sem.org imac xxvii conf s39p003 determining rigid body inertia properties cumbersome systems

7/29/2019 Sem.org IMAC XXVII Conf s39p003 Determining Rigid Body Inertia Properties Cumbersome Systems

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Determining the rigid-body inertia properties of cumbersome systems:comparison of techniques in time and frequency domain

Emiliano Mucchi(*), Stefano Fiorati, Raffaele Di Gregorio, Giorgio Dalpiaz

EnDIF Engineering Department in FerraraUniversit Degli Studi di Ferrara

Via Saragat,1 I-41100 Ferrara ITALYTel: +39 0532 974913Fax: +39 0532 974870

E-mail: [email protected](*) Corresponding author

ABSTRACTThe inertia properties of an engine to be mounted on complex mechanical systems (e.g., cars, boats, etc.)are usually evaluated by considering the engine like a rigid-body (RB). These properties are mainlyemployed to design the geometry and the stiffness of the engine support, so as to build the dynamic model

of the mechanical system the engine has to be mounted on. In this paper, two different techniques formeasuring the RB inertia properties are compared: one in the time domain and the other in the frequencydomain. The time-domain technique is based on the measurement of the oscillation period of a trifilarpendulum; whilst, the frequency-domain technique is based on the measurement of the FrequencyResponse Function (FRF) of the softly suspended RB. These techniques are applied to estimate the RBinertia properties both of an engine block and of a marine diesel engine. Eventually, their pros and contrasare highlighted.

1. INTRODUCTION

The inertia properties of an engine to be mounted on complex mechanical systems (e.g., cars, boats, etc.)are usually evaluated by considering the engine like a rigid-body (RB). These properties are mainlyemployed to design the geometry and the stiffness of the engine support, so as to build the dynamic model

of the mechanical system the engine has to be mounted on.The direct calculation of the RB inertia properties (mass, center of gravity (COG), and inertia tensor) can beimplemented only if the RB mass distribution is completely known. Such a knowledge usually is not availablefor complex geometries and/or devices (e.g., an engine). The experimental evaluation of them tries toovercome this lack.Many researchers have proposed experimental techniques for the indirect measurement of the RB inertiaproperties [13]. Such techniques can be collected into two groups:

9 Time-Domain Methods (TDM)9 Frequency-Domain Methods (FDM)

Time-Domain Methods are based on the measurement of the small-oscillation period of a pendulum [16][15].This conceptually simple measurement hides some traps for unskilled operators. In fact, it requires toimplement a cumbersome set-up procedure. Some researchers have recently proposed a few variants to theclassical TDM. In [14], the estimation of the RB inertia properties comes from an accurate selection of rigid

and flexible body modes. In [7][8] a large test rig can accurately estimate the RB inertia properties of vehiclesand subsystems, and it is able to locate the center of gravity, too. Further proposals can also be found in[17][20][18].Frequency-Domain Methods are all based on the measurement of Frequency Response Functions (FRFs),and can be of three types [2]:

Modal Model Method(MMM) Direct System Identification Method(DSIM) Inertia Restrain Method(IRM)

The Modal Model Method (MMM) [13] uses modal data for calculating the system matrices. The RB modesare extracted from the measured FRFs. Then, the orthogonality condition of the rigid-body modes, whichcontains the systems mass matrix, is used to calculate the unknown RB inertia properties that appear in themass matrix. All the computations reduce themselves to the solution of a suitable equation system.

Proceedings of the IMAC-XXVIIFebruary 9-12, 2009 Orlando, Florida USA

2009 Society for Experimental Mechanics Inc.


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The Direct System Identification Method (DSIM) [5] [6] evaluates the mass, stiffness and damping matrices ofthe test rig through the measured FRFs. In particular, the dynamic model of the test rig is explicitly written,and transformed in the frequency domain. Then, the analytical expressions of the FRFs, as a function of theunknown RB inertia properties and of all the other characteristics of the test rig, are deduced from that model.Eventually, the deduced formulas are used to fit the measured FRFs (i.e., a suitable number of spectral linesis selected to build an over-determined set of equations, which is solved through the least-squares method).

The Inertia Restrain Method (IRM) [11][9][10][12][4] has been used for a variety of specimen types, which

range from simple-geometry types (e.g., beams, plates, simple structures) to complex-geometry and, even,inhomogeneous types [4][7][9] (e.g., large engines). It uses the mass line of the measured FRFs. The test rigconsists in softly suspending the RB, for instance, with bungee cords and/or springs, so that the wholesystem can be modeled as a free body (i.e., the external forces exerted on the RB by the suspension arenegligible when compared with the excitation and inertia forces). The dynamic model of the free body inincipient motion (i.e., null velocities, but non-null accelerations) is used to deduce the analytical expressionsof the FRFs that have to fit the measured ones [10], same as the DSIM does.The IRM simplicity is at the same time its strength and its weakness. In fact, the selection of the responsepoints as well as the excitation points on the RB, in terms of number, location, and response/excitationdirection, considerably influence the IRM accuracy. That is why, Lee et al. [11] used statistical analysis toselect measurement points that minimize the errors on the identified inertia properties.The IRM requires the existence of a flat mass line in a sufficiently wide band between the highest rigid-body-mode frequency and the lowest flexible-mode frequency. The larger the bandwidth of the mass line is, the

lower the interaction between rigid-body modes and flexible modes is. According to the size of the mass-linebandwidth in the measured FRF, three different algorithms have been devised to fit the measuredFRFs[1] [3]:

Unchanged FRFs (to be used when the rigid-body modes and the flexible ones are wellseparated).

Corrected FRFs (to be used when insufficient bandwidth exists between the RB modes and theflexible ones; such an algorithm corrects the measured FRFs, by subtracting the synthesizedFRFs of the flexible modes, before fitting them).

Lower residual (to be used when accurately measured FRFs are not available in the lowfrequency range of the mass line; such an algorithm takes into account the lower residual of thefirst flexible mode in the FRFs analytical expressions that fit the measured FRFs).

The accuracy of the above-mentioned algorithms has been already verified for simple structures whose RBinertia properties are known[1][3]. Moreover, in [13], a few frequency-domain methods have been comparedand their accuracy has been discussed by designing and manufacturing structures specifically for this

purpose.As far as these authors are aware, in the literature, comparisons between TDMs and FDMs have not beenpresented either for simple or for large inhomogeneous structures, yet. This fact motivated this work.The authors have set two separated experimental campaigns devoted to the evaluation of the RB inertiaproperties: one uses a trifilar pendulum (i.e. a TDM) and the other uses the IRM (i.e. a FDM). The accuracyof the methods has been evaluated by using two structures: a 4-cylinder engine block, and a marine dieselengine. For the first structure (the 4-cylinder engine block) the RB inertia properties estimated through thetwo experimental procedures will be compared with the nominal ones computed by using the massdistribution nominal data, provided by the manufacturer who made the 3D-CAD model available. For thesecond structure (the marine diesel engine), the measurements carried out with a third method, calledInTenso [7][8], are also available; hence, the RB inertia properties estimated through the two experimentalprocedures will be compared with the ones obtained through the InTenso method, too.In the following part of the paper, Sections 2 and 3 give a quick theoretical background of the TDM and the

FDM, respectively, and describe the measurement set-up of the tests performed on the 4-cylinder engineblock and on the diesel engine. Section 4 compares the results of the measurements carried out with the twomethods, and discusses pros and contras. Eventually, Section 5 draws the conclusions.

2. TRIFILAR PENDULUM

The trifilar pendulum, also called three-string torsional pendulum, is shown in Figure 1. It consists of twoplatforms, one movable and the other fixed. They are connected through three equally-spaced cables thathave the same length, L. The three points that join the cables to the fixed platform identify a circle (the fixedcircle with radius R1 in Figure 1) that must be perpendicular to the local vertical, so that, when the test rig isat rest, the movable circle (the one with radius R2 in Figure 1), which is identified by the three points that jointhe cables to the movable platform, is horizontally located at a known distance, L0, under the fixed circle, andthe straight line through the centers of the two circles is vertical.


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The center of gravity (COG) of the movable platform must lay on the vertical line through the center of themovable circle, and the specimen, laid on the movable platform, must be so placed that its COG lays on thesame vertical line. Such conditions are identified by three load cells that measure the tension in the cablessince the three tensions must be all equal when they occur. By verifying this tension equalization for threedifferent placements of the specimen on the platform, the COG of the specimen can be located. The sametensions can be used to determine the mass of the specimen and/or of the mobile platform.Such a test rig is a three degrees of freedom (dof) mechanism that, when left to freely oscillate near theequilibrium configuration (i.e., where the motion equations can be linearized), exhibits three oscillation

modes, which can be separately excited by suitably choosing the initial conditions of motion.One out of these modes is an helical oscillation of the movable equipment (mobile platform plus specimen)with helical axis that coincides with the vertical line through the center of the fixed circle. Such an oscillationkeeps the movable circle horizontal and saves the geometric condition on the COGs. For this oscillationmode, the analysis of the linearized motion equations reveals that the moment of inertia, J, of the movableequipment about the straight line through the centers of the mobile and fixed circles is related to theoscillation period, , by the following relationship:

( ) 20 1 20 2

04

s

s

m m gR RJ J J

L

+= + = (1)

where J0 (Js) and m0 (ms) are the moment of inertia and the mass of the movable platform (of the specimen),respectively, and gis the gravity acceleration.

AfterR1, R2 and L0 have been determined, the mass m0 and the oscillation period, 0 , are measured in the

test rig without specimen. J0 is calculated through formula (1) with ms and Js equal to zero and with

replaced by 0 . Then, the specimen is put on its own place, the mass ms is measured, and the final

measurement of allows formula (1) to be used for the determination ofJs.

This simple measurement procedure is affected by the errors carried out when the geometrical and physicalconstants appearing in formula (1) (i.e., R1, R2, L0 and g) are evaluated. To overcome this problem, thebelow-reported calibration procedure can be implemented.The oscillation period of the test rig is firstly measured with a standard specimen (i.e., a RB whose ms and Jsare known with high precision) placed on the movable platform, and then without any specimen. So doing,

the two measured oscillation periods, say s and 0 respectively, satisfy the following equations, directly

deduced from formula (1):

( ) 20 02

0 0 0

s s sJ J k m m

J km

+ = +

=(2)

where

1 2

2

04

gR Rk

L= .

Equations (2) constitute a system of two linear equations in two unknowns: kand J0 (i.e., the instrumentconstants). The solution of system (2) yields the following expressions for the instrument constants:

( )

( )

2 2

0 0 0

2

0 00 2 2

0 0 0

s

s s

s

s s

Jk

m m m

J mJ

m m m

= +

=

+

(3)

It is worth noting that all the moments of inertia measured through the trifilar pendulum refers to axes thatpass through the COG of the specimen.If six moments of inertia, say Jr

(a), Jr

(b) Jr

(f), about six distinct, but concurrent axes are available for a RB,

the calculation of the six entries of the RBs inertia tensor, J , referred to a Cartesian reference with origin atthe common intersection of the six axes, is straightforward.In fact, the moment of inertia about an axis r, passing through the origin of the above-mentioned Cartesian

reference and with direction cosines collected in the unit vector ( ), ,=T

x y z , can be written as

follows [19]:T

rJ = J . (4)

where


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=

J

xx xy xz

xy yy yz

xz yz zz

J J J

J J J

J J J

(5)

Equation (4) can be written for all the six measured Jr (i.e., forJr(a)

, Jr(b)

Jr(f)

), which yields a linear system of

six equations in six unknowns: the entries of J . Thus, the entries of the inertia tensor can be determined bysolving a simple linear system. If three axes are the x, y, and z axes of the above-mentioned Cartesian

reference, and the other three axes are the bisectors of the x-y, x-z and y-z planes of the same reference,the system to solve becomes:

( )

( )

( )

( )

( )

( )

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0.5 0.5 0 1 0 0

0.5 0 0.5 0 1 0

0 0.5 0.5 0 0 1

=

axx r

byy r

czz r

dxy r

exz r

fyz r

J J

J J

J J

J J

J J

J J

(6)

System (6) is simple and has the condition number exactly equal to one [7] [8], which assures that themeasurement errors on the Jrare not amplified by solving the system.The measurements on the 4-cylinder engine block performed through the trifilar pendulum used both theabove-mentioned calibration procedure and no calibration procedure.The nominal inertia properties of the 4-cylinder engine block, obtained by the 3D-CAD, are reported in the2

ndcolumn of Table 1.

All the oscillation periods have been measured by taking the time elapsed during twenty completeoscillations of the movable equipment, each measurement has been repeated five times, and their averagevalue has been considered. On this point, it is worth noting that a relative error in the oscillation period, ,

roughly doubles the relative error in the computed moment of inertia. Indeed, formula (1), written as follows:2J a= (7)

where 1 22

04

mgR Ra

L= , yields the following expression of the moment of inertia, J, computed with an relative

error on :

( )2 2 2 2(1 2 ) (1 2 )J a a J

= + = + + = + + , (8)

and relationship (8) can be transformed as follows:

22

= +J J

J

(9).

The measurements on the engine block have been repeated six times for six different barycentric axes inorder to obtain the six independent moments of inertia necessary to calculate the entries of the inertia tensorreported in the 3

rdcolumn of Table 1. The 4

thcolumn of Table 1 also shows the entries of the inertia tensor

obtained without calibration procedure.The same measurement set-up has been used for the marine diesel engine. In this case, a complex kit ofbrackets was necessary for positioning the engine along the bisectors of the coordinate planes (see Figure 1right), and the presence of these brackets slightly perturbed the measurements in a way that is only roughly

quantifiable. Moreover, each placement of the engine on the mobile platform required much more time thanthe previous case because of the complexity and the high weight of the specimen to move. Thus, thecomplete set of measurements was very time-consuming. The measurement results are reported in the 3

rd

column of Table 2. Engines available data, measured with InTenso method, are reported in the 2nd

columnof Table 2.


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Figure 1 Schematic of the trifilar pendulum with reference frame (left) and photo of the trifilar pendulum and

diesel engine during the measurement of the products of inertia (right).

3. INERTIA RESTRAIN METHOD

The Inertia Restrain Method (IRM) consists on measuring FRFs on the structure under free-free boundaryconditions. Theoretically, six excitation points are required to solve the system of kinematic and dynamicequations; however, experience has shown that using at least nine excitation points yields better results [2];moreover in most cases between eight and twelve response points in three directions are sufficient. Thecalculation is a two step procedure: first, the COG is determined, and, then, the equation system containing

the unknown entries of the RB inertia tensor is stated and solved (see [1][3][21] for details). The test involvesno special equipment and only limited measurement efforts; essential requirements for this method are anaccurate geometrical wire-frame collecting the coordinates of the measurement points and the weight of thestructure.The 4-cylinder engine block was suspended at six connection points as shown in Figure 2 by using softbungee cords. The suspension cords where chosen as soft as possible checking that they were notstretched completely. Fifteen input degrees of freedoms (DOFs) were excited by a large impact hammer withsoft tip (PCB 086D20) measuring the acceleration responses in 24 DOFs by using PCB piezoelectricaccelerometers (frequency range 1 to 10000 Hz). The signals were acquired by using a sampling frequencyof 1600 Hz and a frequency resolution of 0.5 Hz, which allowed the low frequency behavior to be caught.Furthermore, the FRFs were calculated by using the H1 estimator. In addition the measurements of thecoherence function was performed and the reciprocity of the FRFs have been checked in order to have areliable FRF set. The measurement results are reported in the last five columns of Table 1.

Concerning the FRF measurements on the diesel engine, 13 input DOFs were excited and 30 responseDOFs were measured by using the same instrumentations described for the engine block set-up. The enginewas suspended at six connection points by using springs as depicted in Figure 2. The measurement resultsare reported in the last five columns of Table 2.

4. COMPARISON OF TDM AND FDM

Table 1 refers to the measurements on the 4-cylinder engine block. It reports the six moments of inertia andthe COG coordinates of the engine block evaluated through the two above-described methods and their 3D-CAD nominal values. The 3D-CAD nominal values have to be considered as reference.With reference to the TDM measurement results, the analysis of Table 1 reveals that the absolute errors on

the moments of inertia xxJ , yyJ , zzJ , greatly reduces when the calibration procedure is used, and that the

calibrated TDM yields highly accurate results. On the contrary, the measured products of inertia present high

Brackets

Engine

Platform

Load cell


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absolute errors and poor accuracy both for the calibrated TDM and for the not calibrated TDM. The poormeasurement results on the inertia products can be explained by considering that orientation errors may beeasily introduced during the placements of the RB for measuring the inertia moments about the bisectors ofthe coordinate planes. Eventually, the measurement of the COG coordinate is highly accurate.

With reference to the IRM measurement results, Figure 3 shows the FRF-sum (i.e. the complex sum of theFRFs of all the measured structural points). The analysis of Figure 3 (left) reveals that a clear mass lineexists, so the Unchanged FRF algorithm, available in LMS Test.Lab, has been used. For this algorithm, the

influence of the selected frequency band of analysis has been investigated by applying it with three differentbands of analysis: 24-74Hz, 24-157Hz, 24-248Hz. The 5

th, the 6

th, and the 7

thcolumn of Table 1 reports the

results obtained in each case. The analysis of these three columns of Table 1 reveals that an effectiveinfluence of the selected frequency band on the results exists, even though the mass line seems flat.In order to improve the flatness of the mass line, since two input DOFs produced noisy FRFs, they havebeen eliminated and the analysis has been repeated with a set of 13 input DOFs (instead of 15) and 30response DOFs. The results of this further analysis are reported in the 8

thcolumn of Table 1 and the FRF-

sum is shown in Figure 3 (right). In this case, even though the mass line becomes very flat and clear, theevaluation of the RB inertia properties yields absolute error that are comparable with ones obtained withouteliminating measurement points. This fact can be justified by considering that, in the IRM, the estimate of theRB inertia properties is carried out, for each spectral line, with the least-squares method, therefore thegreater the number of measurement points (input and/or response) is, the smaller the influence of an extrameasurement point is.

Finally, with the set of FRFs obtained by eliminating the two noisy inputs, the first two flexible modes havebeen estimated by means of the PolyMAX and LSCF algorithm [22], and the Corrected FRF algorithm hasbeen applied. The so-obtained results are reported in the last column of Table 1. These results are not betterthan the ones obtained by means of the Unchanged FRF algorithm, which can be justified by consideringthat the FRFs show a very flat mass line and the RB modes are well separated from the 1

stflexible mode.

In conclusion, for the 4-cylinder engine block, the calibrated TDM is highly accurate and gives much betterresults than the IRM, even though it has some troubles in evaluating the products of inertia.

Table 2 refers to the measurements on the marine diesel engine. It collects the RB inertia propertiesobtained through three different methods, namely the InTenso Method [7][8] (2

ndcolumn), the trifilar

pendulum (3rd

column) and the IRM (4th

to 8th

columns). The results coming from the InTenso Method andfrom the trifilar pendulum are similar, on the contrary large differences occur in comparison with the IRM.Table 2 also shows the influence of the chosen frequency bands of analysis (12-24Hz, 12-31Hz, 16-29Hz)and the influence of the elimination of three nosy input DOFs. As for the engine block, the selected band of

analysis influences the results, moreover the elimination of a few noisy FRFs to build a new FRF set, called the best FRFs in Table 2, do not led to obtain results closer to the InTenso and trifilar pendulum methods(see the 7

thcolumn of Table 2), in fact the relative FRF-sum depicted in Figure 4 (right) do not show a more

flat and less noisy mass line.Since, in this case, the mass line is not perfectly flat (see Figure 4, left), the contribution of the first twoflexible modes has been taken into account by using the Corrected FRF algorithm, the results of this furtheranalysis, shown in the last column of Table 2, are better than the other IRM results. Nevertheless, theabsolute errors still are worse than the ones of the trifilar pendulum. A source of error that has to be takeninto account for the IRM is the mismatch between the geometrical wire-frame model and the realexcitation/response points. In fact due to practical reasons, the engine was hit at a point that could be up to1-3 cm off compared to the location of the response point and compared to the point on the wire-frame of thegeometry. This might have led to important error in the COG as well as in the moments of inertia.


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Figure 2 Photos of the suspended 4-cylinder engine block (left) and of the marine diesel engine (right).

Figure 3: Amplitude of the FRF-Sum of all the measured FRFs (left) and FRF-sum excluding two noisy input

DOFs (right) for the 4-cylinder engine block.

Figure 4: Amplitude of the FRF-Sum of all the measured FRFs (left) and FRF-sum excluding two noisy input

DOFs (right) for the marine diesel engine.

1st

flexible mode

RB modes

Mass line

1st

flexible mode

RB modes

Mass line

1st

flexible mode

RB modes

Mass line 1st

flexible mode

RB modes

Mass line

Bungee cords

Diesel Engine

Engine block

Springs

SpringsBungee cords


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5. CONCLUDING REMARKS

The comparison between a time-domain method and a frequency-domain method for estimating the RBinertia properties of real structures has been presented.In particular, the comparison has been implemented by applying the two methods to a 4-cylinder engineblock and to a marine diesel engine.This activity has brought to draw the following conclusions:

1. the trifilar pendulum is much more accurate than the IRM for both the tested structures, even though

it shows some limitations when the products of inertia are evaluated;2. the trifilar pendulum is very time-consuming for heavy and irregular object (i.e. an engine), since the

correct placement of the object on the mobile platform might be difficult to perform and often requiresa complex kit of ad hoc designed brackets for each measurement when particular axes areinvolved;

3. the IRM is influenced by the selection of the response/excitation points, by the chosen frequencyband, and by the set of selected FRFs;

4. in the IRM, the mismatch between the geometrical wire-frame model and the realexcitation/response points may lead to consistent error in the RB inertia properties, which cannot beavoided when large hammers are used and complex shape of RB are involved;

5. in the IRM, if the flexible modes are not enough spaced in frequency to avoid contamination of theRB modes, the mass line must be adjusted by using the Corrected FRF algorithm;

6. the costs for the instrumentation and the set-up are comparable in the two methods.

In short, the IRM has to be used with attention in practical cases when the knowledge of the band of analysisand of the number of measured points can not be determined a priori and when mismatch in the geometricdescription of the real structure can occur.

ACKNOWLEDGEMENTS

The authors wish to thank VM Motori S.p.A (Cento, Ferrara, Italy) and the engineers of this Company for co-operation and assistance in the collection of experimental data, moreover a special thanks is due to Stella -Officine Metallurgica Lux srl (Ferrara, Italy) for the experimental equipment preparation. This work has beendeveloped within the laboratory of research and technology transfer InterMech (LAV - Acoustics andVibrations) realized through the contribution of Regione Emilia Romagna - Assessorato Attivita' Produttive,Sviluppo Economico, Piano telematico, PRRIITT misura 3.4 azione A.

REFERENCES

[1] LMS International,Advanced FRF based determination of structural inertia properties, LMS technical report.[2] Lamontia M., On the Determination and Use of Residual Flexibilities, Inertia Restraints and Rigid Body Modes, Proceedings of

IMAC I, pp.153-159, 1982.[3] Leurs W., Gielen L., Brughmans M., Dierckx B, Calculation of rigid body properties from FRF data: practical implementation

and test case, Proceedings of IMAC XV, Tokyo, Japan, 1997.[4] Toivola J., Nuutila O., Comparison of three methods for determining rigid body inertia properties from Frequency Response

Function, Proceedings of IMAC XI, 1993[5] Okuma M., Heylen W., Sas P., Identification of the rigid body properties of 3-d frame structure by MCK identification method,

Proceedings of ISMA25, Leuven, Belgium, September 13-15, 2000.[6] Okuma M., Shi Q., Oho T., Development of the experimental spatial matrix identification method (theory and basic verification

with a frame structure), Journal of Sound and Vibration Vol. 219(1), pp. 5-22, 1999.[7] Previati G., Mastinu G., Gobbi M., Piccardi C., Bolzoni L., Rinaldi S.,A new test rig for measuring the inertia properties of

vehicles and their subsystems, Proceedings of IMECE 2004, Anaheim, California, USA, November 13-20, 2004.[8] Previati G., Gobbi M., Pennati M., Mastinu G., Accurate measurement of mass properties of round vehicles and theirsubsystems, Proceedings of the 7th International Symposium on Advanced Vehicle Control AVEC 2004, Arnhem, theNetherlands, 23-27 August, 2004.

[9] Okubo N., Furukawa T., Measurement of Rigid Body modes for Dynamic Design, Proceedings of IMAC II, pp.545-549, 1984.[10] Fragolent A., Sestieri A., Identification of Rigid Body Inertia Properties from Experimental Data, Mechanical Systems and

Signal Processing, Vol. 10(6), pp. 697-709, 1996.[11] Lee H., Lee Y., Park Y., Response and excitation points selection for accurate rigid-body inertia properties Identification,

Mechanical Systems and Signal Processing, Vol. 13(4), pp. 571-592, 1999.[12] Witter M.C., Brown D.L., Blough J.R., Measuring the six DOF driving point impedance function and an application to RB

inertia property estimation, Mechanical Systems and Signal Processing, Vol. 14(1), pp. 111-124, 2000.[13] Almeida R.A.B., Urgueira A.P.V., Maia N.M.M., Identification of Rigid Body Properties from vibration measurements, Journal

of Sound and Vibration Vol. 299, pp. 884-899, 2007.[14] Pandit S., Hu Z.Q., Yao Y.X., Experimental technique for accurate determination of rigid body characteristics, Proceedings of

IMAC X, , pp. 307-311, 1992.[15] Storozhenko V.A., A technique for identification of the principal central axis of inertia in an inhomogeneous rigid body,

International Applied Mechanics, Vol. 39(12), 2003.


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[16] Ringegni P.L., Actis M.D., Patanella A.J., An experimental technique for determining mass inertial properties of irregularshape bodies and mechanical assemblies, Measurement Vol. 29, pp. 63-75, 2001.

[17] Genta G., Delprete C., Some considerations on the experimental determination of moments of inertia, Meccanica Vol. 29,pp.125-141, 1994.

[18] Da Lio M., Doria, Lot R.,A spatial mechanism for the measurements of the inertia tensor: theory and experimental results,Journal of dynamic systems, measurements, and control, Vol. 121, pp. 111-121, 1999.

[19] Kane T. R., Levinson D. A., Dynamics: Theory and Application, McGraw-Hill Inc., ISBN: 0-07-037846-0, pp. 66-70. 1985.[20] Gentile A., Mangialardi L., Mantriota G., Trentadue A., Measurement of the inertia tensor: an experimental proposal,

Measurements Vol. 14 (3-4), pp. 241-254, 1995.[21] Mucchi E., Bottoni G., Di Gregorio R., Determining the rigid-body inertia properties of a knee prosthesis by FRF

measurements , Proceedings of the IMAC XXVII, Orlando 9-12 February 2009[22] Peeters B., Van der Auweraer H., Guillaume P., Leuridan J., The PolyMAX frequency-domain method: a new standard formodal parameter estimation?, Shock and Vibration Vol. 11(3-4), pp. 395- 410, 2004.

sem.org imac xxvii conf s39p003 determining rigid body inertia properties cumbersome systems

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