Original Research Article

Static condensation in modeling roller guides withpreload

Paweł Dunaj a,*, Stefan Berczyński a, Piotr Pawełko a, Zenon Grządziel b,Marcin Chodźko a

a Institute of Mechanical Technology, West Pomeranian University of Technology, Szczecin, PolandbMaritime University of Szczecin, Poland

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a r t i c l e i n f o

Article history:

Received 24 July 2018

Accepted 8 June 2019

Available online 20 June 2019

Keywords:

Roller guides

Guideway system

Machine tool

FEM

Static reduction

a b s t r a c t

This article presents the methodology for modeling the stiffness of a machine tool load-

bearing system using the finite element method. A new, simplified model of stiffness of the

linear guide with a preload based on equivalent contact model was proposed. An equivalent

contract model was developed where the ball was replaced with four rod elements of

equivalent stiffness, interconnecting the opposite ends of the face of cuboidal finite ele-

ments. To shorten the computation time and facilitate modeling reduction and substructur-

ing methods was used. The accuracy of the proposed model was compared with

experimental results. In addition, the computation times were evaluated by comparing

the simplified model with the full model and other equivalent models. Then, the model was

used to determine the stiffness of the machine load-bearing system on the example of a

machining table using the aforementioned methods and the obtained results were com-

pared in terms of accuracy (less than 1% difference in maximum displacement value) and

computation time with the classic approach (up to 97% in time reduction). This paper

demonstrated the validity of the proposed model, allowing accurate and fast determination

of the stiffness of machine tool load-bearing system.

© 2019 Politechnika Wrocławska. Published by Elsevier B.V. All rights reserved.

Available online at www.sciencedirect.com

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journal homepage: http://www.elsevier.com/locate/acme

1. Introduction

Machine tools are technically complex devices, where devel-opments in technology have driven their development. Onekey determined during the design of a machine tool is thecomplex feature of stiffness, most notably linear guidedeformation which can cause a serious reduction in machine

* Corresponding author.E-mail addresses: pawel.dunaj@zut.edu.pl (P. Dunaj), stefan.berczy

(P. Pawełko), z.grzadziel@am.szczecin.pl (Z. Grządziel), marcin.chodzkhttps://doi.org/10.1016/j.acme.2019.06.0051644-9665/© 2019 Politechnika Wrocławska. Published by Elsevier B.V

precision and efficiency. Therefore, a properly designed load-bearing system is a very important element in the machinetool structure.

Over the years many comprehensive techniques have beendeveloped to model the stiffness of sliding guides in machinetool bearing systems [1,2]. Along with developments intechnology, sliding guides have largely been replaced by rollerguides, which are characterized by a lower resistance to

nski@zut.edu.pl (S. Berczyński), piotr.pawelko@zut.edu.plo@zut.edu.pl (M. Chodźko).

. All rights reserved.

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movement, extended life and practical elimination of stick–slip vibrations. As a result it has become necessary to developnew research methods corresponding to these modernconstruction solutions [3,4]. Despite considerable achieve-ments in contact mechanics, along with progress in researchin the field of tribology and dissemination and development ofthe finite element method, the modeling methods currentlybeing used are far from satisfactory.

Kong et al. [5] established a static model of a linear guide toestimate its stiffness under various external loads. Theauthors presumed that the contact points of the ball bearingswere elastic when analyzing interactions between the ballsand the raceways, and that the carriage and rail wereconsidered rigid when calculating carriage displacementrelative to the rail. Owing to these assumptions, linear guidepreload was modeled as an effect of the initial oversize of theballs, calculated according to Hertzian contact theory [6]. As aresult, those authors obtained a non-linear function ofdisplacement describing the vertical stiffness of a linear guide.

Tao et al. [7] established a stiffness model of a roller linearguide with preload based on the Palmgren equation [8].Assuming that the contact deformation occurring between theroller and raceway could be approximated with a model ofdeformation between an elastic cylinder and an infinitely rigidplate. The correctness of that approach has been verified byexperimental data, concluding that the proposed modelaccurately described the stiffness of a linear roller guide.However, it should be highlighted that there were slightdifferences in torsional stiffness related to the loss of contactbetween the carriage and rollers as a result of the appliedlateral torque.

While the mentioned approach of neglecting the elasticityof the carriage and rail is a common practice. Ohta and Tanaka[9] clearly indicated that incorporating elasticity provides abetter model compliance (by up to 30%). They presentedtheoretical expressions and an iterative method to computelinear guide vertical stiffness, taking into account, i.a.,variation of contact angle, vertical displacement and preload.Construction of the model based on that methodologyconsisted of replacing the balls with concentrated forcesrepresenting the reaction forces. However, the authorsindicated that such an approach caused an overestimationof displacements and ultimately reaction forces were modeledas uniformly distributed pressure acting on the raceways ofthe linear guide. This pressure acts over the entire length of thecarriage and a width equal to the length of the small axis of thecontact ellipse. Additionally, the authors indicated that it isnecessary to fix the upper surface of the carriage in order toeliminate the angular deformations resulting from calculationerrors.

Zou and Wang [10] simplified the Ohta and Tanaka model[9] by ignoring carriage outward deformations and raildeformations. The authors presented two contact modelstaking into account preload effect, frictional heat and thermaldeformation. In summary, they argued that stiffness variationis directly related to initial contact deformation of the rollerball, which depends mainly on thermal and wear effects.

Regardless, the models described above are hardly appli-cable in commercial FEM software packages due to thecomplexity of the functions or iterative methods required to

calculate contact stiffness. To overcome this problem, Lin et al.[11] introduced a stiffness model of a linear guide with preload,based on a finite element method. The authors modeled therail and carriage as solid elements connected by a series ofspring elements, deliberately ignoring the effect of the rollingballs. Stiffness of the springs was determined on the basis ofHertzian contact theory. The authors made a simplificationconsisting of reducing the number of rolling raceways fromfour to two, thus avoiding the complexity in concurrent meshgeneration. As a result, the model obtained good compatibilitywith experimental data.

Pawełko et al. [12] presented a methodological basis ofmodeling the static properties of linear roller guides withpreload in the FEM convention. The authors discretized thegeometrical model using solid elements and modeled thecontact between the ball and raceways as Hertzian contact.Appropriate preload was obtained through an increase in theball size by applying a thermal load to the ball. However, theauthors ascertained that solving such a model is very timeconsuming and memory-intensive, mainly because of thecomplexity of issues associated with the contact (the use of asolver of non-linear statics issues). Therefore, the authorsmade a simplification, replacing the ball with one two-nodelinear ROD element and replacing the model of contactbetween the ball and raceway with a RIGID element lying inthe contact ellipsoid. The authors concluded that the simpli-fied model gave satisfactory compliance of carriage displace-ment with regard to manufacturer data, while shortening thecomputation time.

However, even a simplified FEM model can still havethousands of degrees of freedom and solving it is timeconsuming. It should also be noted that typical machine toolscontain several linear guides, and so the use of complexmodels with a large number of degrees of freedom is notrecommended and ineffective. Furthermore, the stiffnesscharacteristics of the structure vary, depending on changesin machine tool configuration, making it is practicallyimpossible to build a finite element model defining thebehavior of each configuration. The only feasible choice ishence substructuring.

Substructuring is based on the idea that complex structurescan be decomposed into a number of components known assubstructures, which can be analyzed separately. A model ofthe whole structure is obtained by recoupling the resultingelements. Analyzed substructures undergo model reduction,resulting in a reduction in the number of degrees of freedom(DOF) to shorten the computation time.

Substructuring and reducing the number of degrees offreedom have found wide application in machine tooldynamics modeling [13–19]. Garitaonandia et al. [20] presenta robust procedure to develop comprehensive and reliabledynamic models of machine tools, establishing finite elementmodels for each machine tool component and verifyingagainst experimental data, defining substructures and reduc-ing them using the Craig–Bampton method [21]. In the finalstage, the assembly process was conducted to obtain a modelof the entire machine tool, taking into account the flexibleconnections between substructures, where relative motionwas based on sliding guides modeled with the use of linearspring elements.

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The purpose of this present work is to present a simplifiedmodel of stiffness of a linear roller guide for machine toolsbased on substructuring, and presents a new model of contactbetween the ball and raceways, which is equivalent to thesurface-to-surface contact model. The developed modelprovides a significant reduction in computation time (inrelation to the classical approach) and substructuring thanksto the elimination of non-linear elements. Presentation of themodeling methodology is based on a Mannesmann-Rexroth1651-25 guide [22]. In the first stage, a comprehensive model ofthe linear guide was built, taking into account phenomenarelated to the contact between the rolling elements andraceways and preload, obtaining a satisfactory compatibility ofthe model with the real equivalent. Then a simplified modelwas built and both models compared in terms of accuracy ofthe obtained results and computation time. A slight loss inaccuracy was accompanied by a significant shortening of thecomputation time.

The article also presents an example of the application ofthe proposed model, determining the stiffness of the machinetable load-bearing system using substructuring. And theobtained results compared with the conventional analysis.

2. A comprehensive model of a linear guide

Linear guides are the structural equivalents of linear kinematicpairs whose task is to maintain the relative displacement ofindividual machine body elements in a specific direction.These carriage systems are able to absorb the loads caused bycutting forces and gravity. The stiffness of the linear guide is akey factor for the stiffness of the machine tool, which is why itis necessary to define this in the design of the load-bearingsystem.

Using the finite element method (MSC Nastran/Patran) [23],a complex model of stiffness was built for a Mannesmann-Rexroth 1651-25 linear roller guide, implicitly taking intoaccount phenomena such as the change in contact anglebetween the balls and raceways, changes in stiffness depend-ing on the direction of the applied load as well as preload. Theguide consisted of a carriage, a rail and four rows of balls with acontact angle of 458. In each row there were 13 balls in theactive contact area. The construction of the guide and itsstiffness are presented in Figs. 1 and 2, respectively.

Fig. 1 – Structure of

The first stage of building the linear guide stiffness modelwas construction of the geometric model. Due to the fact thatthe manufacturer did not provide full technical documenta-tion, it was necessary to carry out additional geometricmeasurements, especially within the contact surfaces (ballradius, raceway radius).

Then, the geometric model underwent simplifications,consisting of omitting elements that were not significant fromthe point of view of the analyzed phenomenon, such as smallroundings, phases, mounting holes and inclinations.

In the next stage, the model was discretized with the use ofsolid elements (CHEXA type, six-sided isoperimetric elementswith 8 nodes), material data was defined for individualelements (Young's modulus, Poisson's number) and thesurface-to-surface contact between the balls and racewayswas modeled corresponding to Hertzian contact model. Themodel building process is presented in Fig. 3.

An extremely important issue affecting the accuracy ofcomputations in a contact model is selection of the appropri-ate mesh density [24,25]. Bearing in mind both the aforemen-tioned accuracy of computations and the time needed for theirimplementation, it was decided to perform local meshrefinement in the vicinity of the contact areas. Preliminarytests for various finite element mesh densities were per-formed, and a density was chosen that made it possible toobtain best agreement of the actual stress distribution withHertzian theory (the maximum stress in the contact area in theconstructed model was 3620 MPa, compared to the theoreticalvalue of 3660 MPa). The detailed procedure including meshdependency was presented in the paper [12].

The next step was to determine the stiffness characteristicsof the guide, achieved by means of a two-stage non-linearanalysis. In the first stage, the preload was modeled (0.08 ofdynamic load-bearing capacity, C = 22,800 N) [22] by applying a77.8 8C heat load to the balls, resulting in an increase in sizeand thus the creation of stresses with the assumed value.

The results of the calculations from this stage served asboundary conditions imposed in the second step, in which thedeformation of the carriage was determined for an incremen-tally rising load applied to the upper surface of the carriage inten steps from zero to the dynamic load-bearing capacity of22,800 N, while lower surface of the rail was fixed.

As a result of this analysis (MSC Nastran Nonlinear Static –

SOL 106), the stiffness characteristics of the carriage were

the linear guide.

Fig. 2 – Stiffness characteristics of the linear guide [20].

Fig. 3 – Preparation and discretization of the geometric model.

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obtained and compared to the data provided by the manufac-turer, determining that the model was a good representationof its real counterpart. A comparison of the characteristics isshown in Fig. 4.

3. Building a simplified model

Despite the good compatibility of the model presented inSection 2, the long computation time (about 8 h) andnecessarily large processing power was less than desirable,so simplifications which would balance the accuracy of themodel and shorten the computation time, and tools thatwould enable an effective use of the model as an element of a

complex structure. Therefore, it seems justified to usesubstructuring and static reduction, which has a certainlimitation in that all elements of the model within the reducedcomponent must be characterized by a linear model.

Bearing in mind the limitations of the method and thepurpose of the model (determination of stiffness in techno-logical machines), the new model of linear guide stiffness hadthe following assumptions: the linear guide is preloaded,displacements in the system are small, loads acting on thesystem do not cause plastic deformation, there is no kinematicinput (connected with inaccurate mounting of the guide), andthere are no errors in the system related to the geometry of theballs (the balls are perfectly spherical and have the samediameter) [26].

Fig. 4 – Comparison of the stiffness characteristics of the model with the real object.

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This idealization was based on the range of loads duringmachine tool operation. It can be seen that as the load isincreased uniformly, the increasing rate of deformationdeclines. Therefore, above the inflection point of the load-deformation curve, the stiffness characteristic is almost linear.

It should also be noted that an increase in preload isaccompanied by a decrease in the argument of the stiffnessfunction (load value) for which the inflection point occurs[22,27], thereby increasing the range in which stiffnesscharacteristic can be described by a linear model. Intechnological machines, linear guides are usually selected insuch a way that they work in the range of working loads abovethe inflection point of the stiffness characteristic.

The non-linearity of the stiffness characteristic of themodel is caused by the model of contact between the racewaysand the ball. This model can be considered as the contactbetween a deformable ball and cylinder, which is schemati-cally shown in Fig. 5. This issue is described by Hertzianequations, allowing determination of the state of deformationand the state of stress in the areas of contact between twobodies. The contact surface area changes depending on thevalue of the pressing forces, which results in a change in thevalues and distributions of stresses as well as deformationsand displacements in the contact area.

After the deformation of the bodies caused by their mutualpressure (Fig. 5), in a general case the contact area formed is anellipse with semi-axes a and b (a > b), which can be determinedfrom the equations:

a ¼ a3

ffiffiffiffiffiffiffiffiPmn

r; b ¼ b3

ffiffiffiffiffiffiffiffiPmn

r(1)

m ¼ 4ð1=r1Þ þ ð1=r01Þ þ ð1=r2Þ þ ð1=r02Þ

;

n ¼ 83

E1E2

E2ð1�v1Þ þ E1ð1�v2Þ(2)

where a and b – coefficients depending on the value of the ratioB/A where:

A ¼ 2m

;

B ¼ 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1r1

� 1r01

� �2

þ 1r2

� 1r02

� �2

þ 21r1

� 1r01

� �1r2

� 1r02

� �cos2’

s (3)

Based on the above formulas, assuming material dataanalogous to the model developed in Section 2(E1 = E2 = 210,000 N/mm2, v1 ¼ v2 ¼ 0:3, r1 ¼ r01 ¼ 2:375 mm,r2 ¼ 2:5 mm; r02 ¼ 1) and the load P = 10 kN, the dimensionsof the contact ellipse were: a = 0.524 mm, and b = 0.183 mm.

This theoretical contact ellipse was used to eliminate thenon-linear surface-to-surface contact model. A substitutelinear model was developed where the ball was replaced withfour rod elements of equivalent stiffness, stretched betweenthe opposite ends of the face of cuboidal finite elements (thedimensions of the face correspond to the rectangle describedin the contact ellipse). A schematic representation of thereplacement of the contact model is shown in Fig. 6.

For such a developed model, the stiffness characteristicwas determined in a manner analogous to that shown inSection 3. The results are shown in Fig. 7.

In addition, Fig. 8 presents a comparison of displacements ofthe carriage with respect to the rail under load, and forcomparison, results for the methods constructed in accordance

Fig. 5 – Geometry of contacting bodies and the contact ellipse.

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with [12] and [9], taking into account only vertical stiffness for asimilar number of degrees of freedom. Table 1 shows thecomputation times.

It can be concluded that the proposed model shows goodcompatibility with its real counterpart. Replacing the ball withrod elements and the surface-to-surface contact model withan equivalent contact model, resulted in a significant reduc-tion in computation time, with only a slight decrease inaccuracy. In addition, it should be noted that the developedmodel does not include elements with non-linear stiffnesscharacteristic, and unlike other models it does not includemultipoint constraints. Therefore, it is possible to reduce thedegrees of freedom using a Guyan reduction [28] andimplementation in the substructuring method.

4. Simplified model application insubstructuring method

This section presents an example of the developed model indetermining the stiffness of a machine table load-bearingsystem. As mentioned before, machine tools are complexmechanical systems consisting of many elements intercon-nected with a variety of connections forming kinematicconstraints. Due to the complexity of the model, determiningthe stiffness of the machine using the classic finite elementmethod can be very time consuming and hence inefficient.The computation time can be significantly shortened bysubstructuring.

Substructuring is based on the idea of dividing a complexmodel built in the convention of the finite element method(FEM) into smaller component-based models known assubstructures. This approach, apart from significantly short-ening the computation time, has a number of other advan-tages, enabling analysis of individual elements to identify localparameters describing their properties, changes to the model

of a single component without changing the global model,creating a library of the most frequently used components, andthus using the same model of an element occurring severaltimes in the structure (e.g. linear guides). In the case ofstructures whose elements are made by various manufac-turers, it is possible to implement models of individualcomponents provided by them, without the need to build aseparate model, which allows a significant reduction ofpreparation time of the entire structure.

The models presented in the article were established usingthe finite element method (MSC Nastran/Patran) [23]. Theessence of this method is to replace the continuous model ofthe mechanical system described by means of partialdifferential equations, by a discrete model comprising asystem of algebraic equations:

Ku ¼ P (4)

where K – stiffness matrix, u – primary variable and P – appliedload.

Models presented in the form of matrix equations can besubjected to a reduction process. One of the first and also themost popular methods of reduction is static condensation,also known as static reduction or Guyan reduction [28]. Thismethod consists of reducing the degrees of freedom of the FEMmodel to the degrees of freedom located on the boundarynodes, by reduction of the internal degrees of freedom definedas being outside the superelement boundary, while boundaryDOF's remaining after reduction retain the stiffness of themodel. After truncation of the degrees of freedom, on whichboundary conditions and multipoint constraints were im-posed, the following division can be made:

Koo Koa

Kao Kaa

� �uo

ua

� �¼ Po

Pa

� �(5)

Fig. 6 – An equivalent model of contact.

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where uo – internal degrees of freedom, ua – boundary degreesof freedom, Kaa; Pa – sub-matrices representing the associatedmatrices before reduction operation.

Left-multiplying the first row of the matrix Eq. (5) by K�1oo can

be presented as:

K�1oo ½Koouo þ Koaua� ¼ K�1

oo Po (6)

On the basis of Eq. (5), it is possible to define the concept oftransformation matrix between the displacements of theinternal degrees of freedom:

Goa ¼ �K�1oo Koa (7)

The physical interpretation of the matrix is as follows: thecolumns represent the displacements of the internal degreesof freedom caused by applying a load that causes individualdeflection successively to one of the external degrees of

freedom, while the rest of the boundary degrees of freedomremain constrained. In summary, the number of columns inthe matrix corresponds to the number of external degrees offreedom, while the number of rows equals the number ofinternal degrees of freedom.

Eq. (6) can be reduced to the form

utotalo ¼ ufree

o þ ufixedo (8)

where the displacement of internal degrees of freedom isexpressed in the form of a sum of vectors ufree

o representingthe displacement of internal degrees of freedom directly de-pendent on the displacement of external degrees of freedom,defined as follows:

ufreeo ¼ Goaua (9)

and a vector ufixedo representing a static solution for the consid-

ered substructure, when the loads are applied and external

Fig. 7 – Comparison of stiffness characteristics.

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degrees of freedom are fully constrained. The vector ufixedo takes

the form:

ufixedo ¼ uo

o ¼ K�1oo Po (10)

Writing the second row of the matrix Eq. (5) in the form:

KToaGoaua þ uo

o þ Kaaua ¼ Pa (11)

Fig. 8 – Comparison of displacements of the

one can determine a reduced stiffness matrix Kaa and a vectorof reduced forces Pa for the substructure, assuming the follow-ing form:

Kaa ¼ KToaGoa þ Kaa (12)

Paa ¼ GToaPo þ Pa (13)

carriage relative to the rail under load.

Table 1 – Comparison of computation times of individualmodels, in relation to the full model.

Model Time

Full 100%Simplified model 0.05%Pawełko et al. model 7%Ohta and Tanaka model 0.05%

Fig. 9 – The division of degrees of freedom with the presentatioconsidered system.

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The matrices describing the synthesized model can bedetermined using the following relationship:

Kgg ¼ Kjj þX

Kaa (14)

where the matrix Kjj represents the stiffness of the residualstructure, defined as consisting of elements not assigned to

n of subsequent steps of the substructuring process for the

Fig. 10 – Presentation of calculation results for individual models.

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any component as internal degrees of freedom. The globalmodel response is obtained by using synthesis matrices andinverse transformation.

Fig. 9 schematically shows the system for which anexemplary analysis was carried out. It consisted of linearguides and a loaded table. In the first modeling stage it wasdivided into substructures representing the main compo-nents, taking into account the actual connections occurringin the structure. As a result of this division, elementsrepresenting the table and four carriages were obtained.Due to the fact that the models representing the carriages didnot differ from each other, it was decided to use one modelfour times (aggregation of the stiffness matrix according toEq. (14)), which saved time needed to build other models.Next, the degrees of freedom of the presented substructureswere reduced in accordance with the previously presentedprocedure, thus reducing the dimensionality of the guide

Fig. 11 – Comparison of calculation

system model from 156,600 to 48 degrees of freedom (9999%),and from 18,024 to 96 degrees of freedom (9946%) for thetable. The next step was synthesis of the substructures toobtain the global model response. The results are presentedin Fig. 10.

As can be seen in Fig. 10, displacement areas obtained(using MSC Nastran – Linear Static SOL 101) on the basis ofindividual models differ slightly (less than 1% difference inmaximum displacement value). The difference can be noticedwhen using the Ohta and Tanaka model due to the fact thatthis model only takes into account vertical stiffness. The use ofthe substructuring method to calculate the same models doesnot affect the accuracy of the results in any way. It does,however, significantly shorten the calculation time. A com-parison of calculation times of the models built based on theconventional finite element method and substructuring isshown in Fig. 11.

times for individual models.

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5. Conclusion

To sum up, the stiffness model of the linear guide presented inthis article gives an accurate reproduction of vertical andhorizontal stiffness. Thanks to the applied simplifications, itbecomes possible to quickly and efficiently determine thestiffness of the load-bearing system of the machine tool.

Thanks to the developed contact model it is possible to usesubstructuring, which greatly simplifies work on the design ofthe load-bearing system. Thanks to the substructuringmethod, it is easy to determine the stiffness of the machinetool load-bearing system when using different types ofcarriages, by replacing just the model of the respectivesubstructure. This saves time associated with the need torebuild the entire model in the event of a need to change thelinear guide. In addition, it should be noted that when solvingstatic problems, the solution obtained by the classical finiteelement method does not differ from the solution using thesubstructuring method, which is related to the fact that Guyanreduction method is exact.

Despite the fact that this article presents calculations on amilling machine guide system, the proposed model can beused to determine the stiffness of the entire machine tool byusing the substructuring method. The increasing popularity ofsubstructuring seems to be unavoidable due to the prevailingtrend in modular machine tool construction, and the need todetermine the so-called spatial stiffness of machine toolsdepending on the mutual position of their structural parts.

Acknowledgement

The research work reported here was made possible by EU grant:‘‘Light construction vertical lathe’’ POIR.04.01.02-00-0078/16.

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