Proc IMechE Part K: A frequency-based substructuring J ...home.iitk.ac.in/~mlaw/Law_JMBD_paper.pdf · Mohit Law, Department of Machine Tools and Automation, Fraunhofer Institute for

Original Article

A frequency-based substructuringapproach to efficiently modelposition-dependent dynamics inmachine tools

Mohit Law and Steffen Ihlenfeldt

Abstract

Structural deformations of machine tool components that translate and/or rotate relative to each other to realize tool

motion results in tool point dynamics varying along the tool path. These changing dynamics interact with the cutting

process and the control loop of the drives to limit machine performance, making it necessary to virtually characterize

these interactions such as to guide design decisions. To facilitate rapid evaluation of these varying dynamics, this paper

describes a generalized frequency-based substructuring approach that combines the position-invariant component level

receptances at the contacting interfaces between substructures to obtain the position-dependent tool point response.

Receptances at the contacting interfaces are approximated by projecting them to a point to facilitate a multiple point

receptance coupling formulation. Complete machine behavior is represented by just a few sets of receptances, making

the model computationally more efficient than full-order finite element models and other dynamic substructuring

methods. Position-dependent dynamic behavior for a representative three axis milling machine is simulated and numer-

ically verified. Rapid investigations of the varying dynamics assist in virtually characterizing machine performance before

eventual prototyping.

Keywords

Machine tool, dynamic substructuring, receptances, position-dependent dynamics, modular synthesis

Date received: 21 August 2014; accepted: 10 November 2014

Introduction

Machine tools are essentially a collection of intercon-nected deformable bodies that undergo large transla-tional and/or rotational motion with respect to eachother to realize tool motion. Structural deformationsof the machine tool substructures undergoing largemotion results in tool point dynamics varying alongthe tool path. The changing structural dynamics inter-act with the cutting process and the feed drive control.These interactions may limit the machine tool per-formance due to position-varying cutting stability ofthe system and/or due to the position-varying con-trol–structure interactions further limiting the pos-itioning speed and accuracy. To avoid these issues inthe development of high performance machines, it hasbecome increasingly necessary to virtually evaluatethis position-varying performance to guide designimprovements before eventual physical prototyping.

Modeling the position-dependent dynamics oftennecessitates a flexible multibody modeling approach

in which all major machine elements are modeled withfinite elements (FE). Machine FE models are often onthe order of 1,000,000 degrees of freedom (DOF) ormore. This size makes position-dependent responseanalysis that is based on adaptive/re-meshing strate-gies for every discrete position rather cumbersome; seeFigure 1(a) for the sequence of modeling stepsinvolved. The adaptive/re-meshing strategies cantake up significant portion of total manpower andcomputational effort required for the design and ana-lysis of machine tools,1 limiting the analyses to one ortwo design concepts and at only a few discrete

Proc IMechE Part K:

J Multi-body Dynamics

2015, Vol. 229(3) 304–317

! IMechE 2014

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DOI: 10.1177/1464419314562264

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Department of Machine Tools and Automation, Fraunhofer Institute for

Machine Tools and Forming Technology IWU, Chemnitz, Germany

Corresponding author:

Mohit Law, Department of Machine Tools and Automation, Fraunhofer

Institute for Machine Tools and Forming Technology IWU,

Reichenhainer Str. 88, 09126 Chemnitz, Germany.

Email: [email protected]

at INDIAN INST TECH KANPUR on December 3, 2015pik.sagepub.comDownloaded from

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positions. For machine tool manufacturers to betterrespond to the changing manufacturing paradigms,there is need for more efficient models and methodsthat facilitate rapid exploration of design alternativeswhile allowing for modularity and reconfigurability inthe design process. To address this, several dedicatedresearch effort has been expended in recent years;2–8

with the dynamic substructuring approach fast emer-ging as an efficient alternative to facilitate modularityand evaluate position dependency.4–8

Dynamic substructuring facilitates evaluation oflarge and/or complex structures by dividing them

into several smaller substructures for which thebehavior is generally easier to determine.9,10 Assuch, these bottom-up substructuring approachesfacilitate modularity by synthesizing the position-invariant response of the main machine tool compo-nents to obtain the position-dependent tool centerpoint (TCP) response; see Figure 1(b) for the sequenceof modeling steps involved with this approach.

To facilitate such dynamic substructuring, which isakin to a generic component mode synthesis,11 thesubstructures are independently reduced before syn-thesis. Since the quality of the reduced model

Figure 1. Sequence of modeling steps required to obtain the position-dependent response: (a) traditional method; (b) dynamic

substructuring approach, and (c) frequency-based substructuring approach.

Law and Ihlenfeldt 305



influences the assembled response, several recentinvestigations have focused on developing improvedvariants of reduced models that are better able toapproximate the full model behavior.4,7,12

Furthermore, synthesizing these reduced models atthe contacting interfaces is nontrivial, especially sowhen the substructures have been modeled separately(modular design), resulting in different mesh reso-lutions at the contacting interfaces. Assembly ofmodels with nonconforming discretizations has beenshown possible by approximating compatibility bysets of algebraic constraint equations,7,8 or by approx-imating the displacement fields using shape functiondefinitions,13–15 or by interface reduction.16,17

Though the bottom-up dynamic substructuringapproach based on substructural synthesis of reducedmodels has proved effective, its dependence on thequality of the reduced models, numerical challengeswith approximating displacement compatibility fornonconfirming meshes and the need to solve thereduced synthesized equations of motion for everydifferent position, has largely limited its effective dif-fusion in practice. To overcome these issues, thispaper describes an alternate frequency-based dynamicsubstructuring approach to model position depend-ency. As opposed to the ‘‘time-domain’’ dynamic sub-structuring methods used earlier4–8,12 that rely onsolution to the reduced synthesized equations ofmotion described by their spatial mass, damping,and stiffness matrices, the frequency-based substruc-turing method describes each subsystem in terms of itsreceptances, i.e. frequency response functions (FRFs)of the uncoupled systems.18 To obtain the synthesizedposition-dependent TCP response, position-invariantcomponent level receptances are only required at thecoupling locations between the substructures as wellas at any point where the assembly response is to bepredicted.

This frequency-based substructuring method,otherwise also referred to as the receptance couplingsubstructure analysis (RCSA) method been success-fully used to obtain assembled TCP response by com-bining response of arbitrary tools to the machineresponse.19,20 Earlier use of RCSA methods werereported for the simple case of two substructures incontact at a single nodal location, e.g. tool and tool-holder connection and hence are not directly applic-able to modeling substructural contact in machinetools, in which substructures are in contact at multipleinterfaces and nodes simultaneously.

A multiple point RCSA procedure, as is requiredpresently, was successfully formulated by Schmitz andDuncan21 to predict the dynamics of assemblies withcoincident neutral axes that are simultaneously in con-tact at several locations. These formulations were sub-sequently extended by the authors22 to evaluate thedynamics of mobile machine tools that were connectedto different workpiece/base combinations at severallocations simultaneously, such as to investigate

dynamics under varying base/part/contact character-istics. This paper builds on these earlier formula-tions21,22 to model substructures that aresimultaneously in contact over multiple interfaceswith changing compatibility conditions associatedwith tool motion. An overview of the modeling stepsrequired to obtain the position-dependent TCP FRFswith the proposed approach is shown in Figure 1(c).

Proposed approach is demonstrated by modelingthe position-dependent dynamic behavior for a repre-sentative three axis milling machine. Since theapproach is based on approximating point-to-pointcompatibility between substructures, receptances atthe contacting interfaces are projected to a point tofacilitate a multiple point RCSA formulation. Thistreatment tolerates mesh-incompatibility issues, ifany, during synthesis—something that was nontrivialwith earlier methods. The proposed approach doesnot require model reduction of any kind and isshown to be computationally more efficient thanfull-order models and other dynamic substructuringmethods. The framework is first used to evaluate posi-tion-invariant substructural characteristics that aresubsequently synthesized to obtain position-depen-dent behavior. This is followed by discussions onmodeling considerations and the main conclusions.

Generalized frequency-basedsubstructuring for machine tools

A representative three axis milling machine is selectedto allow for easy comparison of its position-depen-dent behavior with other state-of-the-art resultsreported in the literature.7 Since position dependencyfor the machine under consideration is primarily dueto the motion of the spindle–spindle housing (sub-structure I) in contact with and moving over a verticalcolumn (substructure II), at first only these substruc-tures are modeled and combined—as shown schemat-ically in Figure 2.

The spindle assembly that includes the tool, tool-holder, and the spindle shaft are all modeled withTimoshenko beam elements. The spindle housingand column substructural components are modeledwith 10-noded solid tetrahedral elements with threetranslational DOFs at each node. FE models forthese substructures have been generated from theirrespective CAD models. Additional modeling detailsfor the substructures are available elsewhere. 7

Substructure I has four guide-blocks in contactwith the two guide-rails on the substructure II,which corresponds to four pairs of contacting inter-faces between the two substructures. Locations 2–5 inFigure 2 correspond to the guide-block interfaces onthe spindle-spindle housing. Location 2 correspondsto the guide-block interface surface on the top left(þXMT) of substructure I and location 3 correspondsto the top right (�XMT, not visible in Figure 2).Similarly, location 4 corresponds to the bottom left

306 Proc IMechE Part K: J Multi-body Dynamics 229(3)



(þXMT) and location 5 corresponds to the bottomright (�XMT, not visible in Figure 2). Locations 6–9lie on the column substructure and correspond to thesubset of the interface surfaces of the guide rail incontact with locations 2–5 at a particular position.Location 1 corresponds to the TCP location.

After independently obtaining the modal solutionto each substructure from the FE environment, themodal parameters corresponding to each of the inter-face surface DOFs are exported to the MATLAB�

environment for the subsequent frequency-based sub-structuring. Component level receptances at the inter-face surface (or its subset) that is in contact at aparticular instant are constructed with these modalparameters—as discussed in next section. Followingthis, receptances at the contacting interfaces are pro-jected to a point to facilitate a multiple point recep-tance coupling formulation. The point FRFs alongwith cross FRFs are subsequently synthesized toobtain the position-dependent TCP response usingthe multiple point RCSA approach, formulated in‘‘Multiple point receptance coupling substructuralanalysis’’ section.

Constructing receptances at interfaces

Each of the contacting interfaces has a total of nnodes in contact with the adjacent substructure. Thenumber of such interfaces nodes may or not be thesame for the different interface surfaces under consid-eration. Each of these interface nodes has three trans-lational DOFs, making the component receptancesfor any node in compact form to be

ui ¼ hijfj ð1Þ

where u represents the displacement in each of theprincipal x, y, z directions; f represents the force in

these principal directions; h represents the displace-ment-to-force receptance evaluated for each directionat a total of N number of points in the frequencyvector !; and i and j are the respective response andexcitation locations. hij is constructed using the massnormalized eigenvectors and eigenvalues output fromthe FE environment, as

hij ¼XNm

r

�ir�jr

�!2 þ im2�r!!nr þ !2nr

ð2Þ

where !n is the undamped eigenvalue; �i,j is the eigen-vector at the location of interest; ! is the frequencyvector; �r is the modal damping ratio for r modes ofinterest for a total of Nm modes; im is the imaginaryoperator.

Obtaining point receptances at interfacesby projection

Assuming that the interfaces in question are relativelysmall and stiff in comparison to the total substructurebeing considered, the interface behavior may beapproximated by a local rigid section.17 This isequivalent to rigidifying the interface surface or partof the local interface that is in contact by a set ofreceptances that represent the motion of the interface.This approximation is described by a projection of theoriginal boundary receptances on to a single point asshown schematically in the inset in Figure 2. Thisprojection is mathematically described as

h1h2...

hn

8>><>>:

9>>=>>;¼

T1

T2

..

.

Tn

2664

3775 Rq

� �ð3Þ

Figure 2. Schematics of the spindle–spindle housing substructure (substructure I) in contact with and moving over the column

substructure (substructure II) and projection of receptances (inset—detail ‘‘A’’).




where subscripts for the response and excitation loca-tions are dropped. n is the number of interfaces nodesfor the interface surface under consideration; h1...n

represents the generalized set of receptances asso-ciated with every node and in every direction evalu-ated at each frequency point; Rq corresponds to theprojected set of receptances that represent the rigidbody motions of the interface; T1...n corresponds tothe projection matrix per node for the total of nnodes per interface surface. This projection matrixfor each node that has three translational DOFs is17

Tp ¼

1 0 0 0 �dp,z dp,y

0 1 0 dp,z 0 �dp,x

0 0 1 �dp,y dp,x 0

264

375;

p ¼ 1 . . . n ð4Þ

dp within equation (4) is the position vector describingthe position of the nodes under consideration withrespect to the position of the reference node q ontowhich the boundary receptances are projected. Thiscondensation node q is placed at the center of theinterface surface whose motion it is meant to repre-sent. The position vector hence becomes

dp ¼

dp,xdp,ydp,z

8<:

9=; ¼

xpypzp

8<:

9=;�

xqyqzq

8<:

9=;; p ¼ 1 . . . n

ð5Þ

where xp; yp; zp� �

and xq; yq; zq� �

correspond tothe global coordinate locations of the nodes underconsideration.

Using the formulations in equations (1) to (5), theprojected point receptances at the interface (or itssubset) under consideration may be obtained as

Rq

� �8 2�9¼

T1

T2

..

.

Tn

2664

3775

�1 h1h2...

hn

8>><>>:

9>>=>>;

ð6Þ

where Rq comprises sets of three translational(Rqx , Rqy , and Rqz ) and three rotational (Rq� , Rq� ,and Rq� ) point receptances about the principalmachine tool axis. These point receptances are con-structed for each of the interface surfaces in contact,i.e. for locations 2–9 shown in Figure 2.

The formulations in equations (1) to (6) are usedto obtain the direct point receptances at locations rep-resenting the different interface surfaces, i.e. forlocations 2–9. However, since the multiple pointRCSA formulation also requires evaluation of crossreceptances between the TCP and the connectionlocations on substructure I, as well as in between dif-ferent connections locations for either of the substruc-tures—these necessarily need to be evaluated.

Obtaining cross receptances

Since cross receptances cannot be directly evaluatedby projection, they are instead indirectly evaluatedusing the projected direct receptances—as discussedbelow.

Cross receptances between TCP and connection locations.

The projected point receptances at the connectionlocations 2–5 (see Figure 2) for substructure I canbe re-written in their expanded modal form at eachof the interface location(s) as

Rqq ¼XNm

r

�qr�qr


; q ¼ 2 . . . 5

ð7Þ

where !n is the undamped eigenvalue that forms partof this receptance and �q is the eigenvector corres-ponding to the projected point receptance. Equation(7) holds for each of the three translational recep-tances as well as the rotational receptances obtainedfrom equation (6).

Assuming that the modes are well spaced in thefrequency range of interest, equation (8) can be eval-uated at each distinct eigenvalue !nr as

Rqq !¼!nrj ¼�qr�qr

im2�r!2nr

; q ¼ 2 . . . 5 ð8Þ

Rearranging equation (8) gives the eigenvector �qr

at the mode and location of interest as

�qr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRqq !¼!nr

�� im2�r!2nr

q; q ¼ 2 . . . 5 ð9Þ

This eigenvector can now be used to constructthe cross receptances between the TCP (location 1,Figure 2) and the interface locations (locations 2–5,Figure 2) as

R1,q ¼XNm

r

�1r�qr


; q ¼ 2 . . . 5

ð10Þ

where �1 is the mass normalized eigenvector at theTCP that is output from the FE environment. Sincethe tool is modeled with Timoshenko beam elementsthat have six DOFs per node, sets of translationalcross receptances as well as the three rotationalcross receptances can be obtained with equation (10).

Cross receptances between different connection locations.

The eigenvectors evaluated for the mode and locationof interest using equations (7) to (9) are used to con-struct the cross receptances between any of the twodifferent connection locations for either of the




substructures under consideration. For example, thecross receptances between connection location 2 andthe locations 3, 4, and 5 for substructure I areobtained as

R2,q ¼XNm

r

�2r�qr


; q ¼ 3 . . . 5

ð11Þ

where �qr for q ¼ 2 . . . 5 are obtained from equa-tion (9). Other cross receptances between the differentlocations on substructure I are similarly obtained.Construction of cross receptances for substructure IIwill require evaluation of the eigenvector �qr at themode and location of interest, i.e. for locations 6–9(see Figure 2). These are evaluated using the formu-lations in equations (7) to (9) for q ¼ 6 . . . 9.

Multiple point receptance couplingsubstructural analysis

Having formulated and approximated the componentlevel direct and cross receptances, the synthesizedposition-dependent TCP response is obtained byapproximating compatibility and equilibrium criteriaat the four pairs of contacting interfaces for a com-manded position.

The component level receptances for each of thetwo substructures under consideration may be definedin compact form as

q1 ¼ R11f1 þ R12f2 þ R13f3 þ R14f4 þ R15f5





ð12Þ

for substructure I, and

q6 ¼ R66f6 þ R67f7 þ R68f8 þ R69f9

q7 ¼ R77f7 þ R76f6 þ R78f8 þ R79f9

q8 ¼ R88f8 þ R86f6 þ R87f7 þ R89f9

q9 ¼ R99f9 þ R96f6 þ R97f7 þ R98f8

ð13Þ

for substructure II respectively. qi 8 9 and fi 8 9 in equa-tions (12) and (13) are the generalized displacementand force vectors at the contacting interface and TCPlocations respectively describing only the translationalbehavior in the machine’s principal directions. R

within equation (13) represents the generalized recep-tance matrix at the projected locations and describes

behavior only in the machine’s principal directionswithout any cross terms as

R ¼

Rqx 0 00 Rqy 00 0 Rqz

24

35 ð14Þ

The direct TCP receptances, i.e. R11 in equation(12) are obtained directly from the mass normalizedeigenvector at the TCP that is output from the FEenvironment using equation (2). The direct projectedFRFs at each of the interface locations in equations(12) and (13) are obtained using equations (1) to (6).The cross receptances between the TCP and the con-nection location(s) for substructure I, as well as inbetween different connection location(s) for either ofthe substructures I and II are obtained using equa-tions (7) to (11). Due to symmetry and Maxwell’sreciprocity, for all the receptances in equations (12)and (13), Rij ¼ Rji.

The four pairs of contacting interfaces (seeFigure 2) between the two substructures comprise:location 2 on substructure I in contact with location6 on substructure II (pair 1), location 3 on substruc-ture I in contact with location 7 on substructure II(pair 2), location 4 on substructure I in contact withlocation 8 on substructure II (pair 3), and location 5on substructure I in contact with location 9 on sub-structure II (pair 4). When in contact, the equilibriumconditions for a force F1 applied at location 1 in theassembled configuration are

F1 ¼ f1; f2 þ f6 ¼ 0; f3 þ f7 ¼ 0; f4 þ f8 ¼ 0;

and, f5 þ f9 ¼ 0 ð15Þ

The interface compatibility conditions for a flexiblecontact with a viscous damping model are

Kðq6 � q2Þ ¼ �f6; Kðq7 � q3Þ ¼ �f7;

Kðq8 � q4Þ ¼ �f8; and, Kðq9 � q5Þ ¼ �f9ð16Þ

wherein the complex stiffness matrix K for constantlevels of stiffness kx,y,z and damping cx,y,z is

K ¼

kx þ i!cx 0 00 ky þ i!cy 00 0 kz þ i!cz

24

35 ð17Þ

Although a single K matrix is employed in equa-tion (17), each coupling location and/or directioncould have a different stiffness and/or damping.Describing the damping using the modal dampingratio of the joint �j, the damping coefficient becomes:c ¼ 2�j

ffiffiffiffiffiffiffikmp

; where m, the mass of the joint is assumedto be unity.

Having satisfied the compatibility and equilibriumcriteria at the contacting interfaces for a given




position, the desired assembled TCP receptances G11

with brevity can be shown to be21,22

G11 ¼ R11 þ R12f2F1þ R13

f3F1þ R14

f4F1þ R15

f5F1

ð18Þ

wherein G11 has the same structure as R in equation(14). The direct projected receptances in equation (18)are obtained from equations (1) to (6), and the crossreceptances from equations (7) to (11), leaving onlythe following terms that are to be evaluated: f2=F1,f3=F1; f4=F1; and f5=F1. These terms are obtained byfirst substituting the component receptances of equa-tions (12) and (13) into the compatibility condition ofequation (16), which gives

K R66f6 þ R67f7 þ R68f8 þ R69f9 � R22f2�

�R21f1 � R23f3 � R24f4 � R25f5�¼ �f6

K R77f7 þ R76f6 þ R78f8 þ R79f9 � R33f3�

�R31f1 � R32f2 � R34f4 � R35f5�¼ �f7

K R88f8 þ R86f6 þ R87f7 þ R89f9 � R44f4�

�R41f1 � R42f2 � R43f3 � R45f5�¼ �f8

K R99f9 þ R96f6 þ R97f7 þ R98f8 � R55f5�

�R51f1 � R52f2 � R53f3 � R54f4�¼ �f9

ð19Þ

Eliminating f6, f7, f8, and f9 from the above by sub-stituting relations from the equilibrium conditions ofequation (15) into equation (19) and after grouping

the like terms, we get

R22 þR55 þK�1� �

f2 þ R23 þR67ð Þf3

þ R24 þR68ð Þf4 þ R25 þR69ð Þf5 ¼ �R21F1

R32 þR76ð Þf2 þ R33 þR77 þK�1� �

f3

þ R34 þR78ð Þf4 þ R35 þR79ð Þf5 ¼ �R31F1

R42 þR86ð Þf2 þ R43 þR87ð Þf3

þ R44 þR88 þK�1� �

f4 þ R45 þR89ð Þf5 ¼ �R41F1

R52 þR96ð Þf2 þ R53 þR97ð Þf3

þ R54 þR98ð Þf4 þ R55 þR99 þK�1� �

f5 ¼ �R51F1

ð20Þ

The terms in equation (18) are obtained by rearran-ging equation (20) in compact matrix form as

where A½ � is 12 � 3, or 3np� 3 �N matrix (N are thenumber of points in the frequency vector ! and npcorresponds to the number of connection pairs; pres-ently np ¼ 4). The matrix size is 12 � 3 because Rij is a3 � 3 matrix—representing only the translationalDOFs while neglecting the rotational DOFs, see equa-tion (14). The first three rows of A½ � give f2=F1; thenext three rows give f3=F1; the following set of threerows give f4=F1; and the final three rows give f5=F1. Bysubstituting these ratios into equation (18), the desiredreceptance matrix at the TCP in all machine tool prin-cipal directions can be computed for a commandedposition.

If substructures are simultaneously in contact atmore than four points, above formulations can beextended to np contacting pairs by observing therecursive pattern in equation (21). For a similarcoordinate numbering scheme as in Figure 2, the A

matrix in equation (21) can be shown to be21

R22 þ Rnpþ1, npþ1 þ K�1 R23 þ Rnpþ2, npþ3 � � � R2,npþ1 þ Rnpþ2, 2npþ1

R32 þ Rnpþ3, npþ2 R33 þ Rnpþ3, npþ3 þ K�1 � � � R3,npþ1 þ Rnpþ3, 2npþ1

..

. ... . .

. ...

Rnpþ1, 2 þ R2npþ1, npþ2 Rnpþ1, 3 þ R2npþ1, npþ3 . . . Rnpþ1, npþ1 þ R2npþ1, 2npþ1 þ K�1

2666664

3777775

�1

�R21

�R31

..

.

�Rnpþ1, 1

266664

377775¼ A½ �

ð22Þ

f2f3f4f5

8>><>>:

9>>=>>;

F1

F1

F1

F1

8>><>>:

9>>=>>;

�1

¼

R22 þR55 þK�1 R23 þR67 R24 þR68 R25 þR69

R32 þR76 R33 þR77 þK�1 R34 þR78 R35 þR79

R42 þR86 R43 þR87 R44 þR88 þK�1 R45 þR89

R52 þR96 R53 þR97 R54 þR98 R55 þR99 þK�1

2664

3775

�1�R21

�R31

�R41

�R51

2664

3775 ¼ A½ �

ð21Þ




Evaluation of position-dependent dynamics forsubstructures in contact at multiple points is possibleby substituting equation (22) into modified forms ofequation (18), as necessary. The single-stage substruc-turing approach formulated above can also beextended to a multiple-stage approach as necessary.A multiple-stage approach may be required to modelthe dynamics as influenced by several machine toolcomponents simultaneously undergoing a change inposition to realize complex tool motions. The generalidea, as demonstrated elsewhere8,23 would be to firstcombine two or more substructures to obtain theirsynthesized response, followed by combining thissynthesized response with additional substructuresas desired to obtain the global synthesized toolpoint response.

The above proposed frequency-based substructur-ing formulation describes a framework to efficientlyevaluate the position-dependent dynamics in machinetools. Furthermore, the framework makes possibleprediction of assembled response between any twolocations as desired, for example between the motorinput and table position, as may be required for posi-tion-dependent control structure interaction analyses.

Substructural response characteristics

Prior to obtaining the synthesized position-dependentTCP response, position-invariant component levelreceptances for both substructures are compared inthis section to better understand the substructuralcharacteristics and their interaction.

Direct component level receptances for all princi-pal directions at the free end of substructure I (cor-responding to the TCP) as well as at representativecoupling location 2 on substructure I are shown inFigure 3. Also shown in Figure 3 are the crossFRFs between the tool and the coupling location 2as well as between coupling locations 2 and 3. DirectTCP FRFs are obtained using equation (2). Directprojected FRFs at coupling location 2 are obtainedusing equations (1) to (6). Cross receptances betweenTCP and location 2 as well as between locations 2 and3 are obtained using equations (7) to (10) and equa-tion (11), respectively. All receptances are assumed tohave uniform damping of the level of �r ¼ 0:02 for allmodes. Similarly, direct and cross component levelreceptances for substructure II are shown in Figure4, which shows the receptances at the representativecoupling locations 6 and 7.

Receptances for substructure I (Figure 3) containrigid body modes due to this substructure being in anunsupported free–free configuration. Receptances forsubstructure II (Figure 4) are obtained by projectionof modal results obtained after having imposedboundary conditions on the Column as shown inFigure 2. Receptances are shown up to 10 kHz inthe log–log scale for easier interpretation of thehigh-frequency behavior.

The direct TCP FRFs at location 1 (Figure 3(a))show the higher frequency modes corresponding tothe local tool, tool-holder, and spindle shaft to beaxis-symmetric in the XY plane—because of the uni-form cross section of these components. The Z direc-tional response appears sufficiently stiffer than theX/Y response. From the projected receptances at loca-tion 2 (Figure 3(b)), it appears that different modesfor different directions become dominant at differentfrequencies. The frequency spectrum in this case isdominated by the lower frequency modes of the spin-dle housing that are observable at this location. Thecross receptances between the tool and location 2(Figure 3(c)) and between locations 2 and 3(Figure 3(d)) are at least an order of magnitudelesser than the direct responses.

Direct and cross receptances for substructure II(Figure 4(a) to (b)) are dominated by the globallower frequency modes of the column. The Z direc-tional response for substructure II also appears stifferthan the X/Y response. There appears some significantcross-talk between the locations 6 and 7 (Figure 4(b))and this is expected to influence the overall assembledtool point response. Overall, there are several modeson either of the substructures that will interact witheach other during synthesis.

Assembled position-dependent responseand its verification

Assembled TCP response comparisons are made inFigure 5 for the representative tool position shown inFigure 2. Comparisons are made with full-order modelanalyses conducted in ANSYS� and are shown for therepresentative Y direction only. Interfaces were idea-lized as being connected by linear spring elements forwhich the equivalent contact stiffness values areobtained from manufacturers’ catalogues. Joints atthese contacting interfaces were idealized as two trans-lational springs perpendicular to the direction ofmotion with no resistance, i.e. no spring in the direc-tion of motion. Equivalent contact stiffness for each ofthe four guide-block and guide-rail interfaces for bothdirections was assigned as 187N/mm. Joint dampingwas articulated in its FRF form, by assuming modaljoint damping ratio of �j ¼ 0:1.

Response comparisons in Figure 5 are made for theentire frequency range of interest, i.e. up to 1800Hz.The low-frequencymodes (up to 150Hz) correspond tothe global column bending and torsional modes. Themid-frequency modes (from 150 Hz up to 700Hz)correspond to the modes of the spindle housing. Thehigh-frequency modes (from 700 Hz up to 1800Hz)correspond to the local spindle, tool-holder, and toolmodes. As is evident from comparisons in Figure 5, thefrequency-based substructuring method quite reason-ably captures the full-order model response and it doesthis at a fraction of the computational costs associatedwith full-order models—as shown in comparisons




in Table 1. Response comparisons with other reduced‘‘time-domain’’ dynamic substructuring methods7

were found to overlap with full-order model resultsand are hence not shown for brevity.

The proposed method leads to considerable simu-lation time savings; taking �5 s/ position as comparedto �6 h/ position for the full-order model, therebyfacilitating further position-dependent analysis.

Savings in computational effort is mainly due tothe substructures being represented by only a fewsets of position-invariant receptances, which aresynthesized to obtain the position-dependentresponse. Even though the frequency-based approachrelies on modal properties output from the FE envir-onment, modal analysis for each of the substructures

under consideration is carried out offline, prior to syn-thesis. Computational cost of this modal analysis iscomparable to the computational cost of meshing andoutputting the structural matrices required for other‘‘time-domain’’ dynamic substructuring methods.However, since the proposed approach does notinvolve any model reduction, it is computationallymore efficient and elegant than other reduced model‘‘time-domain’’ dynamic substructuring methods.

Evaluation of position-dependentdynamics

Following the sequence of steps outlined inFigure 1(c) and solving equation (18) for every

(a)

(b)

(c)

(d)

Figure 3. Component level receptances for substructure I in all three principal machine tool directions: (a) direct receptances at

TCP location, location 1; (b) projected direct receptances at location 2; (c) cross receptances between locations 1 and 2; (d) cross

receptances between locations 2 and 3.

FRFs: frequency response functions.




discrete position results in the position-dependentresponse. Position-dependent TCP FRFs are evalu-ated at three different representative positions of thetool within the work volume: a top position that cor-responds to the configuration shown in Figure 2 andthe mid and bottom positions corresponding to toolmovement in the Z direction by 0.2m and 0.4m,respectively.

Each time the tool moves to a new position in themachine work volume, set by �Z (Figure 2), a new setof nodes on the interfaces of both substructures comeinto contact. Response of these new set of nodes oneither of the interface surfaces are re-approximated byprojecting their receptances to a single location,using equations (1) to (6). The cross responses arealso re-approximated using equations (7) to (11).

(a)

(b)

Figure 4. Component level receptances for substructure II in all three principal machine tool directions: (a) projected direct

receptances at location 6; (b) cross receptances between locations 6 and 7.

FRFs: frequency response functions.

Figure 5. Comparison of the assembled TCP FRFs between full-order models, and the proposed frequency-based substructuring

method in the Y direction for flexible contact at the interfaces.

TCP; tool center point; FRF: frequency response function.




Compatibility and equilibrium conditions arere-approximated to synthesize these receptancesusing the multiple point RCSA formulation. Theresultant position-dependent TCP FRFs are shownin Figure 6, which compares changes in the represen-tative Y directional response only.

As substructure I travels over substructure II, itresults in locally changing boundary conditions.These changes result in the global structural modescorresponding to the column and the spindle housing(up to 300Hz) exhibiting stronger position depend-ency compared to the higher frequency local spindle,

Figure 6. Comparison of the Y directional position-dependent TCP response at three different tool positions: top, mid, and bottom.

TCP; tool center point; FRFs: frequency response functions.

Table 1. Comparison of model size and computational effort required for full-order model, reduced ‘‘time-domain’’ dynamic

substructuring method, and the proposed frequency-based substructuring method.

Full-order

model

Reduced ‘‘time-domain’’

dynamic substructuring method7Proposed frequency-based

substructuring method

Model size �125,000 DOFs �1500 DOFs 25 sets of receptances

Computational time

(Intel� i7 2.79 GHz

processor 8 GB RAM)

�6 h/position �10 s/position (plus time required

for model reduction, single run off-line: �6 h)

�5 s/position

DOFs: degrees of freedom.




tool-holder, and tool modes. The column bendingmode at �74Hz varies by up to �7% in natural fre-quencies over the full Z stroke of the machine,whereas the dynamic stiffness changes by �40%between the top and mid positions, disappearing com-pletely for when the tool is at the bottom position.The other dominant low-frequency spindle housingmode at �185Hz varies more in dynamic stiffnessthan natural frequency, changing by �15% indynamic stiffness.

Evaluation of these position-dependent dynamicsis automated and can be coupled to input trajectoryof the tool path such that the position-dependent TCPresponse may be evaluated at discrete time and/orposition intervals as desired.

Discussions and modeling considerations

The accuracy of the proposed frequency-based sub-structuring approach was observed to be a strongfunction of the convergence quality of the substruc-tural models. Guaranteeing convergence is oftenaccompanied by an increase in the order of the sub-structural models. An increase in model order haslittle bearing on the proposed frequency-basedapproach, since even with models with largernumber of DOFs, the frequency-based approach stillrequires only a few set of receptances.

Additionally, ability of the projected point recep-tances to approximate motion of the interface surfaceit represents also influences the quality of theassembled model. For a given interface surface area,the quality of projected receptances were observed tobe less dependent on the mesh density at the interfacesurface, than on the size of the interface surface theyapproximate, with larger surfaces resulting in poorerapproximations by the projected receptances.Projected receptances may result in numerical rigidi-fication of the interface surface they are to represent.This rigidification may result in the point receptancesnot being able to completely describe interface surfacemotion undergoing large deformations.

Furthermore, results were also observed to besensitive to modal truncations. To avoid truncationerrors without using residuals,9 the modes weretruncated well beyond the highest frequency ofinterest.

Conclusions and outlook

An efficient, elegant, and generalized methodology ispresented for predicting the position-dependentdynamics in machine tools based on a frequency-based substructuring approach. The methodologywas demonstrated by modeling the position-depen-dent dynamic behavior for a representative threeaxis vertical milling machine. Each of the mainmachine tool components was described by its posi-tion-invariant receptances at its contacting interfaces.

These receptances were projected to a point to facili-tate a multiple point receptance coupling procedure.Position-dependent tool point dynamics wereobtained by combining the substructural position-invariant point receptances with others byapproximating compatibility and equilibrium at thecontacting interfaces between substructures for agiven position. This treatment facilitates modularityin the design process by making possible couplingof substructures with nonuniform mesh distri-butions at their contacting interfaces, somethingthat is nontrivial in commercial FE codes or withother state-of-the-art dynamic substructuringformulations.

Since complete machine behavior was representedby just a few sets of receptances, the model was shownto be computationally more efficient than full-ordermodels and other state-of-the-art dynamic substruc-turing methods. Rapid evaluation of the machine’schanging dynamics and its interaction with the cuttingprocess can characterize the machine’s position-vary-ing cutting performance early in the design stage suchas to guide design improvements.

These computationally more efficient modelsalso aid the development of mechatronic hardware-in-the-loop simulations models that play an importantrole in the design loop of machine tools. These simu-lation models in which the real CNC system isconnected to virtual machine models requires time-deterministic and highly efficient computation of themachine models (states)—facilitated by the methodspresented herein. Moreover, since the time-deterministic model predicted behavior has high-fide-lity, limitations associated with the highly abstractand simplified machine models used elsewhere areovercome. This allows for development of controlprograms and their realistic testing on the real hard-ware without having to use the real machine.Furthermore, since methods presented can efficientlyaccount for the time-and-position-variant nature ofthe machine, advanced robust control strategies forparameter-varying plants can be developed andtested in the virtual environment. These simulationmodels can further facilitate the optimization of con-trol programs and machine cycle times by the use ofin-process adjustment of the tool path based on modelpredicted dynamics.

Relative ease of modeling flexibilities at the con-tacting interfaces makes investigations under varyingcontact characteristics easier than what is possible incommercial FE codes. The ease of introducing andmodeling joint characteristics at the interfaces alsomakes it possible to deploy such methods for jointparameter identification and model updating inmachine tools.

Conflict of interest

None declared.




Funding

This research was supported by the Fraunhofer

Gesellschaft’s ICON Project for Strategic ResearchCo-Operation on Sustainable Energy Technologies.

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Appendix

Notation

A compact matrix form descriptionrequired to compute assembled systemresponse

cx,y,z damping coefficients in the each of theprincipal directions

dp position vector for the node underconsideration

f forcef generalized force vectorF generalized force vector applied in

assembled configurationG represents the assembled structure’s

frequency response function matrixh represents the displacement-to-force

receptancei response locationim imaginary operatorj excitation locationkx,y,z translational spring stiffness in the each

of the principal directionsK complex stiffness matrixm mass of the connection elementn number of nodes in contactnp number of connection pairsN number of points in the frequency

vector, !Nm total number of modes of interestp node counterq node onto which the boundary recep-

tances are projectedq generalized displacement/rotational

vectors at location of interestr counter for number of modesR projected point receptance matrix




Rq generalized set of projected pointreceptances that describes both transla-tional (Rqx , Rqy , and Rqz ) and rota-tional component behavior (Rq� , Rq� ,and Rq� )

T projection matrixux,y,z displacements at point and direction of

interestx, y, z indicate parameters pertaining to the x,

y, and z directions, respectivelyxp, yp, zp global coordinate locations of the node

under considerationxq, yq, zq global coordinate locations of the node

where the boundary receptances areprojected

XMT principal machine direction�Z instantaneous position of substructure

I relative to substructure II

! frequency vector (rad/s)!nr undamped eigenvalue corresponding to

mode r (rad/s)�i,j eigenvector between locations of

interest�q eigenvector corresponding to the pro-

jected point receptancePsum over

�j damping ratio at the joint�r modal damping ratio for mode r8 for each




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