Numerical and Experimental Analysis of Torsion Springs Using NURBS Curves

applied sciences

Article Numerical and Experimental Analysis of Torsion Springs Using NURBS Curves

Young Shin Kim 1, Yu Jun Song 2 and Euy Sik Jeon 3,*

1 Industrial Technology Research Institute, Kongju National University, Cheonan-daero, Seobuk-gu, Cheonan-si 31080, Chungcheongnam-do, Korea; [email protected] 2 International Testing and Evaluation Laboratory (ITEL), 53, Osongsaengmyeong 10-ro, Osong-eup, heungdeok-gu, Cheongju-si 28164, Chungcheongbuk-do, Korea; [email protected] 3 Department of Mechanical Engineering, Graduate School, Kongju National University, Cheonan-daero, Seobuk-gu, Cheonan-si 31080, Chungcheongnam-do, Korea * Correspondence: [email protected]; Tel.: +82-41-521-9284

Received: 28 March 2020; Accepted: 7 April 2020; Published: 10 April 2020

Abstract: Torsion springs, which transfer power through the twisting of their coil, provide advantages such as module simplification and efficient use of space. The design of a torsion spring has been formulated, but it is difficult to determine the local behaviors of torsion springs according to actual load conditions. This study proposes a torsion-spring design method through finite element analysis (FEA) using nonuniform-rational-basis-spline (NURBS) curves. Through experimentation, the angle and displacement values for the actual spring load were converted into useable data. Torsion-spring displacement values were obtained via experimentation and converted into coordinates that may be expressed using NURBS curves. The results of these experiments were then compared to those obtained via FEA, and the validity of this method was thereby verified.

Keywords: torsion springs; FEA; NURBS; applied load; local behaviors

1. Introduction Torsion springs transfer power through the twisting of their coil, and provide advantages such as module simplification, efficient use of space, and reduction of overall product weight. As such, they have been widely used in various electronic products and industrial fields, including automobiles and machinery. Research on these springs began in 1963 with Wahl’s study on isotropic spiral compression and tension, and torsion springs [1,2]. Thus far, the spring design formula proposed by Wahl is used as a standard and has been extensively employed in industrial-machinery applications. This applied design formula can easily calculate the rotation angle of the spring for the applied load. However, as this formula simplifies the problem, it is difficult to determine the behavior or deformation of a torsion spring according to the change in load conditions with different angles and directions. Case studies considering the structural analysis of such springs using finite-element analysis (FEA) have been proposed to solve this design problem [3–6]. This method allows the designer to visually confirm and design these springs considering the stress and displacement that locally occur in the spring. As FEA expresses curves as a sum of subdivided linear elements, many elements are needed to increase the reliability of curve-structure analysis. As springs are curved structures, the number of elements used in these analyses must inevitably be increased to ensure accuracy, thereby lengthening analysis time. Due to this increase in spring-design time, as well as the aforementioned design problem,

Appl. Sci. 2020, 10, 2629; doi:10.3390/app10072629 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 2629 2 of 10 and considering the eﬀective product applications of these springs, a more eﬀective torsion-spring design method is required. This study proposes the evaluation method of deformation behavior of torsion spring via FEA using nonuniform-rational-basis-spline (NURBS) curves [7,8]. The NURBS curve is a model that can accurately express a curved structure with a small amount of information and a simple calculation formula. These curves were applied to FEA, and through experimentation, the angle and displacement values for the actual spring load were converted into useable data. The torsion-spring displacement values measured through experiments were converted into coordinates that could be expressed using NURBS curves. Experiments were then conducted for comparison with the FEA results and to verify the validity of this method.

2. Torsion-Spring Design

2.1. NURBS Curve A range of research has been conducted on curve expression according to polynomial theory. Among the numerous curve-expression methods, NURBS is the most accurate technique used to express complex organic shapes in two and three dimensions [9,10]. Due to its simple calculation method, NURBS is employed in a variety of industrial ﬁelds that require curved shapes. A NURBS curve can be expressed through a combination of parameters, including knot, control point, degree, and weight, in basis functions with a relatively simple calculation algorithm. In NURBS curves, the initial basis function can be expressed as in Equation (1).

Xn N(u) = Ni,p(u), (1) i=0 where Ni,p(u) is the B-spline basis function. The equation applied to the algorithm differs with the number and value of the knot vectors; Equation (1) may also be expressed as either Equation (2) or Equation (3). To define B-spline basis functions, we need one more parameter: the degree of these basis functions, p. The i-th B-spline basis function of degree p, written as Ni,p(u), is defined recursively as follows: ( 1 i f u u u + N (u) = i ≤ ≤ i 1 (2) i,0 0, Otherwise

u ui ui+p+1 u Ni,p(u) = − Ni,p 1(u) + − Ni+1,p 1(u), (3) u + u − u + + u + − i p − i i p 1 − i 1 where i is the degree plus 1 (i = p + 1), ui is the knot, and u is the knot vector. The following basis function expresses the curve of a geometric shape as the degree (p) increases; however, as the function increases in complexity and decreases in accuracy, a suitable degree should be used according to the desired curve shape. The NURBS basis function can express the NURBS curve as a combination of the weight (wi) and control point (Pi), as shown in Equation (4).

n Pn X Ni,pwi j=0 Ni,pwiPi C(u) = P Pi = , (4) n N w Pk i=0 j=0 j,n i j=0 Ni,pwi where Pi is the control point and wi is the weight of control point. This equation can then be simpliﬁed into Equation (5). Xn C(u) = Ri,p(u)Pi (5) i=0 Appl. Sci. 2020, 10, 2629 3 of 10

The NURBS curve expressed by Equation (5) can be freely varied as the weight (wi) increases and the control point (Pi)shifts, demonstrating the relationship between the NURBS curve and these parameters.

2.2. Expression of Torsion Spring Through NURBS Curves Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 10 To express the shape of a torsion spring with a constant curvature using the NURBS curve, Equations2.2. Expression (6) to (8)of Torsion are used Spring for theThroughx, y, andNURBSz planes Curves on the global coordinate system. To express the shape of a torsion spring with a constant curvature using the NURBS curve, Xn Equations (6) to (8) are used for the x, y, and z planes on the global coordinate system. x(u) = Ri,p(u)Px (6) i=0 𝑥(𝑢) = 𝑅,(𝑢)𝑃 (6) n X y(u) = Ri,p(u)Py (7) i=0 𝑦(𝑢) = 𝑅,(𝑢)𝑃 (7) n X z(u) = Ri,p(u)Pz, (8) 𝑧(𝑢) = ∑ 𝑅,(𝑢)𝑃, (8) i=0 where 𝑷 are the coordinates of the interpolation point generated through the initial NURBS curve, where PN are𝑵 the coordinates of the interpolation point generated through the initial NURBS curve, and 𝑷𝑵𝒅𝒊𝒔𝒑 are the coordinates through a combination of the knot, control point, and weight and PNdisp are the coordinates through a combination of the knot, control point, and weight (Equations (6)(Equations to (8)). The (6) spring to (8)). shape The spring can be shape expressed can be on expr threeessed planes on three in 3D planes space, in 3D as shownspace, as in shown Figure in1. Figure 1.

FigureFigure 1. 1.Representation Representation of of nonuniform-rational-basis-spline nonuniform-rational-basis-spline (NURBS) (NURBS) curves curves coupled coupled in 3D in 3Dspace. space. 3. Torsion-Spring Displacement Analysis 3. Torsion-Spring Displacement Analysis 3.1. Torsion-Spring Displacement Analysis 3.1. Torsion-Spring Displacement Analysis ToTo verify verify the the correlation correlation betweenbetween the interpolation interpolation point point and and control control point, point, the coordinates the coordinates of of thethe interpolation interpolation point point generated generated through through thethe NURBSNURBS curve curve are are applied applied to to finite-element finite-element analysis analysis to createto create the torsion-spring the torsion-spring shape shape for for analysis analysis [11 [11–13].–13]. Figure Figure2 2 shows shows thethe finite-elementfinite-element model model of of the torsionthe torsion spring spring that was that usedwas used to derive to derive the displacementthe displacement of theof the control control point point as as it it shifted shifted according to theaccording external forceto the [external14,15]. Tableforce 1[14,15]. shows Table the parameters 1 shows the of parameters the model of used the model for analysis. used for Commercial analysis. softwareCommercial HyperWorks software Optistruct HyperWorks (Altair, Optistruct United (Altai States)r, United was used States) for was finite-element used for finite-element analysis, and the shiftedanalysis, displacement and the shifted data displacement were acquired data for were generated acquired moment for generatedMz. The moment displacement 𝑀. The values shifted afterdisplacement the generation values of theshifted initial after coordinates, the generation and of loads the initial were coordinates, analyzed for and a total loads of were 100 elementsanalyzed (Ni). for a total of 100 elements (𝑁). Appl. Sci. 2020, 10, 2629 4 of 10 Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 10

FigureFigure 2. 2.Finite-element Finite-element model of of torsion torsion spring. spring. Table 1. Design parameters. Table 1. Design parameters. Parameters Unit Dimension Parameters Unit Dimension Spring cross-section diameter (d) mm 3 Spring cross-section diameter (𝑑) mm 3 Spring radius (R) mm 20 SpringSpring radius length (𝑅 (l) mm mm20 125.6 SpringMoment length in the (𝒍)z -axis direction (Mz)mm N m125.6 1000 Elastic modulus GPa 206 Moment in the z-axis direction (𝑴𝒛) N m 1000 Elastic modulus GPa 206 The displacement of the element determined through ﬁnite-element analysis (xdisp, ydisp, zdisp) was added to the coordinates of the interpolation point generated through the NURBS curve to express 𝒙 ,𝒚 ,𝒛 the deformedThe displacement shape of the of spring.the element This determined is expressed through using Equationsfinite-element (9) toanalysis (11). ( 𝒅𝒊𝒔𝒑 𝒅𝒊𝒔𝒑 𝒅𝒊𝒔𝒑) was added to the coordinates of the interpolation point generated through the NURBS curve to express the deformed shape of the spring. This is expressed using Equations (9) to (11). PNdisp(x) = PN(x) + xdisp (9)

𝑃(𝑥) =𝑃(𝑥) +𝑥 (9) PNdisp(y) = PN(y) + ydisp (10) 𝑃(𝑦) =𝑃(𝑦) +𝑦 (10) PNdisp(z) = PN(z) + zdisp (11) 𝑃(𝑧) =𝑃(𝑧) +𝑧 (11) where PN are the coordinates of the interpolation point generated through the initial NURBS curve, where 𝑃 are the coordinates of the interpolation point generated through the initial NURBS curve, and P are the coordinates of the interpolation point of the NURBS curve after deformation. andNdisp 𝑃 are the coordinates of the interpolation point of the NURBS curve after deformation. The coordinates of P (x, y, z), derived from Equations (9) to (11), were next applied to the inverse The coordinates ofNdisp 𝑃(𝑥,𝑦,𝑧), derived from Equations (9) to (11), were next applied to the methodinverse to method determine to determine the displacement the displacement of the control of the control point. point. This processThis process is shown is shown in Equation in (12), whichEquation was derived (12), which with was reference derived towith Equations reference (4) to andEquations (5). Figure (4) and3 shows (5). Figure the control3 shows point the derived usingcontrol Equation point (12),derived using Equation (12), Pn i= 1 Ri,p(u) = ∑ ,() P(x𝑃,y(,,,z) ) = , ,(12) (12) PNdisp((,,)x, y, z) ( ) wherewherePx ,𝑃P𝑥,y 𝑃and andPz 𝑃are are the the coordinates coordinates of of the thecontrol control point, and and 𝑅R,i,P𝑢(u ) isis the the NURBS NURBS curve- curve-based function.based function.

In Figure3, (x, y, z)disp represents the x, y, and z displacements of the point; PN(x, y, z) represents the coordinates of the point before interpolation; and PNdisp(x, y, z) represents the coordinates of the point after interpolation. The inverse method can be applied to determine the displacement and load as KNURBS. 1 a[D] = a[K]− [F], (13) where [D] is the displacement matrix of the control point derived from the existing stiffness matrix, a is 1 the stiffness-correction constant, [K]− is the inverse of the stiffness matrix, and [F] is the external force matrix. Stiffness-correction constant a can be expressed as follows:

T a = [ai] . (14) Appl. Sci. 2020, 10, 2629 5 of 10

Thus, the following is obtained by applying Equation (14) to Equation (13).

T T 1 [ai] [D] = [ai] [K]− [F] (15)

Equation (15) can then be expressed as in Equation (16).

1 [D v] = [KNRBS]− [F] (16) ∈ 1 where [D v] is the control point displacement determined using the inverse method, and [KNRBS]− is ∈ the inverse of the stiﬀness matrix of the control point. Using Equation (16), the Mz load was applied to the NURBS curve-based spring structure that was composed of a total of nine control points, and analysis was conducted. Using this method, the graphs in Figure4 were derived for loads Fx, Fy, Fz, Mx, My, and Mz. Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 10

Figure 3. Derivation of control point using inverse method. Figure 3. Derivation of control point using inverse method. 3.2. Experiment-Equipment Setup In Figure 3, (𝑥, 𝑦,) 𝑧 represents the x, y, and z displacements of the point; 𝑃(𝑥, 𝑦,) 𝑧 representsExperiments the coordinates were conducted of the topoint measure before the interpolation; displacement, and load,𝑃 and(𝑥, angle 𝑦,) 𝑧 represents using a torsion-spring the measuringcoordinates device. of the point Torsion-spring after interpolation. parameters The inverse were selected method forcan abe total applied of six to determine types ofspecimens. the

Fivedisplacement experiments and were load conducted as 𝐾. for each specimen, resulting in a total of 30 experiments. Table2 shows the design parameters of the six spring𝑎𝐷 specimens.=𝑎𝐾𝐹 The, experiment equipment was conﬁgured(13) to measure the torsion spring. Figure5 shows the actual torsion spring and the device used to measure where 𝐷 is the displacement matrix of the control point derived from the existing stiffness the spring displacement, load, and angle. The rotation angle was set to 0◦, 20◦, and 40◦, considering the matrix, 𝑎 is the stiffness-correction constant, 𝐾 is the inverse of the stiffness matrix, and 𝐹 is linear component of the load applied to the torsion spring. To determine torsion-spring displacement, the external force matrix. Stiffness-correction constant a can be expressed as follows: certain sections were marked, and displacement and load values were measured at 20◦ and 40◦. Images were taken of the front of the system to determine𝑎= the𝑎 x and. y coordinates, and the z-axis was measured(14) throughThus, images the offollowing the sides is andobtained Vernier by applying calipers. Equation (14) to Equation (13).

Table𝑎 𝐷 2.Specimen= 𝑎 𝐾 types. 𝐹 (15) Equation (15) can then be expressed as in Equation (16). Type Spring Cross-Section Diameter, d (mm) Spring Diameter, D (mm) Spring Turns, N 1 2.6𝐷∈ = 𝐾 32.6 𝐹 1.5 (16) 2 2.6 32.6 2.5 where 𝐷∈𝒗 is the control point displacement determined using the inverse method, and 𝐾 3 2.8 32.8 1.5 is the inverse of the stiffness matrix of the control point. 4 2.8 32.8 2.5 𝑀 5Using 3.0 Equation (16), the load was applied to the 33.0 NURBS curve-based spring 1.5 structure that was6 composed 2.0 of a total of nine control points, and analysis 33.0 was conducted. 2.5 Using this method, the graphs in Figure 4 were derived for loads 𝐹, 𝐹, 𝐹, 𝑀, 𝑀, and 𝑀.

(a) (b) Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 10

Figure 3. Derivation of control point using inverse method.

In Figure 3, (𝑥, 𝑦,) 𝑧 represents the x, y, and z displacements of the point; 𝑃(𝑥, 𝑦,) 𝑧 represents the coordinates of the point before interpolation; and 𝑃(𝑥, 𝑦,) 𝑧 represents the coordinates of the point after interpolation. The inverse method can be applied to determine the

displacement and load as 𝐾. 𝑎𝐷 =𝑎𝐾𝐹, (13) where 𝐷 is the displacement matrix of the control point derived from the existing stiffness matrix, 𝑎 is the stiffness-correction constant, 𝐾 is the inverse of the stiffness matrix, and 𝐹 is the external force matrix. Stiffness-correction constant a can be expressed as follows:

𝑎=𝑎 . (14) Thus, the following is obtained by applying Equation (14) to Equation (13).

𝑎 𝐷 = 𝑎 𝐾 𝐹 (15) Equation (15) can then be expressed as in Equation (16).

𝐷∈ = 𝐾 𝐹 (16) where 𝐷∈𝒗 is the control point displacement determined using the inverse method, and 𝐾 is the inverse of the stiffness matrix of the control point.

Using Equation (16), the 𝑀 load was applied to the NURBS curve-based spring structure that was composed of a total of nine control points, and analysis was conducted. Appl. Sci. 2020, 10, 2629 6 of 10 Appl. Sci. 2020Using, 10, x FOR this PEER method, REVIEW the graphs in Figure 4 were derived for loads 𝐹, 𝐹, 𝐹, 𝑀, 𝑀6, andof 10 𝑀.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 10 (c) (a) (d) (b)

Figure 4. Direction-displacement graphs for load conditions (a) 𝐹, (b) 𝑀, (c) 𝐹, (d) 𝑀, (e) 𝐹, (f)

and 𝑀.

3.2. Experiment-Equipment Setup Experiments were conducted to measure the displacement, load, and angle using a torsion- spring measuring device. Torsion-spring parameters were selected for a total of six types of specimens. Five experiments were conducted for each specimen, resulting in a total of 30 experiments. Table 2 shows the design parameters of the six spring specimens. The experiment equipment was configured to measure the torsion spring. Figure 5 shows the actual torsion spring and the device(e) used to measure the spring displacement, load, (f) and angle. The rotation angle was set to 0°, 20°, and 40°, considering the linear component of the load applied to the torsion spring. To determineFigure torsion-spring 4. Direction-displacement displacement, certain graphs sections for load were conditions marked, (a and) Fx ,(displacementb) Mx,(c) Fy and,(d) loadMy,( e) Fz, (f) and Mz. values wereFigure measured 4. Direction-displacement at 20° and 40°. Images graphs were for takenload conditions of the front (a) 𝐹of, the(b) system𝑀, (c) 𝐹 to, (determined) 𝑀, (e) 𝐹 the, (f ) x and y coordinates,and 𝑀. and the z-axis was measured through images of the sides and Vernier calipers.

set to 0°, 20°, and (40°,a) considering the linear component of the load (appliedb) to the torsion spring. To determine torsion-spring displacement, certain sections were marked, and displacement and load Figure 5. Springs and experiment setup: (a) torsion springs; (b) experiment setup. values wereFigure measured 5. Springs at 20° and and experiment 40°. Images setup: were (a taken) torsion of the springs; front ( bof) experimentthe system to setup. determine the x and y coordinates, and the z-axis was measured through images of the sides and Vernier calipers.

(a) (b)

Figure 5. Springs and experiment setup: (a) torsion springs; (b) experiment setup. Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 10 Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 10 Table 2. Specimen types. Table 2. Specimen types. Type Spring Cross-Section Diameter, 𝐝 (mm) Spring Diameter, 𝐃 (mm) Spring Turns, 𝑵 Type Spring Cross-Section Diameter, 𝐝 (mm) Spring Diameter, 𝐃 (mm) Spring Turns, 𝑵 1 2.6 32.6 1.5 1 2.6 32.6 1.5 2 2.6 32.6 2.5 2 2.6 32.6 2.5 3 2.8 32.8 1.5 3 2.8 32.8 1.5 4 2.8 32.8 2.5 4 2.8 32.8 2.5 5 3.0 33.0 1.5 5 3.0 33.0 1.5 6 2.0 33.0 2.5 Appl. Sci.6 2020, 102.0, 2629 33.0 2.5 7 of 10 4. Results and Discussion 4. Results and Discussion 4. Results and Discussion 4.1. Experiment-Data Analysis 4.1. Experiment-Data Analysis 4.1. Experiment-DataThe force according Analysis to the rotation angle was measured using a load cell and rotary encoder. The force according to the rotation angle was measured using a load cell and rotary encoder. FigureThe 6 forceshows according images of the to thex- and rotation y-coordinate angle was measurements measured obtained using a loadaccording cell andto the rotary angle encoder.of theFigure torsion 6 spring, shows imagesand Figure of the 7 shows x- and a y-coordinate load graph according measurements to the obtained rotation angle.according This to graph the angle was of Figure6 shows images of the x- and y-coordinate measurements obtained according to the angle of drawnthe torsionusing the spring, average and data Figure from 7 shows the results a load of graphfive replicates. according Analytical to the rotation results angle. indicated This graphthat the was the torsion spring, and Figure7 shows a load graph according to the rotation angle. This graph was torsion-springdrawn using load the wasaverage generated data from nonlinearly the results according of five replicates. to the rotation Analytical angle. resultsThe regression indicated that the drawn using the average data from the results of ﬁve replicates. Analytical results indicated that equationtorsion-spring was derived load using was generated the measured nonlinearly data and according was expressed to the rotationas a quadratic angle. polynomialThe regression in the thegraph. torsion-springequation R-square was valuesderived load of was using the generatedregression the measured nonlinearlyequations data and were according was 98% expressed or more. to the asThis rotationa quadratic regression angle. polynomial equation The regression had in the equationa highgraph. adjustment. was R-square derived In values addition, using of thethe it measuredregressionwas found equations datathat the and load waswere increased expressed98% or more. as asthe aThis cross-sectional quadratic regression polynomial diameterequation had in the graph.(d) aof high the R-square torsionadjustment. values spring In ofincreased, addition, the regression itand was that found equations the loadthat the weredecreased load 98% increased oras the more. nu asmber This the cross-sectional regressionof spring turns equation diameter (𝑁) had a highincreased.( adjustment.d) of the torsion Inaddition, spring increased, it was found and that that the the load load decreased increased as as the the nu cross-sectionalmber of spring diameter turns (𝑁) ( d) of the torsionincreased. spring increased, and that the load decreased as the number of spring turns (N) increased.

(a) (b) (c) (a) (b) (c) Figure 6. Measurement of x- and y-coordinates of torsion springs: (a) 0°; (b) 20°; (c) 40°. FigureFigure 6. Measurement 6. Measurement of x-of andx- and y-coordinates y-coordinates of of torsion torsion springs: springs: ((a) 00°;◦;( (bb)) 20°; 20◦ ;((cc) )40°. 40◦ .

Figure 7. 7. ForceForce vs. vs. degree degree graph graph of oftorsion torsion springs. springs. Figure 7. Force vs. degree graph of torsion springs. 4.2. Comparative Analysis of Analytical Results and Experiment Values We performed comparative analysis of the deformed shape of the torsion spring, measured through the experiments, and the analytical results obtained using Equation (16). Figure8 shows graphs of the analytically and experimentally obtained data. We then performed comparative analysis of the deformed shape of the torsion spring at 20◦ and 40◦, measured through the experiments and Equation (16). Under the same load conditions, the displacement determined by the proposed equation exceeded that of the torsion spring measured through the experiments. However, by modifying the load parameters, similar behaviors and displacements were observed, thereby verifying the use of the proposed equation. The maximal error rate ranged from 5% to 6%, which could be attributed to the error that can occur during experiment measurements. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 10

4.2. Comparative Analysis of Analytical Results and Experiment Values We performed comparative analysis of the deformed shape of the torsion spring, measured through the experiments, and the analytical results obtained using Equation (16). Figure 8 shows graphs of the analytically and experimentally obtained data. We then performed comparative analysis of the deformed shape of the torsion spring at 20° and 40°, measured through the experiments and Equation (16). Under the same load conditions, the displacement determined by the proposed equation exceeded that of the torsion spring measured through the experiments. However, by modifying the load parameters, similar behaviors and displacements were observed, Appl.thereby Sci. 2020 verifying, 10, 2629 the use of the proposed equation. The maximal error rate ranged from 5% to 6%,8 of 10 which could be attributed to the error that can occur during experiment measurements.

(a) (b)

(e) (f)

(g) (h)

Figure 8. Cont. Appl. Sci. 2020, 10, 2629 9 of 10

Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 10

(i) (j)

(k) (l)

Figure 8. Comparative analysis of torsion springs: (a) d = 2.6 mm, D = 32.6 mm, n = 1.5 (20°); (b) d = Figure 8. Comparative analysis of torsion springs: (a) d = 2.6 mm, D = 32.6 mm, n = 1.5 (20◦); 2.6 mm, D = 32.6 mm, n = 1.5 (40°); (c) d = 2.6 mm, D = 32.6 mm, n = 2.5 (20°); (d) d = 2.6 mm, D = 32.6 (b) d = 2.6 mm,D = 32.6 mm, n = 1.5 (40◦); (c) d = 2.6 mm, D = 32.6 mm, n = 2.5 (20◦); (d) d = 2.6 mm, mm, n = 2.5 (40°); (e) d = 2.8 mm, D = 32.8 mm, n = 1.5 (20°); (f) d = 2.8 mm, D = 32.8 mm, n = 1.5 D = 32.6 mm, n = 2.5 (40◦); (e) d = 2.8 mm, D = 32.8 mm, n = 1.5 (20◦); (f) d = 2.8 mm, D = 32.8 mm, (40°); (g) d = 2.8 mm, D = 32.8 mm, n = 2.5 (20°); (h) d = 2.8 mm, D = 32.8 mm, n = 2.5 (40°); (i) d=3.0 n = 1.5 (40 ); (g) d = 2.8 mm, D = 32.8 mm, n = 2.5 (20 ); (h) d = 2.8 mm, D = 32.8 mm, n = 2.5 (40 ); mm, D =◦ 33.0 mm, n = 1.5 (20°); (j) d = 3.0 mm, D = 33.0◦ mm, n = 1.5 (40°); (k) d=3.0 mm, D = 33.0 mm, ◦ (i) d = 3.0 mm, D = 33.0 mm, n = 1.5 (20 ); (j) d = 3.0 mm, D = 33.0 mm, n = 1.5 (40 ); (k) d = 3.0 mm, n = 2.5 (20°); (l) d = 3.0 mm, D = 33.0 mm,◦ n = 2.5 (40°). ◦ D = 33.0 mm, n = 2.5 (20◦); (l) d = 3.0 mm, D = 33.0 mm, n = 2.5 (40◦).

5. Conclusions5. Conclusions This study applied nonuniform-rational-basis-spline (NURBS) curves for the design of torsion This study applied nonuniform-rational-basis-spline (NURBS) curves for the design of torsion springs, analyzed the displacements of these springs using finite-element analysis, and verified the springs, analyzed the displacements of these springs using finite-element analysis, and verified the design of these springs through experimentation. design of these springs through experimentation. (1) A method was proposed for deriving the coordinates of a control point for shifted elements by(1) applying A method the wasinverse proposed method for on derivingthe basis theof data coordinates derived through of a control finite-element point for shifted analysis. elements In by applyingaddition, the the inverse relationship method betwee on then basisthe movement of data derived of the control through point finite-element and stiffness analysis. matrix was In addition, theidentified relationship and betweenformulated the by movement varying the of torsion-spring the control point parameters. and sti ffness matrix was identified and formulated(2) A by method varying was the proposed torsion-spring for deriving parameters. the torsion-spring shape by converting the torsion- spring(2) A displacement method was proposedmeasured forthrough deriving experiments the torsion-spring into coordinates shape bythat converting could be expressed the torsion-spring using displacementNURBS curves. measured through experiments into coordinates that could be expressed using NURBS(3) curves. Comparative analyses between the results of the proposed analytical method and the experiment(3) Comparative measurements analyses demonstrated between the results that the of pr theoposed proposed method analytical is valid method within anda satisfactory the experiment measurementsrange of error. demonstrated that the proposed method is valid within a satisfactory range of error.

AuthorAuthor Contributions: Contributions:Y.S.K., Y.S.K., Y.J.S.,S.Y.J., and E.S.J. E.S.J. conceived conceived and and designed designed the experiments; the experiments; Y.S.K., Y.S.K.,S.Y.J., and Y.J.S., and E.S.J.E.S.J. performed performed the the experiments; experiments; Y.S.K., Y.S.K., Y.J.S., S.Y.J., and and E.S.J.E.S.J. analyzedanalyzed the the data; data; Y.S.K., Y.S.K., S.Y.J., Y.J.S., and and E.S.J. E.S.J. contributed reagentscontributed/materials reagents/materials/ana/analysis tools; andlysis Y.S.K., tools; Y.J.S., and and Y.S.K., E.S.J. S.Y.J., wrote and the E.S.J. paper. wrote All the authors paper. have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest Conflicts of Interest: The authors declare no conflict of interest References References 1. Yuldirim, V. Exact determination of the global tip deflection of both close-coiled and open-coiled 1. Yuldirim,cylindrical V. Exact helical determination compression ofspring the global having tip arbitrary deflection doubly-symmetric of both close-coiled cross-section. and open-coiled Int. J. Mech. cylindrical Sci. helical2016 compression, 115–116, 280–298. spring having arbitrary doubly-symmetric cross-section. Int. J. Mech. Sci. 2016, 115–116, 280–298. [CrossRef] Appl. Sci. 2020, 10, 2629 10 of 10

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