Automatic Expanding Mandrel with Air Sensing Device: Design and Analysis

applied sciences

Article Automatic Expanding Mandrel with Air Sensing Device: Design and Analysis

Enrique Soriano Heras 1,* , Higinio Rubio 1 , Alejandro Bustos 1 and Juan Carlos García Prada 2

1 Departamento de Ingeniería Mecánica, Universidad Carlos III de Madrid, Avda. de la Universidad, 30, 28911 Leganés-Madrid, Spain; [email protected] (H.R.); [email protected] (A.B.) 2 Departamento de Mecánica, Universidad Nacional de Educación a Distancia, Juan del Rosal, 12, 28040 Madrid, Spain; [email protected] * Correspondence: [email protected]; Tel.: +34-636-835501

Received: 18 March 2020; Accepted: 3 April 2020; Published: 8 April 2020

Featured Application: This paper describes the development of an automatic collet chuck holder prototype which was designed considering the proposed methodology and analytical models within novel systems in control of workpiece presence, control of workpiece position, control of clamping force and in its structure transmission. Results provide reliable theoretical and technical supports for optimization of the design and application of a collet chuck holder in high performance machining processes, inspection and quality control.

Abstract: In precision machining, expanding mandrels are used for jobs with close tolerances. An expanding mandrel consists of a tapered arbor or shaft, with a thin-slotted clamping sleeve or collet made of hardened steel. The internal tapered and external cylindrical surfaces are ground to a high degree of accuracy, and the mandrel expands to fit the internal bore of the workpiece. Expanding mandrels are, essentially,wedge mechanisms. This paper proposes an automatic expanding mandrel with a novel force transmission system for high stiffness within a novel air sensing system, which allows detection of the correct part position before starting machining. A computational model for determining the dynamic clamping force of the proposed mechanism is developed and implemented using MATLAB. This model considers the influence of the stiffness behaviors of the collet, force transmission structure and workpiece. Additionally, this paper presents the finite element method analyses which were conducted to check the proposed computational model. The amount of clamping force transmitted by a collet chuck holder depends strongly on: clearances, wedge angle, stiffness of the collet chuck holder and workpiece stiffness.

Keywords: expanding mandrel; clamping force; machine tool ﬁxture; assembly design; air sensing

1. Introduction Expanding mandrels are collet chuck holders, which are often used in turning, milling, grinding and inspection. These ﬁxtures make it possible to clamp parts uniformly from the inside, so the outer surface is free to be machined. Commonly, expanding mandrels attached to a turning machine are used to perform secondary machining operations. High-speed cutting technology has increased the spindle rotational speed of machine tools, which implies that expanding mandrels must achieve high rotational speeds and at the same time maintain good rotational accuracy. Most of the expanding mandrels use solid thin-slotted clamping sleeves (collets) made of hardened steel and ground to a high degree of accuracy on their internal tapered and external cylindrical surfaces.

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Tsutsumi [1] measured the clamping force of manual collet chuck holders in a static state by using a strain–gauges-based detector. Nevertheless, it was shown that manual collet chuck holders, with thin-slotted clamping sleeves, have poor repeatability of the clamping force distribution because they are manually operated. Schulz and Rondé [2] introduced a test stand for measuring torque transmitted by collets. They observed that for a degreased tool and collet, and friction coefficient of about 0.25, its uncertainty can be as high as 20%. For these reasons, collet chuck holders for high-speed or high-precision operation should have a wedge-actuated axial tightening system preventing distortions and thus improving concentricity and balancing conditions, as reported by Ema and Marui [3]. Rivin [4] describes a collet chuck design that uses a NiTi bushing in its clamping mechanism. Recently, Malukhin et al. [5] and Shin et al. [6] developed shape–memory alloy (SMA) tool clamping devices for micro machining. The tool clamp is simplified by using a collet with a SMA ring. The number of parts in the collet chuck is reduced and the errors in spindles due to stack-up tolerances are minimized. The analytical models used to explain the clamping–unclamping mechanisms are based on a conventional elasticity problem. Recent advancement in developments of strengthening models for nanostructured materials provide practical large-scale industrial applications in the field of collet chuck holders [7]. Nyamekye and Mudiam [8], Rahman and Tsutsumi [9] and Walter and Stähl [10] proposed models based on solid rigid theory to establish the loss of clamping force in lathe chucks due to centrifugal force and the main cutting force. According to these computational models, the loss of clamping force equals the total centrifugal force of the jaws. Due to the high centrifugal forces on the jaws that occur in high-speed turning, it is necessary to consider the behavior of the stiffness of the clamping systems when conducting the exact computation of the dynamic clamping force. Feng et al. [11] developed a mathematical model that considers the stiffness behaviors of the workpiece and the jaws to calculate the dynamic clamping force of jaw chucks during high-speed turning. This model was verified by means of finite element method (FEM) analyses and experimental investigations. Because a collet chuck is basically a wedge mechanism, the static clamping force for a given acting force to the mechanism depends on the friction coefficients of the tapered and clamping surfaces, on clearance between the collet and the clamped part, and on collet stiffness [12–17]. Most expanding mandrels have complex parts in their transmission structure. This paper presents an automatic expanding mandrel design based on standard parts in its force transmission system (e.g., screws, rings, pins) with high stiffness behavior. The proposed expanding mandrel design provides an air sensing system that can detect the presence and the correct axial position of the workpiece. As the joining of different materials is influenced by a range of parameters which are better recognized, monitored and optimized through both experimental and analytical methods [18,19], this paper presents an analytical model for determining the dynamic clamping force of the proposed expanding mandrel design based on solid rigid theory, elasticity and mechanical contact. The proposed analytical model takes stiffness behaviors into account and was implemented over Matlab. Finally, we present the FEM analyses that were conducted to check the proposed analytical model.

2. Expanding Mandrel Design

2.1. Acting Mechanisms Automatic collet chucks are driven by two mechanisms: by pushing or by pulling. The operating principle of collet chucks is based on the wedge eﬀect, as shown in Figure1. The axial movement of the collet (2) or the cone (4) makes the jaws of the collet expand or contract, thus creating the clamping movement. Moreover, the applied axial force increases the normal force on the collet and thus the friction force and the maximum transmitted torque are also increased. Appl. Sci. 2020, 10, 2551 3 of 17 Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 18

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FigureFigure 1.1. Collet chuck acting by pushing and byby pulling.pulling. 2.2. Structure of an Automatic Expanding Mandrel with Air Sensing System 2.2. Structure of an automatic expanding mandrel with air sensing system Figure2 shows an exploded view of the set in which standard parts (e.g., screws, cotter pins, Figure 2 shows an exploded view of the set in which standard parts (e.g., screws, cotter pins, gaskets) have been excluded. The section and transparent perspective with air sensing system ducts gaskets) have been excluded. The section and transparent perspective with air sensing system ducts are shown. As shown in Figure3, the acting piston (1) is coupled to a double-action cylinder in the are shown. As shown in Figure 3, the acting piston (1) is coupled to a double-action cylinder in the machine by means of an adapter piece (2); both pieces have a central bore which is crossed by an air machine by means of an adapter piece (2); both pieces have a central bore which is crossed by an air lance (3), which ends in a collecting chamber. Ducts from the collecting chamber allow the passage of lance (3), which ends in a collecting chamber. Ducts from the collecting chamber allow the passage compressed air to be used as a signal in the part presence detection and positioning system. The ducts of compressed air to be used as a signal in the part presence detection and positioning system. The end in three small bores at the front of the body (5), which also acts as a stop. ducts end in three small bores at the front of the body (5), which also acts as a stop. The collet (7) slides along the acting cone (6). The acting system by pulling (see Figure3) consists Figure 1. Collet chuck acting by pushing and by pulling. of 3 extracting rings which are mounted to the inside of the rear part of the acting cone (6) with 3 M8 80 cap screws, 12.9 quality, in accordance with ISO 4762 and ISO 898-1, to the acting ring (10), 2.2. Structure× of an automatic expanding mandrel with air sensing system and of the piston cover (9). The rear cover (4), which is coupled to the machine spindle nose via a short taperFigure in accordance 2 shows withan exploded ISO 702-1, view also of limits the set the in stroke which of standard the piston parts cover (e.g., (9). screws, The rear cotter body pins, (11) coversgaskets) the have acting been system. excluded. The section and transparent perspective with air sensing system ducts are shown.When theAs actingshown force in Figure provided 3, the by acting the machine piston (1) actuator is coupled retracts to thea double-action acting piston cylinder (1), the pistonin the covermachine (9) retractsby means the of screws an adapter (M8 piece80), so (2); the both acting piec ringes have (10) retractsa central the bore collet which (7) andis crossed thus, the by partan air is × clamped.lance (3), Whenwhich theends acting in a forcecollecting pushes chamber. the acting Ducts piston from (1), the the collecting piston cover chamber (9) pushes allow the the extracting passage ringsof compressed (8), so the air acting to be ring used pushes as a signal the collet in the (7) andpart thus,presence the partdetection is unclamped. and positioning system. The ducts end in three small bores at the front of the body (5), which also acts as a stop.

Figure 2. Automatic expanding mandrel design.

Figure 2.2. Automatic expanding mandrel design.

Figure 3. Joint view section.

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Figure 1. Collet chuck acting by pushing and by pulling.

2.2. Structure of an automatic expanding mandrel with air sensing system Figure 2 shows an exploded view of the set in which standard parts (e.g., screws, cotter pins, gaskets) have been excluded. The section and transparent perspective with air sensing system ducts are shown. As shown in Figure 3, the acting piston (1) is coupled to a double-action cylinder in the machine by means of an adapter piece (2); both pieces have a central bore which is crossed by an air lance (3), which ends in a collecting chamber. Ducts from the collecting chamber allow the passage of compressed air to be used as a signal in the part presence detection and positioning system. The ducts end in three small bores at the front of the body (5), which also acts as a stop.

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Figure 2. Automatic expanding mandrel design.

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The collet (7) slides along the acting cone (6). The acting system by pulling (see Figure 3) consists of 3 extracting rings which are mounted to the inside of the rear part of the acting cone (6) with 3 M8 x 80 cap screws, 12.9 quality, in accordance with ISO 4762 and ISO 898-1, to the acting ring (10), and of the piston cover (9). The rear cover (4), which is coupled to the machine spindle nose via a short taper in accordance with ISO 702-1, also limits the stroke of the piston cover (9). The rear body (11) covers the acting system. When the acting force provided by the machine actuator retracts the acting piston (1), the piston cover (9) retracts the screws (M8 x 80), so the acting ring (10) retracts the collet (7) and thus, the part is clamped. When the acting force pushes the acting piston (1), the piston cover (9) pushes the FigureFigure 3. 3. JointJoint view section. section. extracting rings (8), so the acting ring pushes the collet (7) and thus, the part is unclamped. Materials Materials of of the the automatic automatic expanding expanding mandrelmandrel main main components components (see (see Figures Figures 2 and2 and 3)3 are) are summarized in Table 1. summarized in Table1.

TableTable 1. 1.Materials Materials ofof thethe expanding mandrel. mandrel. Component Material Heat treatment Mark Component Material Heat Treatment Mark Acting piston 42CrMo4 Tempering 1 Acting piston 42CrMo4 Tempering 1 Adapter 42CrMo4 Tempering 2 Adapter 42CrMo4 Tempering 2 AirAir lance lance C45E C45E - - 3 3 RearRear cover cover 42CrMo4 42CrMo4 Tempering Tempering 4 4 BodyBody 42CrMo4 42CrMo4 Tempering Tempering 5 5 ActingActing cone cone 18CrMo4 18CrMo4 Tempering Tempering cemented cemented 6 6 Collet 50CrV4 Tempering 7 ExtractingCollet rings 50CrV4 18CrMo4 Tempering Tempering 7 8 ExtractingPiston cover rings 18CrMo4 C45E Tempering - 8 9 ActingPiston ring cover 18CrMo4 C45E Tempering - 9 10 RearActing body ring 18CrMo4 C45E Tempering - 10 11 Rear body C45E - 11 3. Mechanical Analysis 3. Mechanical analysis Figure4 shows the sequence of the proposed mechanical analysis. Figure 4 shows the sequence of the proposed mechanical analysis.

FigureFigure 4. 4.Sequence Sequence ofof the mechanical analysis. analysis.

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3.1. Maximum Acting Force Determination

Appl.As Sci. shown2020, 10, inx FOR Figure PEER5, REVIEW the components which are subjected to the maximum tensile stress are5 theof 18 screws (M8 80). Thus, according to ISO 4762, 4010, 4014, 4017, 4032 and 898, the maximum force, × F3.1.acc,max, Maximumwhich shouldacting force be applied determination to each screw can be calculated by Equation (1), where S is the stress area, and σ is the screw’s tensile stress. As shownmax in Figure 5, the components which are subjected to the maximum tensile stress are the screws (M8 × 80). Thus, according to ISO 4762, 4010, 4014, 4017, 4032 and 898, the maximum Facc,max = σmaxS (1) force, Facc,max, which should be applied to each screw can be calculated by Equation (1), where S is the stress area, and σmax is the screw’s tensile stress.

FigureFigure 5. 5.Components Components subjectedsubjected toto thethe maximum maximum tensile tensile stress. stress.

3.2. Maximum Clamping Force Determination =σ FSacc,max max (1) A two-dimensional model is used to mathematically describe the basic principle of automatic chuck holders (see Figure6). This model assumes a uniform distribution of the contact pressure p, a3.2. condition Maximum that clamping is only force satisﬁed determination when both tapered contact surfaces have the same length. Thus, the normal force Fn can be computed as the integral of the contact pressure p between the maximum A two-dimensional model is used to mathematically describe the basic principle of automatic and the minimum radius of the tapered collet surface, as shown in Equation (2). In this equation, rs is chuck holders (see Figure 6). This model assumes a uniform distribution of the contact pressure p, a the maximum radius, ri is the minimum radius and r is the radius of the area element dA. Normal wear condition that is only satisfied when both tapered contact surfaces have the same length. Thus, the (Sn) is assumed to be uniform for all points of the tapered surface. Consequently, the maximum normal force Fn can be computed as the integral of the contact pressure p between the maximum and pressure occurs for the minimum value of the radius ri in Equation (3). the minimum radius of the tapered collet surface, as shown in Equation (2). In this equation, rs is the Z rs Z maximum radius, ri is the minimum radiusp maxandr ir 2isπ ther radi2usπp ofmax theri area element dA. Normal wear Fn = pdA = dr = (rs ri) (2) (Sn) is assumed to be uniform for all points of the tapered surface. Consequently, the maximum ri r sin α sin α − pressure occurs for the minimum value of the radius ri in Equation (3). r p = p i (3) max r

The acting force Fd can be computed from Figure6 according to Equation (4).

F = Fn sin α = 2πpmaxr (rs r ) (4) d i − i The second contact occurs between the collet and the workpiece as shown in Figure7. In order to compute the forces and the slippage moment, it is considered that the pressure transmitted for each jaw is equal and constant. The clamping force Fs is calculated in Equation (5) and used in Equation (6) to obtain the frictional force FR and in Equation (7) to obtain the total force FT. The slippage moment

Figure 6. Cone–collet contact pressure.

rs pr2πr 2π pr F ==pdAmaxii dr = max () r − r nsiαα (2) ri r sin sin

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3.1. Maximum acting force determination As shown in Figure 5, the components which are subjected to the maximum tensile stress are the screws (M8 × 80). Thus, according to ISO 4762, 4010, 4014, 4017, 4032 and 898, the maximum force, Facc,max, which should be applied to each screw can be calculated by Equation (1), where S is the stress area, and σmax is the screw’s tensile stress.

Appl. Sci. 2020 10 , , 2551 6 of 17 Figure 5. Components subjected to the maximum tensile stress. Mt is calculated by using Equations (8) and (9), where µcw is the friction coeﬃcient between the collet and the workpiece. =σ FSacc,max max (1) Z F(θ) Z F(θ) Z θ Fs = cos βdF = 2 cos βdF = 2 pbR cos βdβ =2pbR sin θ (5) F( θ) 0 0 3.2. Maximum clamping force− determination F = µ F = 2µ pbR sin θ = 2 tan ϕpbR sin θ (6) A two-dimensional modelR is cwuseds to mathematicallycw describe the basic principle of automatic chuck holders (see Figure 6). This modelq assumes a Funiform2 pbRdistributionsin θ of the contact pressure p, a F = F2 + F2 = s = (7) condition that is only satisfied whenT boths taperedR cos contactϕ surfacescos ϕ have the same length. Thus, the normal force Fn can be computed as the integral of the contact pressure p between the maximum and dFR = µcwdFs = 2 tan ϕpbRs cos βdβ (8) the minimum radius of the tapered collet surface, as shown in Equation (2). In this equation, rs is the Z F(θ) Z θ maximum radius, ri is the minimum radius and2 r is the radius of the area element2 dA. Normal wear Appl. Sci. 2020, 10, x FOR PEERMt REVIEW= RdFR = R pb tan ϕ cos βdβ =2 tan ϕpbR sin θ (9)6 of 18 (Sn) is assumed to be uniformF( θ )for all pointsθ of the tapered surface. Consequently, the maximum − − pressure occurs for the minimum value of the radius ri in Equation (3). r pp= i (3) max r

The acting force Fd can be computed from Figure 6 according to Equation (4). ==απ − FFdnsin 2 prrrmax isi() (4)

The second contact occurs between the collet and the workpiece as shown in Figure 7. In order to compute the forces and the slippage moment, it is considered that the pressure transmitted for each jaw is equal and constant. The clamping force Fs is calculated in Equation (5) and used in Equation (6) to obtain the frictional force FR and in Equation (7) to obtain the total force FT. The Figure 6. Cone–collet contact pressure. slippage moment Mt is calculated byFigure using 6. Cone–collet Equations contact (8) and pressure. (9), where μcw is the friction coefficient between the collet and the workpiece. rs pr2πr 2π pr F ==pdAmaxii dr = max () r − r nsiαα (2) ri r sin sin

FigureFigure 7. 7.ContactContact pressure pressure and and forces onon collet collet workpiece. workpiece.

The total clamping force, Fs in Equation (5), needed to clamp the workpiece sufficiently should θθθ also be of suffi===cientFF() magnitudeββ to resist () slippage of the clamped workpiece ββ = underθ the influence of the Fs −θ cos dF 2 cos dF 2 pbR cos d 2 pbR sin (5) torque, Mt in EquationF() (10), generated 0 by the expected 0 main cutting force, Fc in Figure8. The feeding force Fv is mainly absorbed by the mandrel body, which acts as a stop and reference surface, and the force Fp is absorbed by==μμ the collet. In this study,θ =the effectsϕ of theseθ two forces are neglected. FRcwscw F2sin2tansin pbR pbR (6) DA Mt = Fc (10) 2 θ =+==22Fs 2sinpbR FFFTsR (7) cosϕϕ cos

==μϕββ dFRcws dF2tan pbR scos d (8)

θθ F() 22 M== RdF R pbtanϕββ cos d = 2tan ϕ pbR sinθ (9) tRF()−−θθ

The total clamping force, Fs in Equation (5), needed to clamp the workpiece sufficiently should also be of sufficient magnitude to resist slippage of the clamped workpiece under the influence of the torque, Mt in Equation (10), generated by the expected main cutting force, Fc in Figure 8. The feeding force Fv is mainly absorbed by the mandrel body, which acts as a stop and reference surface, and the force Fp is absorbed by the collet. In this study, the effects of these two forces are neglected. D MF= A (10) tc2

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Figure 8. Forces acting in collet chuck holders.

3.3. Tolerance analysis Figure 8. Forces acting in collet chuck holders. The radial deflection ofFigure the 8.springForces collet acting injaws collet is chuckso small holders. that it is comparable to the 3.3.manufacturing Tolerance Analysisanalysis tolerances. Therefore, the tolerances must be known precisely to predict the clamping force. TheThe radial tolerancesradial deﬂection deflection of the of workpiece theof springthe spring and collet the jawscollet collet is jaws so define small is the thatso maximumsmall it is comparable that and it isminimum tocomparable the manufacturing clearances. to the tolerances.manufacturingThe relationships Therefore, tolerances. to calculate the tolerances Therefore, these tolerances must the be tolera known are expressednces precisely must in toEquationsbe predict known the(11) precisely clamping and (12). to However, force. predict thethe clampingoutsideThe diameter tolerances force. of ofthe the collet workpiece must be and equal the to collet the inside deﬁne diameter the maximum of the clamped and minimum workpiece clearances. in its Theworking relationshipsThe tolerancesstate (see to Figure of calculate the 9).workpiece these tolerances and the collet are expressed define the in maximum Equations and (11) minimum and (12). clearances. However, theThe outside relationships diameter to calculate of the collet these must tolerances be equal are to expressed the inside diameterin Equations of the (11) clamped and (12). workpiece However, in the its ()DT+−+() DT workingoutside diameter state (see of Figure the =collet9). pp must,max be equal ww to th ,mine inside diameter of the clamped workpiece(11) in its Cmax working state (see Figure 9). 2 Dp + Tp,max (Dw + Tw,min) ()DT+−+Cmax = () DT − (11) = pp,max ww ,min 2 (11) Cmax ()DT+−+() DT = pp,min2 ww ,max (12) Cmin 2Dp + Tp,min (Dw + Tw,max) Cmin = − (12) +−+2 ()DTpp,min() DT ww ,max C = (12) min 2

FigureFigure 9. Tolerances of the collet and the the workpiece. workpiece.

3.4.3.4. ColletCollet Initialinitial Deﬂectiondeflection

EachEach jawjaw isis fixedfixed toto thetheFigure rootroot 9. of ofTolerances thethe collet,collet, of the so itcollet can andbe considered the workpiece. as as a a cantilever cantilever as as shown shown in in FigureFigure 10 10.. ByBy usingusing classicalclassical elasticityelasticity theories, the maximal deflection deflection εε producedproduced by by the the clamping clamping force3.4.force ColletF Fs sis is initial estimated estimated deflection (see (see Equation Equation (13)). (13)). The The radial radial sti stiffnessffness of of the the collet, collet,kR ,k isR, givenis given by Equationby Equation (14). Equation(14). Equation (15) gives (15) gives the expression the expressi toon compute to compute the clampingthe clamping force forceFs , F whichs2, which is used is used to deformto deform the Each jaw is fixed to the root of the collet, so it can be considered2 as a cantilever as shown in collet.the collet. This This allows allows one one to calculate to calculate the neededthe needed increase increase in the in the acting acting force force using using Equation Equation (4). (4). Figure 10. By using classical elasticity theories, the maximal deflection ε produced by the clamping

force Fs is estimated (see Equation (13)). The radialF L stiffness4 of the collet, kR, is given by Equation ε = s (13) (14). Equation (15) gives the expression to compute8EI the clamping force Fs2, which is used to deform the collet. This allows one to calculate the needed increase in the acting force using Equation (4).

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8EI kR = (14) L4

Fs2 = CmaxkR (15) Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 18

Figure 10. Equivalent representation of a jaw of the collet.

F L4 ε = s (13) 8EI

= 8EI kR (14) L4 Figure 10. Equivalent representation of a jaw of the collet.

3.5. Collet Chuck Stiffness Due to Clamping Force F = Ck (15) s2maxF L4R The total clamping force generates a lateralε = bends and a radial deflection at the clamping(13) position. 8EI Since a collet chuck holder is mainly composed of an acting cone, a collet and a piston, the stiffness behavior3.5. Collet of chuck the colletstiffness chuck due to transmission clamping force system can be described with two parameters each for the collet chuck and the collet stiffness: 8EI The total clamping force generates a klateral= bend and a radial deflection at the clamping R 4 (14) position. Since a collet chuck holder is mainly composedL of an acting cone, a collet and a piston, the 1. Radial stiffness of chuck acting system, kr,s in N/µm, measured at its center position. The chuck stiffnessacting behavior system comprisesof the collet a pistonchuck andtransmission a cone. system can be described with two parameters = each for the collet chuck and the collet stiffness:Fs2maxCk R (15) 2. Bending stiffness of chuck acting system, kf,s in Nm/µm/m. 1. Radial stiffness of chuck acting system, kr,s in N/μm, measured at its center position. The chuck 3. Radial stiffness of collet jaw, kr,p in N/µm, measured at the bottom position of collet jaw. acting system comprises a piston and a cone. 4. Bending stiffness of collet jaw, kf,p in Nm/µm/m. 3.5.2. ColletBending chuck stiffness stiffness of due chuck to clamping acting system,force kf,s in Nm/μm/m. 3. Radial stiffness of collet jaw, kr,p in N/μm, measured at the bottom position of collet jaw. The radialtotal clamping deflection force at the generates clamping a positionlateral bend due toand the a totalradial clamping deflection force at involvesthe clamping four 4. Bending stiffness of collet jaw, kf,p in Nm/μm/m. components,position. Since shown a collet in chuck Figure holder11: is mainly composed of an acting cone, a collet and a piston, the The radial deflection at the clamping position due to the total clamping force involves four stiffness behavior of the collet chuck transmission system can be described with two parameters components,Radial deflection shown in of Figure collet chuck11: holder, ε1. •each for the collet chuck and the collet stiffness: • RadialRadial deflectiondeflection due of collet to the chuck bending holder, of collet ε1. chuck holder, ε2. •1. Radial stiffness of chuck acting system, kr,s in N/μm, measured at its center position. The chuck • RadialRadial deflectiondeflection of due collet to the jaw bending relative of to collet chuck chuck acting holder, system, ε2ε. 3. • acting system comprises a piston and a cone. • RadialRadial deflectiondeflection due of collet to the jaw bending relative of to collet chuck jaw acting relative system, to chuck ε3. acting system, ε4. •2. Bending stiffness of chuck acting system, kf,s in Nm/μm/m. • Radial deflection due to the bending of collet jaw relative to chuck acting system, ε4. 3. Radial stiffness of collet jaw, kr,p in N/μm, measured at the bottom position of collet jaw. 4. Bending stiffness of collet jaw, kf,p in Nm/μm/m. The radial deflection at the clamping position due to the total clamping force involves four components, shown in Figure 11: • Radial deflection of collet chuck holder, ε1. • Radial deflection due to the bending of collet chuck holder, ε2. • Radial deflection of collet jaw relative to chuck acting system, ε3. • Radial deflection due to the bending of collet jaw relative to chuck acting system, ε4.

Figure 11. Deformation of collet chuck due to total clamping force.

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The radial stiﬀness at the clamping position, kr,T, is computed by using the Equation (16).

Fs kr,T = (16) ε1 + ε2 + ε3 + ε4

Deﬂections at the clamping position can be analytically determined by the equations summarized in Table2. The distances dp and dam are also shown in Figure 11.

Table 2. Deﬂections of collet chuck due to clamping force.

Deﬂection Equation

Fs ε1 kr,s F (d +d ) γ s p am 1 k f ,s dp ε2 γ1 2 + dam

Fs ε3 kr,p

Fsdam γ2 k f ,p γ ε4 2dam

Substituting the equations summarized in Table2 into Equation (16) yields Equation (17).

F = s kr,T d (17) p + + Fs 2 dam (dp dam) F d d Fs + + Fs + s p am kr,s k f ,s kr,p k f ,p

3.6. Collet Chuck Deﬂection Due to Centrifugal Force

The variation in the clamping force, Fs3, due to centrifugal force Fcen on each of the collet jaws, and on the transmission system, is calculated by Equation (18), where Fse is the preset static clamping force before the collet chuck has been rotated, calculated using Equations (5), (13) and (15); “+” indicates external clamping and “ “ indicates internal clamping. −

Fs = Fse Fcen (18) 3 ±

As shown in Figure 12, the centrifugal force Fi of the element “i” is computed as a function of the rotational speed n, the mass mi and the radius of the mass center rci, where i = cn, pt or pn (which represent transmission system, piston, and collet, respectively), by Equation (19).

πn2 F = m r (19) i i i 30

Due to the different positions of the components, the centrifugal forces cause different radial stiffness and thus, a radial deflection at the clamping position. As such, the total radial stiffness of the collet chuck for each centrifugal force, kr,Tj, is given by Equations (20)–(22).

Fpn kr,T1 = (20) ε1 + ε2 + ε3 + ε4

Fcn kr,T2 = (21) ε1 + ε2

Fpt kr,T3 = (22) ε1 + ε2 Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 18

Due to the different positions of the components, the centrifugal forces cause different radial stiffness and thus, a radial deflection at the clamping position. As such, the total radial stiffness of the collet chuck for each centrifugal force, kr,Tj , is given by Equations (20)–(22).

Fpn k = rT,1 ε+ε+ε+ε (20) 1234

F k = cn rT,2 ε+ε (21) 12

Appl. Sci. 2020, 10, 2551 10 of 17 Fpt k = rT,3 ε+ε (22) The diﬀerent amounts of deﬂection at each clamping12 position caused by centrifugal forces can be determined by the equations summarized in Table3.

FigureFigure 12. 12.Centrifugal Centrifugal forces forces and and distances distances at at clamping clamping position. position.

Table 3. Deﬂections of collet chuck due to centrifugal forces. The different amounts of deflection at each clamping position caused by centrifugal forces can be determinedDeﬂection by the equations Equation summarized Collet Equationin Table 3. Transmission Equation Piston

Fpt Substituting the equationsFpn summarized in Table F3cn into Equations (20)–(22) yields Equations ε1 kr,pt (23)–(25). kr,pn kr,cn Fpt Fpn(dp+dam) Fcn(dp+dcn) γ k 1 k k f ,pt f ,pn f ,cn dp dp dp γ1 + dam ε2 γ1 2 + dam γ1 2 + dam 2 F ε pn – – 3 kr,pn F d γ pn am – – 2 k f ,pn – ε4 γ2dam – Appl. Sci. 2020, 10, 2551 11 of 17

Substituting the equations summarized in Table3 into Equations (20)–(22) yields Equations (23)–(25). 1 kr,T1 = (23) d2+d d +2d d +2d d 1 + p pn p pn am p am + 1 + dpndam kr,s 2k f ,s kr,p k f ,p 1 kr,T2 = (24) d2+d d +2d d +2d d 1 + p cn p cn am pn am kr,s 2k f ,s 1 kr,T3 = (25) d2+2d2 1 + p am + 3damdp kr,pt k f ,pt 2k f ,s

By assuming the loss of clamping forces ∆Fs and the centrifugal forces Fi are always equal on the collet side and on the workpiece, we have Equation (26).

∆F Fpn F Fpt F s = + cn + + se (26) kw kr,T1 kr,T2 kr,T3 kr,T

Figure 13 shows the diagram used to determine the variance in the clamping force depending on the stiﬀness of the collet chuck or expanding mandrel and the workpiece. Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 18

FigureFigure 13.13. Diagram forfor determiningdetermining thethe variancevariance inin thethe clampingclamping force.force.

InCollet, Equation transmission, (26) and in piston Figure and 13, k contactw is the radialstiffnes workpieces between sti theffness. workpiece In addition and tocollet the ideal jaws radial were stideterminedffness kw1, using the contact the finite stiff nesselement between (FE) models the collet proposed jaws and in the this workpiece paper. kk must be considered as shown in Equation (27). 4. FE analysis of clamping force 1 kw = (27) k 1 + k 1 For performing the simulations using FEM,w− 12 commercialk− packs (ALGOR and CATIA) were usedCollet, considering transmission, all parts, piston the collet, and acting contact cone stiff andness workpiece between the shown workpiece in Figure and 14a. collet The jaws 3D model were determinedwas homogeneously using the meshed finite element with 10-node (FE) models tetrahedra proposedl elements in this paper.shown in Figure 14b. The types of steel chosen are summarized in Table 1. For the sake of simplicity, we considered the materials were 4.perfectly FE Analysis plastic of and Clamping used the Force von Mises criterion. A Cartesian coordinate system (X, Y, Z) was used as shownFor performing in Figure 14c,d. the simulations All simulations using were FEM, performed 2 commercial considering packs (ALGOR an expanding and CATIA) mandrel were driven used consideringby pulling, with all parts, dimensions the collet, shown acting in Figure cone and15. workpiece shown in Figure 14a. The 3D model was homogeneouslyTwo different types meshed of withsimulations 10-node were tetrahedral carried elements out. The shown first case, in Figure without 14b. rotation, The types was of steelanalyzed chosen where are summarizedthe collet and in acting Table cone1. For were the sakeplaced of in simplicity, the final weclamped considered position. the The materials simulations were perfectlywere carried plastic out and considering used the vonCoulomb Mises criterion.friction between A Cartesian the two coordinate bodies, systemcharacterized (X, Y, Z) by was a dynamic used as friction coefficient, μ = 0.09. Thus, nonlinear analysis must be considered in numerical simulations due to the contact nature of the problem or the possibility of plastic deformation during the clamping process. The second type of simulation, shown in Figure 16, corresponds to the dynamic process with rotational speed n = 2000 r/min. For both types of simulations, the acting forces Fj, applied in the longitudinal direction of the collet, varied from 25 to 9000 N, and the end of the acting cone was fixed as shown in Figure 14c. The 3D FE models consisted of a workpiece, collet and cone. The workpiece stiffness could be changed according to its material properties. The contact stiffness between the workpiece and collet jaws was computed and simulated using contact elements.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 18

Figure 13. Diagram for determining the variance in the clamping force.

Collet, transmission, piston and contact stiffness between the workpiece and collet jaws were determined using the finite element (FE) models proposed in this paper.

4. FE analysis of clamping force For performing the simulations using FEM, 2 commercial packs (ALGOR and CATIA) were used considering all parts, the collet, acting cone and workpiece shown in Figure 14a. The 3D model was homogeneously meshed with 10-node tetrahedral elements shown in Figure 14b. The types of steel chosen are summarized in Table 1. For the sake of simplicity, we considered the materials were perfectly plastic and used the von Mises criterion. A Cartesian coordinate system (X, Y, Z) was used as shown in Figure 14c,d. All simulations were performed considering an expanding mandrel driven by pulling, with dimensions shown in Figure 15. Two different types of simulations were carried out. The first case, without rotation, was analyzed where the collet and acting cone were placed in the final clamped position. The simulations were carried out considering Coulomb friction between the two bodies, characterized by a dynamic friction coefficient, μ = 0.09. Thus, nonlinear analysis must be considered in numerical simulations due to the contact nature of the problem or the possibility of plastic deformation during the clamping process. The second type of simulation, shown in Figure 16, corresponds to the dynamic process with rotational speed n = 2000 r/min. For both types of simulations, the acting forces Fj, applied in the Appl.longitudinal Sci. 2020, 10 direction, 2551 of the collet, varied from 25 to 9000 N, and the end of the acting cone12 ofwas 17 fixed as shown in Figure 14c. The 3D FE models consisted of a workpiece, collet and cone. The workpiece stiffness could be shownchanged in according Figure 14c,d. to its All material simulations properties. were performed The contact considering stiffness between an expanding the workpiece mandrel and driven collet by pulling,jaws was with computed dimensions and simulated shown in Figureusing contact 15. elements.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 18

FigureFigure 14. 14.Proposed Proposed FiniteFinite Element model model in in ALGOR. ALGOR.

FigureFigure 15. 15.Special Special expanding expanding mandrelmandrel dimensions:dimensions: (a) (a) collet, collet, (b) (b )acting acting system. system.

Two different types of simulations were carried out. The first case, without rotation, was analyzed where the collet and acting cone were placed in the final clamped position. The simulations were carried out considering Coulomb friction between the two bodies, characterized by a dynamic friction coefficient, µ = 0.09. Thus, nonlinear analysis must be considered in numerical simulations due to the contact nature of the problem or the possibility of plastic deformation during the clamping process. The second type of simulation, shown in Figure 16, corresponds to the dynamic process with rotational speed n = 2000 r/min. For both types of simulations, the acting forces Fj, applied in the longitudinal direction of the collet, varied from 25 to 9000 N, and the end of the acting cone was fixed as shown in Figure 14c. The 3D FE models consisted of a workpiece, collet and cone. The workpiece stiffness could be changed according to its material properties. The contact stiffness between the workpiece and collet jaws was computed and simulatedFigure 16. using Proposed contact finite elements. element model in CATIA.

5. Evaluation Several experiments were conducted to verify the feasibility of the automatic expanding mandrel.

5.1. Collet initial deflection Figure 17 compares the amount of the acting force needed to deform the collet obtained from the FE model and from Equations (11) and (12). The value of the collet radial stiffness, kR, can be obtained from Equation14 or from the slope of the resulting straight.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 18

Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 18 Figure 14. Proposed Finite Element model in ALGOR.

Figure 14. Proposed Finite Element model in ALGOR.

Figure 15. Special expanding mandrel dimensions: (a) collet, (b) acting system. Appl. Sci. 2020, 10, 2551 13 of 17 Figure 15. Special expanding mandrel dimensions: (a) collet, (b) acting system.

Figure 16. Proposed finite element model in CATIA. ® Figure 16. Proposed finiteFigure element 16. Proposed model in finite CATIA element. Results model of in (a CATIA.) stress and (b) displacements. 5. Evaluation 5. Evaluation 5. EvaluationSeveral experiments were conducted to verify the feasibility of the automatic expanding mandrel.Several experiments were conducted to verify the feasibility of the automatic expanding mandrel. Several experiments were conducted to verify the feasibility of the automatic expanding mandrel. 5.1. Collet initial Initial deflection Deflection 5.1. ColletFigure initial 1717 comparescompares deflection thethe amount amount of of the the acting acting force forc needede needed to to deform deform the the collet collet obtained obtained from from the theFE modelFE model and fromand from Equations Equations (11) and (11) (12). and The (12). value The ofvalue the colletof the radial collet sti radialffness, stiffness,kR, can be kR obtained, can be Figure 17 compares the amount of the acting force needed to deform the collet obtained from obtainedfrom Equation from Equation14 (14) or from or the from slope the of slope the resulting of the resulting straight. straight. the FE model and from Equations (11) and (12). The value of the collet radial stiffness, kR, can be obtained from Equation14 or from the slope of the resulting straight.

Figure 17. Collet deﬂections: analytical and ﬁnite element analysis.

5.2. Expanding Mandrel Deflection The loss of clamping force, taking into account the influence of the collet chuck and workpiece stiffness derived from Equations (26) and (27), is given by Equation (28).

F ∆F = se (28) s k 1 r,T − kw A dimensionless parameter ψ is deﬁned according to Equation (29) for better comprehension of the analysis results. kr,T ψ = (29) kw Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 18

Figure 17. Collet deflections: analytical and finite element analysis.

5.2. Expanding mandrel deflection The loss of clamping force, taking into account the influence of the collet chuck and workpiece stiffness derived from Equations (26) and (27), is given by Equation (28).

Fse Δ=F s k 1− rT, (28) kw

A dimensionless parameter ψ is defined according to Equation (29) for better comprehension of the analysis results.

krT, ψ= (29) Appl. Sci. 2020, 10, 2551 kw 14 of 17

5.3.5.3. Variance Variance of of the the clamping Clamping force Force

FigureFigure 1818 shows the influenceinfluence ofof thethe expanding expanding mandrel mandrel sti stiffnessffness kr ,kTr,T. An . An increase increase in inkr ,Tkr,Tresults results in inan an increase increase in radialin radial stiff nessstiffness at the at clamping the clamping position position and therefore, and therefore, the chucking the chucking accuracy isaccuracy improved. is improved.Meanwhile, Meanwhile, greater expanding greater mandrel expanding stiffness mandrel contributes stiffness to a contributes greater PSI factor,to a greater resulting PSI in afactor, minor resultingeffective centrifugalin a minor forceeffective at the centrifugal clamping force position. at the As clamping such, it is position. obvious As that such, an increase it is obvious in kr,T thatalways an increasehas influence in kr,Tin always reducing has theinfluence variance in reducing of clamping the force.variance The of radial clamping stiffness force. of theThe collet radial chuck stiffness can of be theeffectively collet chuck changed can by be optimizing effectively the changed chuck structure,by optimizing particularly the chuck the structure structure, of theparticularly transmission the structuresystem. However, of the transmission it should be system. taken into However, account it that should the lower be taken the workpiece into account stiff ness,that the the lower higher the the workpiecevariance of stiffness, the clamping the higher force. the variance of the clamping force.

FigureFigure 18. 18. InfluenceInﬂuence of of the the stiffness stiﬀness of of expanding expanding mandrel mandrel and and workpiece. workpiece.

FigureFigure 1919 shows shows the the computation computation of the of clampingthe clampi forceng variance,force variance, taking intotaking account into the account expanding the expandingmandrel prototype mandrel stiprototypeffness as stiffness shown inas Figureshown 20 in. Figure Low workpiece 20. Low workpiece stiffness and stiffness high expanding and high expandingAppl.mandrel Sci. 2020 sti mandrel,ff 10ness, x FOR provide stiffness PEER REVIEW a reduction provide a in reductio the variancen in the of variance the clamping of the force. clamping force. 15 of 18

Figure 19. Variance of clamping force. Figure 19. Variance of clamping force.

Figure 20. Expanding mandrel prototype within the air sensing system.

5.4. Stress throughout contact between cone and collet A dimensionless parameter t is defined according to Equation (30) for a better comprehension of the analysis results, where La is the cone length in contact with the collet at any time, and Lb is the total length of the collet. Obviously, 0 ≤ t ≤ 1.

La τ= (30) Lb

The maximum value of the ratio, defined by Equation (31), where σvm is the von Mises stress and Si is the cone and collet yield stress, would not be more than 1. σ vm,max ≤ 1 (31) Si

As can be observed from the results of the stress experiment shown in Figure 21, at points placed at the edge of the collet´s lateral holes, there is a sudden increase in their von Mises ratio. At other points, the collet and the cone do not exhibit such sharp changes.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 18

Appl. Sci. 2020, 10, 2551 15 of 17 Figure 19. Variance of clamping force.

FigureFigure 20. 20. ExpandingExpanding mandrel mandrel prototype prototype within within the the air air sensing sensing system. system.

5.4.5.4. Stress Stress throughout throughout contact Contact between between cone Cone and and collet Collet AA dimensionless dimensionless parameter parameter tt is defined deﬁned according toto EquationEquation (30)(30) for for a a better better comprehension comprehension of ofthe the analysis analysis results, results, where whereLa Lais theis the cone cone length length in contactin contact with with the the collet collet at any at any time, time, and andLb is L theb is totalthe totallength length of the of collet.the collet. Obviously, Obviously, 0 t0 ≤1. t ≤ 1. ≤ ≤ L τ = a (30) La L τ= b (30) L The maximum value of the ratio, deﬁnedb by Equation (31), where σvm is the von Mises stress and Si is the cone and collet yield stress, would not be more than 1. The maximum value of the ratio, defined by Equation (31), where σvm is the von Mises stress σvm,max and Si is the cone and collet yield stress, would not be more1 than 1. (31) S ≤ σ i vm,max ≤ 1 As can be observed from the results of the stress experiment shown in Figure 21, at points(31) placed at Si the edge of the collet’s lateral holes, there is a sudden increase in their von Mises ratio. At other points, Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 18 the colletAs can and be the observed cone do from not exhibit the results such sharpof the changes.stress experiment shown in Figure 21, at points placed at the edge of the collet´s lateral holes, there is a sudden increase in their von Mises ratio. At other points, the collet and the cone do not exhibit such sharp changes.

Figure 21. von Mises stress distribution ratio for cone–collet. Figure 21. von Mises stress distribution ratio for cone–collet.

5.5. Stress throughout contact between collet and workpiece For the von Mises stress, the simulation has led to the distribution shown in Figure 22. For the workpiece, the evolution of the von Mises stress with the clamping process is highly uniform, and the maximum values are far from the material yield stress at any time. For the collet, the stresses generated by the clamping process appear at the closest zone to the extreme of the collet and at the extremes of the collet holes. Stresses are not significant at any point by the end of the clamping process.

Figure 22. von Mises stress throughout contact between collet and workpiece.

6. Conclusions In this paper, we present an automatic expanding mandrel with a novel force transmission system, with high stiffness using a novel air sensing system. We have also developed a computational model implemented in Matlab for determining the dynamic clamping force of the proposed design, which takes into account the influence of the stiffness behaviors of the collet, force transmission structure and workpiece.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 18

Appl. Sci. 2020, 10, 2551 Figure 21. von Mises stress distribution ratio for cone–collet. 16 of 17

5.5. Stress throughout contact between collet and workpiece 5.5. Stress throughout Contact between Collet and Workpiece For the von Mises stress, the simulation has led to the distribution shown in Figure 22. For the workpiece,For the the von evolution Mises stress, of the the von simulation Mises stress has ledwith to the the clamping distribution process shown is inhighly Figure uniform, 22. For and the workpiece,the maximum the evolutionvalues are of far the from von Misesthe material stress withyield the stress clamping at any process time. For is highly the collet, uniform, the andstresses the maximumgenerated valuesby the areclamping far from process the material appear yield at the stress closest at any zone time. to Forthe theextreme collet, of the the stresses collet and generated at the byextremes the clamping of the processcollet holes. appear Stresses at the closestare not zone significant to theextreme at any point of the by collet the and end at of the the extremes clamping of theprocess. collet holes. Stresses are not signiﬁcant at any point by the end of the clamping process.

Figure 22. von Mises stress throughout contact between colletcollet andand workpiece.workpiece. 6. Conclusions 6. Conclusions In this paper, we present an automatic expanding mandrel with a novel force transmission In this paper, we present an automatic expanding mandrel with a novel force transmission system, with high stiffness using a novel air sensing system. We have also developed a computational system, with high stiffness using a novel air sensing system. We have also developed a model implemented in Matlab for determining the dynamic clamping force of the proposed design, computational model implemented in Matlab for determining the dynamic clamping force of the which takes into account the influence of the stiffness behaviors of the collet, force transmission proposed design, which takes into account the influence of the stiffness behaviors of the collet, force structure and workpiece. transmission structure and workpiece. The amount of clamping force transmitted by a collet chuck holder depends strongly on:

Clearances determined by the tolerances of the collet and the workpiece as well as of the collet’s • initial static stiffness. Wedge angle. Decreasing the wedge angle will increase the mechanical advantage and thus the • transmitted clamping force; however, this will result in an exponential increase in the tension experienced by the collet. Stiffness of the collet chuck holder. A collet chuck holder with higher structural stiffness, • particularly in its force transmission system, requires less acting force and therefore has a more effective force transmission system. Workpiece stiffness. Workpieces with lower stiffness reduce the loss in clamping force provided • by collet chuck holders.

7. Patents A Spanish patent resulting from the air sensing device reported in this manuscript is under number ES2413910. Appl. Sci. 2020, 10, 2551 17 of 17

Author Contributions: The authors made most of the contributions regarding conceptualization, development of theory, validation, verification of the analytical models, discussion of the results, as well as the final manuscript. Individual contributions are as follows: introduction, methodology, visualization and original draft preparation: E.S.H., H.R. and J.C.G.P.; review, editing, validation and formal analysis: E.S.H., H.R. and A.B.; supervision: J.C.G.P. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.

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