Analysis of reaction forces in fixture locating points: An Analytical, numerical, and experimental study

Reaction forces are important parameters in fixture design. They are generated by the clamping forces and machining loads at the fixture locating points. These forces are used as input values in the determination of clamping forces, fixture stiffness, and workpiece deformation. In this paper, an analytical model based on the minimum norm principle was developed to calculate these forces. Numerical simulations and experimental tests were performed on a 3D polyhedral workpiece to validate the model. The simulations were conducted using Abaqus® software and the experimental tests used a fixture and a 3D polyhedral workpiece. The theoretical, numerical, and experimental results showed good agreement for the normal component of reaction forces. The maximum errors of 3.9% and 15% were observed between the theoretical predictions compared to the numerical and experimental results, respectively. The model was also used to study the effects of two influential parameters, the coefficient of friction and clamping force, on the reaction forces. The good agreement between the theoretical, numerical, and experimental results demonstrated the efficiency of the proposed model in the rapid calculation of reaction forces for fixturing 3D polyhedral workpieces.


Introduction
Fixture design is usually performed in four steps, including setup planning, xture planning, unit design, and veri cation.The magnitude of the reaction forces is used as a key parameter in the last three steps.In xture planning, decisions about the overall design of the xture are made depending on the geometry of the workpiece, its degree of rigidity, and the facilities available in the workshop, based on which the locating, clamping, and supporting systems are designed.The position of locators, the position and orientation of clamps, and consequently the workpiece deformation due to the application of clamping and machining loads are among the parameters that are directly related to the magnitude of reaction forces.For example, the position and orientation of the resultant clamping force must be chosen such that the normal component of reaction forces at the contact points remain positive in the operating conditions.The magnitudes of reaction forces at the locating points act as inputs to the xture unit design process.The geometry and material of the locators are designed based on the magnitude and direction of application of the reaction forces.Also, the design of xture body plates is directly dependent on the values of reaction forces since locators apply these loads to the xture body plates in the form of forces and moments.In the veri cation step, which is the last step of the xture design procedure, the reaction forces play an important role in investigating the workpiece stability, accuracy of the locating system, and avoiding the occurrence of jamming during workpiece loading into the xture.It can be concluded that the reaction forces are used in almost all steps of the xture design procedure.Therefore, any effort to reach the exact value of these forces in a reasonable time would be of great help to the xture designer.
The development of an analytical model to calculate the reaction forces in the contact points of a xture is di cult since the utilization of static equilibrium equations is impossible because the problem of xturing a 3D workpiece in a xture with six locating points is a statically indeterminate con guration.In a 3D xturing system, each reaction force has three components at each contact point, including one in normal and two in tangential directions.So, there will be a total of 18 unknown components in a 3-2-1 locating system that may not be calculated by having in hand only six static equilibrium equations.
Performing nite element analysis seems to be the rst solution to this problem; however, not only this type of analysis is usually expensive but also the results are strictly dependent on the contact geometry and boundary conditions [1,2].In some special con gurations, the FEM software can't even model the contacts between the workpiece and locators, especially when a exible workpiece rests on the sphericalhead locators in the xture.On the other hand, xture planning is usually performed in the early stages of the xture design procedure, in which the xture designer has insu cient knowledge about the contact behavior between the workpiece and the xturing system.The xture designer usually takes advantage of his/her experience to achieve a xture plan in which the reaction forces have preferably an almost uniform distribution at all contact points.So, a fast and agile tool to calculate the reaction forces at the contact points is required for the xture designer in different steps of the xture design procedure.In the present paper, a fast and accurate analytical model was developed based on the minimum norm principle to calculate the magnitude of these forces in xturing 3D polyhedral workpieces.The position and orientation of the locating and clamping elements are the only inputs to the developed model.The results of the previous research were used to initially validate the theoretical predictions.Also, the results of nite element analysis were used to further verify the predictions of the proposed model.The nal validation was performed using experimental results.
In reviewing the research background, papers that have calculated the reaction forces at the locating points are investigated.Also, the numerical or experimental studies that have used reaction forces in any step of the xture design procedure are included in this section.Chou et al. [3] proposed an analytical model based on the screw theory to design locating system for polyhedral parts.The reaction forces were calculated in the locating points of a 2D workpiece using the static equilibrium equations.The positive values of the normal components of reaction forces were introduced as the main criteria for keeping contact between the workpiece and locating agents.Lee and Cutkosky [4] calculated the tangential component of reaction forces at the locating points using a de nition called the slip limit surface.It was stated that there is a direct relationship between the slip speed of the workpiece on the locators and the tangential component of the reaction force at the contact point.They used this rule to calculate the tangential components of reaction forces in the contact points.Cai et al. [5] used the values of reaction forces at the locating points to investigate the stability of the non-rigid workpiece in the xture.It was proposed that more than three locators must be used on the base surfaces of these types of workpieces.The stability of such con gurations was investigated with different quantities of the base locators.In [6], the screw theory was used to model the external forces applied to the workpiece in the form of perturbation twists.The positions of the locators were designed such that their wrenches had the maximum contrariety conditions against the perturbation twists.A 3D polyhedral workpiece was used to prove the capabilities of the proposed model.Hurtado and Melkote [7] analyzed the stability of a workpiece with free-form geometry that was located in a xture using the pin-array con guration.A multi-objective optimization model was proposed and solved to calculate the best combination of pins, achieve the optimal conformation between the workpiece surface and the pins, and minimize the maximum deformation of the workpiece.In [8], the reaction forces were calculated on the locators in order to study the workpiece stability in the xture.The stiffness matrix of the xture-workpiece assembly was derived by the nite element analysis software.The vector of the external forces was used and the equation [K] [X]=[F] was solved to calculate the displacement vector at the contact points.The reaction forces were then obtained at the locating points and their values were used in the stability analysis.In [9], a nite element analysis was performed to calculate normal components of reaction forces at the contact points between the workpiece and locators by taking into account the machining loads.The main purpose of this analysis was to optimize the locating and clamping layouts.The proposed model suffered from shortcomings including the utilization of the expensive nite element analysis in each iteration of the optimization process and evaluation of model capabilities using a 2D workpiece.In a similar study, Kaya [10] used the Genetic algorithm to optimize the position of locators, clamps, and supports by considering the clamping and machining loads as external forces.The calculated machining loads in [9] were applied to the workpiece.Finite element analysis was performed in each iteration of the optimization process to calculate the maximum deformation of the workpiece.The workpiece stability in the xture was investigated with each new solution that was generated by the genetic algorithm.Satyanarayana and Melkote [1] developed theoretical and numerical models to study the behavior of a single contact that was established between the workpiece and a locator.They applied different boundary conditions to the con guration and studied the effects of the contact geometry on the contact force by investigating the at-at and spherical-at contact con gurations.The output parameters were the maximum elastic deformation of the workpiece and the reaction force in the contact point.By assuming that frictionless contacts were established between the workpiece and locators, Wang et al. [11] proposed a multiobjective optimization model to maximize the workpiece positioning accuracy and locating reproducibility.The stability of the workpiece was evaluated on the locating system at each step of the optimization process.Chen et al. [12] proposed a model to minimize the magnitude of clamping forces by considering the effects of the frictional contacts and chip removal process on the results.The nite element analysis was performed to calculate the deformation of the workpiece in each iteration.The genetic algorithm was utilized to optimize the locating layout.In [13], the supporting system was designed by performing nite element analysis on a workpiece with polyhedral geometry located in the xture.The weight of the workpiece and clamping forces were considered as the external loads.The maximum deformation of the workpiece (the position to which the supporting element should be applied) was obtained by ignoring the friction between the workpiece and locators.Jiang et al. [14] proposed a multi-objective optimization method to design locating system for a workpiece in a checking xture.The main aim of the proposed model was to simultaneously maximize the workpiece locating accuracy, maintain the stability of the workpiece in the xture, and quick and easy loading/unloading of the workpiece into/from the xture.The proposed search algorithm used the continuous search method on the workpiece surfaces to eliminate the discretization error.Maintaining workpiece stability in the locating system was mentioned as a requirement in each iteration of the optimization process.Xiong et al. [15] suggested a genetic algorithm-based optimization model to calculate the optimum locating layout for non-rigid parts used in the aerospace industry.The deformation of the workpiece in the regions close to the active machining area was determined using nite element analysis.The effects of the geometry of contact between the workpiece and locators were studied on the workpiece deformation.Parvaz and Nategh [16,17] proposed models for designing locating and clamping systems for workpieces with freeform geometry.The values of the reaction forces were used in the locating layout design to analyze workpiece stability and avoid the occurrence of jamming during workpiece loading into the xture.Also, positive values of the normal component of reaction forces were used to ensure the contact keeping condition between the workpiece and locating agents.Recently, Parvaz et al. [18] used the values of reaction forces to predict jamming occurrence conditions in the xture design application.The capabilities of the proposed model were evaluated using the peg-in-hole and block and palm case studies.The results of the proposed model were validated using numerical analysis and experimental tests.
According to the literature survey, several studies have been performed in the eld of using nite element analysis to perform xture design activities by taking into account the frictional contacts and chip removal effects.Finite element analysis requires heavy processing and thorough knowledge of the contact conditions between the workpiece and the xturing elements.Also, the results of nite element analysis are severely dependent on the boundary conditions applied to the problem, which can even lead to incorrect results in some contact con gurations.Furthermore, to the best knowledge of the authors, there is no comprehensive research in the literature that measures the reaction forces at the locating points of a xture dedicated to a three-dimensional workpiece simultaneously by the theoretical, numerical, and experimental methods.Therefore, an agile tool capable of calculating reaction forces accurately in the contact points between workpiece and locators without the need for detailed contact information is very useful to the xture designer.In this paper, such a tool is developed based on the minimum norm principle, the predictions of which were compared to the published results in the previous studies.Also, a nite element analysis was performed using Abaqus ® software to validate the theoretical predictions.Finally, experimental tests were performed using a special setup that was designed and fabricated to measure reaction forces in the real-world 3D xturing application.The main novelties of the present paper include the development of an analytical model for rapid calculation of reaction forces, verifying the accuracy of its predictions in both normal and tangential directions, studying the effects of in uential parameters on the results, validating the analytical predictions with the numerical results, and conducting experimental tests on a 3D workpiece to verify the theoretical and numerical data.

Theoretical Modeling
The design of locating system is one of the main steps of the xture planning process.Six locators are usually used in the design of xtures for non-exible parts.Since there are three components of a reaction force at each contact point, there will be a total of 18 unknown reaction force components in the problem.So, the problem of xturing a non-exible workpiece is statically indeterminate which can't be solved by considering the six static equilibrium equations.An analytical model based on the minimum norm principle is used to solve the problem, similar to the theoretical modeling used by the authors in [18] to model the jamming phenomenon in xturing applications.This principle states that between several solutions to a statically indeterminate problem, the best is the one with the minimum intensity.Utilization of this principle to calculate reaction forces in xturing a non-exible workpiece leads to a nonlinear quadratic optimization problem.The objective function is to minimize the magnitude of the resultant reaction force vector, which consists of six reaction force components.Each of these components has, in turn, one normal and two tangential components.The optimization problem can be stated as: Such that: Jacobian matrix: External wrench: In Eq. (1a), R is the resultant reaction force vector, which has six components R i .R i is the reaction force vector at the ith contact point, which has three components including one in normal R i n and two in tangential (R i t1 and R i t2 ) directions.J is the Jacobian matrix that maps the positions and orientations from the local coordinate system (n − t 1 − t 2 ) to the global one (X − Y − Z).W e is the resultant wrench resulting from the external forces and moments acting on the workpiece.The rst constraint in Eq. ( 1a) is included in the problem in order for the reaction forces to satisfy the static equilibrium equations between the internal and external forces in the global coordinate system.The second constraint has two equations.The rst is the requirement of a positive value for the normal components of reaction forces at all contact points (a necessary condition for the stability of the workpiece in the xture).The second condition states that Coulomb's law is used to model friction at the contact points between the workpiece and locators.The expression for this constraint adds nonlinearity to the optimization problem.The Jacobian matrix J can be calculated from Eq. (1b), in which J i is the component of the Jacobian matrix for the ith locating point being used to map positions and orientations from the local to the global coordinate system.Each J i has three components (J i n , J i t1 , and J i t2 ) in the normal, rst, and second tangential directions.r i is the position of the ith contact point in the global coordinate system.n i , t 1i , and t 2i are the unit vectors in the normal, rst, and second tangential directions at the ith contact point.The resultant external wrench (W e ) can be calculated from Eq. (1c), in which W g , W c , and W m are the wrenches that are applied to the workpiece due to part weight, clamping force, and machining loads, respectively.The machining forces are not accounted for in the present study because the main aim of the paper is to validate the predictions obtained from the proposed theory with the results of numerical analysis and experimental tests.Machining forces and moments can be calculated from the theories developed for calculating machining loads (such as the models suggested in [19]) and applied as W m in Eq. (1c).The reaction forces can then be calculated by solving the optimization problem in short time intervals.This suggestion is currently under experimental investigation by the authors.
The optimization problem stated in Eq. (1a) can be modeled and solved using Matlab ® software with the techniques provided in it for solving quadratic optimization problems.If the optimization problem could not be solved, it means that at least one of the constraints stated in Eqs.(1a) is not satis ed.In another word, either the workpiece has lost contact with at least one of the locators or the magnitude of tangential forces at the contact points has increased to such an extent that it reached the maximum value.Usually, in 3D xturing applications, the workpiece keeps contact with the locators if the locating system is designed correctly and the clamping forces are applied to the workpiece in the appropriate positions and orientations.Therefore, the lack of a feasible solution to the optimization problem is due to dissatisfaction with Coulomb's friction law.If the optimization problem achieved a feasible solution, the Coulomb's friction constraint is satis ed and the tangential forces at all contacts coincided inside their corresponding friction cones.In such a con guration, the reaction forces can be calculated at all contact points.

Numerical Analysis
Figure 1 shows the model used to perform nite element analysis.This workpiece is a bulk polyhedral part with dimensions 150 mm × 200 mm × 75 mm.The A, B, and C surfaces were chosen as the base, second, and third locating planes.It has six locators L 1 to L 6 , which are arranged on the workpiece using the well-known 3-2-1 principle.The workpiece clamping system is also designed with a single clamp, applied to the workpiece in the position and orientation shown in Fig. 1.The clamping system is intentionally designed with a single clamp to ease the experimental testing and also eliminate the effects of the clamping sequence on the results.A global X-Y-Z coordinate system is de ned at the center of the base locating plane of the workpiece.Also, six local coordinate systems (n − t 1 − t 2 ) are de ned at the six contact points.The position and orientation of the locating and clamping elements for this workpiece are presented in Table 1 (refer to section 4).The simulation was performed in Abaqus ® V2016 software to calculate reaction forces in the contact points.For this purpose, the workpiece model was imported from CAD software as a 3D deformable part.The base locators were de ned as rigid parts with a radius and length of 5 mm and 10 mm, respectively.It was assumed that the workpiece was made of AL7075-t6 with Young's modulus of E = 71.7 GPa and a Poisson ratio of ν = 0.33 [20].The simulation model was then assembled by positioning locators on the workpiece surfaces.The surface-to-surface frictional contacts were de ned between the workpiece and locators.In general-purpose operations and not under extreme normal force, the coe cient of friction between the mating surfaces depends on the materials in contact and their surface roughnesses.The coe cient of friction may have different values at each contact point, depending on the material of the locators and the roughness of their surfaces.
The effects of the coe cient of friction, as an in uential parameter, were studied on theoretical predictions and simulation results.The range [0.2-0.5] was chosen for studying these effects on the steel-aluminum pair (as materials of the locators and workpiece, respectively) in different conditions.This interval was chosen to cover all possible scenarios for workpiece-locator materials and their surface roughnesses.The boundary conditions were de ned as constraining all degrees of freedom of the locators.The clamping force was applied as pressure to the workpiece in the position and orientation stated in Table 1.The workpiece meshed with a mesh size of 5 mm using the well-known C3D20RH standard solid element, which is a 20-node quadratic continuum hexagonal element with hybrid formulation and reduced integration.The chamfered region of the workpiece meshed with the C3D10 element, which is a 10-node quadratic tetrahedron-shaped element.Also, a range of [4][5][6][7][8][9][10] mm was chosen for mesh size to perform the mesh-independency analysis.This interval was equivalent to the number of elements [16914 − 1582].The standard solver was used to solve the problem with the simulation time, initial, minimum, and maximum increment sizes equal to 1 s , 0.05 s , 10 − 5s , and 0.1 s , respectively.The assembled and meshed models with the applied loads and boundary conditions are shown in Fig. 2. The model was solved on a PC with a Core-i7 CPU and 16 GB of RAM in less than 30 s .The results are compared with the theoretical predictions and experimental data in Section 5.

Experiments
Experiments were performed on a special setup that was designed and fabricated to measure reaction forces at the contact points between the workpiece and locators.Figure 3a shows the CAD model of the experimental setup which consists of the workpiece, xture with locating and clamping systems, loadcells, and data gathering system.The workpiece was made of Aluminum with dimensions 100mm × 150mm × 75mm.The casting process was used to produce the initial shape of the workpiece, which was followed by machining processes to achieve the nal shape.
The setup was designed such that applying only one clamp to the workpiece was enough to keep the contacts between the workpiece and locators.A 3D chamfer with an angle and edge length of 45 ∘ and 20 mm, respectively was applied to the workpiece at its vertex such that the clamp may rest on the created surface.This speci c design made it possible for all locators to have positive normal reaction force vectors.The xture was made of St37 steel plates with a thickness of 20 mm.This thickness was chosen so that the assembled xture had enough rigidity to withstand the external loads.Three sheets with dimensions of 260mm × 280mm, 170mm × 260mm, and 170mm × 260mm were assembled mutually perpendicular to each other using twelve M16 bolts.The bending beam loadcells (Zemic ® H8C-250 Kg) were used to measure reaction forces at the locating points.They were installed on the base, second, and third plates of the xture body using two M14 bolts and a spacer with a thickness of 5 mm.
The spacers were used so that the free side of the loadcell, which should be bent to measure the reaction force, had an appropriate distance from the xture body plates.The position and orientation of the locating and clamping elements are presented in Table 1.These values are used as inputs to the proposed theoretical and numerical models.The spherical-head locators were installed on the free side of the loadcells, making contact with the workpiece at the tip of the spheres.The heads of these locators were ground to a spherical shape by a bench grinding machine.They were made of M12 bolts and fastened to the loadcells using two nuts.
(1, 0, 0) (0, 1, 0) (0, -1, 0) (1, 0, 0) (0, 0, 1) (0, -1, 0) (1, 0, 0) (0, 0, 1) (-1, 0, 0) (0, 0, 1) (0, 1, 0) The heights of the locators were measured and equalized using a digital caliper with an accuracy of 0.01 mm.The xture was located at the test table and leveled using an accurate 300 mm level.The workpiece was then loaded into the xture ensuring that it made contact with all locators.A speci c stand, which was fabricated with a U-shape pro le, was used to apply the clamping force to the workpiece at an angle of 45 ∘ , perpendicular to the chamfered surface of the workpiece.It was installed on the base plate using three M16 bolts.The clamping force was applied using an S-type loadcell (Zemic ® H3-100 Kg) and its mounting (Zemic ® HJ-8-201-2 t).The spherical joint inside the mounting suppressed the torque that tends to be applied to the workpiece during applying force by the loadcell.As a result, the force was applied to the workpiece as pressure with an intensity that is measured by the reader (Additel ® ADT 221A multifunction calibrator with a minimum resolution of 0.0001 mV).It should be mentioned that all loadcells were calibrated before being used in the experimental setup.The calibration process was performed by hanging several standard weights from the loadcells and reading the output voltage by the reader system.The additional weights, that are applied to the loadcell during the calibration process, were the mass of the belt, chain, and the holding structure of the calibration weights.They were measured and accounted for in the calibration process.Figure 3c represents the calibration process that was performed in a standard testing laboratory.
The workpiece was loaded into the xture after nalizing the assembly of the experimental setup.The mass of the workpiece was measured using a digital weighing scale with an accuracy of 0.001 g.It is considered as a dead load being applied to the workpiece affecting reaction forces at the contact points.The loadcell orientation was adjusted using an industrial protractor.Initially, no clamping force was applied to the workpiece such that it rested only on the base locators by its weight.The reaction forces on the base locators were then measured from the bending type loadcells by connecting their connection sockets to the reader, one by one.At the next step, the clamping force was applied to the workpiece with the intensities of 398 N and 913 N. It should be noted that the top nut of the loadcell was used to apply clamping forces such that no torque was applied to the workpiece.The reaction forces were again measured in all six locators.These forces were measured once immediately after the application of clamping forces and once after a time interval of 5 min.The reason for waiting for this interval during the tests was to give the workpiece-xture system a time to release potential jams that may occur between the in-contact surfaces of the parts.Each experiment was repeated three times and the mean value was recorded as output data.The experimental results were gathered and noted in an Excel worksheet, being compared to the theoretical predictions and numerical results in section 5.

Results And Discussion
The analytical model proposed in Section 2 was implemented in MATLAB software and solved using the fmincon function and the Active-set algorithm.The predictions of reaction forces from the suggested analysis were rst compared to the results presented in a previous study [21].A 3D polyhedral model, which made contact with seven locators within a 4-2-1 layout, was used in the mentioned research to evaluate the capabilities of the suggested theoretical model.Two clamps were applied to the workpiece simultaneously in X and Y directions.The values of the reaction forces in the base locators were very low since the workpiece weight was the only force that was applied to the workpiece in the Z direction.As a result, the workpiece would not have good stability against any potential disturbance that may be applied to it by any external load in the + Z direction (such as the axial component of machining forces in milling operation).In [21], the normal components of reaction forces at the contact points were calculated using an elastic contact model.Also, experimental tests were performed to ensure the accuracy of the predictions of the analytical model.The material of the workpiece, required coe cients, intensity of clamping forces, and position and orientation of application of locators and clamps have been included in [21].These parameters were substituted in the analytical model proposed in the present paper and reaction forces were obtained in the locating points by solving the optimization problem stated in Eq. ( 1).The outcome theoretical predictions were then compared with the analytical and experimental results published in the mentioned research.Figure 4 shows the results of this comparison.A good agreement can be observed between the predictions of reaction forces from the proposed analysis in the present paper with the theoretical and experimental results presented in [21].The low values of the normal components of reaction forces on the base locators are quite reasonable since no clamp was applied to the workpiece in the -Z direction.
Excluding the base locators, the maximum errors in calculating the normal components of reaction forces from the proposed theoretical model in the present paper and the analytical model presented in [21], for three levels of clamping force 127.2 N, 370.6 N, and 631.5N were obtained as 12.8%, 20.3%, and 13.1%, respectively.Also, the maximum error in calculating the same component of reaction forces calculated from the proposed analysis in the present paper and the experimental results reported in [21], with the same levels of clamping forces, were obtained as 18.6%, 11.4%, and 14.8%, respectively.It should be mentioned that the proposed theoretical model in the mentioned research is based on the elastic contact theory, which assumed that the workpiece is a defomable part.In the propsed model of the present study, the workpiece was assumed to be rigid.So, a percentage of error between the predictions of the theoretical models is related to these assumptions.It should be mentioned that the proposed model in the present study outperforms the mentioned model in the previous study from the viewpoints of simplicity, limited number of input parameters, and ease of use for the unexperienced xture designer.The absolute value of error is used to report the errors in calculating the values of reaction forces in the base locators, by paying attention to their low values.The maximum absolute errors in calculating reaction forces on the base locators from the proposed analysis in the present paper and the theoretical predictions presented in [21], with the same levels of clamping forces, were equal to 1.37 N, 2.08 N, and 2.1 N, respectively.The error values in calculating the same parameters between the proposed analysis in the present paper and the experimental results presented in [21] were equal to 0.77 N, 2.82 N, and 3.46 N, respectively.
The normal components of reaction forces obtained from the proposed theoretical model converged to the results presented in [21] by decreasing the value of the coe cient of friction in the suggested model.So, a hypothesis may come to mind that states that the reaction forces predicted by the proposed model always converge to the experimental results by decreasing the coe cient of friction.To validate this hypothesis, it should be noted that the predictions of the theoretical model, presented in Fig. 4, were obtained by substituting µ = 0.01 in the optimization problem presented in Eq. ( 1).The low values of errors between the predicted values with the results of [21] con rmed the correctness of the mentioned hypothesis.As a result, it can be stated that the value of the coe cient of friction must be as low as possible in the utilization of the proposed model to calculate reaction forces with acceptable accuracy.In calculating reaction forces from the proposed theoretical model, the resultant tangential component of the reaction force at each contact point (i.e. ) almost reached its maximum value (μ × R i n ) by substituting the coe cient of friction (µ = 0.25), which is suggested in [21].It indicates that the slip occurred between the workpiece and locators at the contact points.On the other hand, slip can't occur at the contact points of a xture designed for bulk polyhedral parts by paying attention to the high rigidity of the workpiece.It seems that the tangential components of reaction forces calculated from the proposed theoretical model in the present paper are not acceptable, which will be further veri ed using the workpiece model depicted in Fig. 1.
To further study the validity of the mentioned hypothesis, numerical analysis was performed on the workpiece model shown in Fig. 1.The suggested theoretical model was also applied to this workpiece and its predictions were compared with the numerical results.The interval [0.2-0.5] with a step of 0.05 was used to study the effect of the coe cient of friction on the results.The values less than 0.2 were also added to the mentioned range in order to further study the validity of the above-mentioned hypothesis.Three levels of clamping force (0 N, 398 N, and 913 N) were considered in the theoretical model and numerical analysis to study the effect of clamping force on the results.These values were chosen for the clamping force intensity based on the clamping loads that are going to be applied to the workpiece in the experiments.The mass of the workpiece was measured as 2.48 Kg using a digital scale and applied to the theoretical model and numerical analysis.The other input values were used as mentioned in Table 1. Figure 5a shows the comparison between the theoretical predictions and the numerical results with different values of the coe cient of friction.The curves of the theoretical prediction and its corresponding numerical result have been drawn with the same color and marker.
The normal components of reaction forces on locators No. 1, 3, and 5 are lower than 100 N, even by applying the clamping force with the maximum intensity.As a result, the stability of the workpiece is decreased at these contact points.It can be justi ed by taking into account the geometry of the workpiece, the position of these locators, and the position and orientation of application of the clamping force to the workpiece.A large portion of clamping force is tolerated by locators No. 2, 4, and 6 because, in comparison to other locators, the position and orientation of these locators are such that they are opposite to the position and orientation of the clamping force.Applying the second and even third clamping forces to the workpiece in other positions and orientations may increase workpiece stability; however, in addition to adding the effects of the clamping sequence on the values of reaction forces to the problem, it would not be very cost-effective.

√
The theoretical predictions converged to the numerical results as the coe cient of friction decreased in the theoretical model.The theoretical predictions almost fully matched the numerical results with the assumption of frictionless contacts (µ = 0.01) at all locating points.The same results were obtained for the case in which clamping force was applied to the workpiece with an intensity of 398 N. Assuming that the actual coe cient of friction between the workpiece and locators is 0.2, the maximum and average errors of 13% and 10% were obtained between the theoretical prediction (by substituting µ = 0.01) and numerical results.As a result, the normal components of the reaction forces can be calculated accurately from the proposed analysis with the assumption of µ = 0.01.It is worth noting that the theoretical predictions fully matched the numerical results when the clamping force was decreased to zero.In this case, the normal components of reaction forces were obtained the same as 7.86 N, 3.87 N, and 12.54 N from the proposed theoretical model and numerical analysis for the rst, second, and third locators, respectively.
It was stated that the tangential components of the reaction forces obtained from the proposed theoretical model tend to reach the maximum possible values.It should be proved by calculating these components on the workpiece depicted in Fig. 1. Figure 5b shows the numerical results that were obtained for the tangential components of reaction forces on six locating points.The magnitudes of the resultant tangential force vectors tend to reach the maximum possible values in almost all locators, meaning that slip occurred at almost all six locating points.Since slip may rarely happen in xturing a bulk polyhedral part, the tangential components of reaction forces can't be calculated from the proposed theoretical model.The magnitudes of the tangential components of reaction forces are usually low.For the present case study, the numerical results showed that the ratio of the tangential to the normal component is always below 5%.In terms of the design of the xture elements, the normal components of reaction forces apply a high-intensity bending moment on the xture body plates.The bending moments caused by the tangential forces can be neglected by taking into account the low magnitude of these components.
The accuracy of the theoretical prediction was con rmed through the numerical results as well as the results published in the previous studies.Now, the effect of the in uential parameters, including the coe cient of friction and intensity of clamping force, can be studied on the results using the proposed The experimental tests were performed according to the procedure stated in section 4. Figure 6 shows the results of experimental tests with three repetitions, the average value, and an extra re-assembly test with two levels of clamping forces.According to the results, the maximum standard deviations of 10 N and 17 N (equivalent to 1 Kg and 1.7 Kg) were obtained for the data gathered from the initial three tests with the clamping forces 398 N and 913 N, respectively.Also, the average standard deviation in the same results was determined as 8 N for both levels of clamping forces which represents the low scattering of the data gathered from the experiments.The average difference between the results of the re-assembly test and the average value of the initial three tests was equal to 5% and 3% with the clamping forces of 398 N and 913 N, respectively.These values indicate the repeatability of the results of the initial and re-assembly tests.
The average of the three initial tests was used as the main result.Table 2 represents the comparison between the results of the theoretical model, numerical analysis, and experimental tests.The coe cient of friction between the workpiece and locators should be used as input data into the theoretical model and numerical analysis.The measurement of this coe cient between the workpiece and locators is di cult and almost impossible when a spherical-head locator makes contact with the at surface of the workpiece.The range [0.2-0.3] was chosen for the friction coe cient in calculating the reaction forces from the numerical analysis.The value of this coe cient was assumed to be equal to 0.01 in the theoretical model.From the results reported in Table 2, the values of reaction forces obtained from the numerical analysis had very low variations by changing the coe cient of friction in the mentioned range.So, the average value between the three cases was used as the numerical result.The reaction forces were rst measured in the base locators when no clamping force was applied to the workpiece.The experimental results were well matched to the theoretical and numerical results.The maximum errors of 0.48 N and 0.47 N were obtained between the theoretical predictions and numerical results in comparison with the experimental data.The values of reaction forces in the rst, third, and fth locators are low, similar to what was observed in the theoretical predictions and numerical results.It should be noted that these low values are usually unacceptable by the xture designer and an improved clamping layout should be used to increase the reaction force and hence workpiece stability at these points.The adjustment of the height of these locators is of great importance in experiments since any slight variation in the height may change the value of their reaction forces.These locators are named loose locators based on their low values of reaction forces.With the clamping force of 398 N, the maximum errors of 7 N and 10 N were obtained for these locators from the theoretical model and numerical analysis in comparison with the experimental data, respectively.These errors were increased to 19 N and 13 N when the intensity of the clamping force was increased to 913 N. As a whole index, the average errors in predicting the reaction forces in loose locators from the theoretical model and numerical analysis were obtained as 5 N and 4 N, respectively.
The values of reaction forces on locators No. 2, 4, and 6 are high enough such that the workpiece keeps its stability on these contact points.They are named robust locators.Figure 7a represents the error curves between the theoretical, numerical, and experimental results for the robust locators with two levels of clamping forces.The maximum error between the theoretical predictions and experimental results was obtained as 15%.This value was changed to 17% between the numerical and experimental results.The average value of error between the theoretical predictions and numerical results was obtained as 8.5% for the robust locators, being the same between the numerical and experimental results.
Figure 7b shows the results of the mesh independency analysis.A range of [4][5][6][7][8][9][10] mm (equivalent to the number of elements [16914 − 1582]) was chosen for the mesh size to conduct this analysis.The accuracy of the results decreased as the mesh size increased to more than 10 mm.Also, the CPU usage increased drastically by decreasing the mesh size to lower than 4 mm.The curves of Fig. 7b indicate that the numerical results are almost independent of the mesh size of the model.The maximum standard deviation of the analysis results was obtained as 14.7 N with different mesh sizes.The static (or quasistatic) nature of the problem should also be investigated in the numerical analysis.The kinetic and total energy curves in the simulation are shown in Fig. 7c with a clamping force, coe cient of friction, and mesh size of 913 N, 0.25, and 5 mm, respectively.The value of kinetic energy is less than 5% of the total energy throughout the process.Therefore, it can be concluded that the simulation was performed in static conditions and the use of a standard solver seems to be reasonable.
As a result, it can be concluded that the proposed theoretical model could well predict the normal component of reaction forces in the locating points of the xture that was designed for the bulk polyhedral parts.The accuracy of theoretical predictions was veri ed by numerical analysis and experimental tests.The main sources of error in the analytical model were the application of the clamping force as a concentrated load to the workpiece and an assumption that states that the workpiece is rigid.There are some sources of error in the experiments including the adjustment accuracy of the locators' height, the error in positions of locating points, and the accuracy of applying clamping force in the exact position and orientation.Some of these errors may be reduced by repeating tests and re-adjusting the important parameters, while others may affect the measurement results.

Conclusions
In this paper, an agile theoretical model was developed based on the minimum norm principle to calculate the reaction forces in the locating points of the xtures dedicated to the bulk polyhedral parts.The predictions of the theoretical model were validated with the results of previous research, numerical analysis, and experimental tests.A good agreement was observed between the theoretical prediction, the analytical and experimental results of a previous study, numerical results, and the experimental results obtained in the present paper.The theoretical model of the present study outperformed the analytical model in the previous study in terms of ease of use in the early stages of the xture design procedure, prompt and accurate response, and no need for accurate information and geometry of the contact between the workpiece and locators.The main results of the present paper are summarized as follows: The predictions of the proposed theoretical model converged to the numerical results by decreasing the coe cient of friction.A perfect agreement between theoretical predictions and numerical results was obtained when the coe cient of friction μ = 0.01 was used in the proposed theoretical model.
Assuming that the actual coe cient of friction between the workpiece and the locators was equal to 0.2, the maximum and average errors in predicting the normal component of the reaction force from the proposed theoretical model were obtained as 13% and 10% in comparison to the numerical results.
The prediction of the tangential components of reaction forces from the theoretical model was not accurate.Based on the low values of these components, they were less important than the normal ones from the viewpoint of the design of xture elements.
The values of the coe cient of friction (in the range [0.2-0.3]) did not affect the magnitude of the normal components of reaction forces in xturing bulk polyhedral parts.A difference of 3.8% was observed between the numerical results obtained for tangential components by variating the coe cient of friction in the mentioned range.
Experimental ndings con rmed that the locators may be classi ed into loose and robust categories in the xture.The loose locators generated low-intensity reaction forces based on their position and orientation in the xture.The adjustment of the height of these locators is of great importance to measure their reaction forces with appropriate accuracy.The robust locators increased the stability of the workpiece by generating reaction forces with appropriate intensities.
The maximum errors between the theoretical predictions and experimental results were obtained as 19 N and 15% for the loose and robust locators, respectively.These errors were obtained as 13 N and 17% between the numerical and experimental results.
It was nally concluded that the proposed theoretical model can be used as an assistant to the xture designer to quickly calculate the reaction forces on the locating points of xtures for bulk polyhedral parts.The results may be used in locating/clamping layout design, calculating the workpiece deformation, analyzing the workpiece stability, and performing workpiece jamming analysis in the xture.
Further studies may consider evaluating the capabilities of the proposed model in predicting the online values of reaction forces in the locating points by applying the machining loads as dynamic external The authors have no relevant nancial or non-nancial interests to disclose.The assembled and meshed model of the workpiece in nite element software theoretical model.The coe cient of friction does not have a signi cant effect on the normal component of reaction forces in the range of coe cient of friction [0.2-0.3].Figure5ashows that the curves of the theoretical and numerical results in this range, where the coe cient of friction in the xturing systems generally coincides in, are almost linear.Five levels of intensity (200 N, 400 N, 600 N, 800 N, and 1000 N) were chosen to study the effects of the intensity of clamping force on the results.The theoretical model was used to calculate the results for each of these levels, the results of which are represented in Fig.5c.It was assumed that the coe cient of friction was constant and equal to 0.3.The results indicate that the intensity of clamping force has an almost linear effect on the values of the reaction forces.For example, by increasing the clamping forces from level one (200 N) to level three (600 N), an almost three-fold increase has been applied to the normal component of reaction force in the rst locator.This behavior is almost the same in all locators.Therefore, the reaction forces may be calculated from either the proposed theoretical model or the numerical analysis by performing only one run with any reasonable value of the intensity of clamping force.The results can then be used to immediately obtain the reaction forces at all contact points by multiplying the reaction forces by the ratio of new to old clamping force intensities.

Figures
Figures

Table 1
The position and orientation of the locators and clamps in the xture

Table 2
The normal components of reaction forces obtained from the theoretical model, numerical analysis, and experimental tests