generally deterining the dynaic architecture of the syste [4]. This ay iply design choices like the selection of the vibration isolation syste, achine frae ass, achine frae support stiffness and internal achine dynaics characteristics. Although there is aple attention for the role of the floor vibration level, often specified in ters of socalled VC-curves (Vibration Criteria) [3], the attention for the dynaic behavior of the floor itself, possibly in cobination with the achine dynaics, is only gradually growing. In precision engineering practice, the design of the achine is often guided by dynaic odeling of the achine itself, either luped ass approach or using finite eleents. The floor is assued either infinitely stiff, or in better cases approxiated with a finite stiffness at the interface locations between achine and floor. 3501

2 3502 PROCEEDINGS OF ISMA2012-USD2012 One of the reasons behind is, that the equipent developent industry has less (access to) knowledge of building dynaics, and oreover it is not a priori known on which floor the achine will be placed. Nevertheless, fro a dynaics point of view, it is straightforward to see the iportance of including the effect of floor dynaics in the achine dynaics analyses during the design. Firstly, the achine dynaics will alter due to the fact that the achine is placed on a non-rigid floor. And secondly, the floor dynaics, and therewith the effective floor vibration level, will alter due to the placeent of a (heavy) achine on the floor. Therefore, like the widespread use of VC-curves to specify floor vibration levels, it is considered to be useful to have ethods to further quantify and possibly specify floor dynaics, preferably based on experiental data, in order to predict the coupled achine dynaics behavior. This paper intends to present and deonstrate a ethod, which allows for incorporation of floor dynaics in the analysis of high precision equipent accuracy. In section 2, the selected approach will be otivated and worked out in ore detail, including a suary of the underlying theory. After that, section 3 describes how the approach is applied to a siplified but experiental test case. The results are discussed in section 4 and section 5 finally gives the conclusions. 2 Approach to incorporate ground vibrations 2.1 Motivation for the chosen approach High precision otion systes can be represented by a siplified achine odel like in figure 1. The achine is connected to the floor with a finite stiffness; the big frae generally has a significant ass. A guided otion syste (sall oving ass, s ) is driven by an actuation syste, which has its reaction forces f exerted on the achine frae. A third part of the achine ( r ), which is connected with a finite stiffness (relatively copliant spring) to the achine frae, acts as a reference frae, used to easure the position U of the otion syste relative to it. k f r U f s frae k ount k ount Figure 1: Scheatic representation of siplified achine odel When analyzing the effect of ground vibrations on achine perforance during syste design, obviously with ore sophisticated odels of the achine to be designed than depicted above, there are a few rather siplistic approaches for incorporating the contribution of the floor, as indicated in figure 2. Infinitely stiff floor. This approach is only valid if the achine is relatively light weight and the achine connection stiffness is relatively low as copared to the (local) floor stiffness. This is generally not true for high precision equipent. Floor stiffness added to achine odel. This approach is widely used in precision engineering practice. It takes the finite stiffness of the floor into account, which iproves the prediction of achine vibration odes due to the fact that the floor stiffness is in the sae order of agnitude as the internal stiffness of the achine, and soeties even lower. An advantage of this ethod is that in any cases

3 SUBSTRUCTURING AND COUPLING 3503 it is possible to experientally identify the (local) floor stiffness with a haer ipact easureent. However, the ethod still does not account for specific dynaic characteristics of the floor, which ay especially be of iportance if the internal floor resonances are in the sae frequency region as the achine resonances. Single Degree of Freedo oscillator to represent floor dynaics. Fro literature on floor dynaics in relation to achine perforance [5], it is clear that a floor is better characterized with a ass, spring and daper than with only a spring. Although this is not (yet) coon practice in precision engineering, it will be shown that this rather siple approach would already iprove prediction perforance for achine vibrations. U U U f f f k ount k ount k ount k ount k ount k ount k floor k floor floor k floor c floor Figure 2: State of the art approaches for incorporating floor vibrations in achine odeling: 1) iposing floor vibrations fro an infinitely stiff floor (left), 2) taking finite floor stiffness into account (iddle), 3) single degree of freedo dynaic odel of floor (right) Although fro the approaches presented here, the third ethod is preferred, it still requires a proper quantification of the floor dynaics, e.g. by experiental identification or by specification to end-users. Furtherore, it still does not incorporate ore coplex floor dynaics including ulti degree of freedo ode shapes. Therefore, a ore generic approach is chosen to cope with the coupling between the achine dynaics and the floor dynaics, as scheatically depicted in figure 3. In this figure, the solid black rectangle represents the floor dynaics (i.e. is a dynaic syste in itself) and the upper part represents the achine dynaics. Note furtherore, that a ore generic approach will not restrict the achine odel to the usual structure of high precision equipent as sketched in figure 1, but will also be applicable to other systes like edical iaging equipent, etrology systes, robots and other positioning devices that ay suffer fro floor vibrations. U f floor Figure 3: Scheatic representation of achine on the floor As explained in the following sections in ore detail, we utilize the ethod of Dynaic Substructuring to properly describe the dynaic coupling between the achine dynaics and the floor dynaics. Starting point is firstly, that a achine odel is assued to be available, generally as a result of dynaic odeling during the achine design phase. The achine odel is presued to have free interface conditions, i.e.

4 3504 PROCEEDINGS OF ISMA2012-USD2012 the achine is odeled without a floor. In case of existing equipent or prototypes, these odels ay be obtained fro (odal analysis) easureents, although special attention is required to get proper easureents. Especially with heavy achines, it is hardly possible to obtain free interface easureents; the achine should be softly suspended for that. An alternative is to easure the achine dynaics on an extreely rigid floor, or to use (piezo-electric) shakers with ipedance sensors. Secondly, the floor dynaics are generally not available as a nuerical odel, so it is assued to be obtained fro experiental responses (easured without the achine present). Note here, that the experiental responses fro the floor is typically obtained as free interface responses, as it will be ipractical to provide a rigid fixation of the building at the interface location. As a third ingredient in the chosen approach, it is assued that the floor vibration level of the building is characterized with power spectru easureents before the achine is installed. An alternative is, that a vibration criterion (e.g. VC-D) is taken as input spectru for the coupled syste analysis. The ultiate goal of the proposed approach is, to translate the easured ground vibration level (obtained in free response of the floor dynaics, i.e. without achine present), into achine response, taking the coupling between achine and floor dynaics into account. 2.2 Dynaic Coupling Dynaic Substructuring allows analyzing of coplex systes by analyzing its substructures. The description of the substructures can be in three different doains, the physical doain, the odal doain and the frequency doain. The technique is equivalent for each doain. In this paper only the frequency doain will be discussed. For ore inforation on all three doains see [6]. To apply Dynaic Substructuring to the total syste as shown in figure 3, the syste will be divided into two substructures. One substructure will represent the achine and the other will represent the floor, as shown in figure 4. The region between these two substructures represents the interface, which will be discussed into ore detail. The atheatical background of the approach is worked out below. u o U u i f i u c λ u c λ floor Figure 4: Scheatic representation of achine on the floor Consider the total syste as given by u i u o u c u f c u f i = Yii Yoi Yci Yio Yoo Y co Y Y Y Y f cc Y ic ic 0 0 oc 0 0 cc 0 0 Y f ci Yii f c fi fo + g c f f c + g f c f f i (1)

5 SUBSTRUCTURING AND COUPLING 3505 Where the superscript indicates that these entries are related to the achine and f indicates that these entries are related to the floor. The subscript i denotes that these entries are related to the input DOF, o denotes that these entries are related to the output DOF and c denotes that these entries are related to the coupling DOF, as indicated in figure 4. The forces f are the externally iposed forces and g are the reaction forces related to the coupling between these substructures, in figure 4 indicated with the blue arrows. For the substructures in equation 1 to act like the total syste, it ust fulfill two conditions. The first condition is known as the copatibility condition, which ensures that the distance between the interfaces of the coupled substructures should be zero. The second condition is known as the equilibriu condition, which ensures that the interface between each substructure is in equilibriu. Matheatically this is given by u c u f c = 0 g c + g f c = 0 Because the reaction forces should be equal, but with an opposite sign, a Lagrange ultiplier λ is used, so gc f equals λ and gc equals +λ. Next the third and fourth row of equation 1 are substituted into the copatibility condition and solved for λ. Now λ can be substituted into equation 1 and the total syste can be solved when it is assued that all the transfer functions and applied forces are known. For this application either or both the applied forces and the transissibility function related to the floor are not known, so these will be oitted here. These will be dealt with in the next section. So finally the dynaic coupling is given by u i 0 u o = Y f Y 0 ( ) 1 [Y Y u cc + Ycc f ci Yco Y ] f i cc f o (3) c I fc Note that fro this equation it follows that the dynaic coupling now only depends on the achine and the interface flexibility of the floor. (2) 2.3 Transfer Path Analysis For soe applications it is not possible to either easure the operational forces or the transissibility fro the point where the operational forces are applied to the interface. To still be able to calculate the response of the total syste to these forces, a technique called Transfer Path Analysis (TPA) can be used. In 2010 De Klerk and Rixen [1] published a TPA ethod based on a fixed interface. For this ethod the interface of one substructure is fixed and the reaction forces are easured while this substructure was operational. When this substructure is then assebled to a second substructure and the reaction forces are applied as external forces on the interface, it can be shown that the response of the second substructure to the applied reaction force is equivalent to the response of the coupled syste to the original operational forces. The response of the substructure where the excitation is located will however be different. This ethod was developed to analyze how the gear forces of the differential of a car would change the noise inside the car. This ethod is coonly known as the blocked force ethod. Because it is quite ipractical to fix the interface of the floor rigidly, another ethod should be used. Recently Rixen et al. [7] published an analogous ethod, based on the free interface. So instead of a fixed interface, the interface is now copletely free and instead of reaction forces, the free vibrations are easured. Reconsider the syste of equation 1. When the third and fourth row are substituted in the copatibility condition and the Lagrange ultiplier λ is substituted for the applied forces on the interface, the following equation is obtained u i u o 0 u f i = Yii Yoi Yci Yio Yoo Y co Yic 0 Yoc 0 Y cc + Y f cc 0 0 Y f ic Y f fo λ ci Y f ii f i f f i (4)

6 3506 PROCEEDINGS OF ISMA2012-USD2012 Note that there are no externally applied loads on the interface. These can however be applied easily, see [7]. Next suppose that there is an iposed excitation, which can either be a force or a displaceent, on the internal DOF of the floor. Then the last row of equation 4 can be eliinated. Which in the case of an iposed force results in u i u o Y f ci f f i = Yii Yoi Yci Yio Yoo Y co Yic Yoc Y cc + Y f cc f i f o (5) λ Let us now define a new proble where there is a displaceent differential δ c is iposed on the interface DOF. The copatibility condition is then defined as u c u f c = δ c (6) In other words, the distance between the interface of the substructures is no longer zero, but a certain interface gap is iposed. In this alternative proble, coparing to 4, the total set of equations is given by u i u o δc u f i = Yii Yoi Yci Yio Yoo Y co Yic 0 Yoc 0 Y cc + Y f cc 0 0 Y f ic Y ci Y f ii f i f o λ 0 (7) Fro equation 5 and 7 it can be easily seen that these are equivalent for what concerns the achine, when the interface gap δ c is equal to Y f ci f f i. If the sae force is applied on the internal DOF of the floor when the interface is free, it is easily shown that the free vibrations at the interface are equal to Y f ci f f i. In other words, with reference to figure 5, if the free substructure A has a vibration level at the interface due to internal forces like depicted in the left part of the figure, then the resulting vibrations of the coupled syste, as far as substructure B is concerned, are equal to those of the syste depicted at the right, where the corresponding interface gap displaceents are iposed between the two substructures at the interface. B 2 u B o B 2 u B o k B 1 δ c k B 1 B 1 u B c δ c B 1 u B c A 2 u A c A 2 u A c A 2 u A equ c k A 2 k A 2 k A 2 f A i A 1 u A i f A i A 1 u A i A 1 u A equ i k A 1 k A 1 k A 1 Figure 5: Equivalence of interface gap with internal floor disturbance: Free interface vibrations (left), original syste (center) and equivalent syste (right). This basically suarizes the approach to predict achine vibrations under the presence of floor vibrations, as they are easured under free interface conditions.

7 SUBSTRUCTURING AND COUPLING Application to test case For the test case described in Section 3.1, as a first step the floor dynaics have been characterized in absence of the test set-up, as described in the Section 3.2. After that, the test-case was placed and the coupled response was easured, as reported in Section 3.3. Finally, it will be described in Section 3.4 how the vibration levels have been easured, first of the free floor vibrations and second of the test case placed on the floor with the sae disturbance, on which a Transfer Path Analysis was applied. 3.1 Description of test case With the scheatic achine odel as indicated in figure 1 in ind, and considering that the free oving body of the guided otion syste is not the priary interest in this study, a test case has been designed which basically siulates the achine. A scheatic representation is shown in figure 6. This test case consists of u r o k t ref. frae k f i c k c ount Figure 6: Scheatic representation of test case three rigid bodies, one represents the ount which will be coupled to the floor, the second represents the frae on which the reaction forces are applied and the third represents the reference frae. In this setting, the ain variables of interest are the vibration levels u r o of the reference, while incorporating the dynaic behavior of the floor, as a result of two disturbances: (reaction) forces on the achine frae. The vibration levels due to this disturbance are represented as a frequency response function, by the accelerance of the syste. (free) vibration levels of the floor. The vibration levels due to this disturbance are represented as a power spectru, given the presence of a certain floor disturbance. The realized experiental set-up for the test case is depicted in figure 7. In this setup the frae is represented by a large granite block of 1150 kg which is suspended fro the floor by four copressed disc springs. The total stiffness is about 10 7 N/, which results in a vertical vibration ode at 15 Hz. The second ass, a block of 110 kg, represents the reference. This ass is suspended on passive air ounts, which should provide a suspension frequency of about 6 Hz. Although experiental odels (describing the ulti DOF dynaics by frequency responses) of the realized test set-up have been obtained based on accelerance easureents on an extreely rigid floor, these odels had liited quality for a nuber of reasons, like non-linearity in the disc springs of the suspension to the floor. More details can be found in [8]. To deonstrate the applicability of the ethods presented in section 2, the dynaics of the test set-up will be represented by a relatively siple nuerical odel in the reainder of this section. The odel is chosen like depicted in figure 6, but excluding the ass for the reference frae. The paraeters (ass, daping and stiffness) have been fitted on experiental responses, when the test setup was placed on an extreely rigid floor. Although the odel has vertical translations and two tilt rotations as degrees of freedo, only the vertical translations will be analyzed further in the next sections.

8 3508 PROCEEDINGS OF ISMA2012-USD2012 Figure 7: Realized experiental set-up of test case 3.2 Experiental characterization of floor dynaics To experientally deterine the response of a floor, there are several techniques available. In this case it is chosen to obtain the response of the floor fro ipact easureents, using a large ipact haer fro PCB, type 086D50. A building typically has a lot of daping, so a couple of things should be taken care of when perforing ipact easureents. Because of the large daping, the frequency resolution ight be a proble. The response in this case lasted less than half a second, resulting in a frequency resolution of 2. For a typical factory floor the eigenfrequency is between 10 and 20 Hz, so a uch better frequency resolution is needed. When longer easureent blocks are used, ore background noise is easured. If this noise is haronic, for instance caused by a pup, it will appear as a resonance peak. To iniize the response caused by the background noise, an exponential window can be used. Generally an exponential window is known to increase the easured daping. For a floor easureent, another proble occurs. Because floor easureents are always done in-situ, there are no rigid body odes, so the acceleration at 0 Hz should be zero and the response for low frequencies is also very sall. For the easureent syste used in this case, even at AC settings, there was still a sall constant signal present. Without an exponential window, this would not cause a proble, only a peak at 0 Hz. When an exponential window is applied, this signal now depends on tie. In the response function this will appear as if there are rigid body odes. By reoving the constant before the window is applied, this can be easily solved. Finally a response as shown in figure 8 is obtained. Besides the easured floor accelerance, figure 8 also gives the approxiative floor characteristics according to the ethods discussed in section 2.1. It is clear, that the low frequency behavior of the floor is well approxiated by the stiffness characteristic (in this case a stiffness of N/), while the first resonance of the floor at 12 Hz can be approxiated by the SDOF oscillator response. On the other hand, it is also clear that ore odes are present in the floor (e.g. 15 Hz) that are not captured by the SDOF oscillator. 3.3 Experiental results of coupled response To experientally validate the ethod as explained in section 2.2, the odel as described in section 3.1 is used (see also figure 6). For this validation, the translational response of the frae caused by the input force f i is predicted using the odel and validated with experiental responses fro the realized test case. To

9 SUBSTRUCTURING AND COUPLING 3509 accelerance (s 2 /N) easureent linearized stiffness SDOF response frequency (Hz) 50 Figure 8: Free interface floor dynaics easureent. Experiental accelerance of floor (solid); fitted single degree of freedo oscillator response (dotted); approxiate linearized stiffness (dashed) obtain a representative odel, the test case was first placed on a very rigid floor, on which the test case is assued to behave as if it is fixed at the ounts. On this easureent, the paraeters for the odel are fitted. Next this odel is coupled to the floor easureents as obtained in section 3.2, which results in a prediction of the response on this non-rigid floor. The actual response of the test case on this non-rigid floor is also easured and provides a validation response. These three responses are shown in figure 9. accelerance (s 2 /N) fixed prediction validation frequency (Hz) 50 Figure 9: Machine dynaics accelerance response including coupling effects. Rigid floor (dotted), predicted response on non-rigid floor (solid) and validated response on non-rigid floor (dashed). Fro this figure it can be concluded that there is a lot of dynaic coupling around the first eigenfrequency of the floor. Furtherore the aplitude of the first ode of the frae is less for the coupled response. It is found that this technique is able to predict both these eigenfrequencies properly. For frequencies above 18 Hz, the predicted response is alost the sae as the fixed response, which indicates that there is no dynaic coupling above 18Hz. It was found that the experiental responses for the rigid floor and the non-rigid floor are also equal, which indeed indicates that there is no dynaic coupling above 18 Hz. The difference between the predicted and the easured response on the non-rigid floor (beyond 18 Hz) ay be explained by the isatch between the fitted odel and the experiental frequency responses of the test set-up. 3.4 Experiental results of Transfer Path Analysis To validate the technique fro section 2.3 (equation 7 and figure 5), both ingredients fro the previous two sections are needed, as well as the free vibration level of the (non-rigid) floor. To provide a constant

10 3510 PROCEEDINGS OF ISMA2012-USD2012 disturbance for this easureent, a shaker with a sall reaction ass was placed on the floor, near to test case. This shaker was excited with a two-tone signal at 12 and 16 Hz. To easure the floor vibrations, four acceleroeters are placed next to the four ounts of the test case and a tie trace of 1600 seconds is recorded. The frequency resolution should of course be equal to the frequency resolution of the other easureents, which is 0.25 Hz, so 400 blocks can be ade fro this easureent. This easureent is done without the test case, to obtain the free vibration level δ c, and with the test case for a validation easureent. The response of all four signals is averaged to obtain only the vertical translational vibrations. The free vibrations are used in equation 7 to predict the new vibration level of the floor. The results are shown in figure 10. power ( 2 s 4 ) free prediction validation frequency (Hz) Figure 10: Validation of Transfer Path Analysis. Measured free floor vibrations (dotted), predicted vibrations with coupling (solid) and validated vibrations with coupling (dashed) Fro these results it appears that in the easureents there is hardly any change between the vibration levels of the free interface floor and the coupled syste where the test case is placed on the floor. The only difference is at the two excited frequencies, which sees slightly aplified by the test case. There is however a lot of difference predicted by the transfer path analysis ethod applied here. Only at 12 Hz the vibrations are predicted accurately, but at 16 Hz the vibrations are underestiated. Overall it can be concluded that the ethod fro section 2.3 is not yet validated with this experient. A possible explanation is that the floor ay behave non-linear. The response of the floor has been obtained with an ipact force input of about 5 kn, whereas the forces in the validation experient have been orders of agnitude saller. Apparently, the floor behaves uch stiffer at these sall signal levels, causing a strongly reduced ipact of the coupling between floor and achine dynaics due to the presence of the test case. 4 Coparison of floor coupling techniques Although the proposed Transfer Path Analysis ethod could not yet be fully validated on the selected test case, the experiental results on Frequency Based Substructuring (FBS) gave good confidence in being able to predict the effect of floor dynaics on the coupled response [8]. Therefore, this ethod will now be exploited to ake a coparison between the ethods described in section 2.1 at one hand, and the FBS approach on the other hand. This will be done for dynaic coupling in section 4.1 and for the ground vibrations in section 4.2. Hereby, the achine dynaics will be represented by the odel of figure 6, now including the additional DOF for the reference frae. For the floor dynaics, the experiental floor characteristics fro section 3.2 will be used.

11 SUBSTRUCTURING AND COUPLING Coupled achine / floor odel In this section the three ethods fro section 2.1 will be copared to the ethod fro section 2.2. Fro the experiental results it was found that the Frequency Based Substructuring ethod was able to predict the actual response fairly well [8] (see also figure 9), so this will be the reference case for what the actual response will be and this will be referred to as the full coupling case. In this section the response is considered fro a force on the frae to accelerations of the reference frae, as shown in figure 6. The fixed response of this odel is very easily obtained by fixing the coupling DOF. For the ethod where only the floor stiffness is taken into account, a spring is added for each physical coupling DOF, four in this case, with a stiffness of N/, which is the sae value as shown in figure 8. The three responses are shown in figure 11. accelerance (s 2 /N) fixed floor stiffness full coupling frequency (Hz) 50 Figure 11: Transfer function in the achine, fro frae to the reference: fixed response (dotted), only floor stiffness (dashed) and actual response (solid) Fro this figure it can be concluded that for this case the response of the achine has not iproved, when only the floor stiffness incorporated into the odel. Because the floor stiffness is placed at each ount, the total stiffness is four ties the actual stiffness and therefore the response with the floor stiffness is alost identical to the fixed response. Next the ass and the daping of the floor is added to the odel. It is assued that the area under the achine behaves rigidly for the frequency range of interest, so all four ounts are rigidly attached to the rigid body, which represents the floor. The ass of the floor is around kg and the daping is 10%, such that a response as shown in figure 8 is obtained. The response of the achine with the floor odeled as an SDOF oscillator, together with a fixed and actual response is shown in figure 12. Fro this figure it can be concluded that for this case the SDOF approxiation is a uch better assuption. The daping of the floor is apparently not copletely well estiated with a viscous daping odel, hence the slightly deeper anti-resonance, but the three eigenodes of this syste are very well estiated. The peak at 15 Hz originates fro a second ode of the floor, which is not present in an SDOF approxiation. 4.2 Transfer Path Analysis Although the TPA ethod fro section 2.3 is not validated with an experient yet, it is still interesting to see what the vibrations of the reference frae will be as predicted with this ethod, copared to a ore classical approach, directly iposed vibrations on the ounts of the achine. For this coparison the achine odel of figure 6 is used again, and a flat power spectru is assued for the floor. In the first case, these vibrations are directly applied as iposed accelerations on the ounts of the achine and the vibrations of the reference frae are coputed. For the second case, these vibrations are used as the interface

12 3512 PROCEEDINGS OF ISMA2012-USD2012 accelerance (s 2 /N) fixed SDOF floor full coupling frequency (Hz) 50 Figure 12: Transfer function in the achine, fro frae to the reference: fixed response (dotted), SDOF floor (dashed) and actual response (solid) gap δ c and then the vibrations of the reference frae are coputed. These two vibration levels, together with the original free vibration level of the floor are shown in figure floor direct transission power ( 2 s 4 ) frequency (Hz) Figure 13: Modeled transission of floor vibrations to a siple achine odel. Free floor vibrations (dotted), directly applied vibrations (dashed) and transfer path analysis vibrations (solid) Fro this figure it is clear that the directly applied vibrations do not take the coupled dynaics into account, whereas the TPA ethod clearly shows the dynaic coupling in the vibration level. It can also be concluded that when there is no dynaic coupling, both ethods are equal. 5 Conclusions and outlook 5.1 Conclusions In current precision engineering practice, prediction of high precision equipent accuracy relies on rather crude approxiations of floor characteristics. In this paper, ore advanced ethods for incorporating the coupling of the achine dynaics with (easured) floor dynaics have been presented, including ipleentation on an experiental test case. The ethod presented also allows for predicting achine vibration levels on a specific floor, of which both the dynaic characteristics and the (free interface) vibration levels are known. This approach is especially useful in engineering applications, where the achine is being designed while the floor characteristics, both dynaics and vibration levels, are known a priori, either fro easureents or by eans of specification.

13 SUBSTRUCTURING AND COUPLING 3513 Related to the effect of coupling between achine dynaics and floor dynaics, it can be concluded that coupling of (nuerical) achine odels with (experiental) floor characteristics, based on Frequency Based Substructuring (FBS) ethods, is successfully ipleented. Coparing the different approaches available for dealing with this coupling, ore advanced ethods will typically better approach the real coupling. Assuing a rigid floor is only valid if the achine support stiffness is relatively low with respect to the actual floor stiffness. Assuing a finite floor stiffness can be a better approxiation, but will oit the influence of any dynaics, that is generally present in non-rigid floors. A better approxiation is then to use a Single Degree of Freedo (SDOF) oscillator odel for the floor, although that still oits other ode shapes of the floor. In fact, full coupling via e.g. the FBS approach is the only way to correctly incorporate floor dynaics in achine dynaics odelling for high precision applications. This is especially true for applications where perforance is deterined by first achine resonances that are typically in the sae frequency range as floor dynaics resonances, e.g. fro Hz, which is quite realistic, like in the presented test case. Related to the prediction of achine vibrations based on easured (free interface) floor vibrations, it can be concluded that the FBS approach provides a good fraework to do so, with a so called Transfer Path Analysis (TPA), which directly incorporates coupling effects. Although the ethod is succesfully ipleented in practice, the results could not yet be validated by real life experients on the test case. A possible reason for this is non-linearity, whereas the odels have been obtained under different signal conditions (ipact easureent) than the validation experient that was perfored to apply the TPA ethod (steady state ulti-sine). Despite the fact that the TPA ethod was not validated on the experiental test case at hand, the experiental results of the FBS approach give good confidence, and it is recoended to further introduce the presented approach in dynaic analysis of the effect of floor vibrations in high precision equipent. 5.2 Outlook Based on the results presented in this paper, it is recoended to further validate the presented approach. Firstly, that would iply a further analysis of possible non-linear behavior in either the achine dynaics or the floor dynaics, by using different signal types and aplitudes. Secondly, an even ore siplified test case ight be realized to validate the the Transfer Path Analysis ethod in practice. Against the background of current precision engineering practice, where only the floor vibration level and possibly floor stiffness is specified to predict achine accuracy, the presented work leads to the insight that at least a Single Degree of Freedo oscillator odel should be introduced to represent the floor dynaics. If only the first resonance frequency and the effective odal ass of the floor are known, this would already allow for a uch ore accurate prediction of the coupled dynaics. In ore advanced applications, the introduction of full coupling with experiental floor odels by Frequency Based Substructuring will allow for even ore accurate perforance prediction of (high precision) equipent on factory floors. This way, one ay potentially avoid costly and conservative design choices in the achine design. References [1] D. de Klerk and D.J. Rixen, Coponent transfer path analysis ethod with copensation for test bench dynaics, Mechanical Systes and Signal Processing, Vol. 24, No. 6, (2010), pp [2] E.I. Rivin, Vibration isolation of precision equipent, Precision Engineering, Vol. 17, No. 1, (1995), pp [3] C.G. Gordon, Generic criteria for vibration-sensitive equipent. International Society for Optical Engineering (SPIE), Vol. 1619, (1992), pp

14 3514 PROCEEDINGS OF ISMA2012-USD2012 [4] R.M. Schidt, G. Schitter, J. van Eijk, The Design of High Perforance Mechatronics, Delft University Press, Delft (2011). [5] H. Aick, S. Hardash, P. Gillett, and R.J. Reaveley, Design of stiff, low-vibration floor structures, International Society for Optical Engineering (SPIE), Vol. 1619, (1992), pp [6] D. de Klerk, D.J. Rixen and S.N. Vooreeren, General fraework for dynaic substructuring: history, review and classification of Techniques, AIAA Journal, Vol. 46, No. 5, (2008), pp [7] D.J. Rixen, M.A. Boogaard, G. van Schothorst and G.W. van der Poel, Blocked forces and free displaceents approaches for vibration transission analysis, Mechanical Systes and Signal Processing, (subitted). [8] M.A. Boogaard, Machines with high accuracy on factory floors, Master of Science thesis, TU Delft (2012)