COMPENSATION FOR DISTORTION OF DYNAMIC SENSOR DATA USING RECEPTANCE COUPLING

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COMPENSATION FO DISTOTION OF DYNAMIC SENSO DATA SIN ECEPTANCE COPIN Tony. Schmitz niversity o Forida, Department o Mechanica and Aerospace Engineering, ainesvie, F 36.

Appications o this sensor data incude process metroogy, condition-based maintenance, genera aut diagnosis, and nondestructive evauation, or exampe.

1 COMPENSATION FO DISTOTION OF DYNAMIC SENSO DATA SIN ECEPTANCE COPIN Tony. Schmitz niversity o Forida, Department o Mechanica and Aerospace Engineering, ainesvie, F 36. INTODCTION Sensors can be used to record a number o phenomena on mechanica structures. Appications o this sensor data incude process metroogy, condition-based maintenance, genera aut diagnosis, and nondestructive evauation, or exampe. In many cases, the recorded inormation is dynamic in nature and minimum requirements or the sensor requency response, in addition to accuracy, resoution, and durabiity, are required. However, even i a satisactory sensor (or sensors) is seected, i the sensor cannot be paced directy at the point o interest, the structura dynamics between the two ocations can adversey aect the measured signa through both scaing errors and distortion o the requency content. In this work, a uniied mathematica approach or assemby dynamics prediction using receptance couping techniques is presented. The predicted modes can be appied to remove the eects o structura dynamics on the measured data and seect high signa-to-noise ratio sensor mounting ocations.. METHOD DESCIPTION. Compensation or non-idea sensor ocation Substructure anaysis, or component mode synthesis, methods have been used or severa decades to predict the dynamic response o assembies using measurements and/or modes o the individua components, or substructures. These components can be represented by spatia mass, stiness, and damping data; moda data; or receptances [e.g., -7]. The atter representation is generay preerred in situations where the assemby receptances are the desired anaysis output, as is the case here. This research buids on the previous receptance couping work by incuding both transationa and rotationa degrees o reedom and appying the assemby dynamics modes to sensor data compensation. To demonstrate the method, consider the prismatic cantiever beam-cyinder assemby shown in Fig., where a harmonic orce, F (t), is appied to the ree end o the cyinder, coordinate X, and the resuting dispacement X (t) is to be determined. The preerred soution is to pace an appropriate inear transducer at this ocation and record the required dispacement directy. However, i access to the cyinder is obstructed, it may be required to pace the sensor remotey at coordinate X, or exampe. Due to the change in cross-section and increased stiness, there is reduced sensitivity to the X vibrations or X measurements. Additionay, the vibration measured at X can be inuenced by the assemby s structura response between coordinates X and X. Finay, X may be ocated at a node o one o the assemby mode shapes. In order to determine X (t) rom X measurements, the oowing three steps are carried out.. Measure the time-domain response X (t) and Fourier transorm to obtain the requency-domain resut X (ω).. The reationship between X (ω) and F (ω) can be expressed by: ( ω) H F ( ω) X, () where H is the assemby dispacement-to-orce cross receptance between coordinates X and X. Soving or - F gives F ( ω) H X (ω). This resut can be substituted in the equation reating X (ω) and F (ω) to obtain: - ( ω) H F ( ω) H H X ( ω) X, () where H is the assemby dispacement-to-orce direct receptance at coordinate X. 3. Inverse Fourier transorm X (ω) to obtain X (t). This resut now provides the required inormation and removes the inuence o the structura dynamics between X and X. Note that the time-domain orce, F (t), is aso avaiabe by inverse Fourier transorming F (ω). To carry out these steps, the assemby dispacement-toorce receptances H (ω) and H (ω) must be known. The straightorward approach is to measure them directy using some externa mechanica excitation, such as a hammer or shaker, and record the response using a

2 inear transducer, e.g., acceerometer, vibrometer, or dispacement probe. However, there may be situations where either the measurements are diicut to carry out (due to mechanica obstruction) or the assemby changes reguary (or exampe, mutipe toos and hoders may be used in a singe miing spinde [8-3]). In these cases, an accurate mode is required. One choice or deveoping the mode is inite eement anaysis, or FEA, which can provide quaity estimates o natura requency and mode shapes. A current imitation o this method, however, is the treatment o joints, particuary the energy dissipation, or damping, at contact interaces. This is criticay important in the sensor data compensation proposed here because propery scaed assemby receptances, which depend strongy on damping, are required. In many cases, an FEA a priori prediction o assemby dynamics rom design drawings is diicut or impossibe. As an aternative, receptance couping oers a hybrid method where measured receptances can be couped to modeed receptances through appropriate connections, which can incude both inite stiness and non-zero damping. The genera approach is to measure those components with diicut-to-predict behavior and mode those components that are more suited to successu prediction using modeing techniques. To determine the dynamics or the assemby shown in Fig. by receptance couping, severa considerations must be made. First, ony dispacement-to-orce receptances, H ij, have been discussed so ar. However, i accurate assemby predictions are to be made, a our bending receptances must be incuded in the component descriptions (i.e., dispacement-to-orce, H ij, dispacement-to-moment, ij, rotation-to-orce, N ij, and rotation-to-moment, P ij ). Second, the compatibiity conditions at the component interaces must be seected; these can be rigid, exibe with no damping, or exibe with damping. Third, the component receptances must be determined. I measurements are competed, care must be taken to accuratey record a our receptances with adequate bandwidth and resoution. The steps required to predict the Fig. assemby receptances oow.. Deine the components and coordinates or the mode. In this simpe exampe, two components can be deined: a prismatic cantiever and ree-ree cyinder (see Fig. ).. Describe the component receptances. Both measurements and modes can be used. For the modes, the cosed-orm receptance unctions presented by Bishop and Johnson [] or the anaysis o exura vibrations o uniorm Euer-Bernoui beams with ree, ixed, siding, and pinned boundary conditions are appied here. For exampe, the direct and cross receptances or the ree-ree cyinder due to the component orces (t) and (t), appied at coordinates x (t) and x (t), respectivey, and moments m (t) and m (t), appied at θ (t) and θ (t), respectivey, are compacty represented in Eq. (3) (ower case notation is used to designate component variabes). x h θ or { } n p m x h θ n x h p m u [ ]{ } q or { u } [ ]{ } q θ or { } n p m u [ ]{ q } x h θ or { } n p m u [ ]{ q },, (3) where ij is the generaized receptance matrix that describes both transationa and rotationa component behavior []. The individua entries in these matrices depend on the boundary conditions and incude contributions rom both the rigid body (i appicabe) and exura modes. For exampe, at the right end o the ree-ree cyinder, the matrix terms are: h ( ω) x x ( ω) n ( ω) p m ( ω) θ m ( cosλsinh λ sin λcosh λ) 3 EI( + η) λ ( cosλcosh λ ) θ EI (4) ( sin λ sinh λ) ( + η) λ ( cos λ coshλ ) ( cosλsinh λ + sin λcosh λ) EI( + η) λ( cosλcosh λ ) 4 mω where λ, m is the beam mass EI ( ) (kg), + iη ω is the requency (rad/s), is the beam ength (m), E is the eastic moduus (N/m ), I is the nd area moment o inertia,

3 (m 4 ), and η is the structura damping actor (unitess). [Damping was not incuded in reerence [], but has been added as part o this anaysis.] These receptances can be used to coupe components at their end points in order to determine assemby dynamics. 3. Based on the seected mode rom step, express the assemby receptances (see Eq. (5), where upper case notation is used to designate assemby variabes). These receptances are determined by irst writing the component dispacements/ rotations as shown in Eq. (6). i, where X i H ij ij Fi ij i Θi, N ij Pij, and M i (5) u + u q q u b b bqb q + q (6) I the prismatic cantiever beam-cyinder assemby is considered to be a soid body with the cyindrica portion ormed by turning one end o the beam down to its ina diameter, or exampe, a rigid connection is appied at the component interace and the compatibiity conditions are: actua measurement data (no curve itting or moda parameter estimation is required). u H N ( + ) u bb q + ( + ) bb q + q q H N P P (0) (). Sensor ocation seection or high signa-to-noise The receptance couping approach described in the previous section can aso be used to predict assemby mode shapes and, thereore, enabe the seection o sensor ocations with improved signa-to-noise ratios. Again considering the assemby in Fig., the previous two-coordinate mode is most ikey insuicient to generate usabe mode shapes. However, any number o coordinates can be added to improve the mode shapes spatia resoution (within the imitations o Euer- Bernoui beam theory). The mode is redeined in Fig. 3 to incude seven assemby coordinates, or exampe. u u b 0 and u i i, i to, (7) where the atter expression speciies that the component and assemby coordinates are deined at the same spatia positions. The equiibrium conditions vary with the externa orce/moment ocation. To determine the irst coumn o the assemby receptance matrix in Eq. (5), is appied to coordinate. In this case, the equiibrium conditions are: q + q b 0 and q. (8) Substitution o the component dispacements/rotations and equiibrium conditions into the compatibiity conditions yieds q (see Eq. (9)). The expression or is then given by Eq. (0). The other irst coumn receptance is determined in a simiar manner; is shown in Eq. (). To ind the receptances in the second coumn, must be appied to. q ( + bb ) (9) Equations (0) and () contain the assemby receptances required to sove Eq. () or X (t) rom sensor data recorded at X. In this exampe, these receptances were deveoped anayticay using materia properties and component geometry. However, any o the component response terms coud aso be repaced by For this mode, the assemby receptance matrix (Eq. (5)) is rewritten as shown in Eq. (). To determine the 49 ij entries, step 3 rom the previous section is repeated mutipe times or each coumn. For exampe, to determine the irst coumn mode shapes, is appied to as shown in Fig. 3. The and terms are again cacuated using Eqs. (0) and (). However, the assemby is now sectioned at coordinate to deine the two components. To ind 3, the assemby is sectioned at 3 and the resuting equation is ( b3b ) 3. The recursive pattern can be observed to ind the remaining terms in the coumn.

4 M 7 M 7 M 7 O 7 7 M M 77 7 () compared to experimenta resuts or the irst three reeree bending modes in Fig. 6. In each case, the mode shape maximum ampitude was normaized to one. Note that this resut diers rom traditiona moda anaysis where the number o natura requencies and mode shapes must be equa to the number o mode spatia coordinates in order to obtain square matrices. In Eq. (4), a 49 entries in the ij matrix are by by N matrices, where N is the number o points in the requency vector, ω. Each o the receptances in these matrices incudes a modes within the requency range o interest. 3. EXPEIMENTA EVAATION To veriy the receptance couping approach, requency response measurements were carried out on a 87.9 mm x 76. mm x 6. mm pastic beam using a sma impact hammer (PCB 086D80) to excite the beam and ow mass acceerometer (PCB 35B0) to record the response [4]. Free-ree boundary conditions were approximated by suspending the beam verticay rom a thin exibe wire (a mm diameter hoe was dried at one end). The receptance couping mode o the constant cross-section prismatic beam was veriied by measuring the direct requency response unction (FF) at the ree end o the beam (away rom the wire suspension) and comparing to the predicted response. The predicted and measured direct FF are shown in Fig. 5. To popuate the receptance mode, the beam mass (944.3 g) and dimensions were recorded and the nd area moment o inertia was cacuated. The moduus and structura damping actor were considered ree parameters to enabe a good it between the mode and data (E.37x0 9 N/m and η 0.035). 3. Force prediction using acceeration data As described in Section., the externay appied orce can be determined rom the measured response and system FF mode. To veriy this approach experimentay, the suspended pastic beam was excited at its ree end using the impact hammer and the response was measured at the same ocation using the acceerometer. Both signas were anaog ow-pass itered with a cuto requency o khz prior to digita samping at 50 khz or sec. The measured acceeration, a (t), was then Fourier transormed and pre-mutipied by the inverted direct acceerance, equa to the product o -ω and the modeed direct receptance H (ω) - (see Fig. 5), to ind the requency domain - orce, F ( ω) ( ω H) A ( ω). This resut was then inverse Fourier transormed and compared to the input orce due to the hammer impact. The orce resuts and measured acceeration are shown in Fig. 7. The reader may note that the ringing in the hammer orce is due to the anaog ow pass itering. Once the beam mode was estabished, cross FF measurements were competed and compared to the cross receptances determined according to the approach described in Section.. The anaytica mode shapes are

3. Correction or acceerometer pacement The exercise in Section 3. was repeated, except the acceerometer was not paced at the hammer input ocation.

5 3. Correction or acceerometer pacement The exercise in Section 3. was repeated, except the acceerometer was not paced at the hammer input ocation. In this case, the hammer impact occurred 54 mm rom the beam ree end. I this position is reerred to as X, then the receptance H (ω) can be used to determine the orce, F. The ω ω H A ω H (ω) receptance is ound by artiiciay separating the beam into two components at X and then joining the components as described in Section.. See Eq. (3), where the and cross and direct component receptances are determined using a 54 mm beam ength, whie the bb direct receptances are cacuated using a beam ength o ( ) mm. See Fig. 8. Note that incorrect resuts are obtained i ony the dispacement-to-orce receptances are used, i.e., H h h + h. ( ) hbb bb - ( ) ( ) ( ) previous moduus, density, and structura damping vaues, the modiied beam mode was produced by rigidy couping the 5 constant geometry sections shown in Fig. 9. The measured and predicted direct receptance at the ree tapered end are shown in Fig. 0. easonabe agreement is seen, but the predicted natura requencies or the higher order modes are increasingy too arge. H N ( + ) bb bb P (3) The orce input was again reconstructed using acceerometer measurements at the tapered end and hammer impacts at: ) the tapered end; and ) mm rom the ree end, in conjunction with the corresponding mode receptances. The measured and predicted cross receptance at the mm ocation are provided in Fig.. The mode divergence at higher requencies is again observed. 3.3 Modiied beam geometry For the next experiments, the beam geometry was modiied by cutting away portions near the beam midde and tapering the beam ree end (Fig. 9). sing the The measured and predicted orce signas or the modiied beam are shown in Figs. and 3. Figure shows the resuts or the orce input at the beam tapered end, whie Fig. 3 dispays the resuts or hammer excitation mm rom the ree end. In both cases, additiona content is seen in the predicted signa ater the initia hammer impact. This extra (incorrect) orce is due to the sight mismatch in natura requencies between the mode and actua beam. However, the

6 reconstructed orces sti provide reasonabe representations o the actua impacts. Measurements and predictions are shown or both direct and cross receptances. Force reconstruction rom acceerometer data, again using both direct and cross mode receptances, is aso provided. 5. ACKNOWEDMENTS The author grateuy acknowedges partia inancia support or this research rom the Nationa Science Foundation (DMI-03809) and Oice o Nava esearch (003 Young Investigator Program). The author aso thanks. S. Duncan or contributions to this work. 4. CONCSIONS This paper describes the use o receptance couping techniques to buid dynamic modes o assembies in order to compensate or scaing and requency distortion errors arising rom non-idea sensor pacement on mechanica structures. The receptance couping equations provided here incude both transationa and rotationa degrees o reedom, as we as the potentia or non-rigid connections between substructures. Experimenta resuts are provided or a pastic prismatic beam and it is shown that orce signas can be recovered rom acceeration measurements at: ) the orce input position; and ) other ocations. An impact orce was appied so that a broad range o structura modes woud be excited simutaneousy and, thereore, provide a rigorous test or the approach. Experimenta resuts are aso shown or a modiied beam geometry, where the mechanica properties determined rom a it to the direct dispacement-to-orce receptance at the end o the reeree origina beam are used to deveop a receptance couping mode or the new beam geometry. 6. EFEENCES. Bishop,.E.D. and Johnson, D.C., 960, The Mechanics o Vibration, Cambridge niversity Press, Cambridge.. Hurty, W.C., 965, Dynamic Anaysis o Structura Systems using Component Modes. AIAA Journa 3/4: Kosterman, A.., and emon, J.., 969, Buiding Bock Approach to Structura Dynamics, American Society o Mechanica Engineering Annua Vibration Conerence, pubication VIB Jetmundsen, B., Bieawa,.., and Fanney, W.., 988, eneraized Frequency Domain Substructure Synthesis. Journa o the American Heicopter Society 33: en, Y. and Beards, C.F., 995, On Substructure Synthesis with FF Data, Journa o Sound and Vibration 85: Ewins, D.J., 000, Moda Testing: Theory, Practice and Appication, nd Edition, esearch Studies Press, Phiadephia, PA. 7. ui, W. and Ewins, D.J., 00, Substructure Synthesis via Eastic Media, Journa o Sound and Vibration 57/: Schmitz, T.. and Donadson,., 000, Predicting High- Speed Machining Dynamics by Substructure Anaysis, Annas o the CIP 49/: Schmitz, T.., Davies, M.A., and Kennedy, M., 00, Too Point Frequency esponse Prediction or High- Speed Machining by CSA, ASME Journa o Manuacturing Science and Engineering 3: Schmitz, T.., Davies, M.A., Medicus, K., and Snyder, J., 00, Improving High-Speed Machining Materia emova ates by apid Dynamic Anaysis, Annas o the CIP 50/: Park, S.S., Atintas, Y., and Movahhedy, M., 003, eceptance Couping or End Mis, Internationa Journa o Machine Toos and Manuacture, 43: Yigit, A.S. and soy, A.., 00, Dynamic Stiness Evauation or econigurabe Machine Toos incuding Weaky Non-inear Joint Characteristics, Proceedings o the I MECH E Part B, Journa o Engineering Manuacture, 6/: Schmitz, T. and Burns, T., 003, eceptance Couping or High-Speed Machining Dynamics Prediction, Proceedings o the st Internationa Moda Anaysis Conerence, February 3-6, Kissimmee, F (on CD). 4. Equipment manuacturers are speciied or competeness. This does not impy endorsement by the author or the niversity o Forida.

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