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Theoretical and experimental determination of capstan drive stiffness
Precision Engineering 31 (2007) 55–67
Theoretical and experimental determination of capstan drive stiffness Jaime Werkmeister, Alexander Slocum ∗ Department of Mechanical Engineering, MIT, Rm 3-445 77 Massachusetts Ave., Cambridge, MA 02139, USA Received 13 December 2004; received in revised form 10 February 2006; accepted 3 March 2006 Available online 5 May 2006
Abstract Cable or metal band capstan drives are used as rotary transmission elements for their very low (nominally zero) backlash and high stiffness properties. Cable drives, in particular, are found in many types of equipment, and to obtain high stiffness, the cable is typically wrapped around the input and output drum in a figure-eight pattern. This paper develops analytical methods for predicting the torsional stiffness of capstan drives to enable designers to better assess machine performance in the design study phase. Experiments validated the analysis. © 2006 Elsevier Inc. All rights reserved. Keywords: Wire capstan drive; Torsional stiffness; Transmission
1. Introduction Capstan drives have many uses in products, such as printers, plotters, copiers/scanners and tape recorders. For example, in printers, the head that supports the ink cartridge is typically actuated by a cable driven by a capstan1 [1]. Another use in printers is the feeding system; paper from the tray is fed onto the platen by a rotating capstan2 [2]. Tape recorders use a capstan that supports and controls the speed of the tape3 [3]. In precision machines, capstans can be used as rotary power transmission elements. For example, a device from SensAble Technologies4 uses a combination of two input drums and one output drum to aid in a 3 degree-of-freedom touch based application, as shown in Fig. 1. Capstans in these applications are typically configured with a cable wrapped in a figure-eight pattern around input and output drums. Multiple wrappings give the drive high stiffness and reduce radial loads. Two input drums are
∗
Corresponding author. Tel.: +1 617 452 2275; fax: +1 617 258 6427. E-mail address: [email protected] (A. Slocum). 1 Commercial Printing and Pre-Press Services: http://www.oneals.com. Last visited 12/04. 2 Arts56 Unit 11: Printing, created 12/98 by Claris Home Page. Last modified 12/00: http://lore.fhda.edu/Faculty/andreatta/arts56/Unit11.1.html. Last visited 12/04. 3 “I collect it: How Does an 8-Track Tape Work”. Article by Abigail Lavine, icollectit, 2000: http://www.icollectit.co.uk/icollectit.cfm?template name=a show detail.cfm&a id=168. Last visited 12/04. 4 SensAble Technologies: http://www.sensable.com/. Last visited 12/04. 0141-6359/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.precisioneng.2006.03.001
used to obtain rotary motion, about Z, and linear motion about Y of the hepatic system tip. A third input drum, located on the right-hand side of Fig. 1 is used to give the tip rotary motion about Y. Despite the widespread use of capstan drives and their availability as modular components,5 a review of the literature6 [4,5] did not provide a common theory of attainable drive stiffness and no data was given to validate different approaches. In this paper, a theory of wire capstan drive stiffness is presented along with experimental verification. 2. Capstan engagement angle A wire (cable) capstan drive is composed of an output drum, diameter Doutput , and an input drum, diameter Dinput , linked by cable segments. The cable can be wrapped multiple times between the input and output transmission drums in a figureeight pattern. Fig. 2 shows the basic representation of this drive. Initially tension exists in the cable, which is the preload force, Tpreload . If an external torque, Γ drum , is applied to the input drum Dinput , part of the cable extends due to increased tension,
5 Sagebrush Technology, Inc. “The Roto-Lok Drive Technology: A Revolutionary Innovation”, Sagebrush Technology, Inc. 2002. http://www. sagebrushtech.com/tech/technology.html. Last visited 12/04. 6 Stephen W. Attaway, The Mechanics of Friction in Rope Resuce, International Technical Rescue Symposium, 1999: http://www.amrg.org/ Mechanics of Friction in Rope Rescue.pdf. Last visited 12/04.
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Nomenclature Æ Doutput Dinput dF dL dN dT dθ dδ Kfree Ko slip1 Ki slip1 Ko slip2 Ki slip2 K1 K2 Ktotal Ktorsion L Lfree L rdrum routput rinput rcable T Tpreload TLoad T
stiffness/length of cable (N) output drum diameter (m) input drum diameter (m) differential force (N) differential element of cable wound around drum (m) differential normal force (N) differential tension (N) differential angle (rad) differential element of the elongation of cable (m) free length cable stiffness (N/m) output drum cable stiffness slip area 1 (N/m) input drum cable stiffness slip area 1 (N/m) output drum cable stiffness slip area 2 (N/m) input drum cable stiffness slip area 2 (N/m) slip side 1 cable stiffness (N/m) slip side 2 cable stiffness (N/m) total cable stiffness (N/m) torsional stiffness of the capstan (N m/rad) original free length of cable in tensile test measurement (m) free length of cable not in contact with either drum (m) length deflection (m) radius of drum, input or output (m) radius of output drum (m) radius of input drum (m) radius of cable (m) cable tension (N) cable preload tension (N) tension due to applied load (N) additional tension = TLoad (N)
Greek letters Γ drum general torque acting on a drum (N m) Γ output torque acting on output drum (N m) Γ input torque acting on input drum (N m) α general angle introduced with an applied torque (rad) δ elongation in cable on drum (m) ε strain in cable μ coefficient of friction between cable and drum θ generic angle (rad) θ slip deformation region of the loaded cable on the drum where slip can occur (rad) θ slip1 angle of cable in the slip 1 region (rad) θ slip2 angle of cable in the slip 2 region (rad) θ no-slip region of no deformation and no slip (rad) θ wrap total angle of cable wrapped around drum (rad) θ cable-encoder encoder position of cable as rotated (rad) θ lever angular deflection of the input drum (rad)
while the other part contracts due to tension being relieved. For example, if the input drum is rotated counter-clockwise in Fig. 2, a reasonable assumption is the tension in one cable will increase by some amount T, which will be called TLoad and the tension in the other cable will be reduced by some amount TLoad . For illustrative purposes, this can be demonstrated with a rubber band encircling a cylinder (soda can). Inserting a finger under the rubber band, twisting and pulling outward creates a device that looks like a capstan drive. As the cylinder is rotated counterclockwise, the rubber band only stretches on the right-hand side. Coloring the portion of the rubber band on the upper and lower surface of the cylinder helps with visualization. This difference in extension and contraction of the rubber band is enabled by friction between the band and the cylinder. As shown in Fig. 3, the active tractive region of interest, where cable extension on the drum dynamically occurs, can be defined as acting over an angle θ slip , which is the section of the cable where some slip can occur while the cable is loaded. This angle is the angle one would derive from the classic capstan equation as being required to hold the applied load, given the tensions in the two cable segments: θ slip1 + θ no-slip + θ slip2 = θ wrap . For a robust design, θ no-slip must be greater than zero and if θ wrap is greater than 2π, it indicates that there are multiple wraps on a drum. From a torque balance, TLoad can be determined: Γdrum = (Tpreload + TLoad − (Tpreload − TLoad ))rdrum ⇒ TLoad =
Γdrum 2rdrum
(1)
The input and output drum torques create an elongation in the cable. Note the cable was already slightly extended from the initial cable length due to the preload force. Since the system is in equilibrium with the preload force, its current length will be considered the initial length and any additional extension is what is of interest with respect to determining the stiffness. Fig. 4 shows the free body diagram of a small cable segment of the stretched cable in traction with the drum; dN is the normal force from the cable in contact with the drum and μdN is the traction force due to friction between the cable and the drum: dθ dθ FX = T cos +μ dN − (T + dT ) cos = 0 (2) 2 2 dθ dθ FY = dN − T sin − (T + dT ) sin = 0 (3) 2 2 Since dθ is infinitesimally small: dT = μ dθ (4) T Integrating from T(θ) to a tension T and 0 to θ slip , where θ slip is the angle-of-slip, not the angle of entire contact, as defined in Fig. 3: T θslip dT =μ dθ (5) 0 T (θ) T T (θ) = T e−μθslip
(6)
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Fig. 1. PhantomTM haptic input device from SensAble Technologies Corp. Two input drums on one output drum shown on the left and one input drum on one output drum shown on the right. Bottom picture displays the overall device.
since the cross-sectional area of a cable is difficult to define: ε=
T Æ
(9)
The load strain can also be stated as the amount of deflection divided by the unstretched length of the segment under consid-
Fig. 2. Representation of capstan drive where arrows represent direction of drum and cable motion.
Expression (6) is the classic capstan equation used to find the two regions θ slip1 and θ slip2 where deformations in the cable occur as the cable transfers torque-induced loads to the drum: Tpreload + TLoad 1 θslip1 = ln μ Tpreload ⇒ T = e−μθ (Tpreload + TLoad )
θslip2
−1 = ln μ
Tpreload − TLoad Tpreload
(7)
⇒ T = eμθ (Tpreload − TLoad )
(8)
3. Cable deformation From Hooke’s Law, the term Æ for the cable is defined as the product of the modulus of elasticity E and the effective area A
Fig. 3. Nomenclature for the stiffness derivation model for the two sides of the capstan drive.
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modulus of elasticity, coefficient of friction and radius of the drum: Tpreload rdrum (Tpreload + TLoad ) 1− δ= Æμ Tpreload + TLoad Tpreload + TLoad Tpreload rdrum 1 ln − (15) μ Tpreload Æ Tpreload + TLoad rdrum TLoad − Tpreload ln δ= Æμ Tpreload
Fig. 4. Differential cable element.
eration ε=
dδ dδ = dL rdrum dθ
(10)
The stiffness is determined from (1/K) = compliance (dδ/dTLoad ): 1/Tpreload dδ rdrum = 1 − Tpreload dTLoad μÆ (Tpreload + TLoad )/Tpreload Tpreload + TLoad μÆ (16) ⇒ Ko slip1 = 2 Doutput TLoad
Hence dδ =
Trdrum dθ Æ
(11)
where dδ is the elongation in the small segment dL under tension T at that point. The deflection of interest is the extension due to just the load. The preload is even everywhere in the cable, and it is also represented in the expression for tension T when Eq. (11) is integrated; hence the elongation component due to preload force must be subtracted: dδ = dδTpreload +TLoad − dδTpreload θ Tpreload rdrum θslip Trdrum = dθ − Æ Æ 0
(12)
Likewise for the input drum the stiffness is: Tpreload + TLoad μÆ Ki slip1 = 2 Dinput TLoad 3.2. Cable deformation on slip side 2 In a manner like that for slip side 1: θslip2 δ (Tpreload − TLoad )rdrum μθ dδ = e dθ Æ 0 0
−
Tpreload rdrum θslip2 Æ
3.1. Cable deformation on slip side 1 The relation between the deflection in the cable, tension, drum radius, effective modulus, angle-of-slip, and Eqs. (7) and (11), produces the elongation of the cable on slip side 1 of the drum: δ θslip1 (Tpreload + TLoad )rdrum −μθ dδ = dθ e Æ 0 0 −
Tpreload rdrum θslip1 Æ
(13)
Following through with the integration produces the basic form of the elongation of the cable: δ=
Tpreload rdrum θslip1 rdrum (Tpreload + TLoad ) (1 − e−μθslip1 ) − Æμ Æ (14)
where δ is the elongation of the cable for the region of θ slip1 and represents the extension in the cable due to the external torque on slip side 1 of the drum. Substituting for the exponential from Eq. (7), and combining like terms, the deflection is determined as a function of TLoad , cross-sectional area of the cable,
(17)
Fig. 5. Determining the free length Lfree of the cable.
(18)
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Fig. 6. Spring model of capstan drive output drum.
Following through with the integration produces the basic form of the elongation of the cable:
and radius of the drum:
Tpreload rdrum (Tpreload − TLoad ) 1− Æμ Tpreload − TLoad Tpreload − TLoad Tpreload rdrum 1 ln − Æ μ Tpreload Tpreload − TLoad rdrum δ=− TLoad + Tpreload ln Æμ Tpreload
δ= δ=
rdrum (Tpreload − TLoad ) (1 − eμθslip2 ) Æμ −
Tpreload rdrum θslip2 Æ
(19)
where δ is the elongation of the cable for the region of θ slip2 and represents the extension in the cable due to the external torque on slip side 2 of the drum. Again, substituting for the exponential from Eq. (7), and combining like terms, the deflection is determined as a function of TLoad , cross-sectional area of the cable, modulus of elasticity, coefficient of friction
(20)
The stiffness is determined from (1/K) = compliance (dδ/dTLoad ): −1/Tpreload rdrum dδ = − 1 + Tpreload Tpreload − TLoad /Tpreload dTLoad μÆ Tpreload − TLoad μÆ (21) ⇒ Ko slip2 = 2 Doutput TLoad
Fig. 7. Partition of cable.
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Likewise for the input drum the stiffness is: Tpreload − TLoad μÆ Ki slip2 = 2 Dinput TLoad
(22)
Eqs. (16) and (21) are the limits of the total deflection on the output drum due to the extension in the cable from the external torque and preload and Eqs. (17) and (22) are the limits of the total deflection due to the extension in the cable from the external torque and preload on the input drum. 3.3. Cable deformation in free length section The free length deflection and stiffness of the cable are determined from Hooke’s law: l =
TLfree Æ
Kfree =
(23)
Æ Lfree
(24)
An expression for Lfree can be obtained from the capstan’s geometry shown in Fig. 5. The distance between the input and output drum, radius of the two drums and distance between the two drums gives Lfree : Lfree =
routput + rinput tan(θ/2)
(25)
4. System torsional stiffness The stiffness of the output drum can be modeled as effectively having three springs on either side as shown in Fig. 6. These springs represent each section of the cable for each of the two sides of the drums: a section on the output drum, the freelength section between the drums and a section on the input drum. Ko slip2 and Ko slip1 includes the stiffness associated with cable on the output drum over regimes θ slip1 and θ slip2 , where some deformations can occur. Kfree is the stiffness of each cable segment between drums. Ki slip1 and Ki slip2 are the stiffnesses associated with the cable on the input drum over region θ slip1 and θ slip2 , respectively. Determining the stiffness on each side of the drum, as indicated in Fig. 7, produces the first and second side of the stiffness, respectively. K1 =
1 (1/Ko slip1 ) + (1/Kfree ) + (1/Ki
=
1 (1/Ko slip2 ) + (1/Kfree ) + (1/Ki Ko
The two sides act in parallel thus the cable’s total longitudinal stiffness is the addition of these two springs in parallel. Ktotal = K2 + K1
(28)
The torsional stiffness of the output drum is determined from: slip1 )
Ko slip1 Kfree Ki slip1 = Ko slip1 Kfree + Ko slip1 Ki slip1 + Kfree Ki
K2 =
Fig. 8. Cable segment in tensile test machine.
δ= (26) slip1
α=
δ Ddrum /2
=>
(29) 4Γ 2 Ddrum Ktotal
(30)
Since Γ drum = Ktorsion α, when one drum is rigidly held, the torsional stiffness of the other drum will appear to be:
slip2 )
Ko slip2 Kfree Ki slip2 K slip2 free + Ko slip2 Ki slip2 + Kfree Ki
2Γdrum Ddrum Ktotal
(27) slip2
Ktorsion =
2 Ktotal Ddrum 4
(31)
J. Werkmeister, A. Slocum / Precision Engineering 31 (2007) 55–67
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Fig. 9. Experiential system for measuring capstan drive performance.
5. Experimental procedure
5.2. Torsional stiffness
5.1. Æ value
The experimental system for determining the torsional stiffness of a capstan drive consisted of a 1.5 mm steel cable wrapped in a figure-eight pattern around two drums with diameters of 50 and 280 mm. The input drum was fixed as would be the case if a motor servo was loaded in position. Torque was applied to the output drum to simulate a load being applied. The rotations of the input and output drums were measured and the difference
A section of cable, according to ASTM, was mounted and pulled to failure in an InstronTM tensile test machine as shown in Fig. 8 and in accordance with ASTM standards [6]. Once testing was complete, the stress strain curve was produced and the effective modulus was calculated.
Fig. 10. Rear view of experiential system for measuring capstan drive performance.
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was due to the capstan drive compliance. To measure this rotation, a lever arm was attached to each drums’ axis. As torque was applied to the output drum, any rotation encountered in the input drum was subtracted to give the net deflection of the capstan drive: δ = θcable
encoder routput
− θlever rinput
(32)
Capacitance probes detected these motions at the end of the lever arms. A load cell was spliced between the cable to determine the cable preload. Figs. 9 and 10 show the overall setup. Rotating the output drum 15–20 times back and forth ensures the system runs smoothly and the system has achieved steady state with the entire cable under the preload tension, Tpreload , before measurements were taken. Weights were added to the end of the load arm and the motion of the locked input and output shafts were recorded with values of Tpreload and TLoad . The full procedure was: 1. Wrap cable in a figure-eight pattern once around the input and output drums. 2. Tighten the cable to desired preload. 3. Rotate output drum back and forth 15–20 times. 4. Record initial reading of capacitance probes and load cell with no preload. 5. Check desired preload and record readings from capacitance probes and load cell. 6. Add weight to the torque arm in increments of 1 kg until cable starts to slip. 7. Record data from both capacitance probes for every 1 kg added and load cell. 8. Repeat two more times. 9. After third run, increase cable preload and repeat procedure starting from step 2. 6. Results 6.1. Effective modulus The effective cable modulus, Æ, is determined by loading the cable to 25% of its rated tensile strength and noting the displacement of the load. Æ FL δ => Æ = L δ where L is the original length of the testing sample. F=
Fig. 11. Applied force vs. strain which is displacement divided by original length, for the 1.5 mm diameter cable.
Fig. 12. One kiloNewton cyclic tensioning of the 15 mm cable results in a Æ value of 1.2 × 105 N.
not from 0 N because an initial preload must be maintained in the cable while in the machine. Lastly, the discontinuity in the loading chart was due to the cable adjusting itself on the machine and was not seen after the second cycle.
(33)
6.1.1. Cable seasoning The cable has two values for Æ, one before proofloading and one after. Proofloading is the process of stretching a new cable beyond its intended working load. Fig. 11 shows initial load displacement curve of the cable. Proofloading the cable is necessary to remove slack between cable strands. After tensile testing a section of cable to determine its ultimate strength, the cable to be used in the capstan drive was then cycled with the proofloading force. A 467 mm long section of 1.5 mm cable was cycled 10 times to 1 kN and gave the Æ to be 1.2 × 105 N as shown in Fig. 12. The cycling ranges from 175 to 1000 N and
Fig. 13. Relative theoretical stiffness magnitudes given Dinput = 50.8 mm, Doutput = 279 mm, Lfree = 28.5 mm, coefficient of friction = 0.1 and Æ = 1.2 × 105 N.
J. Werkmeister, A. Slocum / Precision Engineering 31 (2007) 55–67
6.2. Torsional stiffness Before experimental results are analyzed, a general understanding of the torsional stiffness equation is desired. First the major parameters around the drum: Ko slip1 , Ki slip1 , Ko slip2 , Ki slip2 and Kfree , will be analyzed. Then four major parameters in the torsional stiffness equation will be converted to sensitivities to see how they affect the torsional stiffness equation. After
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the theory is analyzed, experimental results will be analyzed along with the hysteresis encountered. In all the experiments, the parasitic motion in the input drum varied by only 1–2% of the total system deflection. 6.2.1. Relative theoretical stiffness magnitudes According to the analysis, as the amount of applied load increases on the capstan drive the torsional stiffness decreases as shown in Fig. 13. The graph shows that the largest stiffness on the capstan drive resides for the lowest load for Ki slip1 and Ki slip2 . As the load increases, the stiffness decreases. This is verified by the experiments. 6.2.2. Parameter sensitivities Four cases will be discussed for determining the sensitive parameters in the torsional stiffness equation of a capstan drive. These four cases are Tpreload , TLoad , μ and Lfree . After substitution of known values into Eqs. (26)–(28), and (31) the overall torsional stiffness equation is: 2 A = μÆDoutput 2 B = Tpreload Doutput TLoad + 2Lfree μTpreload 2 + Tpreload Dinput TLoad − 2Lfree μTLoad
C = Doutput TLoad + 2Lfree μTpreload + 2Lfree μTLoad + Dinput TLoad
(34)
D = Doutput TLoad + 2Lfree μTpreload − 2Lfree μTLoad + Dinput TLoad B Ktotal = A CD As Tpreload is increased, the torsional stiffness increases. When the load increases in addition to the preload, the stiff-
Fig. 14. (bottom) Torsional stiffness sensitivity to Tpreload and TLoad ; (top) torsional stiffness vs. Tpreload and TLoad . Dinput = 50.8 mm, Doutput = 279 mm, Lfree = 28.5 mm, coefficient of friction = 0.1, Æ = 1.2 × 105 N, TL min = 5 N, TL max = 25 N and TP range = 0–25 N.
Fig. 15. Torsional stiffness sensitivity to Tpreload . Dinput = 50.8 mm, Doutput = 279 mm, Lfree = 28.5 mm, coefficient of friction = 0.1 and Æ = 1.2 × 105 N.
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ness does not increase as rapidly as the no load situation, as shown in Fig. 14. Fig. 14 results from taking the partial derivative of the torsional stiffness equation with respect to Tpreload , with Dinput = 50.8 mm, Doutput = 279 mm, Lfree = 28.5 mm, coefficient of friction = 0.1 and Æ = 1.2 × 105 N: ∂Ktotal = 9.3 × 102 TLoad ∂Tpreload ⎛ 2 +1.1 × 10−5 T 2 3.6×10−2 TLoad prelaod ⎜ ⎜ + 1.2 × 10−3 T Load Tprelaod ⎜ ×⎜ 2 −3 ⎜ (0.33T Load + 5.7 × 10 Tprelaod ) ⎝ 2 (0.32TLoad + 5.7 × 10−3 Tprelaod )
⎞
The graph shows a large TLoad has little affect on torsional stiffness for any Tpreload . For smaller TLoad values, the torsional stiffness increases greatly for smaller values of Tpreload . Thus, Tpreload is more affected by a low TLoad . A 3D graph portrays this as shown in Fig. 15. The graph shows for a TLoad value of 0.5 N and Tpreload of 0 N the maximum stiffness is seen. As Tpreload increases and TLoad remains as 0.5 N, the stiffness value decreases. As TLoad increases, stiffness decreases. If the Æ value increases the amount of load it carries increases as well.
⎟ ⎟ ⎟ ⎟ (35) ⎟ ⎠
Fig. 16. (bottom) Torsional stiffness sensitivity to TLoad and Tpreload ; (top) torsional stiffness vs. TLoad and Tpreload . Dinput = 50.8 mm, Doutput = 279 mm, Lfree = 28.5 mm, coefficient of friction = 0.1, Æ = 1.2 × 105 N, TP min = 5 N, TP max = 25 N and TL range = 0–25 N.
Fig. 17. (bottom) Torsional stiffness sensitivity to coefficient of friction and Tpreload ; (top) torsional stiffness vs. coefficient of friction and Tpreload . Dinput = 50.8 mm, Doutput = 279 mm, Lfree = 28.5 mm, TLoad = 50 N, Æ = 1.2 × 105 N, TP min = 5 N, TP max = 25 N and μrange = 0–1.
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The next parameter is TLoad . Fig. 16 (top) shows as TLoad increases, the torsional stiffness decreases. For larger values of Tpreload , the torsional stiffness still tends to 0 but at a slower rate. Taking the partial derivative of the torsional stiffness equation with respect to TLoad , with Dinput = 50.8 mm, Doutput = 279 mm, Lfree = 28.5 mm, coefficient of friction = 0.1
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and Æ = 1.2 × 105 N: ∂Ktotal = −4.8 × 104 Tpreload ∂TLoad ⎛ 2 3.6×10−2 TLoad + 1.2×10−3 Tpreload TLoad ⎜ ⎜ +1.1 × 10−5 T 2 preload ⎜ ×⎜
2 ⎜ 0.33TLoad + 5.7 × 10−3 Tprelaod ⎝
2 0.32TLoad + 5.7 × 10−3 Tprelaod
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (36)
The resulting Fig. 16 shows TLoad has little affect on large Tpreload . Comparing the sensitivity of Tpreload with TLoad , Tpreload has more of an affect on the torsional stiffness than TLoad . Not until Tpreload is small will TLoad be a factor. The next parameter to be investigated is the coefficient of friction. Taking the partial derivative of the torsional stiffness equation with respect to the coefficient of friction, with Dinput = 50.8 mm, Doutput = 279 mm, Lfree = 28.5 mm, TLoad = 50 N and Æ = 1.2 × 105 N: A = 1.4 × 105 B = 0.625μ3 + 8.1Tpreload − 0.482Tpreload μ2 3 2 C = −41.95μ + 0.18Tpreload μ + 1.07 × 10−3 Tpreload μ2 2 2 D = −2.7 × 10−3 μ3 Tpreload + 3.2 × 10−2 μ2 Tpreload
+ 1.4Tpreload μ E = −4.1 × 10−2 μ3 Tpreload − 3.62μ2 + 1.85
(37)
3 × 10−4 μ3 Tpreload
F = 4.9 + 5.7 × 10−2 μTpreload + 0.85μ G = 4.9 + 5.7 × 10−2 μTpreload − 0.85μ ∂Ktotal B+C+D+E =A ∂μ F 2 G2 Fig. 17 shows the effect of the coefficient of friction, μ, on the torsional stiffness equation. The graphs shows as μ increases the torsional stiffness increases but for higher values of Tpreload , the coefficient has a greater affect on the torsional stiffness. Lastly the free length of the cable has similar results as for TLoad as shown in Fig. 18. As the free length increases, tor-
Fig. 18. (bottom) Torsional stiffness sensitivity to Lfree and Tpreload ; (top) torsional stiffness vs. to Lfree and Tpreload . Dinput = 50.8 mm, Doutput = 279 mm, coefficient of friction = 0.1, TLoad = 50 N, Æ = 1.2 × 105 N, TP min = 5 N, TP max = 25 N and Lfree range = 0–50.8 mm.
Fig. 19. Logarithmic curve plots to determine capstan drive torsional stiffness.
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sional stiffness decreases, but larger values of Tpreload can help to compensate. Taking the partial derivative of the torsional stiffness equation with respect to Lfree , with Dinput = 50.8 mm, Doutput = 279 mm, coefficient of friction = 0.1, TLoad = 50 N and Æ = 1.2 × 105 N: A = −1.8 × 102 2 4 B = 24.5Tpreload + 5.5 × 103 + 4.0 × 10−2 Tpreload L2free
C = 2.0 × 103 L2free − 4.4 × 102 Lfree Tpreload 2 − 18.0Tpreload L2free 3 D = 1.9Tpreload Lfree
(38)
E = 4.9 + 0.2Lfree Tpreload + 3.0Lfree F = 4.9 + 0.2Lfree Tpreload − 3.0Lfree B+C+D ∂Ktotal =A ∂Lfree E2 F 2 Out of the four parameters, Tpreload and the coefficient of friction have a larger affect on the torsional stiffness equation than does TLoad and Lfree . For the experimental results shown below, a Tpreload of 67 N was used and the coefficient of friction was 0.1. Different values of Tpreload was measured and experimented with, but for simplicity 67 N was used unless stated otherwise. 6.2.3. Torsional stiffness—experimental Measurements were run with Æ equal to 1.2 × 105 N for a 1.5 mm cable. Initial results gave the torsional stiffness of the capstan drive to have a logarithmic fitting as shown in Fig. 19. The curve fits agree with experimental values within a few percent, while the curve fits for theory agree less. However, both curve fits capture the dominant trends and can be useful for design guidance and initial performance estimates. This data is preliminary for a single case and additional experiments and dimensional variants are required to completely characterize the capstan’s behavior. Results from the 1.5 mm cable diameter was also compared to those from a 7 × 19 stranded 3 mm diameter cable. Unfortunately, the 3 mm cable did not produce good results because it was too thick and never properly wrapped around the small input drum. Ideally, the maximum cable diameter of the drum should be 40–45 times the cable diameter [7]. 6.2.4. Hysteresis As the weights were added onto the loading arm to create the desired torque, the preload value would change by 1–1.5 N. As the weights were removed a new Tpreload value was measured as shown in Fig. 20. The beginning Tpreload was measured as 88.6 N and after adding and removing the weights, the preload was 89.1 N. The preload remained constant because measurements were taken on the side of the drum that reads Tpreload . This region was defined above as θ no-slip and Table 1 gives the results. Hysteresis is present in the system, and thus an encoder should be used on the output drum. However, the performance is linear enough that any robust control system would be able to provide excellent performance.
Fig. 20. Torque vs. rotation measurements (note the hysteresis). Table 1 No-slip zone load cell data Load cell reading (N) Preload (up) Weight of arm (up) Weight of arm + 1 kg (up) Weight of arm + 2 kg (up) Weight of arm + 3 kg (up) Weight of arm + 4 kg (up) Weight of arm + 3 kg (down) Weight of arm + 2 kg (down) Weight of arm + 1 kg (down) Weight of arm (down) Preload (down)
88.6 88.6 88.6 88.7 88.7 88.9 88.9 88.9 89.1 89.2 89.1
Table 2 Slip 1 side of drum load cell data Load cell reading (N) Preload Weight of arm Weight of arm + 1 kg Weight of arm + 2 kg Weight of arm + 3 kg
77.9 79.4 80.7 82.7 84.7
For regions outside of the no-slip zone, the known TLoad increased for θ slip1 and decreased for θ slip2 just as predicted, as shown in Tables 2 and 3. These regions were not used because the true preload could only be read where there was slip and the cable was always moving due to friction in slip 1 and 2 regions. Table 3 Slip 2 side of drum load cell data Load cell reading (N) Preload Weight of arm Weight of arm + 1 kg Weight of arm + 2 kg Weight of arm + 3 kg
70.9 69.9 66.4 61.4 56.6
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7. Conclusions Analytical methods were successfully developed to model the torsional stiffness of a capstan drive. Modeling the cable (or band) as having two regions of slip, θ slip1 and θ slip2 , considers the total elongation of the cable wrapped around the drum due to external torque and initial preload. Experiments showed the torsional stiffness of a capstan to be 1.0 × 104 N m/rad and agreed with the theory to within 5% for larger angles. Friction and microslip between the cable and capstan are the biggest sources of hysteresis and variation. The resulting analytical model provides a tool for designers to predict the torsional stiffness of capstan drives, which will allow them to be sized in conjunction with the design of servo systems. This will allow capstan drive actuated machines’ performance to be predicted and optimized before they are built. One of the most important parameters is preload, yet practically, it is the most difficult to deterministically set during manufacturing. There is a need for the creation of deterministic cable or band preloading anchoring methods on drums, for example, where a screw could be turned with a torque wrench to accurately set the preload. In addition, once the slack is removed from the cable and the preload applied, any creep that might occur in the cable, which should be minimal if the cable has been pre-seasoned, should be compensated for by a strong spring element. Further investigation would be needed to determine: • Torsional stiffness in relation to multiple wraps of the cable around the input drum.
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• Appropriate cable preload. • Effect of cable diameter/drum diameter on accuracy of the theory and drive performance. • Relative performance between cables and metal bands. Acknowledgements The authors wish to thank Thomas Massie of SensAble Technologies for donating a capstan drive and Sava Industries for donating the steel cable. The authors would also like to thank Dr. Brad Damazo and Dr. Alkan Donmez at NIST for their help during this project. This work was funded by the Cambridge MIT Institute and the MIT-Microsoft iCampus Alliance. The authors would also like to thank the reviewers for their excellent comments and suggestions. References [1] [2] [3] [4]
Seu. US Patent 5,779,376; July 14, 1998. Murcia, et al. US Patent 6,565,173; May 20, 2003. Stephens, et al. US Patent 5,878,934; March 9, 1999. Shigley J, Mischke C. Standard handbook of machine design. 2nd ed. NY: McGraw-Hill; 1996. p. 31.5–31.9. [5] Slocum AH. Precision machine design. Society of Manufacturing Engineers; 1992. p. 691–693. [6] ASTM “E8 Standard Tests Methods of Tension Testing of Metallic Material.” Annual Book or ASTM Standards, vol. 3.01. American Society for Testing and Materials. [7] Machinery’s Handbook, 26th ed. NY, NY: Industrial Press. p. 342.
Theoretical and experimental determination of capstan drive stiffness Jaime Werkmeister, Alexander Slocum ∗ Department of Mechanical Engineering, MIT, Rm 3-445 77 Massachusetts Ave., Cambridge, MA 02139, USA Received 13 December 2004; received in revised form 10 February 2006; accepted 3 March 2006 Available online 5 May 2006
Abstract Cable or metal band capstan drives are used as rotary transmission elements for their very low (nominally zero) backlash and high stiffness properties. Cable drives, in particular, are found in many types of equipment, and to obtain high stiffness, the cable is typically wrapped around the input and output drum in a figure-eight pattern. This paper develops analytical methods for predicting the torsional stiffness of capstan drives to enable designers to better assess machine performance in the design study phase. Experiments validated the analysis. © 2006 Elsevier Inc. All rights reserved. Keywords: Wire capstan drive; Torsional stiffness; Transmission
1. Introduction Capstan drives have many uses in products, such as printers, plotters, copiers/scanners and tape recorders. For example, in printers, the head that supports the ink cartridge is typically actuated by a cable driven by a capstan1 [1]. Another use in printers is the feeding system; paper from the tray is fed onto the platen by a rotating capstan2 [2]. Tape recorders use a capstan that supports and controls the speed of the tape3 [3]. In precision machines, capstans can be used as rotary power transmission elements. For example, a device from SensAble Technologies4 uses a combination of two input drums and one output drum to aid in a 3 degree-of-freedom touch based application, as shown in Fig. 1. Capstans in these applications are typically configured with a cable wrapped in a figure-eight pattern around input and output drums. Multiple wrappings give the drive high stiffness and reduce radial loads. Two input drums are
∗
Corresponding author. Tel.: +1 617 452 2275; fax: +1 617 258 6427. E-mail address: [email protected] (A. Slocum). 1 Commercial Printing and Pre-Press Services: http://www.oneals.com. Last visited 12/04. 2 Arts56 Unit 11: Printing, created 12/98 by Claris Home Page. Last modified 12/00: http://lore.fhda.edu/Faculty/andreatta/arts56/Unit11.1.html. Last visited 12/04. 3 “I collect it: How Does an 8-Track Tape Work”. Article by Abigail Lavine, icollectit, 2000: http://www.icollectit.co.uk/icollectit.cfm?template name=a show detail.cfm&a id=168. Last visited 12/04. 4 SensAble Technologies: http://www.sensable.com/. Last visited 12/04. 0141-6359/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.precisioneng.2006.03.001
used to obtain rotary motion, about Z, and linear motion about Y of the hepatic system tip. A third input drum, located on the right-hand side of Fig. 1 is used to give the tip rotary motion about Y. Despite the widespread use of capstan drives and their availability as modular components,5 a review of the literature6 [4,5] did not provide a common theory of attainable drive stiffness and no data was given to validate different approaches. In this paper, a theory of wire capstan drive stiffness is presented along with experimental verification. 2. Capstan engagement angle A wire (cable) capstan drive is composed of an output drum, diameter Doutput , and an input drum, diameter Dinput , linked by cable segments. The cable can be wrapped multiple times between the input and output transmission drums in a figureeight pattern. Fig. 2 shows the basic representation of this drive. Initially tension exists in the cable, which is the preload force, Tpreload . If an external torque, Γ drum , is applied to the input drum Dinput , part of the cable extends due to increased tension,
5 Sagebrush Technology, Inc. “The Roto-Lok Drive Technology: A Revolutionary Innovation”, Sagebrush Technology, Inc. 2002. http://www. sagebrushtech.com/tech/technology.html. Last visited 12/04. 6 Stephen W. Attaway, The Mechanics of Friction in Rope Resuce, International Technical Rescue Symposium, 1999: http://www.amrg.org/ Mechanics of Friction in Rope Rescue.pdf. Last visited 12/04.
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Nomenclature Æ Doutput Dinput dF dL dN dT dθ dδ Kfree Ko slip1 Ki slip1 Ko slip2 Ki slip2 K1 K2 Ktotal Ktorsion L Lfree L rdrum routput rinput rcable T Tpreload TLoad T
stiffness/length of cable (N) output drum diameter (m) input drum diameter (m) differential force (N) differential element of cable wound around drum (m) differential normal force (N) differential tension (N) differential angle (rad) differential element of the elongation of cable (m) free length cable stiffness (N/m) output drum cable stiffness slip area 1 (N/m) input drum cable stiffness slip area 1 (N/m) output drum cable stiffness slip area 2 (N/m) input drum cable stiffness slip area 2 (N/m) slip side 1 cable stiffness (N/m) slip side 2 cable stiffness (N/m) total cable stiffness (N/m) torsional stiffness of the capstan (N m/rad) original free length of cable in tensile test measurement (m) free length of cable not in contact with either drum (m) length deflection (m) radius of drum, input or output (m) radius of output drum (m) radius of input drum (m) radius of cable (m) cable tension (N) cable preload tension (N) tension due to applied load (N) additional tension = TLoad (N)
Greek letters Γ drum general torque acting on a drum (N m) Γ output torque acting on output drum (N m) Γ input torque acting on input drum (N m) α general angle introduced with an applied torque (rad) δ elongation in cable on drum (m) ε strain in cable μ coefficient of friction between cable and drum θ generic angle (rad) θ slip deformation region of the loaded cable on the drum where slip can occur (rad) θ slip1 angle of cable in the slip 1 region (rad) θ slip2 angle of cable in the slip 2 region (rad) θ no-slip region of no deformation and no slip (rad) θ wrap total angle of cable wrapped around drum (rad) θ cable-encoder encoder position of cable as rotated (rad) θ lever angular deflection of the input drum (rad)
while the other part contracts due to tension being relieved. For example, if the input drum is rotated counter-clockwise in Fig. 2, a reasonable assumption is the tension in one cable will increase by some amount T, which will be called TLoad and the tension in the other cable will be reduced by some amount TLoad . For illustrative purposes, this can be demonstrated with a rubber band encircling a cylinder (soda can). Inserting a finger under the rubber band, twisting and pulling outward creates a device that looks like a capstan drive. As the cylinder is rotated counterclockwise, the rubber band only stretches on the right-hand side. Coloring the portion of the rubber band on the upper and lower surface of the cylinder helps with visualization. This difference in extension and contraction of the rubber band is enabled by friction between the band and the cylinder. As shown in Fig. 3, the active tractive region of interest, where cable extension on the drum dynamically occurs, can be defined as acting over an angle θ slip , which is the section of the cable where some slip can occur while the cable is loaded. This angle is the angle one would derive from the classic capstan equation as being required to hold the applied load, given the tensions in the two cable segments: θ slip1 + θ no-slip + θ slip2 = θ wrap . For a robust design, θ no-slip must be greater than zero and if θ wrap is greater than 2π, it indicates that there are multiple wraps on a drum. From a torque balance, TLoad can be determined: Γdrum = (Tpreload + TLoad − (Tpreload − TLoad ))rdrum ⇒ TLoad =
Γdrum 2rdrum
(1)
The input and output drum torques create an elongation in the cable. Note the cable was already slightly extended from the initial cable length due to the preload force. Since the system is in equilibrium with the preload force, its current length will be considered the initial length and any additional extension is what is of interest with respect to determining the stiffness. Fig. 4 shows the free body diagram of a small cable segment of the stretched cable in traction with the drum; dN is the normal force from the cable in contact with the drum and μdN is the traction force due to friction between the cable and the drum: dθ dθ FX = T cos +μ dN − (T + dT ) cos = 0 (2) 2 2 dθ dθ FY = dN − T sin − (T + dT ) sin = 0 (3) 2 2 Since dθ is infinitesimally small: dT = μ dθ (4) T Integrating from T(θ) to a tension T and 0 to θ slip , where θ slip is the angle-of-slip, not the angle of entire contact, as defined in Fig. 3: T θslip dT =μ dθ (5) 0 T (θ) T T (θ) = T e−μθslip
(6)
J. Werkmeister, A. Slocum / Precision Engineering 31 (2007) 55–67
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Fig. 1. PhantomTM haptic input device from SensAble Technologies Corp. Two input drums on one output drum shown on the left and one input drum on one output drum shown on the right. Bottom picture displays the overall device.
since the cross-sectional area of a cable is difficult to define: ε=
T Æ
(9)
The load strain can also be stated as the amount of deflection divided by the unstretched length of the segment under consid-
Fig. 2. Representation of capstan drive where arrows represent direction of drum and cable motion.
Expression (6) is the classic capstan equation used to find the two regions θ slip1 and θ slip2 where deformations in the cable occur as the cable transfers torque-induced loads to the drum: Tpreload + TLoad 1 θslip1 = ln μ Tpreload ⇒ T = e−μθ (Tpreload + TLoad )
θslip2
−1 = ln μ
Tpreload − TLoad Tpreload
(7)
⇒ T = eμθ (Tpreload − TLoad )
(8)
3. Cable deformation From Hooke’s Law, the term Æ for the cable is defined as the product of the modulus of elasticity E and the effective area A
Fig. 3. Nomenclature for the stiffness derivation model for the two sides of the capstan drive.
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modulus of elasticity, coefficient of friction and radius of the drum: Tpreload rdrum (Tpreload + TLoad ) 1− δ= Æμ Tpreload + TLoad Tpreload + TLoad Tpreload rdrum 1 ln − (15) μ Tpreload Æ Tpreload + TLoad rdrum TLoad − Tpreload ln δ= Æμ Tpreload
Fig. 4. Differential cable element.
eration ε=
dδ dδ = dL rdrum dθ
(10)
The stiffness is determined from (1/K) = compliance (dδ/dTLoad ): 1/Tpreload dδ rdrum = 1 − Tpreload dTLoad μÆ (Tpreload + TLoad )/Tpreload Tpreload + TLoad μÆ (16) ⇒ Ko slip1 = 2 Doutput TLoad
Hence dδ =
Trdrum dθ Æ
(11)
where dδ is the elongation in the small segment dL under tension T at that point. The deflection of interest is the extension due to just the load. The preload is even everywhere in the cable, and it is also represented in the expression for tension T when Eq. (11) is integrated; hence the elongation component due to preload force must be subtracted: dδ = dδTpreload +TLoad − dδTpreload θ Tpreload rdrum θslip Trdrum = dθ − Æ Æ 0
(12)
Likewise for the input drum the stiffness is: Tpreload + TLoad μÆ Ki slip1 = 2 Dinput TLoad 3.2. Cable deformation on slip side 2 In a manner like that for slip side 1: θslip2 δ (Tpreload − TLoad )rdrum μθ dδ = e dθ Æ 0 0
−
Tpreload rdrum θslip2 Æ
3.1. Cable deformation on slip side 1 The relation between the deflection in the cable, tension, drum radius, effective modulus, angle-of-slip, and Eqs. (7) and (11), produces the elongation of the cable on slip side 1 of the drum: δ θslip1 (Tpreload + TLoad )rdrum −μθ dδ = dθ e Æ 0 0 −
Tpreload rdrum θslip1 Æ
(13)
Following through with the integration produces the basic form of the elongation of the cable: δ=
Tpreload rdrum θslip1 rdrum (Tpreload + TLoad ) (1 − e−μθslip1 ) − Æμ Æ (14)
where δ is the elongation of the cable for the region of θ slip1 and represents the extension in the cable due to the external torque on slip side 1 of the drum. Substituting for the exponential from Eq. (7), and combining like terms, the deflection is determined as a function of TLoad , cross-sectional area of the cable,
(17)
Fig. 5. Determining the free length Lfree of the cable.
(18)
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59
Fig. 6. Spring model of capstan drive output drum.
Following through with the integration produces the basic form of the elongation of the cable:
and radius of the drum:
Tpreload rdrum (Tpreload − TLoad ) 1− Æμ Tpreload − TLoad Tpreload − TLoad Tpreload rdrum 1 ln − Æ μ Tpreload Tpreload − TLoad rdrum δ=− TLoad + Tpreload ln Æμ Tpreload
δ= δ=
rdrum (Tpreload − TLoad ) (1 − eμθslip2 ) Æμ −
Tpreload rdrum θslip2 Æ
(19)
where δ is the elongation of the cable for the region of θ slip2 and represents the extension in the cable due to the external torque on slip side 2 of the drum. Again, substituting for the exponential from Eq. (7), and combining like terms, the deflection is determined as a function of TLoad , cross-sectional area of the cable, modulus of elasticity, coefficient of friction
(20)
The stiffness is determined from (1/K) = compliance (dδ/dTLoad ): −1/Tpreload rdrum dδ = − 1 + Tpreload Tpreload − TLoad /Tpreload dTLoad μÆ Tpreload − TLoad μÆ (21) ⇒ Ko slip2 = 2 Doutput TLoad
Fig. 7. Partition of cable.
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J. Werkmeister, A. Slocum / Precision Engineering 31 (2007) 55–67
Likewise for the input drum the stiffness is: Tpreload − TLoad μÆ Ki slip2 = 2 Dinput TLoad
(22)
Eqs. (16) and (21) are the limits of the total deflection on the output drum due to the extension in the cable from the external torque and preload and Eqs. (17) and (22) are the limits of the total deflection due to the extension in the cable from the external torque and preload on the input drum. 3.3. Cable deformation in free length section The free length deflection and stiffness of the cable are determined from Hooke’s law: l =
TLfree Æ
Kfree =
(23)
Æ Lfree
(24)
An expression for Lfree can be obtained from the capstan’s geometry shown in Fig. 5. The distance between the input and output drum, radius of the two drums and distance between the two drums gives Lfree : Lfree =
routput + rinput tan(θ/2)
(25)
4. System torsional stiffness The stiffness of the output drum can be modeled as effectively having three springs on either side as shown in Fig. 6. These springs represent each section of the cable for each of the two sides of the drums: a section on the output drum, the freelength section between the drums and a section on the input drum. Ko slip2 and Ko slip1 includes the stiffness associated with cable on the output drum over regimes θ slip1 and θ slip2 , where some deformations can occur. Kfree is the stiffness of each cable segment between drums. Ki slip1 and Ki slip2 are the stiffnesses associated with the cable on the input drum over region θ slip1 and θ slip2 , respectively. Determining the stiffness on each side of the drum, as indicated in Fig. 7, produces the first and second side of the stiffness, respectively. K1 =
1 (1/Ko slip1 ) + (1/Kfree ) + (1/Ki
=
1 (1/Ko slip2 ) + (1/Kfree ) + (1/Ki Ko
The two sides act in parallel thus the cable’s total longitudinal stiffness is the addition of these two springs in parallel. Ktotal = K2 + K1
(28)
The torsional stiffness of the output drum is determined from: slip1 )
Ko slip1 Kfree Ki slip1 = Ko slip1 Kfree + Ko slip1 Ki slip1 + Kfree Ki
K2 =
Fig. 8. Cable segment in tensile test machine.
δ= (26) slip1
α=
δ Ddrum /2
=>
(29) 4Γ 2 Ddrum Ktotal
(30)
Since Γ drum = Ktorsion α, when one drum is rigidly held, the torsional stiffness of the other drum will appear to be:
slip2 )
Ko slip2 Kfree Ki slip2 K slip2 free + Ko slip2 Ki slip2 + Kfree Ki
2Γdrum Ddrum Ktotal
(27) slip2
Ktorsion =
2 Ktotal Ddrum 4
(31)
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Fig. 9. Experiential system for measuring capstan drive performance.
5. Experimental procedure
5.2. Torsional stiffness
5.1. Æ value
The experimental system for determining the torsional stiffness of a capstan drive consisted of a 1.5 mm steel cable wrapped in a figure-eight pattern around two drums with diameters of 50 and 280 mm. The input drum was fixed as would be the case if a motor servo was loaded in position. Torque was applied to the output drum to simulate a load being applied. The rotations of the input and output drums were measured and the difference
A section of cable, according to ASTM, was mounted and pulled to failure in an InstronTM tensile test machine as shown in Fig. 8 and in accordance with ASTM standards [6]. Once testing was complete, the stress strain curve was produced and the effective modulus was calculated.
Fig. 10. Rear view of experiential system for measuring capstan drive performance.
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was due to the capstan drive compliance. To measure this rotation, a lever arm was attached to each drums’ axis. As torque was applied to the output drum, any rotation encountered in the input drum was subtracted to give the net deflection of the capstan drive: δ = θcable
encoder routput
− θlever rinput
(32)
Capacitance probes detected these motions at the end of the lever arms. A load cell was spliced between the cable to determine the cable preload. Figs. 9 and 10 show the overall setup. Rotating the output drum 15–20 times back and forth ensures the system runs smoothly and the system has achieved steady state with the entire cable under the preload tension, Tpreload , before measurements were taken. Weights were added to the end of the load arm and the motion of the locked input and output shafts were recorded with values of Tpreload and TLoad . The full procedure was: 1. Wrap cable in a figure-eight pattern once around the input and output drums. 2. Tighten the cable to desired preload. 3. Rotate output drum back and forth 15–20 times. 4. Record initial reading of capacitance probes and load cell with no preload. 5. Check desired preload and record readings from capacitance probes and load cell. 6. Add weight to the torque arm in increments of 1 kg until cable starts to slip. 7. Record data from both capacitance probes for every 1 kg added and load cell. 8. Repeat two more times. 9. After third run, increase cable preload and repeat procedure starting from step 2. 6. Results 6.1. Effective modulus The effective cable modulus, Æ, is determined by loading the cable to 25% of its rated tensile strength and noting the displacement of the load. Æ FL δ => Æ = L δ where L is the original length of the testing sample. F=
Fig. 11. Applied force vs. strain which is displacement divided by original length, for the 1.5 mm diameter cable.
Fig. 12. One kiloNewton cyclic tensioning of the 15 mm cable results in a Æ value of 1.2 × 105 N.
not from 0 N because an initial preload must be maintained in the cable while in the machine. Lastly, the discontinuity in the loading chart was due to the cable adjusting itself on the machine and was not seen after the second cycle.
(33)
6.1.1. Cable seasoning The cable has two values for Æ, one before proofloading and one after. Proofloading is the process of stretching a new cable beyond its intended working load. Fig. 11 shows initial load displacement curve of the cable. Proofloading the cable is necessary to remove slack between cable strands. After tensile testing a section of cable to determine its ultimate strength, the cable to be used in the capstan drive was then cycled with the proofloading force. A 467 mm long section of 1.5 mm cable was cycled 10 times to 1 kN and gave the Æ to be 1.2 × 105 N as shown in Fig. 12. The cycling ranges from 175 to 1000 N and
Fig. 13. Relative theoretical stiffness magnitudes given Dinput = 50.8 mm, Doutput = 279 mm, Lfree = 28.5 mm, coefficient of friction = 0.1 and Æ = 1.2 × 105 N.
J. Werkmeister, A. Slocum / Precision Engineering 31 (2007) 55–67
6.2. Torsional stiffness Before experimental results are analyzed, a general understanding of the torsional stiffness equation is desired. First the major parameters around the drum: Ko slip1 , Ki slip1 , Ko slip2 , Ki slip2 and Kfree , will be analyzed. Then four major parameters in the torsional stiffness equation will be converted to sensitivities to see how they affect the torsional stiffness equation. After
63
the theory is analyzed, experimental results will be analyzed along with the hysteresis encountered. In all the experiments, the parasitic motion in the input drum varied by only 1–2% of the total system deflection. 6.2.1. Relative theoretical stiffness magnitudes According to the analysis, as the amount of applied load increases on the capstan drive the torsional stiffness decreases as shown in Fig. 13. The graph shows that the largest stiffness on the capstan drive resides for the lowest load for Ki slip1 and Ki slip2 . As the load increases, the stiffness decreases. This is verified by the experiments. 6.2.2. Parameter sensitivities Four cases will be discussed for determining the sensitive parameters in the torsional stiffness equation of a capstan drive. These four cases are Tpreload , TLoad , μ and Lfree . After substitution of known values into Eqs. (26)–(28), and (31) the overall torsional stiffness equation is: 2 A = μÆDoutput 2 B = Tpreload Doutput TLoad + 2Lfree μTpreload 2 + Tpreload Dinput TLoad − 2Lfree μTLoad
C = Doutput TLoad + 2Lfree μTpreload + 2Lfree μTLoad + Dinput TLoad
(34)
D = Doutput TLoad + 2Lfree μTpreload − 2Lfree μTLoad + Dinput TLoad B Ktotal = A CD As Tpreload is increased, the torsional stiffness increases. When the load increases in addition to the preload, the stiff-
Fig. 14. (bottom) Torsional stiffness sensitivity to Tpreload and TLoad ; (top) torsional stiffness vs. Tpreload and TLoad . Dinput = 50.8 mm, Doutput = 279 mm, Lfree = 28.5 mm, coefficient of friction = 0.1, Æ = 1.2 × 105 N, TL min = 5 N, TL max = 25 N and TP range = 0–25 N.
Fig. 15. Torsional stiffness sensitivity to Tpreload . Dinput = 50.8 mm, Doutput = 279 mm, Lfree = 28.5 mm, coefficient of friction = 0.1 and Æ = 1.2 × 105 N.
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ness does not increase as rapidly as the no load situation, as shown in Fig. 14. Fig. 14 results from taking the partial derivative of the torsional stiffness equation with respect to Tpreload , with Dinput = 50.8 mm, Doutput = 279 mm, Lfree = 28.5 mm, coefficient of friction = 0.1 and Æ = 1.2 × 105 N: ∂Ktotal = 9.3 × 102 TLoad ∂Tpreload ⎛ 2 +1.1 × 10−5 T 2 3.6×10−2 TLoad prelaod ⎜ ⎜ + 1.2 × 10−3 T Load Tprelaod ⎜ ×⎜ 2 −3 ⎜ (0.33T Load + 5.7 × 10 Tprelaod ) ⎝ 2 (0.32TLoad + 5.7 × 10−3 Tprelaod )
⎞
The graph shows a large TLoad has little affect on torsional stiffness for any Tpreload . For smaller TLoad values, the torsional stiffness increases greatly for smaller values of Tpreload . Thus, Tpreload is more affected by a low TLoad . A 3D graph portrays this as shown in Fig. 15. The graph shows for a TLoad value of 0.5 N and Tpreload of 0 N the maximum stiffness is seen. As Tpreload increases and TLoad remains as 0.5 N, the stiffness value decreases. As TLoad increases, stiffness decreases. If the Æ value increases the amount of load it carries increases as well.
⎟ ⎟ ⎟ ⎟ (35) ⎟ ⎠
Fig. 16. (bottom) Torsional stiffness sensitivity to TLoad and Tpreload ; (top) torsional stiffness vs. TLoad and Tpreload . Dinput = 50.8 mm, Doutput = 279 mm, Lfree = 28.5 mm, coefficient of friction = 0.1, Æ = 1.2 × 105 N, TP min = 5 N, TP max = 25 N and TL range = 0–25 N.
Fig. 17. (bottom) Torsional stiffness sensitivity to coefficient of friction and Tpreload ; (top) torsional stiffness vs. coefficient of friction and Tpreload . Dinput = 50.8 mm, Doutput = 279 mm, Lfree = 28.5 mm, TLoad = 50 N, Æ = 1.2 × 105 N, TP min = 5 N, TP max = 25 N and μrange = 0–1.
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The next parameter is TLoad . Fig. 16 (top) shows as TLoad increases, the torsional stiffness decreases. For larger values of Tpreload , the torsional stiffness still tends to 0 but at a slower rate. Taking the partial derivative of the torsional stiffness equation with respect to TLoad , with Dinput = 50.8 mm, Doutput = 279 mm, Lfree = 28.5 mm, coefficient of friction = 0.1
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and Æ = 1.2 × 105 N: ∂Ktotal = −4.8 × 104 Tpreload ∂TLoad ⎛ 2 3.6×10−2 TLoad + 1.2×10−3 Tpreload TLoad ⎜ ⎜ +1.1 × 10−5 T 2 preload ⎜ ×⎜
2 ⎜ 0.33TLoad + 5.7 × 10−3 Tprelaod ⎝
2 0.32TLoad + 5.7 × 10−3 Tprelaod
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (36)
The resulting Fig. 16 shows TLoad has little affect on large Tpreload . Comparing the sensitivity of Tpreload with TLoad , Tpreload has more of an affect on the torsional stiffness than TLoad . Not until Tpreload is small will TLoad be a factor. The next parameter to be investigated is the coefficient of friction. Taking the partial derivative of the torsional stiffness equation with respect to the coefficient of friction, with Dinput = 50.8 mm, Doutput = 279 mm, Lfree = 28.5 mm, TLoad = 50 N and Æ = 1.2 × 105 N: A = 1.4 × 105 B = 0.625μ3 + 8.1Tpreload − 0.482Tpreload μ2 3 2 C = −41.95μ + 0.18Tpreload μ + 1.07 × 10−3 Tpreload μ2 2 2 D = −2.7 × 10−3 μ3 Tpreload + 3.2 × 10−2 μ2 Tpreload
+ 1.4Tpreload μ E = −4.1 × 10−2 μ3 Tpreload − 3.62μ2 + 1.85
(37)
3 × 10−4 μ3 Tpreload
F = 4.9 + 5.7 × 10−2 μTpreload + 0.85μ G = 4.9 + 5.7 × 10−2 μTpreload − 0.85μ ∂Ktotal B+C+D+E =A ∂μ F 2 G2 Fig. 17 shows the effect of the coefficient of friction, μ, on the torsional stiffness equation. The graphs shows as μ increases the torsional stiffness increases but for higher values of Tpreload , the coefficient has a greater affect on the torsional stiffness. Lastly the free length of the cable has similar results as for TLoad as shown in Fig. 18. As the free length increases, tor-
Fig. 18. (bottom) Torsional stiffness sensitivity to Lfree and Tpreload ; (top) torsional stiffness vs. to Lfree and Tpreload . Dinput = 50.8 mm, Doutput = 279 mm, coefficient of friction = 0.1, TLoad = 50 N, Æ = 1.2 × 105 N, TP min = 5 N, TP max = 25 N and Lfree range = 0–50.8 mm.
Fig. 19. Logarithmic curve plots to determine capstan drive torsional stiffness.
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sional stiffness decreases, but larger values of Tpreload can help to compensate. Taking the partial derivative of the torsional stiffness equation with respect to Lfree , with Dinput = 50.8 mm, Doutput = 279 mm, coefficient of friction = 0.1, TLoad = 50 N and Æ = 1.2 × 105 N: A = −1.8 × 102 2 4 B = 24.5Tpreload + 5.5 × 103 + 4.0 × 10−2 Tpreload L2free
C = 2.0 × 103 L2free − 4.4 × 102 Lfree Tpreload 2 − 18.0Tpreload L2free 3 D = 1.9Tpreload Lfree
(38)
E = 4.9 + 0.2Lfree Tpreload + 3.0Lfree F = 4.9 + 0.2Lfree Tpreload − 3.0Lfree B+C+D ∂Ktotal =A ∂Lfree E2 F 2 Out of the four parameters, Tpreload and the coefficient of friction have a larger affect on the torsional stiffness equation than does TLoad and Lfree . For the experimental results shown below, a Tpreload of 67 N was used and the coefficient of friction was 0.1. Different values of Tpreload was measured and experimented with, but for simplicity 67 N was used unless stated otherwise. 6.2.3. Torsional stiffness—experimental Measurements were run with Æ equal to 1.2 × 105 N for a 1.5 mm cable. Initial results gave the torsional stiffness of the capstan drive to have a logarithmic fitting as shown in Fig. 19. The curve fits agree with experimental values within a few percent, while the curve fits for theory agree less. However, both curve fits capture the dominant trends and can be useful for design guidance and initial performance estimates. This data is preliminary for a single case and additional experiments and dimensional variants are required to completely characterize the capstan’s behavior. Results from the 1.5 mm cable diameter was also compared to those from a 7 × 19 stranded 3 mm diameter cable. Unfortunately, the 3 mm cable did not produce good results because it was too thick and never properly wrapped around the small input drum. Ideally, the maximum cable diameter of the drum should be 40–45 times the cable diameter [7]. 6.2.4. Hysteresis As the weights were added onto the loading arm to create the desired torque, the preload value would change by 1–1.5 N. As the weights were removed a new Tpreload value was measured as shown in Fig. 20. The beginning Tpreload was measured as 88.6 N and after adding and removing the weights, the preload was 89.1 N. The preload remained constant because measurements were taken on the side of the drum that reads Tpreload . This region was defined above as θ no-slip and Table 1 gives the results. Hysteresis is present in the system, and thus an encoder should be used on the output drum. However, the performance is linear enough that any robust control system would be able to provide excellent performance.
Fig. 20. Torque vs. rotation measurements (note the hysteresis). Table 1 No-slip zone load cell data Load cell reading (N) Preload (up) Weight of arm (up) Weight of arm + 1 kg (up) Weight of arm + 2 kg (up) Weight of arm + 3 kg (up) Weight of arm + 4 kg (up) Weight of arm + 3 kg (down) Weight of arm + 2 kg (down) Weight of arm + 1 kg (down) Weight of arm (down) Preload (down)
88.6 88.6 88.6 88.7 88.7 88.9 88.9 88.9 89.1 89.2 89.1
Table 2 Slip 1 side of drum load cell data Load cell reading (N) Preload Weight of arm Weight of arm + 1 kg Weight of arm + 2 kg Weight of arm + 3 kg
77.9 79.4 80.7 82.7 84.7
For regions outside of the no-slip zone, the known TLoad increased for θ slip1 and decreased for θ slip2 just as predicted, as shown in Tables 2 and 3. These regions were not used because the true preload could only be read where there was slip and the cable was always moving due to friction in slip 1 and 2 regions. Table 3 Slip 2 side of drum load cell data Load cell reading (N) Preload Weight of arm Weight of arm + 1 kg Weight of arm + 2 kg Weight of arm + 3 kg
70.9 69.9 66.4 61.4 56.6
J. Werkmeister, A. Slocum / Precision Engineering 31 (2007) 55–67
7. Conclusions Analytical methods were successfully developed to model the torsional stiffness of a capstan drive. Modeling the cable (or band) as having two regions of slip, θ slip1 and θ slip2 , considers the total elongation of the cable wrapped around the drum due to external torque and initial preload. Experiments showed the torsional stiffness of a capstan to be 1.0 × 104 N m/rad and agreed with the theory to within 5% for larger angles. Friction and microslip between the cable and capstan are the biggest sources of hysteresis and variation. The resulting analytical model provides a tool for designers to predict the torsional stiffness of capstan drives, which will allow them to be sized in conjunction with the design of servo systems. This will allow capstan drive actuated machines’ performance to be predicted and optimized before they are built. One of the most important parameters is preload, yet practically, it is the most difficult to deterministically set during manufacturing. There is a need for the creation of deterministic cable or band preloading anchoring methods on drums, for example, where a screw could be turned with a torque wrench to accurately set the preload. In addition, once the slack is removed from the cable and the preload applied, any creep that might occur in the cable, which should be minimal if the cable has been pre-seasoned, should be compensated for by a strong spring element. Further investigation would be needed to determine: • Torsional stiffness in relation to multiple wraps of the cable around the input drum.
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• Appropriate cable preload. • Effect of cable diameter/drum diameter on accuracy of the theory and drive performance. • Relative performance between cables and metal bands. Acknowledgements The authors wish to thank Thomas Massie of SensAble Technologies for donating a capstan drive and Sava Industries for donating the steel cable. The authors would also like to thank Dr. Brad Damazo and Dr. Alkan Donmez at NIST for their help during this project. This work was funded by the Cambridge MIT Institute and the MIT-Microsoft iCampus Alliance. The authors would also like to thank the reviewers for their excellent comments and suggestions. References [1] [2] [3] [4]
Seu. US Patent 5,779,376; July 14, 1998. Murcia, et al. US Patent 6,565,173; May 20, 2003. Stephens, et al. US Patent 5,878,934; March 9, 1999. Shigley J, Mischke C. Standard handbook of machine design. 2nd ed. NY: McGraw-Hill; 1996. p. 31.5–31.9. [5] Slocum AH. Precision machine design. Society of Manufacturing Engineers; 1992. p. 691–693. [6] ASTM “E8 Standard Tests Methods of Tension Testing of Metallic Material.” Annual Book or ASTM Standards, vol. 3.01. American Society for Testing and Materials. [7] Machinery’s Handbook, 26th ed. NY, NY: Industrial Press. p. 342.