program 60-740—steel involute spur spline load and stress analysis introduction€¦ · ·...
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Program 60-740—Steel Involute Spur Spline Load and Stress Analysis
Introduction This model will calculate the stress and/or spline life from the geometry and load data for steel spur involute splines. Fixed, flexible straight and flexible crowned splines are included. Fixed splines have no relative motion between the external and internal members and no misalignment. The effective fit may have clearance or interference. Fixed splines usually fail from: Shaft Shear Tooth Shear Bursting of the Internal Ring Fretting Corrosion (Loose fits with vibration) Flexible splines have a clearance fit and the shafts are misaligned. There is always relative motion between the external and internal members. Flexible splines usually fail from: Shaft Shear Tooth Wear Bursting of the Internal Ring The life of the spline determined by shear cycles (shaft or tooth) and compressive cycles on fixed splines depends on the number of on-off torque cycles applied, not on the number of revolutions. The life of a flexible spline determined by compressive cycles depends on the number of revolutions of the spline. The load transmitted by a splined connection is not always uniform and an application factor (Ka on the TK Solver model Variable Sheet) must be applied. Table 1 contains some suggested factors that may be used if more accurate information for the application is not available. (Table 1 is contained in the TK Solver model on the table sheets, “Ka”.)
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Table 1
The ability of a spline to withstand transmitted load is dependent on the materials from which the splines are made. The model contains allowable stress data for five different materials, Tables 2 (psi) and 3 (kPa) show the allowable stresses used in the model. This table is also in the model with the name “matl”.
60-740—Steel Involute Spur Spline Load And Stress Analysis
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Table 2
Table 3
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Examples
Example 1
This is a fixed, 26 tooth, 10/20 pitch, 30 deg pressure angle with a face width of 1.5 inches. The spline must transmit 2500 horsepower at 3000 RPM. The spline members will be made of surface hardened steel at 48 to 53 Rc/C. The power source is a turbine and the driven load will produce light shock loading. An application factor of 1.2 will be applied. The load is always in the same direction and does not reverse.
Open a new analysis in 60-740 and enter the data in the data entry form as shown in Figure 1-1. The solved model is shown in Report 1-1.
Fig. 1-1
60-740—Steel Involute Spur Spline Load And Stress Analysis
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Report 1-1
Model Title : Program 60-740 Unit System: US GEOMETRY
Number of Teeth 26
Diametral Pitch 10.000000 1/in `
Module 2.540000 mm `
Pressure Angle 30.000000 deg
Pitch Diameter 2.600000 in
Radial Contact Height 0.0980 in
Face Width 1.5000 in
Effective Face Width 1.5000 in
Type of Spline Connection Fixed
Misalignment 0.0000 in/in
Internal Spline
Outside Dia of Hub 3.2000 in
Major Diameter 2.8000 in
Effective_Minor (Inside) Diameter 2.5020 in
Wall Thickness 0.2000 in
Tooth Thickness @ PD 0.1547 in
Space Width @ PD 0.1595 in
External Spline
Effective_Major (Outside) Diameter 2.6980 in
Minor (Root) Diameter 2.3990 in
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Model Title : Program 60-740 Unit System: US Inside (Bore) Diameter of Shaft 1.0000 in
Wall Thickness 0.6995 in
Tooth Thickness @ PD 0.1545 in
Height of Crown on Tooth 0.0000 in
Crown Radius _ in
Axial Plane Generating Radius _ in
LOAD
Power 2500.00 hp
Torque 52520.83 lbf-in
Unidirectional / Fully-reversed u
Rotation Speed 3000 rpm
Application Factor, Ka 1.20
Face_Load Distribution Factor, Km 1.0000 STRESS/LIFE INTERNAL
Material # 4.0000
Allowable_Shear Stress 40000.0000 psi
Allowable_Compressive Stress 4000.0000 psi
Max Allowable Tensile Stress 45000.0000 psi STRESS/LIFE
Tooth_Shear Stress 13404.0000 psi
Life_Factor 0.4000
Shear On-Off Torque Cycles 950000.0000
Tooth_Compressive Stress 10571.0000 psi
Fixed_Spline Life Factor 0.3500
60-740—Steel Involute Spur Spline Load And Stress Analysis
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Model Title : Program 60-740 Unit System: US Compressive On-Off Torque Cycles 3000000.0000
Flexible/Crowned Spline Life Fact
Revolutions
Wear Life hr
Hub_Shear Stress (Torsion) 19726.0000 psi
Life_Factor 0.5900
Shear On-Off Torque Cycles 66000.0000
Bursting Stress (Tensile) 31601.0000 psi STRESS/LIFE EXTERNAL
Material # 4.0000
Allowable_Shear Stress 40000.0000 psi
Allowable_Compressive Stress 4000.0000 psi STRESS/LIFE
Tooth_Shear Stress 13418.0000 psi
Life_Factor 0.4000
Shear On-Off Torque Cycles 940000.0000
Tooth_Compressive Stress 10571.0000 psi
Fixed_Spline Life Factor 0.3500
Compressive On-Off Torque Cycles 3000000.0000
Flexible/Crowned Spline Life Fact
Revolutions
Wear Life hr
Shaft_Shear Stress (Torsion) 19977.0000 psi
Life_Factor 0.6000
Shear On-Off Torque Cycles 63000.0000
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The limiting stress for the internal member is the hub shear stress produced by the torque carried. The number of on-off torque cycles that the hub will withstand is about 6.6 x 104. The limiting stress for the external member is the shaft shear stress. The number of on-off torque cycles that the shaft will withstand (assuming that the smallest shaft diameter is the same as the spline root diameter is about 6.3 x 104.
Example 2 This flexible 20 pitch spline has 6 teeth, a pressure angle of 45 degrees and a face of 0.25 inch. It is required that the life be at least 1000 hours with a misalignment of 0.003 in/in. The transmitted power is 2.25 HP at 3600 RPM. The power source and the driven equipment produce no shock load and the application factor is 1.0. The load is applied on one direction only. The material has not been specified. We will first try a through hardened steel with a hardness of 302 to 351 Brinell (Rockwell C 33 to 38). Enter the data as shown in Figure 2-1. Report 2-1 shows the solved model.
60-740—Steel Involute Spur Spline Load And Stress Analysis
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Fig. 2-1
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Report 2-1
Model Title : Program 60-740 Unit System: US GEOMETRY
Number of Teeth 6
Diametral Pitch 20.000000 1/in `
Module 1.270000 mm `
Pressure Angle 45.000000 deg
Pitch Diameter 0.300000 in
Radial Contact Height 0.0400 in
Face Width 0.2500 in
Effective Face Width 0.2500 in
Type of Spline Connection Flexible
Misalignment 0.0030 in/in
Internal Spline
Outside Dia of Hub 0.5000 in
Major Diameter 0.3800 in
Effective_Minor (Inside) Diameter 0.2700 in
Wall Thickness 0.0600 in
Tooth Thickness @ PD 0.0682 in
Space Width @ PD 0.0889 in
External Spline
Effective_Major (Outside) Diameter 0.3500 in
Minor (Root) Diameter 0.2400 in
60-740—Steel Involute Spur Spline Load And Stress Analysis
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Model Title : Program 60-740 Unit System: US Inside (Bore) Diameter of Shaft 0.0000 in
Wall Thickness 0.1200 in
Tooth Thickness @ PD 0.0835 in
Height of Crown on Tooth 0.0000 in
Crown Radius _ in
Axial Plane Generating Radius _ in
LOAD
Power 2.25 hp
Torque 39.39 lbf-in
Unidirectional / Fully-reversed u
Rotation Speed 3600 rpm
Application Factor, Ka 1.00
Face_Load Distribution Factor, Km 1.0000 STRESS/LIFE INTERNAL
Material # 3.0000
Allowable_Shear Stress 40000.0000 psi
Allowable_Compressive Stress 3000.0000 psi
Max Allowable Tensile Stress 45000.0000 psi STRESS/LIFE
Tooth_Shear Stress 5180.0000 psi
Life_Factor 0.1300
Shear On-Off Torque Cycles 10000000.0000
Tooth_Compressive Stress 4377.0000 psi
Fixed_Spline Life Factor
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Model Title : Program 60-740 Unit System: US Compressive On-Off Torque Cycles
Flexible/Crowned Spline Life Fact 1.4600
Revolutions 8000000.0000
Wear Life 36.9000 hr
Hub_Shear Stress (Torsion) 2408.0000 psi
Life_Factor 0.0600
Shear On-Off Torque Cycles 10000000.0000
Bursting Stress (Tensile) 7461.0000 psi STRESS/LIFE EXTERNAL
Material # 3.0000
Allowable_Shear Stress 40000.0000 psi
Allowable_Compressive Stress 3000.0000 psi STRESS/LIFE
Tooth_Shear Stress 4248.0000 psi
Life_Factor 0.1100
Shear On-Off Torque Cycles 10000000.0000
Tooth_Compressive Stress 4377.0000 psi
Fixed_Spline Life Factor
Compressive On-Off Torque Cycles
Flexible/Crowned Spline Life Fact 1.4600
Revolutions 8000000.0000
Wear Life 36.9000 hr
Shaft_Shear Stress (Torsion) 14512.0000 psi
Life_Factor 0.3600
Shear On-Off Torque Cycles 2400000.0000
60-740—Steel Involute Spur Spline Load And Stress Analysis
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With this material the wear life is only about 37 hours. We will try a surface hardened steel with a hardness of Rockwell C 48 to 53. Change the material numbers for both internal and external members to 4 and solve the model again. The completed data input form is shown in Figure 2-2, the solved model in Report 2-2. Fig. 2-2
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Report 2-2
Model Title : Program 60-740 Unit System: US GEOMETRY
Number of Teeth 6
Diametral Pitch 20.000000 1/in `
Module 1.270000 mm `
Pressure Angle 45.000000 deg
Pitch Diameter 0.300000 in
Radial Contact Height 0.0400 in
Face Width 0.2500 in
Effective Face Width 0.2500 in
Type of Spline Connection Flexible
Misalignment 0.0030 in/in
Internal Spline
Outside Dia of Hub 0.5000 in
Major Diameter 0.3800 in
Effective_Minor (Inside) Diameter 0.2700 in
Wall Thickness 0.0600 in
Tooth Thickness @ PD 0.0682 in
Space Width @ PD 0.0889 in
External Spline
Effective_Major (Outside) Diameter 0.3500 in
Minor (Root) Diameter 0.2400 in
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Model Title : Program 60-740 Unit System: US Inside (Bore) Diameter of Shaft 0.0000 in
Wall Thickness 0.1200 in
Tooth Thickness @ PD 0.0835 in
Height of Crown on Tooth 0.0000 in
Crown Radius _ in
Axial Plane Generating Radius _ in
LOAD
Power 2.25 hp
Torque 39.39 lbf-in
Unidirectional / Fully-reversed u
Rotation Speed 3600 rpm
Application Factor, Ka 1.00
Face_Load Distribution Factor, Km 1.0000 STRESS/LIFE INTERNAL
Material # 4.0000
Allowable_Shear Stress 40000.0000 psi
Allowable_Compressive Stress 4000.0000 psi
Max Allowable Tensile Stress 45000.0000 psi STRESS/LIFE
Tooth_Shear Stress 5180.0000 psi
Life_Factor 0.1300
Shear On-Off Torque Cycles 10000000.0000
Tooth_Compressive Stress 4377.0000 psi
Fixed_Spline Life Factor
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Model Title : Program 60-740 Unit System: US Compressive On-Off Torque Cycles
Flexible/Crowned Spline Life Fact 1.0900
Revolutions 58000000.0000
Wear Life 269.2000 hr
Hub_Shear Stress (Torsion) 2408.0000 psi
Life_Factor 0.0600
Shear On-Off Torque Cycles 10000000.0000
Bursting Stress (Tensile) 7461.0000 psi STRESS/LIFE EXTERNAL
Material # 4.0000
Allowable_Shear Stress 40000.0000 psi
Allowable_Compressive Stress 4000.0000 psi STRESS/LIFE
Tooth_Shear Stress 4248.0000 psi
Life_Factor 0.1100
Shear On-Off Torque Cycles 10000000.0000
Tooth_Compressive Stress 4377.0000 psi
Fixed_Spline Life Factor
Compressive On-Off Torque Cycles
Flexible/Crowned Spline Life Fact 1.0900
Revolutions 58000000.0000
Wear Life 269.2000 hr
Shaft_Shear Stress (Torsion) 14512.0000 psi
Life_Factor 0.3600
Shear On-Off Torque Cycles 2400000.0000
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A change to this material increased the wear life, but it still is only about 270 hours. As a last effort, before looking at redesigning the spline, we will try a case hardened steel with a hardness of Rockwell C 58 to 63. Change the material numbers for both internal and external members to 5 and solve the model again. The completed data input form is shown in Figure 2-3, the solved model in Report 2-3. Fig. 2-3
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Report 2-3
Model Title : Program 60-740 Unit System: US GEOMETRY
Number of Teeth 6
Diametral Pitch 20.000000 1/in `
Module 1.270000 mm `
Pressure Angle 45.000000 deg
Pitch Diameter 0.300000 in
Radial Contact Height 0.0400 in
Face Width 0.2500 in
Effective Face Width 0.2500 in
Type of Spline Connection Flexible
Misalignment 0.0030 in/in
Internal Spline
Outside Dia of Hub 0.5000 in
Major Diameter 0.3800 in
Effective_Minor (Inside) Diameter 0.2700 in
Wall Thickness 0.0600 in
Tooth Thickness @ PD 0.0682 in
Space Width @ PD 0.0889 in
External Spline
Effective_Major (Outside) Diameter 0.3500 in
Minor (Root) Diameter 0.2400 in
60-740—Steel Involute Spur Spline Load And Stress Analysis
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Model Title : Program 60-740 Unit System: US Inside (Bore) Diameter of Shaft 0.0000 in
Wall Thickness 0.1200 in
Tooth Thickness @ PD 0.0835 in
Height of Crown on Tooth 0.0000 in
Crown Radius _ in
Axial Plane Generating Radius _ in LOAD
Power 2.25 hp
Torque 39.39 lbf-in
Unidirectional / Fully-reversed u
Rotation Speed 3600 rpm
Application Factor, Ka 1.00
Face_Load Distribution Factor, Km 1.0000 STRESS/LIFE INTERNAL
Material # 5.0000
Allowable_Shear Stress 50000.0000 psi
Allowable_Compressive Stress 5000.0000 psi
Max Allowable Tensile Stress 55000.0000 psi STRESS/LIFE
Tooth_Shear Stress 5180.0000 psi
Life_Factor 0.1000
Shear On-Off Torque Cycles 10000000.0000
Tooth_Compressive Stress 4377.0000 psi
Fixed_Spline Life Factor
Compressive On-Off Torque Cycles
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Model Title : Program 60-740 Unit System: US Flexible/Crowned Spline Life Fact 0.8800
Revolutions 260000000.0000
Wear Life 1205.2000 hr
Hub_Shear Stress (Torsion) 2408.0000 psi
Life_Factor 0.0500
Shear On-Off Torque Cycles 10000000.0000
Bursting Stress (Tensile) 7461.0000 psi STRESS/LIFE EXTERNAL
Material # 5.0000
Allowable_Shear Stress 50000.0000 psi
Allowable_Compressive Stress 5000.0000 psi
STRESS/LIFE
Tooth_Shear Stress 4248.0000 psi
Life_Factor 0.0800
Shear On-Off Torque Cycles 10000000.0000
Tooth_Compressive Stress 4377.0000 psi
Fixed_Spline Life Factor
Compressive On-Off Torque Cycles
Flexible/Crowned Spline Life Fact 0.8800
Revolutions 260000000.0000
Wear Life 1205.2000 hr
Shaft_Shear Stress (Torsion) 14512.0000 psi
Life_Factor 0.2900
Shear On-Off Torque Cycles 10000000.0000
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We now have about 1200 hours life, and a spline made with this material should be satisfactory. Example 3 This is a 10 tooth, 2 mm module, 45 degree pressure angle spline. It is to be designed as a crowned flexible spline to handle a misalignment up to 0.005 radians (mm/mm). The material is to be surface hardened steel with a hardness of Rockwell C 48 to 53. We will first try a crown of 0.025 mm (about half the face width times the misalignment). Make certain you have selected metric units. The completed data input form is shown in Figure 3-1. Fig. 3-1
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When the model is solved we get a message that the specified crown is not enough, but that it should be greater than .032 mm. Change the crown to 0.035 mm and solve again. The completed data input form is shown in Figure 3-2, the solved model in Report 3-2. Fig. 3-2
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Report 3-2
Model Title : Program 60-740 Unit System: Metric GEOMETRY
Number of Teeth 10
Diametral Pitch 12.700000 1/in `
Module 2.000000 mm `
Pressure Angle 45.000000 deg
Pitch Diameter 20.000 mm
Radial Contact Height 1.350 mm
Face Width 12.000 mm
Effective Face Width 12.000 mm
Type of Spline Connection Crowned
Misalignment 0.0050 mm/mm
Internal Spline
Outside Dia of Hub 27.000 mm
Major Diameter 22.560 mm
Effective_Minor (Inside) Diameter 18.600 mm
Wall Thickness 2.220 mm
Tooth Thickness @ PD 3.103 mm
Space Width @ PD 3.180 mm
External Spline
Effective_Major (Outside) Diameter 21.300 mm
Minor (Root) Diameter 17.380 mm
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Model Title : Program 60-740 Unit System: Metric Inside (Bore) Diameter of Shaft 0.000 mm
Wall Thickness 8.690 mm
Tooth Thickness @ PD 3.038 mm
Height of Crown on Tooth 0.035 mm
Crown Radius 514.290 mm
Axial Plane Generating Radius 514.290 mm
LOAD
Power 10.00 kW
Torque 19.10 N-m
Unidirectional / Fully-reversed r
Rotation Speed 5000 rpm
Application Factor, Ka 1.00
Face_Load Distribution Factor, Km 1.00 STRESS/LIFE INTERNAL
Material # 4.0000
Allowable_Shear Stress 275790.2917 kPa
Allowable_Compressive Stress 110316.1167 kPa
Max Allowable Tensile Stress 310264.0782 kPa STRESS/LIFE
Tooth_Shear Stress 15448 kPa
Life_Factor 0.06
Shear On-Off Torque Cycles 10000000.0
Tooth_Compressive Stress 99729 kPa
Fixed_Spline Life Factor
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Model Title : Program 60-740 Unit System: Metric Compressive On-Off Torque Cycles
Flexible/Crowned Spline Life Fact 0.90
Revolutions 210000000.0
Wear Life 696.3 hr
Hub_Shear Stress (Torsion) 9641 kPa
Life_Factor 0.03
Shear On-Off Torque Cycles 10000000.0
Bursting Stress (Tensile) 24991 kPa STRESS/LIFE EXTERNAL
Material # 4.0000
Allowable_Shear Stress 275790.2917 kPa
Allowable_Compressive Stress 110316.1167 kPa STRESS/LIFE
Tooth_Shear Stress 15777 kPa
Life_Factor 0.06
Shear On-Off Torque Cycles 10000000.0
Tooth_Compressive Stress 99729 kPa
Fixed_Spline Life Factor
Compressive On-Off Torque Cycles
Flexible/Crowned Spline Life Fact 0.90
Revolutions 210000000.0
Wear Life 696.3 hr
Shaft_Shear Stress (Torsion) 18528 kPa
Life_Factor 0.07
Shear On-Off Torque Cycles 10000000.0
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We now have enough crown to accommodate a misalignment of 0.005 mm/mm. The wear life is about 700 hours. References: 1. Darle W. Dudly, How to Design Involute Splines. 2. Darle W. Dudly, “When Splines Need Stress Control”, in N.P. Chironis, editor, Gear Design and Application. McGraw-Hill Book Co., New York, 1967.