art%3a10.1007%2fs001700300050

Int J Adv Manuf Technol (2003) 21:426–437Ownership and Copyright 2003 Springer-Verlag London Limited

Machining Fixture Verification for Nonlinear Fixture Systems

J. J.-X. Liu and D. R. StrongDepartment of Mechanical and Industrial Engineering, The University of Manitoba, Winnipeg, Manitoba, Canada

This paper presents a fixture configuration verification method-ology for nonlinear fixture systems, which is developed on thebasis of optimal clamping forces and total restraint. Thismethod can be applied for validating the feasibility of a fixturewith point, line and area contacts in two stages: fixturing andmachining. The “�-�-�” principle for nonlinear fixturelocation is proposed. The automatic fixture verification systemis modelled as a nonlinear optimisation problem with respectto minimum clamping forces. The method provides a simpleand effective means for: (a) verifying whether a particularfixturing configuration is valid with respect to locating stability,deterministic workpiece location, clamping stability and totalrestraint and (b) determining minimum variable clampingforces over the entire machining time. Two case studies arepresented to demonstrate the effectiveness and the capabilitiesof the methodology.

Keywords: Fixture analysis; Fixture design; Fixture verifi-cation

1. Introduction

The function of a machining fixture is to establish the requiredposition and orientation of a workpiece with respect to themachining tool or cutter and maintain its position duringmachining through a set of fixture elements in contact withthe workpiece. Common contact types between the workpieceand fixture elements can be reduced to point, line and surfacecontacts, determined by the various sizes and shapes of work-piece and fixturing elements. A viable fixture configurationverification system should be able to analyse a fixture systemwith these types of location contacts. Round pin, vee-blockand plate locators are examples of point, line and surfacecontacts in a fixture system.

Several fixture verification approaches were developed in thepast. In the frictionless case, Chou et al. [1] developed a

Correspondence and offprint requests to: Prof. D. R. Strong,Department of Mechanical and Industrial Engineering, The Universityof Manitoba, Winnipeg, Manitoba, Canada R3T 5V6. E-mail:[email protected]

methodology for determining the locating and clamping pointsand clamping forces, based on state space representation andlinear programming. Using a nonlinear programming method,Trappey and Liu [2] derived a quadratic model for the verifi-cation of the fixture configuration. Fuh et al. [3] developed afixture analysis module for verification and rationalisation of afixturing scheme. To identify adequate clamping forces aniterative method was used. These approaches treated machiningconditions as static and considered point contact for mechan-ical restraint.

Taking dynamic machining conditions into account, Meyerand Liou [4] proposed a fixture generation methodology forprismatic workpieces based on linear programming techniqueswithout considering frictional forces. The fixture layout wasgenerated on the basis of part stability, deterministic pos-itioning, good accessibility, positive clamping sequence andtotal restraint. Their method found that a valid fixture has sixlocators. Tao et al. [5] presented an approach for verifying thetotal restraint of the workpiece based on force closure andclamping equilibrium under a dynamic external load. A frictionmethodology of clamping analysis was developed based onnonlinear programming. Gravity and the centre of gravityunder dynamic machining in both approaches were consideredconstant for determining whether a set of point contactsprovides total restraint.

Using a minimum energy principle, Li et al. [6] developeda model for analysing fixtures with large contact areas suchas mechanical vices. The contact area between the vice andworkpiece is approximated by four squares, each having auniform pressure distribution. Static machining conditions wereassumed in their method. Taking into account line and surfacecontacts for fixturing DeMeter [7] presented a linear programwhich uses static equilibrium constraints to prove the existenceof the restraint of workpiece motion in the absence of externalloads, i.e. machining forces. Utilising the surface contactwrench representation proposed by DeMeter, Tao et al. believedthat their proposed method is readily applicable to the fixturingproblems with polygonal supports. Demeter and Tao et al.assumed that the positions of point contact forces are at theend points of the line segments for line contacts and at thevertices of a polygon for planar contacts.

The verification method presented in this paper demonstratesthat the positions of reactions on locators are not at the end

Machining Fixture Verification for Nonlinear Systems 427

points for these contacts. Figure 1(a) illustrates a fixture systemin 2-D with an edge locator. The part is subjected to amachining force of [�100 N, �150 N]T. If two reactions R1and R2 on an edge locator are assumed, the positions of thesereactions are not at the end points of the line segments asshown in Fig. 1(b).

In many fixture applications, line and surface contacts arerelied upon for location and restraint. In the same operation,a fixture configuration for the machining process may not havethe same configuration in the fixturing process, e.g. removablelocating pins. Thus, it is desirable to develop a fixture verifi-cation system that can validate the feasibility of fixture con-figurations with these contacts for location throughout theoperation in both the fixturing and machining stages. Thispaper addresses the problem of validating fixture locationconfigurations with point, line and surface contacts based onlocating stability, deterministic workpiece location, clampingstability, total restraint, and minimum clamping forces. Thefirst three functions are verified at the fixturing stages. Thefourth function, total restraint, that requires the minimumclamping force is verified in the machining stages. The fixtureverification methodology also takes into account dynamicmachining conditions, including the effect of dynamic gravi-tational forces on the fixture system. Vibration under dynamicmachining is not considered in this paper. Both the fixtureand the workpiece system are assumed to be rigid bodies inthis research.

2. Development of a Nonlinear FixtureVerification System

2.1 Workpiece Static Equilibrium Constraints

Dynamic machining conditions occur when the machiningforces and moments travel or change in magnitude or directionwith respect to time. The variation of machining forces maybe due to the alteration of the depth of the cut, the feedrates,and the machining directions. As a result, the magnitudes ofall fixturing forces, including locating, clamping, friction, andgravitational, will vary with respect to machining time. Thefixture–workpiece system must be in static equilibrium for a

Fig. 1. Positions of reaction forces for a line contact. (a) A fixture configuration in 2-D. (b) Resulting fixture forces and positions.

stable fixture configuration to be realised over the machiningtime. The static equilibrium equations for every force andmoment involved in a fixture–workpiece system, located inCartesian coordinate space, over machining time, can be math-ematically expressed:

�F→

(t) = 0 (1)

� r→

(t) � F→

(t) = 0 (2)

where t denotes the machining time; F→

(t) denotes the forceswith respect to machining time, which comprise the locating,clamping, friction, machining, and gravitational forces; and

r→

(t) denotes the moment arms of the forces at time t.Locating, clamping, and frictional forces are the fixturing

forces to be determined. When the moment arms of the fixtur-ing forces in the equation are known, the fixture system isdefined as a linear system because it consists of six linearequations. Point contacts such as pins for locations are anexample of a linear system. When the moment arms of thefixturing forces in the equation are unknown, the fixture systemis defined as a nonlinear system because the equation becomenonlinear. Fixture systems with line or area contacts such asvee-blocks and plate locators for locations are examples ofnonlinear systems.

2.2 “�-�-�” Principle for Nonlinear FixtureLocation

The purpose of location is to achieve the desired positionaland orientational relationship between the workpiece and themachine tool or cutter, by depriving the workpiece of its sixdegrees of freedom in space. Various locating methods havebeen developed based on this purpose, which can be boileddown to point, line and area contacts for location. One degreeof freedom of the workpiece can be constrained by a roundpin–point contact. Two degrees of freedom of the workpiececan be constrained by an edge locator–line contact. Threedegrees of freedom of the workpiece can be constrained by aplate locator–area contact.

428 J. J.-X. Liu and D. R. Strong

The most basic form of workpiece location is the six-pointor the “3-2-1” locating method [8]. With increasing numbersof pins in the base (primary datum plane) from 3 to N, the“3-2-1” locating principle may be expanded to the “N-2-1”principle. If the locating plane configured by N pins is withinthe requirements of the locating accuracy, which is obtainable,the “N-2-1” principle for location is tenable. Using the samemethod for secondary and tertiary datum planes the “N-2-1”principle for location may be expanded to the “N-N-N” prin-ciple. When the number of pins used in the datum plane tendsto approach infinity, an edge or plate locator may replace thelocation pins. A line contact for location can be modelled asa workpiece supported by an infinite number of pins along theline. The area contact for location can be modelled as aworkpiece supported by an infinite number of pins in 2-D.Hence, the “N-N-N” principle for location may be expandedto the “�-�-�” principle for nonlinear fixture location.

2.3 Positive Reaction Constraints

According to Newton’s third law [9], the forces of action andreaction between bodies in contact have the same magnitude,same line of action, and opposite sense. This law may beapplied to the fixture–workpiece system where locators withpoint, line and area contacts are involved. The locators forpoint contacts may be involved in locating the workpiece withrough or machined surfaces. The locators for line or areacontacts may be involved in locating the workpiece withmachined surfaces. As proposed, a locating element with lineor area contacts for location, can be viewed as an elementmade of an infinite number of tiny pins, each having a reaction.The reaction distribution within a contact region is usuallyunknown and completely dependent upon the magnitude andpoint of application of all the acting forces. To simplify fixtureanalysis, a resultant reaction of all the infinite number ofreaction forces is assumed for a fixture with line or areacontacts in the research. If the resultant reaction is positive, itmeans that two rigid bodies (workpiece and locator) remainin contact.

When the reaction is resolved into two components, oneperpendicular and the other tangent to the contact surface, theperpendicular force is the normal force and the componenttangent to the surface is the frictional force. The frictionalforce is then resolved into two components with respect to twoaxes. The maximum frictional force which can be developed isindependent of the size of the contact area. To maintain theworkpiece stability during the fixturing and machining pro-cesses, positive reactions on the locators are required. This isrealised by the normal vector at the contact being defined asdirected towards the workpiece. The normal force of the Car-tesian component of the reaction must remain positive toensure the workpiece stays in contact with the locators. Fixtureverification checks if the total restraint of the workpiece indynamic machining is maintained at all times. The positionvariables of the reaction can be determined through the staticequilibrium equations of the fixture system.

2.4 Fixturing Force Magnitude Constraints

When a particular clamp or locator is selected during thefixture design, the fixture element has a certain holding orsupporting capacity. Hence, the maximum allowable fixturingforces that locators and clamps can withstand are specified asconstraints. The maximum allowable clamping or locating forcemay be defined as

Fmax = s � Flimit, 0 � s � 1 (3)

where s is a safety factor, and Flimit is the largest possibleclamping or locating force a clamp or a locator could providein terms of its specifications.

During fixture setup and machining, the orientation of thefixture placement might change from a horizontal position toa vertical position or vice versa. As a result, friction forcesmay become the only holding forces for the part, as the weightof the workpiece may no longer be supported by the fixturebase, or clamps or locators. The fixtured workpiece might alsoexperience unexpected disturbing forces. Hence, the minimumclamping force is set in such a way that the magnitudes of thefriction forces involved in holding the part must be sufficient tomaintain the stability of the workpiece in the fixture when itis subject to the weight of the workpiece and unexpecteddisturbing forces. The minimum allowable clamping Fmin,C maybe defined as:

Fmin,C =W

2nf � �+ Fd (4)

where W is the weight of the workpiece, nf is the number ofclamps involved in holding the part with friction forces, � isstatic friction coefficient, and Fd is the disturbing force.

For the case where the weight of the workpiece is supportedby the fixture element other than the base, the minimumallowable clamping force Fmin,C may be defined as:

Fmin,C =W

(ns + 2nf � �)+ Fd (5)

where ns denotes the number of clamps involved in supportingthe weight.

To ensure the workpiece stays in contact with the locatorduring machining, the minimum allowable locating force Fmin,l

(10–20 N) is specified.

2.5 Position Variable Constraints

To ensure that the reaction force on the locators is originatingfrom the locating element in contact with the workpiece, theposition variables for locators must lie on the contact line orwithin the contact region. Figure 2 illustrates common line orarea contacts, and constraints.

2.6 Fixture Verification with Respect to OptimalClamping Forces

If a resultant reaction force on the locator with line or areacontacts is assumed in a fixture–workpiece system, a validfixture configuration usually consists of an infinite number of


Fig. 2. Position variable constraints for common line and area contacts.

feasible solutions to the static equilibrium equations of thefixture–workpiece system with regard to reaction forces, clamp-ing forces, and the positions of reactions on the locators.However, there exists one solution for the minimum clamp-ing force.

Theorem 1. There must exist one solution to the static equilib-rium equations of a nonlinear fixture system under externalloads if the position variables of the reaction are determinedbased on the minimum clamping force.

Proof: Fig. 3(a) illustrates a fixture configuration of a 2-D partwith line contact for location. The following inequalities can bederived based on static equilibrium equations of the fixture systemand the relations between the friction force and normal force:

C �Fx

2��

12

(Fy + W) (6)

Fig. 3. Optimisation of clamping forces for fixtures. (a) A 2-D part with an edge locator. (b) A 3-D part with a plane locator.

C �Fy(xF � xc + � � yc) + FxyF + W(xw � xc + � � yc)

xN � xc + � � yc

(7)

� (Fy + W)

where Fx, Fy, C, and W are the x, y components of machiningforce, clamping force and weight, respectively; xF, yF, xC, yC,xw, and xN are the coordinates for those forces; and � is thefriction coefficient.

For each given xN, there are infinite numbers of solutionsof C which satisfy the inequalities, but there exists a minimumclamping force Cmin and an xN, correspondingly:

Cmin =Fx

2��

12

(Fy + W) (8)

xN =Fy(xF � xc + � � yc) + FxyF + W(xw � xc + � � yc)

Fx

2�+

12

(Fy + W)(9)

+ xc � � � yc


Therefore, the position variable xN of the reaction for linecontact can be determined based on the minimum clampingforce.

Figure 3(b) illustrates a fixture configuration of a 3-D partwith area contact for location. The following inequalities canbe derived also based on the static equilibrium equations ofthe fixture system, and the relations between the friction forceand normal force:

C �1

2��F2

x + F2y �

12

(Fz + W) (10)

C �(FyxF � FxyF) � �(W + Fz)(xN cos� � yN sin �)�(xN cos � � yN sin �) + �(xc cos � � yc sin �)

(11)

C �(yN + xN)(W + Fz) � W(yw + xw) + zF(Fx + Fy) � Fz(yF � xF)

� � zc(cos � + sin �) + (yc + xc) � (yN + xN)

(12)

where sin � = Fx(F2x + F2

y)�1/2, cos � = Fy(F2x + F2

y)�1/2; Fx, Fy,Fz, C, and W are the machining force components, clampingforce and weight, respectively; xF, yF, zF, xC, yC, zC, xw, yw,xN and yN are the coordinates for those forces; and � is thefriction coefficient.

For each given xN and yN, there are infinite numbers offeasible solutions of C which satisfy the inequalities, but thereexists a minimum clamping force Cmin:

Cmin =1

2��F2

x + F2y �

12

(Fz + W) (13)

According to Eq. (13), the position variables xN and yN of thereaction N can be determined through the inequalities (11) and(12) by substituting the C’s in the inequalities with Cmin.

Therefore, there is one optimal solution to the static equilib-rium equations of a nonlinear fixture system under externalloads, based on the minimum clamping force.

If multiple clamps (Ci, i = 1, 2, %, n) are exerted on the3-D part in the same direction, but at a different locations, thefollowing inequalities can be similarly derived.

�n

i=1

Ci �1

2��F2

x + F2y �

12

(Fz + W) (14)

min�n

i=1

Ci =1

2��F2

x + F2y �

12

(Fz + W) (15)

From Eq. (15), it can be seen that the optimisation problemis to minimise the summation of the magnitudes of all clampingforces acting on the workpiece.

2.7 Fixture Verification Model for Nonlinear FixtureSystems

Following the theoretical discussion in previous sections, thefixture verification is modelled as a nonlinear optimisationproblem. The optimal solution is the one that minimises themagnitudes of all of the clamping forces acting on the work-piece during the machining process. The objective function Vis to minimise the sum of the clamping force magnitudes Cj

with respect to machining time t:

MinV(t) = �n

j=1

Cj(t) (j = 1, 2, 3, %, n) (16)

If Fi(t) denotes a fixture force (locating, clamping or friction)magnitude at certain time t, and Fj(t) denotes a clamping forcemagnitude at certain time t when i = j % n, the objectivefunction V(t) can be rewritten as follows:

MinV(t) = �n

i=j

Fi(t) (i = 1, 2, 3, % j, %, n) (17)

Subject to:

1. Static equilibrium constraints – the fixture–workpiece systemmust be in static equilibrium for a stable fixture configur-ation to be realised over the machining time.

�a11a12 % a1n 00 % 0 00 % 0

00 % 0 b21b22 % b2n 00 % 0

00 % 0 00 % 0 c31c32 % c3n

00 % 0 b41b42 % b4n c41c42 % c4n

a51a52 % a5n 00 % 0 c51c52 % c5n

a61a62 % a6n b61b62 % b6n 00 % 0

� � (18)

�Fx,1(t)Fx,2(t) % Fx,n(t)

Fy,1(t)Fy,2(t) % Fy,n(t)

Fz,1(t)Fz,2(t) % Fz,n(t)�= � �

Rx(t)

Ry(t)

Rz(t)

Mx(t)

My(t)

Mz(t)

�where t is the machining time when the cutter is removingthe material from the part; aij, bij and cij are the x, y andz components of the direction and position vectors for eachfixture element (when i � 3, they are the direction coef-ficients; when i 3, they are the position variables); Fx,1(t)%, Fy,1(t) %, and Fz,1(t) % are components of the nfixturing forces with respect to time, which comprise thelocating, clamping and frictional forces; Rx(t), Ry(t) and Rz(t)are the axial components of the resultant cutting force andweight at time t; Mx(t), My(t) and Mz(t) are the componentsof the resultant moment of cutting forces and weight attime t.

2. Coulomb friction constraints – the static friction forces fx,i,fy,i, and fz,i must be positive over machining time t as:

fx,i(t)�0; fy,i(t)�0; and fz,i(t) (19)�0 (i = 1, 2, 3, % n)

If the � is the coefficient of static friction, the resultantmagnitude of the friction force can be less than or equalto the magnitude of the fractional normal force (��Fi(t)�)over time t.


Fig. 4. A flowchart of the fixture configuration verification program.

�f2x,i(t) + f2

y,i(t) + f2z,i(t) (20)

��|Fi(t)| (i = 1, 2, 3, %, n)

3. Fixturing force magnitude constraints:

Fmin,i � Fi(t) � Fmax,i (i = 1, 2, 3, %, n) (21)

4. Positive normal reaction constraints:

Fi(t) � 0 (i = 1, 2, 3, %, n) (22)

5. Position variable constraints:

ALB,ij � aij � AUB,ij; BLB,ij � bij (23)� BUB,ij; and CLB,ij � cij � CUB,ij

where ALB,ij, BLB,ij, and CLB,ij denote lower bounds, andAUB,ij, BUB,ij, and CUB,ij denote upper bounds for theposition variables.

2.8 Fixture Verification for the Fixturing Process

From a mechanistic point of view [1], fixtures must satisfy thefollowing four functional requirements for holding workpiece:locating stability, deterministic workpiece location, clampingstability and total restraint. The first three functions are requiredat the fixturing stages. The last function is required in themachining stages. Using the fixture configuration verificationmodel developed previously in Section 2.7, locating stability,deterministic workpiece location and clamping stability in thefixturing process for a given fixture configuration is firstvalidated by running the verification model on LINGO7.0 [10],a commercial nonlinear programming package.

During the fixturing process, machining time and machiningforces in the model are set to zero. Checking for locatingstability is accomplished by setting all the allowable clampingforces to zero and the minimum locating forces to a givenvalue in the model. If a feasible solution to the model exists,the workpiece is at equilibrium resting in the fixture. Physically,


this means that static equilibrium of the fixture system isachieved and no detaching forces on the locators are found.The magnitudes of the locating forces can also be determinedby the model. If a feasible solution cannot be found, it isverified that the workpiece is unstable due to its gravity actingout of the fixture support. When there is no physical contacton the line of action of gravity, three reactions on the contactarea are assumed in the model. If a feasible solution exists, itis verified that locating stability of the workpiece is achieved.

Checking for deterministic workpiece location and clampingstability is accomplished by setting all minimum and maximumallowable clamping and locating forces to a known value. Ifa feasible solution to the model exists, the workpiece isdeterministically located and the clamping forces do not upsetthe stable and accurate position previously determined by thelocators. The result demonstrates that workpiece equilibrium ismaintained when the workpiece is subject to clamping forcesand weight. The solutions of the positive locating forces onthe locators in the model can be found. The magnitudes ofthe minimum clamping forces to maintain workpiece stabilityin the fixture can also be found. If a feasible solution to themodel cannot be found, the fixture in the fixturing stages isverified to be invalid.

2.9 Fixture Verification for a Machining Process

Fixture verification for a machining process consists of twomain tasks: total restraint of the workpiece and determinationof minimum clamping forces. During machining, the fixtureshould completely restrain the workpiece to counter dynamicand unpredicted machining forces and moments. To ensuretotal restraint of the workpiece under dynamic machining, thefixture configuration is validated for all possible machiningpaths or for the worst case, which is the machining path with

Fig. 5. A fixture configuration with point, line and area contacts. (All coordinates in centimetres.)

cutting forces acting against the clamps. Since total restraintthat determines the minimum clamping force is verified in themachining stages, one of outputs from the model is a setof variable clamping forces that is just enough to alwayscounterbalance the machining forces over the entire machin-ing time.

During the verification of the machining processes, allunknown forces such as reaction, gravitational, friction, andclamping as well as position variables of reaction and gravi-tational forces are determined, over the machining time.The discrete dynamic fixturing solutions to the model overmachining time can be obtained by running the model onLINGO7.0. If a feasible solution to the model is found, thefixture configuration is verified to be valid because all fixtureforces on the locators are positive. This means that theworkpiece maintains contact with fixture elements during theentire machining process. A set of variable clamping forcesis also determined to be sufficient to counterbalance thedynamic machining forces. If a feasible solution to the modelcannot be found, the fixture is verified to be invalid becauseit has not restrained the workpiece completely. To identifywhich locator yields a negative value, all the reactions andallowable locating forces would be reset to allow them tohave negative values and then the model is run again. Thisassists in identifying which locator would actually detachfrom the workpiece so that the invalid fixture configurationcan be corrected. As long as the position variables of theresultant reaction force on edge or plate locators are con-strained within the area in contact with the workpiece duringthe verification process, the validation of the fixture systemis performed correctly.


3. Implementation

The automatic fixture configuration verification system hasbeen implemented on a Pentium III Dell laptop computer. Anoverall outline of the methodology is shown in Fig. 4. Theoptimisation program for verification is run in a commercialoptimisation tool, LINGO7.0, which can solve linear, nonlinear,and integer optimisation models. Data input and output as well,as other models, are run in an Excel spreadsheet that is linkedto LINGO7.0. If the fixture configuration is valid, a feasiblesolution is output with optimal values, in less than 30 seconds.If there is no feasible solution, the fixture configuration beingverified is invalid. By resetting the reaction force constraintsto allow negative values, the system will identify that theworkpiece is detaching from a specific locator in a few seconds.Results of the two case studies are summarised in the followingthe section.

4. Fixture Configuration Verification CaseStudies

4.1 Case Study One

A fixture configuration, as shown in Fig. 5, consists of threelocators, pin, edge, and plate for location as well as twohorizontal clamps for clamping to secure a prismatic part witha step feature on its top and with an elliptical pocket featureon its base. A slot milling operation is going to be performedon the part to produce a through slot.

During the first half of the cutting path, the depth of cut is1 cm and the part is to undergo a machining force of [100N, �100 N, �100 N]T at the middle of the axial depth of thecut. During the second half of the cutting path, the depth ofcut is 2 cm and the part is to undergo a machining force of

Fig. 6. Normal reactions and clamping forces over time.

[200 N, �200 N, �200 N]T at the middle of the axial depthof the cut. The diameter of the end mill is 2 cm. The cuttertravels along the positive x-direction with a feedrate of 0.5cm/s. The total machining time for the entire cutting path is40 seconds with machining forces and moments calculatedevery 2 seconds, which translates into 21 dynamic fixturingsolutions. The part is subjected to an initial body force of [0,0, �48 N]T at its centroid (10.3, 5, 5.1). It is required to verifywhether the fixture configuration is valid and to determine theminimum clamping forces over the entire machining time.

Fixture verification for the fixturing process was performedas follows. Under a gravitational force of 48 N, a feasiblesolution to the verification model is found when three reactionson the plate locator are assumed. All three reaction forces onthe plate locator are found to be positive: 11.4 N at (0, 10,0), 24.7 N at (20, 3.5, 0) and 11.9 N at (0, 3.4, 0), respectively.Therefore, the locating stability for the fixture configurationis verified.

The minimum and maximum allowable locating forces wereset to 10 N and 800 N. The minimum and maximum allowableclamping forces were set to 40 N and 600 N. It is verifiedthat the workpiece is deterministically located, and the clamp-ing forces do not upset the stable and accurate position pre-viously determined by locators when one reaction force on theplate locator is assumed. A feasible solution to the nonlinearverification model under the gravitational and clamping forceswas found, since all fixturing forces C1 = C2 = 40 N, L1 =20.6 N at (8.8, 8.9, 0), L2 = 32.2 N at (11.3, 10, 8), and L3= 40 N at (0, 2, 8) are positive. Friction forces involved inthe y-direction are 4.1 N at locator 1, and 3.7 N at locator 3.Friction forces involved in the z-direction are 6.4 N at locator2, 5.1 N at locator 3, 8 N at clamp 1, and 7.99 N at clamp 2.

Fixture verification for the machining process was carriedout as follows. Dynamic gravitational forces of the workpieceare taken into account in the verification process. One reaction


Fig. 7. Positions of the reaction on plate locator over time (Z = 0).

force on the plate locator and one reaction force on the edgelocator are assumed. Since feasible solutions to the nonlinearverification model were found, total restraint of the workpiecefor a given machining direction in positive x is achieved.Figure 6 illustrates that reaction forces on all three locatorsunder the variable clamping forces are positive. Figures 7 and8 show the positions of reaction forces on the plate andedge locators over time. It can be concluded that the fixtureconfiguration is valid for machining the slot. The requiredminimum variable clamping forces C1 and C2 over time aredetermined as shown in Fig. 6. If a dynamic clamp isemployed, the clamp should provide variable clamping forceson the workpiece with respect to machining time. If a constantclamping force rather than a variable clamping setting isemployed, the static minimum clamping forces for clamps C1and C2 throughout the operation should be set to be 469.3 Nand 167.1 N, respectively. Under the proposed clampingforces, the part stability under the dynamic machining willbe maintained.

Fig. 8. Positions of the reaction of edge locator over time (Y = 10).

4.2 Case Study Two

A fixture configuration as shown in Fig. 9 is to undergo aperipheral milling operation. Figure 9(a) shows the fixture set-up during the fixturing process with two removable pins usedfor locating the part. Figure 9(b) illustrates the fixture set-upduring the machining process in the absence of the two remov-able locating pins. Four clamps are applied on the workpieceto maintain workpiece stability under the milling operations.

The workpiece is subjected to an initial body force of [0N, 0 N, �56.4 N]T at its centroid (20, 5, 4.3), and machiningforces of [100 N, 50 N, �100 N]T, [�50 N, 100 N, �100N]T, [�100 N, �50 N, �100 N]T, and [50 N, �100 N, �100N]T for cutting paths 1, 2, 3, and 4, respectively, at the middleof the axial depth of the cut. The cutter travels: path 1 from(6, 2, 9) to (34, 2, 9), path 2 from (34, 3, 9) to (34, 8, 9),path 3 from (33, 8, 9) to (6, 8, 9), and path 4 from (6, 7, 9)to (6, 3, 9) with a feedrate of 0.25 cm/s during machining.The total machining time is 268 seconds with machining forces


Fig. 9. Fixture configuration with removable locators: (a) fixturing process; (b) machining process. (All coordinates in centimetres.)

and moments calculated every 4 seconds, which produces 68dynamic fixturing solutions. It is required to verify whetherthe fixture configuration is valid, and to determine the minimumclamping forces to hold the workpiece during machining.

Fixture verification for the fixturing process was performedas follows. Under a gravitational force of 56.4 N, locatingstability for the given fixture configuration is achievable, sincea feasible solution to the nonlinear verification model was


Fig. 10. Normal reactions on plane locators 1 and 2 over time.

found when three reactions on two plate locators are assumed.Positive locating forces L1, L2 and L3 are found, which are19.1 N at (0, 4.1, 0), 27 N at (30.3, 3.8, 0), and 10.3 N at(30.3, 10, 0), respectively. The minimum and maximum allow-able locating forces were set to 10 N and 600 N. The minimumand maximum allowable clamping forces were set to 40 Nand 400 N. The coefficient of static friction is 0.2 for allcontacts between the workpiece and the fixture element. It hasbeen verified that the workpiece is deterministically locatedand the clamping forces do not upset the stable and accurateposition previously determined by locators. This is because a

Fig. 11. Minimum clamping forces over time.

feasible solution to the model under the gravitational andclamping forces was found. All fixturing forces C1 = C2 =C3 = C4 = 40 N, L1 = 108.2 N at (9.9, 9.9, 0), and L2 =108.2 N at (30, 0, 0) are positive.

Fixture configuration for the machining process is verifiedto be viable because all reaction forces on the plane locatorsunder the variable clamping forces are positive as shown inFig. 10. The variable minimum clamping forces required tomaintain the part stability over time are determined as shownin Fig. 11. Figure 12 shows the positions of reactions on theplane locators. If a dynamic clamp is employed, the clamp


Fig. 12. Positions of reactions on plane locators over time (Z = 0).

should provide variable clamping forces on the workpiece withrespect to machining time. If a constant clamping force ratherthan a variable clamping setting is employed, the static mini-mum clamping forces for clamps C1, C2, C3 and C4 through-out the operation should be set to 139.7 N, 101 N, 142.4 N,and 102.4 N, respectively.

5. Conclusions

This paper has presented a fixture configuration verificationapproach for nonlinear fixture systems where edge locators,inclined locators, and plate locators are involved in locatingthe part. A validation of the fixture configuration has beenperformed in two stages: both in fixturing and in machiningto ensure the fixture is viable throughout the operation. Thefixture verification model can be embedded into an automaticfixture design system. The proposed method can be appliedin validating the feasibility of a fixture with point, lineand surface contacts for location. A “�-�-�” principle fornonlinear fixture location was proposed based on the locatingline or area represented by an infinite number of pins. Thefixture verification method provides a simple and effectivemeans for: (1) validating a fixture configuration with respectto locating stability, deterministic workpiece location, clamp-ing stability and total restraint and (2) for determiningminimum variable clamping forces over the entire machiningtime. Two case studies have demonstrated the effectivenessand the capabilities of the methodology for verifying nonlin-ear fixture systems.

Acknowledgements

This research was supported by the Natural Sciences andEngineering Research Council of Canada, and ManitobaHydro.

References

1. Y.-C. Chou, V. Chandru and M. M. Barash, “A mathematicalapproach to automatic configuration of machining fixtures: analysisand synthesis”, Journal of Engineering for Industry, 111, pp. 299–306, 1989.

2. A. J. C. Trappey and C. R. Liu, “An automatic workholdingverification system”, Journal of Robotics and Computer-IntegratedManufacturing, 9(4/5), pp. 321–326, 1992.

3. J. Y. H. Fuh, C-H. Chang and M. A. Melkanoff, “An integratedfixture planning and analysis system for machining process”,Journal of Robotics and Computer-Integrated Manufacturing,10(5), pp. 339–353, 1993.

4. R. T. Meyer and F. W. Liou, “Fixture analysis under dynamicmachining”, International Journal of Production Research, 5(5),pp. 1471–1489, 1997.

5. Z. J. Tao, A. S. Kumar and A. Y. C. Nee, “Automatic generationof dynamic clamping forces for machining fixtures”, InternationalJournal of Production Research, 37(12), pp. 2755–2776, 1999.

6. B. Li, S. N. Melkote and S. Y Liang, “Analysis of reactions andminimum clamping force for machining fixtures with large contactareas”, International Journal of Advanced Manufacturing tech-nology, 16, pp. 79–84, 2000.

7. E. C. DeMeter, “Restraint analysis of fixtures which rely onsurface contact”, Journal of Engineering for Industry, 116, pp.207–215, 1994.

8. E. K. Henriksen, Jig and Fixture Design Manual, Industrial Press,New York, 1973.

9. F. P. Beer and E.R. Johnston, Jr. Vector Mechanics for Engineers:Statics, SI metric edn, McGraw-Hill Ryerson, Canada, 1981.

10. LINDO Systems Inc. LINGO User’s Guide, Chicago, 2001.

art%3a10.1007%2fs001700300050

Documents