Speed Reducers for Precision Motion Control Reducer Catalog FR
Excellent Technology for Evolving Industries Harmonic Drive actuators utilize high-precision, zero-backlash Harmonic Drive precision gears and play critical roles in robotics, semiconductor manufacturing equipment, factory automation equipment, medical diagnostics and surgical robotics. Additionally, our products are frequently used in mission-critical spaceflight applications which capture the human spirit. With over years of experience, our expert engineering and production teams continually develop enabling technologies for the evolving motion control market. We are proud of our outstanding engineering capabilities and successful history of providing customer specific solutions to meet their application requirements. Harmonic Drive LLC continues to develop enabling technologies for the evolving motion control market, which drives the pace of global innovation. C. Walton Musser Patented Strain Wave Gearing in 1955 2
Operating Principle of Gears A simple three-element construction combined with the unique operating principle puts extremely high reduction ratio capabilities into a very compact and lightweight package. The high-performance attributes of this gearing technology including, zero-backlash, high-torque-to-weight ratio, compact size, and excellent positional accuracy, are a direct result of the unique operating principles. Wave Generator The Wave Generator is a thin, raced-ball bearing fitted onto an elliptical hub. This serves as a high-efficiency torque converter and is generally mounted onto the input or motor shaft. Flexspline The Flexspline is a non-rigid, thin cylindrical cup with external teeth on the open end of the cup. The Flexspline fits over the Wave Generator and takes on its elliptical shape. The Flexspline is generally used as the output of the gear. Circular Spline The Circular Spline is a rigid ring with internal teeth. It engages the teeth of the Flexspline across the major axis of the Wave Generator ellipse. The Circular Spline has two more teeth than the Flexspline and is generally mounted onto a housing. Circular Spline 0 90 1 360 Wave Generator Flexspline The Flexspline is slightly smaller in diameter than the Circular Spline and usually has two fewer teeth than the Circular Spline. The elliptical shape of the Wave Generator causes the teeth of the Flexspline to engage the Circular Spline at two opposite regions across the major axis of the ellipse. As the Wave Generator rotates the teeth of the Flexspline engage with the Circular Spline at the major axis. For every 1 degree clockwise movement of the Wave Generator, the Flexspline rotates counterclockwise by one tooth in relation to the Circular Spline. Each complete clockwise rotation of the Wave Generator results in the Flexspline moving counterclockwise by two teeth from its original position, relative to the Circular Spline. Normally, this motion is taken out as output. Development of HarmonicDrive Speed Reducers Harmonic Drive gears have been evolving since the strain wave gear was first patented in 1955. Our innovative development and engineering teams have led us to significant advances in our gear technology. In 1988, Harmonic Drive successfully designed and manufactured a new tooth profile, the "S" tooth. Since implementing the "S" tooth profile, improvement in life, strength and torsional stiffness have been realized. In the 1990s, we focused engineering efforts on designing gears featuring space savings, higher speed, higher load capacity and higher reliability. Then in the 2000s, significant reduction in size and thickness were achieved, all while maintaining high precision specifications. 3
Component Set FB FR Series Component Set FR Features Ordering Code Rotational direction and reduction ratio Technical data Design guide Rating table Outline drawing and dimensions Efficiency No-load running torque, starting torque, overdrive starting torque Lost motion and the spring constant Assembly tolerances Precautions on assembly Lubrication 112 113 113 114 115 116 119 120 121 122 122 111
Component Set FR Features FR series component type FR is a heavy duty pancacke gear that uses a double wave generator bearing. It consists of four parts like the FB series and operates using the same principle as the cup type. It is basically structured in the same way as the FB series and supports high torque capacity by arranging the wave generator bearings in two lines and widening the tooth width of the circular spline and the flexspline. Features Flat and thin shape High torque capacity Compact and simple design High positional and rotational accuracies Coaxial input and output Structure of the FR series component type Fig. 112-1 Circular spline D Circular spline S Wave generator bearings Circular spline D It has the same number of teeth as the flexspline. As it generates no relative rotation with the flexspline, it rotates at the same speed as the flexspline. Circular spline S It has two more teeth than the flexspline like the cup-type circular spline. Wave generator Flexspline Wave generator Flexspline * How to tell circular spline D from circular spline S The peripheral chamfering of circular spline D is larger than that of circular spline S. 112
Ordering Code FR - 20 - - 2 - GR Component Set FR Table 113-1 Series FR Size 14 20 25 32 40 65 78 78 88 96 110 104 120 120 120 128 128 128 * The reduction ratio value is based on the following configuration: Input: wave generator, fixed: circular spline, output: flexspline 131 Ratio* 157 132 158 194 200 200 200 200 200 208 242 242 258 258 260 260 320 320 2= Component type Model R = Size 14 GR = Size 20- Rotational direction and reduction ratio 1 D S 2 S D 3 S D Fig. 113-1 Input Output Fixed Output Input Output Input Output Input (Note) Contact us if you use the product as Accelerator (5) and (6). I = Input R = Reductin Ratio (1) Reducer Input: Wave Generator Output: Circular Spline D Fixed: Circular Spline S i= ー 1 R (2) Reducer Input: Wave Generator Output: Circular Spline S Fixed: Circular Spline D i= ー 1 R+1 (3) Reducer Input: Circular Spline D Output: Circular Spline S Fixed: Wave Generator i= ー R R+1 4 D S Output Input 5 S D 6 D S Input Output Input Output 7 D S (4) Overdrive Input: Circular Spline S Output: Circular Spline D Fixed: Wave Generator i= ー R+1 R (5) Overdrive Input: Circular Spline S Output: Wave Generator Fixed: Circular Spline D i=r+1 (6) Overdrive Input: Circular Spline D Output: Wave Generator Fixed: Circular Spline S i= R (7) Differential When all of the Wave Generator, the Circular Spline S and the Circular Spline D rotate, Combinations (1) through (6) are available. 113
114 Nm kgfm Nm kgfm Nm kgfm Nm kgfm rpm I 10 4 kgm 2 J 10 5 kgfms 2 14 20 25 32 40 65 2000 2000 2000 2000 2000 1700 1400 1200 0 6000 6000 00 40 4000 30 3000 20 2000 3600 3600 3600 3600 3300 3000 2200 2000 1700 4000 3600 3000 20 2000 1700 1400 1200 0 20 20 20 2300 2000 1700 1400 1200 0 0.060 0.32 0.7 2.6 6.8 21 76 213 635 0.061 0.33 0.71 2.61 6.9 21 78 217 648 88 110 128 120 200 78 131 157 200 260 128 200 258 120 200 242 78 104 132 158 208 260 96 128 194 258 320 120 200 242 320 1.4 2.0 2.0 2.0 7.0 7.3 9.6 10.4 8.8* 11.0 12.4 16.3 19.4 17.6 17.6 22 25 33 46 51 38 38 36 44 56 70 83 76 76 104 125 1 144 144 165 241 295 352 264 264 293 366 6 606 704 528 528 514 706 857 1117 1269 960 960 13.7 19.6* 19.6* 19.6* 69 72 94 102* 86* 108 122 190 172* 172* 216 245 323 451 0* 372* 372* 353 431 549 686 813 745 745 784 1019 1225 1470 1411* 1411* 1617 2360 2890 34* 2590* 2590* 2870 3590 4960 5940 6900* 5170* 5170* 40 6920 8400 109 12440 9410* 9410* 0.55 1.0 1.0 1.0 3.5 4.2 5.0 5.0 5.0 5.6 7.0 9.3 11.0 11.0 11.0 11 14 18 22 22 22 22 20 25 32 40 46 46 46 45 59 71 85 86 86 94 137 167 209 289 319 319 319 319 293 402 488 584 584 584 584 5.4 9.8 9.8 9.8 34 41 49 49 49 55 69 91 108 108 108 108 137 176 216 216 216 216 196 245 314 392 451 451 451 441 578 696 833 843 843 921 1340 1570 1570 1570 1570 1640 20 2830 3130 3130 3130 3130 2870 3940 47 5720 5720 5720 5720 0.55 1.0 1.4 1.4 3.5 4.2 5.4 6.8 7.9 5.6 7.0 9.3 11.0 13.8 15.0 11 14 18 26 30 32 32 20 25 32 40 47 54 64 45 59 71 85 98 120 94 137 168 201 222 224 167 209 288 345 439 444 444 293 402 488 636 723 812 812 5.4 9.8 13.7 13.7 34 41 53 67 77 55 69 91 108 135 147 108 137 176 255 294 314 314 196 245 314 392 461 529 627 441 578 696 833 960 1176 921 1340 16 1970 21 2200 1640 20 2820 33 4300 43 43 2870 3940 47 6230 7090 7960 7960 4.4 5.9 7.8 7.8 25 34 40 40 40 39 56 67 67 67 67 76 108 137 137 137 137 137 137 196 255 294 294 294 294 363 470 559 559 559 559 745 1070 1070 1070 1070 1070 1320 1660 2300 23 23 23 23 2330 3200 3890 4470 4470 4470 4470 0.45 0.6 0.8 0.8 2.5 3.5 4.1 4.1 4.1 4.0 5.7 6.8 6.8 6.8 6.8 7.8 11 14 14 14 14 14 14 20 26 30 30 30 30 37 48 57 57 57 57 76 109 109 109 109 109 135 169 235 240 240 240 240 238 327 397 456 456 456 456 Ratio Rated torque at 2000rpm Repeated Peak Torque Max. Average Load Torque Max. Momentary Torque Max. Input Speed, rpm Limit for Average Input Speed, rpm Moment of Inertia Table 114-1 Rating table Size Rated input rotational speed Oil lubricant Grease lubricant Oil lubricant Grease lubricant Technical Data Torque value limited by ratcheting. 1. Moment of inertia: I= GD 2 2. See Rating Table Definitions on Page 12 for details of the terms. Load inertia = J 1 4 Component Set FR
Component Set FR Outline dimensions Fig. 115-1 * * H-I evenly spaced * * H-I evenly spaced FR-14 Dimensions Table 115-1 Unit: mm Symbol Size 14 20 25 32 40 65 ØA(h6) B C* D* 0 E -0.1 8.5 1 18 70 12 1 25 17.3 85 14 1 29 20 110 18 1 37 25.9 135 21 1 43 31.5 170 26 1 53 39.1 215 35 1 71.5 265 41 1 83 62 330 1 101 77.2 F* ØG H I 44 6 M3 6 3.85 60 6 M3 6 4.5 75 6 M4 8 5.55 6 M5 10 5.75 120 6 M6 12 6.95 1 6 M8 16 10.25 195 6 M10 20 10.5 240 8 M10 20 11.9 290 8 M12 24 ØJ(H7) Standard 6 9 11 14 14 19 24 28 28 Max. size 8 11 11 17 20 26 26 32 33 K(JS9) +0.1 L 0 3 10.4 4 12.8 5 16.3 5 16.3 6 21.8 8 27.3 8 31.3 8 31.3 M N a ØU ØV W X Z Mass kgf c1 c0.2 29 0.2 c1 c0.2 42 R0.08 to 0.16 0.5 c1.5 c0.2 53 22 32 4.8 1.6 R0.08 to 0.16 0.8 c1.5 c0.2 69 28 42 6.1 1.9 R0.08 to 0.25 1.7 c1.5 c0.4 84 32 52 7.6 2.5 R0.08 to 0.25 3.0 c1.5 c0.4 105 38 62 9.8 3.2 R0.08 to 0.25 6.0 c1.5 c0.4 138 44 86 12.6 4.4 R0.08 to 0.25 12.0 c2 c0.4 169 52 16 5.1 R0.08 to 0.25 22.3 c2 c0.4 211 58 128 19.7 6.3 R0.08 to 0.25 42.6 (Note) For Circular spline D, the peripheral chamfering is M. *The C, D and F values indicate relative position of individual gearing components (wave generator, flexspline, circular spline). Please strictly adhere to these values when designing your housing and mating parts. Four parts (wave generator, flexspline, circular spline D, circular spline S) are not assembled when delivered. 115
Component Set FR Efficiency The efficiency varies depending on the following conditions. Reduction ratio Input rotational speed Load torque Temperature Lubrication (Type and quantity) Efficiency compensation coefficient If the load torque is lower than the rated torque, the efficiency will be lower. Calculate the compensation coefficient Ke from Graph 116-1 to calculate the efficiency using the following example. Calculation Example Efficiency η (%) under the following condition is calculated from the example of FR-20--2GR. Input rotational speed: 0 rpm Load torque: 19.6 Nm Lubrication: Grease lubrication (Harmonic Grease SK-1A) Lubricant temperature: 20 o C Since the rated torque of size 20 with a reduction ratio of is 34 Nm (Ratings: Page 114), the torque ratio α is 0.58. (α=19.6/34=0.58) The efficiency compensation coefficient is Ke=0.86 from Graph 116-1. Efficiency η at load torque 19.6 Nm: η=ke ηr=0.86 x 65=56% Measurement condition Installation Load torque Harmonic Grease SK-1A Grease Name Harmonic Grease SK-2 Lubricant Oil Industrial gear oil class-2 Quantity Recommended quantity (see page 122) * Contact us for oil lubrication. Efficiency compensation coefficient 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Torque ratio* Table 116-1 Graph 116-1 Compensation coefficient Ke Based on recommended tolerance. The rated torque shown in the rating table (see page 114) η =Ke ηr ηr = Efficiency at the rated torque Load torque Torque ratio α = Rated torque * Efficiency compensation coefficient Ke=1 holds when the load torque is greater than the rated torque. 116
Component Set FR Efficiency at rated torque (oil lubrication) Input speed: 0rpm Input speed: 0rpm Graph 117-1 Graph 117-2 Efficiency (%) 70 60 40 30 20 Ratio, 120 200 260 320 Efficiency (%) 70 60 40 30 20, 120 200 260 320 10 10-10 0 10 20 30 40 Ambient Temperature (ºC) -10 0 10 20 30 40 Ambient Temperature (ºC) Input speed: 2000rpm Input speed: 30rpm 70 60 40 30 Ratio, 120 200 260 320 70 60 40 30 Ratio, 120 200 260 320 20 10-10 0 10 20 30 40 Ambient Temperature (ºC) 20 10-10 0 10 20 30 40 Ambient Temperature (ºC) Ratio Graph 117-3 Graph 117-4 Efficiency (%) Efficiency (%) 117
Component Set FR Efficiency at rated torque (grease lubrication) Input speed: 0rpm Input speed: 0rpm Graph 118-1 Graph 118-2 Efficiency (%) 70 60 40 30 20 Ratio, 120 200 260 320 Efficiency (%) 70 60 40 30 20 Ratio, 120 200 260 320 10 10-10 0 10 20 30 40-10 0 10 20 30 40 Ambient Temperature (ºC) Ambient Temperature (ºC) Input speed: 2000rpm Efficiency (%) 70 60 40 30 Graph 118-3 Ratio, 120 200 260 320 Input speed: 30rpm 70 60 40 30, 120 200 260 320 20 10-10 0 10 20 30 40 20 10-10 0 10 20 30 40 Efficiency (%) Graph 118-4 Ratio Ambient Temperature (ºC) Ambient Temperature (ºC) 118
No-load running torque, starting torque, backdriving torque Values indicated are from actual tests with the component sets assembled in their housings, and inclusive of friction resistance of oils seals, and churning of oil. Running torque, starting torque, backdriving torque 300 200 Component Set FR Fig. 119-1 (1) No-load running torque No-load running torque is the torque which is required to rotate the input side (high speed side), when there is no load on the output side (low speed side). The value in the graph indicates the condition when the input rotational speed is 10 rpm and the oil temperature is about 40ºC. (2) Starting torque This is the static torque required to start the high-speed shaft in a no-load condition. (3) Backdriving torque This is the static torque required to start the low-speed shaft in a no-load condition. Torque 10 5.0 1.0 0.5 0.1 14 20 25 32 40 65 Starting torque (kgcm) Backdriving torque (kgm) Size No-load running torque (kgcm) 119
Component Set FR Lost motion and the spring constant Lost motion and the spring constant of the pancake gear is the value when the wave generator or one circular spline is fixed and when a torque is applied to the dynamic spline. Table 120-1 Size 14 20 25 32 40 65 0.04 0.12 0.23 0.46 0.92 1.73 3.9 7.4 14.4 Lost motion (arc min) max. 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 1.26 3.69 7.20 15.78 29. 57.60 126.7 236.2 460.8 Spring constant (kgm/min) ± Load (kgm) Standard product Load (kgm) Spring constant 0.3 0.9 2.1 4.4 7.8 16 27 52 Description on lost motion and spring constant When assembled, rotation of the Wave Generator as a high speed input member imparts a rotating elliptical shape to the Flexspline. This causes progressive engagement of its external teeth with the internal teeth of the Circular Spline. The fixed Circular Spline, having a larger number of teeth than the Flexspline causes the latter to precess at a rate determined by the ratio of tooth difference to the total number of teeth. With the same number of teeth as the Flexspline, The Dynamic Spline rotates with, and at the same speed as, the Flexspline and is the output member of the drive. Lost motion Torsional angle Load torque Fig. 120-1 (1) Lost motion (L/M) The lost motion is the total value of rotational angle of low-speed shaft when the high-speed shaft is fixed in rotational direction with the drive installed and when slight load torque (see Table 120-1) is applied to the low-speed shaft the other way round. Torque (2) Spring constant By increasing the load torque gradually in the same manner as the lost motion and applying the load the other way round, "load torque - torsional angle" diagram emerges as shown in Fig. 120-2. The average spring constant obtained by this diagram is shown in Table 120-1. (This value is only for the HarmonicDrive components.) Example of calculation Use model number FR-40--2A-GR to fix the input shaft in rotational direction, and apply the load (30kgfm) rated in the catalog to the output shaft, and then obtain the torsional angle. L M 1 Torsional angle θ= + (T-T L M) 2 K 1 =1.5+ (30-0.92) 7.8 Torsional angle b Spring constant a Load torque Fig. 120-2 b K= kgfm/arc min a Fig. 120-3 =5.23arc min Maximum value θmax when rotated the other way round is θmax=2 θ=10.46arc min Spring constant by lost motion Torsional angle Torsional angle by lost motion Torsional angle L M 2 T L M T Load torque 120
Design Guide Recommended tolerances for assembly Maintain the recommended assembly tolerances shown in Figure 121-1 and Table 121-1 for maximum performance of your FR gear. Component Set FR Recommended tolerances for assembly Fig. 121-1 Table 121-1 Recommended tolerances for assembly Unit: mm Size 14 20 25 32 40 65 Symbol a b c d e f g h 0.013 0.015 0.016 0.013 0.015 0.016 0.011 0.007 0.017 0.016 0.020 0.017 0.016 0.020 0.013 0.010 0.024 0.016 0.029 0.024 0.016 0.029 0.016 0.012 0.026 0.017 0.031 0.026 0.017 0.031 0.016 0.012 0.026 0.019 0.031 0.026 0.019 0.031 0.017 0.012 0.028 0.024 0.034 0.028 0.024 0.034 0.021 0.015 0.034 0.027 0.041 0.034 0.027 0.041 0.025 0.015 0.043 0.033 0.052 0.043 0.033 0.052 0.030 0.015 0.057 0.038 0.068 0.057 0.038 0.068 0.035 0.015 Installation of the circular spline Conduct design and part control corresponding to the load condition for installation of the circular spline. Transmission torques by the recommended bolts and tightening torques are shown in the following table. Installation with bolts Table 121-1 Size Item 14 20 25 32 40 65 Number of bolts 6 6 6 6 6 6 6 8 8 Bolt size M3 M3 M4 M5 M6 M8 M10 M10 M12 Pitch Circle Diameter mm 44 60 75 120 1 195 240 290 Clamp torque Nm kgfm Torque Nm transmission kgfm 2.0 0.20 54 5.5 2.0 0.20 74 7.5 (Table 121-1/Notes) 1. The material of the thread must withstand the clamp torque. 2. Recommended bolt: JIS B 1176 socket head cap screw / Strength range: JIS B 1051 over 12.9. 3. Torque coefficient: K=0.2 4. Clamp coefficient: A=1.4 5. Tightening friction coefficient μ=0.15 4.5 0.46 159 16 9.0 0.92 338 34 15.3 1.56 573 58 37 3.8 1300 132 74 7.5 26 273 74 7.5 4410 4 128 13.1 77 790 121
Component Set FR Precautions on assembly Maintain the recommended tolerances shown in Figure 122-1 and Table 122-1 for optimal performance. Lubrication There are two types of lubrication; oil lubrication and grease lubrication. Although oil lubrication is common, grease lubrication is applicable to intermittent operation. Oil lubrication 1. Types of Oil The specified standard lubricant is Industrial gear oil class-2 (extreme pressure) ISO VG68. (Page 18). 2. Oil quantity The recommended oil level is shown in Table 122-1. Fig. 122-1 Oil level Table 122-1 Unit: mm A Size A 14 7 20 12 25 15 32 31 Grease lubrication Different from oil lubrication, as a cooling effect is not expected from grease lubrication, it is only available for short operation. 40 38 44 65 62 75 94 Operating condition: ED% 10% or less, continuous operation for 10 minutes or less, the maximum allowable input rotational speed in Table 114-1 or less Recommended grease: Harmonic Grease SK-1A for sizes 20 to Harmonic Grease SK-2 for size 14 (Note) If you use the product over ED% or the maximum allowable rotational speed, the grease will deteriorate, will not work as a lubricating mechanism and will result in damaging the reducer earlier. Extreme care should be taken. 122
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Tooth profile Rotational direction and reduction ratio Rating table definitions Life Torque limits Product sizing and selection Lubrication Torsional stiffness Positional accuracy Vibration Starting torque Backdriving torque No-load running torque Efficiency Design guidelines Assembly guidelines Checking output bearing S tooth profile Cup style Silk hat style Pancake style Grease lubricant Precautions on using Harmonic Grease 4B No.2 Oil lubricant Lubricant for special environments Design guideline Bearing support of the input and output shafts Wave Generator Sealing Assembly Precautions "dedoidal" state Checking procedure How to calculate the maximum moment load How to calculate the average load How to calculate the radial load coefficient (X) and axial load coefficient (Y) How to calculate life How to calculate the life under oscillating movement How to calculate the static safety coefficient 009 010 010 011 012 012 013 014 016 018 018 019 020 021 021 022 022 023 023 024 025 026 028 028 029 030 030 031 031 032 033 034 8
Tooth Profile S tooth profile Harmonic Drive developed a unique gear tooth profile that optimizes the tooth engagement. It has a special curved surface unique to the S tooth profile that allows continuous contact with the tooth profile. It also alleviates the concentration of stress by widening the width of the tooth groove against the tooth thickness and enlarging the radius on the bottom. This tooth profile (the S tooth ) enables up to 30% of the total number of teeth to be engaged simultaneously. Additionally the large tooth root radius increases the tooth strength compared with an involute tooth. This technological innovation results in high torque, high torsional stiffness, long life and smooth rotation. *Patented Engaged route of teeth Conventional tooth profile Fig. 009-1 Engaged area of teeth Fig. 009-2 S tooth profile Beginning of engagement Optimum engaged status 9
Rotational direction and reduction ratio Cup Style Series: CSG, CSF, CSD, CSF-mini Rotational direction Fig. 010-1 1 2 3 Input * R indicates the reduction ratio value from the ratings table. Output (Note) Contact us if you use the product as Accelerator (5) and (6). FS CS (1) Reducer Input: Wave Generator (WG) Output: Flexspline (FS) Fixed: Circular Spline (CS) WG i= ー 1 R (2) Reducer Input: Wave Generator Output: Circular Spline Fixed: Flexspline i= ー 1 R+1 (3) Reducer Input: Flexspline Output: Circular Spline Fixed: Wave Generator i= ー R R+1 4 5 6 7 (4) Overdrive Input: Circular Spline Output: Flexspline Fixed: Wave Generator i= ー R+1 R (5) Overdrive Input: Flexspline Output: Wave Generator Fixed: Circular Spline i= R (6) Overdrive Input: Circular Spline Output: Wave Generator Fixed: Flexspline i=r+1 (7) Differential When all of the wave generator, the flexspline and the circular spline rotate, combinations (1) through (6) are available. Silk hat Series: SHG, SHF, SHD Rotational direction Fig. 010-2 1 2 3 Input * R indicates the reduction ratio value from the ratings. table Output (Note) Contact us if you use the product as an overdrive of (5) or (6). (1) Reducer Input: Wave Generator Output: Flexspline Fixed: Circular Spline i= ー 1 R (2) Reducer Input: Wave Generator Output: Circular Spline Fixed: Flexspline i= ー 1 R+1 (3) Reducer Input: Flexspline Output: Circular Spline Fixed: Wave Generator i= ー R R+1 4 5 6 7 (4) Overdrive Input: Circular Spline Output: Flexspline Fixed: Wave Generator i= ー R+1 R (5) Overdrive Input: Flexspline Output: Wave Generator Fixed: Circular Spline i= R (6) Overdrive Input: Circular Spline Output: Wave Generator Fixed: Flexspline i=r+1 (7) Differential When all of the wave generator, the flexspline and the circular spline rotate, Combinations (1) through (6) are available. 10
Pancake Series: FB and FR Rotational direction Fig. 11-1 1 2 3 Input Output (Note) Contact us if you use the product as Accelerator (5) and (6). Output (1) Reducer Input: Wave Generator Output: Circular Spline D Fixed: Circular Spline S Input i= ー 1 R Output (2) Reducer Input: Wave Generator Output: Circular Spline S Fixed: Circular Spline D Input i= ー 1 R+1 Output Input (3) Reducer Input: Circular Spline D Output: Circular Spline S Fixed: Wave Generator i= ー R R+1 4 5 6 7 Output Input (4) Overdrive Input: Circular Spline S Output: Circular Spline D Fixed: Wave Generator i= ー R+1 R Input Output Input Output (5) Overdrive Input: Circular Spline S Output: Wave Generator Fixed: Circular Spline D i=r+1 (6) Overdrive Input: Circular Spline D Output: Wave Generator Fixed: Circular Spline S i= R (7) Differential When all of the Wave Generator, the Circular Spline S and the Circular Spline D rotate, Combinations (1) through (6) are available. Reduction ratio The reduction ratio is determined by the number of teeth of the Flexspline and the Circular Spline Number of teeth of the Flexspline: Number of teeth of the Circular Spline: Input: Wave Generator Output: Flexspline Fixed: Circular Spline Reduction ratio Zf Zc 1 i1 = = Input: Wave Generator Reduction 1 Output: Circular Spline i2 ratio = = Fixed: Flexspline R 2 R1 indicates the reduction ratio value from the ratings table. R 1 Zf-Zc Zf Zc-Zf Zc Example Number of teeth of the Flexspline: 200 Number of teeth of the Circular Spline: 202 Input: Wave Generator Output: Flexspline Fixed: Circular Spline Input: Wave Generator Output: Circular Spline Fixed: Flexspline Reduction ratio Reduction ratio 1 200-202 i1 = = = 200 R 1 1 202-200 i2 = = = R 2 202-1 1 101 11
Rating Table Definitions See the corresponding pages of each series for values. Rated torque Rated torque indicates allowable continuous load torque at rated input speed. Limit for Repeated Peak Torque (see Graph 12-1) During acceleration and deceleration the Harmonic Drive gear experiences a peak torque as a result of the moment of inertia of the output load. The table indicates the limit for repeated peak torque. Limit for Average Torque In cases where load torque and input speed vary, it is necessary to calculate an average value of load torque. The table indicates the limit for average torque. The average torque calculated must not exceed this limit. (calculation formula: Page 14) Limit for Momentary Peak Torque (see Graph 12-1) The gear may be subjected to momentary peak torques in the event of a collision or emergency stop. The magnitude and frequency of occurrence of such peak torques must be kept to a minimum and they should, under no circumstance, occur during normal operating cycle. The allowable number of occurrences of the momentary peak torque may be calculated by using formula 13-1. Maximum Average Input Speed Maximum Input Speed Do not exceed the allowable rating. (calculation formula of the average input speed: Page 14). Example of application motion profile + Load torque + Wave Generator rotational speed Start Steady Stop (Speed cycle) Start Abnormal impact torque Time Load Torque Repeated Peak Torque Time Graph 012-1 Momentary Peak Torque Moment of Inertia The rating indicates the moment of inertia reflected to the gear input. Life Life of the wave generator The life of a gear is determined by the life of the wave generator bearing. The life may be calculated by using the input speed and the output load torque. Calculation formula for Rated Lifetime Ln Tr Nr Tav Nav Series name L10 CSF, CSD, SHF, SHD, CSF-mini 7,000 hours 35,000 hours 3 Tr Lh=Ln Tav Life Nr Nav CSG, SHG 10,000 hours,000 hours L (average life) * Life is based on the input speed and output load torque from the rating table. Table 012-1 Formula 012-1 Life of L10 or L Rated torque Rated input speed Average load torque on the output side (calculation formula: Page 14) Average input speed (calculation formula: Page 14) Table 012-2 Relative torque rating 17 16 Load torque (when the rated torque is 1) 10 9 8 7 6 5 4 3 Momentary peak torque Graph 012-2 Buckling torque Racheting torque Life of wave generator (L10) Fatigue strength of the flexspline 2 Repeated peak torque 1 Rated torque 0 10 5 10 6 10 7 10 8 10 9 10 10 Total number of input rotations * Lubricant life not taken into consideration in the graph described above. * Use the graph above as reference values. 12
Torque Limits Strength of flexspline The Flexspline is subjected to repeated deflections, and its strength determines the torque capacity of the Harmonic Drive gear. The values given for Rated Torque at Rated Speed and for the allowable Repeated Peak Torque are based on an infinite fatigue life for the Flexspline. The torque that occurs during a collision must be below the momentary peak torque (impact torque). The maximum number of occurrences is given by the equation below. Allowable limit of the bending cycles of the flexspline during rotation of the wave generator while the impact torque is applied: 1.0 x 10 4 (cycles) The torque that occurs during a collision must be below the momentary peak torque (impact torque). The maximum number of occurrences is given by the equation below. Calculation formula Caution N= 1.0 10 4 n 2 t 60 Formula 013-1 Allowable occurances N occurances Time that impact torque is applied t sec Rotational speed of the wave generator n rpm The flexspline bends two times per one revolution of the wave generator. If the number of occurances is exceeded, the Flexspline may experience a fatigue failure. Ratcheting torque When excessive torque (8 to 9 times rated torque) is applied while the gear is in motion, the teeth between the Circular Spline and Flexspline may not engage properly. This phenomenon is called ratcheting and the torque at which this occurs is called ratcheting torque. Ratcheting may cause the Flexspline to become non-concentric with the Circular Spline. Operating in this condition may result in shortened life and a Flexspline fatigue failure. * See the corresponding pages of each series for ratcheting torque values. * Ratcheting torque is affected by the stiffness of the housing to be used when installing the circular spline. Contact us for details of the ratcheting torque. Caution Caution When ratcheting occurs, the teeth may not be correctly engaged and become out of alignment as shown in Figure 013-1. Operating the drive in this condition will cause vibration and damage the flexspline. Once ratcheting occurs, the teeth wear excessively and the ratcheting torque may be lowered. Circular Spline Figure 013-1 Buckling torque When a highly excessive torque (16 to 17 times rated torque) is applied to the output with the input stationary, the flexspline may experience plastic deformation. This is defined as buckling torque. * See the corresponding pages of each series for buckling torque values. "Dedoidal" condition. Flexspline Warning When the flexspline buckles, early failure of the HarmonicDrive gear will occur. 13
Product Sizing & Selection In general, a servo system rarely operates at a continuous load and speed. The input rotational speed, load torque change and comparatively large torque are applied at start and stop. Unexpected impact torque may be applied. These fluctuating load torques should be converted to the average load torque when selecting a model number. As an accurate cross roller bearing is built in the direct external load support (output flange), the maximum moment load, life of the cross roller bearing and the static safety coefficient should Flowchart for selecting a size Please use the flowchart shown below for selecting a size. Operating conditions must not exceed the performance ratings. also be checked.+ Checking the application motion profile Review the application motion profile. Check the specifications shown in the figure below. Load torque Output rotational speed ーT1 T2 T3 T4 t1 t2 t3 t4 tn n1 n2 n3 n4 * n1, n2 and nn indicate the average values. nn Tn Time Time Graph 14-1 Calculate the average load torque applied on the output side from the application motion profile: Tav (Nm). Tav = 3 n 1 t 1 T 1 3 +n 2 t 2 T 2 3 + n n t n T n 3 n 1 t 1 +n 2 t 2 + n n t n Make a preliminary model selection with the following conditions. Tav Limit for average torque torque (See the rating table of each series). Calculate the average output speed: no av (rpm) Obtain the reduction ratio (R). A limit is placed on ni max by motors. Calculate the average input rotational speed from the average output rotational speed (no av) and the reduction ratio (R): ni av (rpm) Calculate the maximum input rotational speed from the max. output rotational speed (no max) and the reduction ratio (R): ni max (rpm) Check whether the preliminary model number satisfies the following condition from the rating table. Ni av n 1 t 1 +n 2 t 2 + n n t n no av = t 1 + t 2 + t n ni max R no max ni av = no av R ni max = no max R Limit for average speed (rpm) Ni max Limit for maximum speed (rpm) NG OK Obtain the value of each application motion profile. Load torque Tn (Nm) Time tn (sec) Output rotational speed nn (rpm) Check whether T1 and T3 are less than the repeated peak torque specification. OK NG Normal operation pattern Starting (acceleration) Steady operation (constant velocity) Stopping (deceleration) Dwell Maximum rotational speed Max. output speed Max. input rotational speed (Restricted by motors) Emergency stop torque When impact torque is applied T1, t1, n1 T2, t2, n2 T3, t3, n3 T4, t4, n4 no max ni max Ts, ts, ns Check whether Ts is less than the the momentary peak torque specification. Calculate (Ns) the allowable number of rotations during impact torque. OK 10 N 4 S = N S 1.0 10 4 n S R 2 t 60 OK NG NG Review the operation conditions and model number Required life L10 = L (hours) Calculate the lifetime. L 10 = 7000 ( ) ( ) (hours) OK Tr Tav 3 nr ni av Check whether the calculated life is equal to or more than the life of the wave generator (see Page 13). The model number is confirmed. NG 14
Example of model number selection Value of each application motion profile Load torque T(Nm) n Time t(sec) n Output speed n(rpm) n Maximum rotational speed Max. output speed Max. input speed (Restricted by motors) no max = 14 rpm ni max = 10 rpm Normal operation pattern Starting (acceleration) T1 = 400 Nm, t1 = 0.3sec, n1 = 7rpm Steady operation (constant velocity) T2 = 320 Nm, t2 = 3sec, n2 = 14rpm Stopping (deceleration) T3 = 200 Nm, t3 = 0.4sec, n3 = 7rpm Dwell T4 = 0 Nm, t4 = 0.2 sec, n4 = 0 rpm Emergency stop torque When impact torque is applied Required life Ts = 0 Nm, ts = 0.15 sec, ns = 14 rpm L 10 = 7000 (hours) Calculate the average load torque to the output side based on the application motion profile: Tav (Nm). Tav = 3 7 rpm 0.3 sec 400Nm 3 +14 rpm 3 sec 320Nm 3 +7 rpm 0.4 sec 200Nm 3 7 rpm 0.3 sec+14 rpm 3 sec+7 rpm 0.4 sec Make a preliminary model selection with the following conditions. Tav = 319 Nm 620 Nm (Limit for average torque for model number CSF-40-120-2A-GR: See the rating table on Page 39.) Thus, CSF-40-120-2A-GR is tentatively selected. Calculate the average output rotational speed: no av (rpm) Obtain the reduction ratio (R). Calculate the average input rotational speed from the average output rotational speed (no av) and the reduction ratio (R): ni av (rpm) Calculate the maximum input rotational speed from the maximum output rotational speed (no max) and the reduction ratio (R): ni max (rpm) 7 rpm 0.3 sec+14 rpm 3 sec+7 rpm 0.4 sec no av = = 12 rpm 0.3 sec + 3 sec + 0.4 sec + 0.2 sec 10 rpm = 128.6 120 14 rpm ni av = 12 rpm 120 = 1440 rpm ni max = 14 rpm 120 = 16 rpm Check whether the preliminary selected model number satisfies the following condition from the rating table. Ni av = 1440 rpm 3600 rpm (Max average input speed of size 40) Ni max = 16 rpm 5600 rpm (Max input speed of size 40) OK NG Check whether T1 and T3 are equal to or less than the repeated peak torque specification. T1 = 400 Nm 617 Nm (Limit of repeated peak torque of size 40) T3 = 200 Nm 617 Nm (Limit of repeated peak torque of size 40) OK NG Check whether Ts is equal to or less than the momentary peak torque specification. Ts = 0 Nm 11 Nm (Limit for momentary torque of size 40) Calculate the allowable number (Ns) rotation during impact torque and confirm 1.0 10 4 Calculate the lifetime. OK OK OK 10 N 4 S == 1190 1.0 10 4 14 rpm 120 2 0.15 sec 60 L 10 = 7000 ( ) 294 Nm 3 319 Nm ( ) 2000 rpm 1440 rpm (hours) Check whether the calculated life is equal to or more than the life of the wave generator (see Page 12). L 10 =7610 hours 7000 (life of the wave generator: L10) The selection of model number CSF-40-120-2A-GR is confirmed from the above calculations. NG NG NG Review the operation conditions, size and reduction ratio 15
Lubrication : CSD-2A, CSF-2A, CSG-2A, FB-2, FB-0, FR-2, SHF-2A, SHG-2A and SHD and SHG/SHF -2SO and -2SH gear units: Grease lubricant and oil lubricant are available for lubricating the component sets and SHD gear unit. It is extremely important to properly grease your component sets and SHD gear unit. Proper lubrication is essential for high performance and reliability. Harmonic Drive component sets are shipped with a rust- preventative oil. The characteristics of the lubricating grease and oil types approved by Harmonic Drive are not changed by mixing with the preservation oil. It is therefore not necessary to remove the preservation oil completely from the gear components. However, the mating surfaces must be degreased before the assembly. : CSG/CSF 2UH and 2UH-LW; CSD-2UF and -2UH; SHG/SHF-2UH and 2UH- LW; SHG/SHF-2UJ; CSF Supermini, CSF Mini, and CSF-2UP. Grease lubricant is standard for lubricating the gear units. You do not need to apply grease during assembly as the product is lubricated and shipped. See Page 19 for using lubricant beyond the temperature range in table 16-2. * Contact us if you want consistency zero (NLGI No.0) for maintenance reasons. Grease lubricant Types of lubricant Harmonic Grease SK-1A This grease was developed for Harmonic Drive gears and features good durability and efficiency. Harmonic Grease SK-2 This grease was developed for small sized Harmonic Drive gears and features smooth rotation of the Wave Generator since high pressure additive is liquefied. Harmonic Grease 4B No.2 This has been developed exclusively for the CSF and CSG and features long life and can be used over a wide range of temperature. (Note) 1. Grease lubrication must have proper sealing, this is essential for 4B No.2. Rotating part: Oil seal with spring is needed. Mating part: O ring or seal adhesive is needed. 2. The grease has the highest deterioration rate in the region where the grease is subjected to the greatest shear (near wave generator). Its viscosity is between JIS No.0 and No.00 depending on the operation. NLGI consistency No. 0 00 Mixing consistency range SK-1A SK-2 4B No.2 Table 016-3 Name of lubricant Table 016-1 Harmonic Grease SK-1A Grease Harmonic Grease SK-2 Harmonic Grease 4B No.2 Oil Industrial gear oil class-2 (extreme pressure) ISO VG68 Temperature Table 016-2 SK-1A 0ºC to + 40ºC Grease SK-2 0ºC to + 40ºC 4B No.2 10ºC to + 70ºC Oil ISO VG68 0ºC to + 40ºC * The hottest section should not be more than 40 above the ambient temperature. Note: The three basic components of the gear - the Flexspline, Wave Generator and Circular Spline - are matched and serialized in the factory. Depending on the product they are either greased or prepared with preservation oil. Then the individual components are assembled. If you receive several units, please be careful not to mix the matched components. This can be avoided by verifying that the serial numbers of the assembled gear components are identical. Compatible grease by size Compatible grease varies depending on the size and reduction ratio. See the following compatibility table. We recommend SK-1A and SK-2 for general use. Ratios 30:1 Size SK-1A SK-2 4B No.2 SK-1A SK-2 4B No.2 SK-1A SK-2 4BNo.2 8 - Ratios :1* and above Size Size 8 - - 11 14 17 20 25 32-11 - - - - - - - 14 17 20 25 32 - - 40 45 58 65 90 - - - SK-1A - - SK-2 Table 016-5 Table 016-6 : Standard grease : Semi-standard grease : Recommended grease for long life and high load * Oil lubrication is required for component-sets size or larger with a reduction ratio of :1. Grease characteristics Grease specification Table 016-4 Table 016-7 Grease Base oil Refined oil Refined oil Base Viscosity cst (25ºC) Thickening agent NLGI consistency No. Additive Storage life Lithium soap base Extreme-pressure additive, others 5 years in sealed condition Lithium soap base Extreme-pressure additive, others 5 years in sealed condition 355 to 385 400 to 430 Composite hydrocarbon oil 265 to 295 265 to 295 290 to 320 Urea No. 2 No. 2 No. 1.5 Extreme-pressure additive, others Drop Point 197ºC 198ºC 247ºC Appearance Yellow Green Light yellow 5 years in sealed condition Grease Durability Fretting resistance Low-temperature performance Grease leakage Excellent : Good : Use Caution : - - 4B No.2 16
When to replace grease The wear characteristics of the gear are strongly influenced by the condition of the grease lubrication. The condition of the grease is affected by the ambient temperature. The graph 017-1 shows the maximum number of input rotations for various temperatures. This graph applies to applications where the average load torque does not exceed the rated torque. Note: Recommended Grease: SK-1A or SK-2 When to replace grease: LGTn (when the average load torque is equal to or less than the rated torque) Graph 017-1 10 10 Grease Life 4B No.2 Number of input rotations 10 9 10 8 SK-1A SK-2 Wave Generator Life 10 7 20 40 60 120 Grease temperature ( o C) Calculation formula when the average load torque exceeds the rated torque Formula 017-1 Tr LGT=LGTn Tav Other precautions 1. Avoid mixing different kinds of grease. The gear should be in an individual case when installed. 3 Formula Symbols Table 017-1 LGT LGTn Tr Tav Grease change (if average load torque exceeds rated torque) Grease change (if average load torque is equal to or less than rated torque) Rated torque Average load torque input revolutions input revolutions See the Graph 017-1. (From Graph) See the "Ratings Table" Nm of each series. Nm Calculation formula: See Page 014. 2. Please contact us when you use HarmonicDrive gears at constant load or in one direction continuously, as it may cause lubrication problems. 3. Grease leakage. A sealed structure is needed to maintain the high durability of the gear and prevent grease leakage. See the corresponding pages of the design guide of each series for Recommended minimum housing clearance, Application guide and Application quantity. 17
Precautions on using Harmonic Grease 4B No.2 Harmonic Grease 4B No.2 lubrication is ideally suited for Harmonic Drive gears. (1) Apply the grease to each contacting joint at the beginning of operation. (2) Remove any contaminents created by abrasion during running-in period. See the corresponding pages of the design guide of each series for recommended minimum housing clearance, Application guide and Application quantity. Precautions (1) Stir Grease When storing Harmonic Grease 4B No.2 lubrication in the container, it is common for the oil to weep from the thickener. Before greasing, stir the grease in the container to mix and soften. (2) Aging (running-in) The aging before the main operation softens the applied grease. More effective greasing performance can be realized when the grease is distributed around each contact surface. Therefore, the following aging methods are recommended. Keep the internal temperature at ºC or cooler. Do not start the aging at high temperature rapidly. Input rotational speed should be 0rpm to 3000rpm. However, the lower rotational speed of 0rpm is more effective. Set the speed as low as possible within the indicated range. The time required for aging is 20 minutes or longer. Operation range for aging: Keep the output rotational angle as large as possible. Contact us if you have any questions for handling Harmonic Grease 4B No.2 lubrication. Note: Strict sealing is required to prevent grease leakage. Oil lubricant Types of oil The specified standard lubricant is Industrial gear oil class-2 (extreme pressure) ISO VG68. We recommend the following brands as a commercial lubricant. Table 018-1 Standard Industrial gear oil class-2 (extreme pressure) ISO VG68 Mobil Oil Mobilgear 600XP68 Exxon Spartan EP68 Shell Omala Oil 68 COSMO Oil Cosmo gear SE68 Japan Energy ES gear G68 NIPPON Oil Bonock M68, Bonock AX68 Idemitsu Kosan Daphne super gear LW68 General Oil General Oil SP gear roll 68 Klüber Syntheso D-68EP When to replace oil First time hours after starting operation Second time or after Every 0 operation hours or every 6 months Note that you should replace the oil earlier than specified if the operating condition is demanding. See the corresponding pages of the design guide of each series for specific details. Other precautions 1. Avoid mixing different kinds of oil. The gear should be in an individual case when installed. 2. When you use size or above at max allowable input speed, please contact us as it may cause lubrication problems. * Oil lubrication is required for component-sets size or larger with a reduction ratio of :1. 18
Lubricant for special environments When the ambient temperature is special (other than the temperature range of the operating environment on Page 016-2), you should select a lubricant appropriate for the operating temperature range. Harmonic Grease 4B No.2 Type of lubricant Grease Operating temperature range 10 C to + 110 C Table 019-1 Available temperature range C to + 130 C Harmonic Grease 4B No.2 The operating temperature range of Harmonic Grease 4B No.2 lubrication is the temperature at the lubricating section with the performance and characteristics of the gear taken into consideration. (It is not ambient temperature.) High temperature lubricant Type of lubricant Grease Oil Lubricant and manufacturer Table 019-2 Available temperature range Low temperature lubricant Table 019-3 Type of lubricant Mobil grease 28: Mobil Oil Mobil SHC-626: Mobil Oil Lubricant and manufacturer 5 C to + C 5 C to + 140 C Available temperature range As the available temperature range indicates the temperature of the independent lubricant, restriction is added on operating conditions (such as load torque, rotational speed and operating cycle) of the gear. When the ambient temperature is very high or low, materials of the parts of the gear need to be reviewed for suitability. Contact us if operating in high temperature. Harmonic Grease 4B No.2 can be used in the available temperature range shown in table 019-1. However, input running torque will increase at low temperatures, and grease life will be decreased at high temperatures due to oxidation and lubricant degradation. Grease Oil Multemp SH-KII: Kyodo Oil Isoflex LDS-18 special A: KLÜBER SH-200-CS: Toray Silicon Syntheso D-32EP: KLÜBER 30 C to + C 25 C to + C 40 C to + 140 C 25 C to + 90 C 19
Torsional Stiffness Stiffness and backlash of the drive system greatly affects the performance of the servo system. Please perform a detailed review of these items before designing your equipment and selecting a model number. Stiffness Fixing the input side (wave generator) and applying torque to the output side (flexspline) generates torsion almost proportional to the torque on the output side. Figure 018-1 shows the torsional angle at the output side when the torque applied on the output side starts from zero, increases up to +T0 and decreases down to T0. This is called the Torque torsion angle diagram, which normally draws a loop of 0 A B Aʼ Bʼ A. The slope described in the Torque torsion angle diagram is represented as the spring constant for the stiffness of the HarmonicDrive gear (unit: Nm/rad). As shown in Figure 020-1, this Torque torsional angle diagram is divided into 3 regions, and the spring constants in the area are represented by K1, K2 and K3. Hysteresis loss (Silk hat and cup style only) As shown in Figure 020-1, when the applied torque is increased to the rated torque and is brought back to [zero], the torsional angle does not return exactly back to the zero point This small difference (B B') is called hysteresis loss. See the corresponding page of each series for the hysteresis loss value. Torque - torsion angle diagram Torsion angle Hysteresis loss B T 0 0 +T 0 A Figure 20-1 Torque K1 The spring constant when the torque changes from [zero] to [T1] K2 The spring constant when the torque changes from [T1] to [T2] K3 The spring constant when the torque changes from [T2] to [T3] B' See the corresponding pages of each series for values of the spring constants (K1, K2, K3) and the torque-torsional angles (T1, T2, - θ1, θ2). Example for calculating the torsion angle The torsion angle (θ) is calculated here using CSF-25--2A-GR as an example. A' Spring constant diagram Torsion angle Figure 20-2 When the applied torque is T1 or less, the torsion angle θl1 is calculated as follows: When the load torque TL1=2.9 Nm θl1 =TL1/K1 =2.9/3.1 10 4 =9.4 10-5 rad(0.33 arc min) K 3 θ 2 K 2 When the applied torque is between T1 and T2, the torsion angle θl2 is calculated as follows: When the load torque is TL2=39 Nm θl2 =θ1+(tl2 T1)/K2 =4.4 10-4 +(39-14)/5.0 10 4 =9.4 10-4 rad(3.2 arc min) When a bidirectional load is applied, the total torsion angle will be 2 x θlx plus hysteresis loss. * The torsion angle calculation is for the gear component set only and does not include any torsional windup of the output shaft. Note: See p.120 for torsional stiffness for pancake gearing. θ 1 K 1 0 T 1 T 2 Torque Backlash (Silk hat and cup style only) Hysteresis loss is primarily caused by internal friction. It is a very small value and will vary roughly in proportion to the applied load. Because HarmonicDrive gears have zero backlash, the only true backlash is due to the clearance in the Oldham coupling, a self-aligning mechanism used on the wave generator. Since the Oldham coupling is used on the input, the backlash measured at the output is extremely small (arc-seconds) since it is divided by the gear reduction ratio. 20
Positional Accuracy Positional Accuracy values represent the difference between the theoretical angle and the actual angle of output for any given input. The values shown in the table are maximum values. See the corresponding pages of each series for transmission accuracy values. Example of measurement Graph 021-1 θer θ 1 θ 2 R Transmission accuracy Input angle Actual output angle Reduction ratio θ1 θer=θ2 R Table 021-1 Formula 021-1 θer Vibration The primary frequency of the transmission error of the HarmonicDrive gear may cause a vibration of the load inertia. This can occur when the driving frequency of the servo system including the HarmonicDrive gear is at, or close to the resonant frequency of the system. Refer to the design guide of each series. How to the calculate resonant frequency of the system f = 1 2π K J Formula 021-3 The primary component of the transmission error occurs twice per input revolution of the input. Therefore, the frequency generated by the transmission error is 2x the input frequency (rev / sec). If the resonant frequency of the entire system, including the HarmonicDrive gear, is F=15 Hz, then the input speed (N) which would generate that frequency could be calculated with the formula below. Formula variables f K J The resonant frequency of the system Spring constant Load inertia Hz Nm/rad kgm 2 Table 021-2 See pages of each series Formula 021-2 15 N = 60 = 4 rpm 2 The resonant frequency is generated at an input speed of 4 rpm. 21