Shaft Stress Calculations



CALCULATIONS AND ANALYSIS

See Stress Calculation Spreadsheet for sources of equations, sources of constants and material properties, and additional calculations

[pic][pic]

Impact Analysis

Direct wheel impact at max speed

▪ By using the deflection equation, [pic] (based upon two fully constrained rod ends), solving for F, and using a basic kinematic equation ([pic] ) to solve for s in terms of F, the force of impact can be determined (227505 N)

▪ Utilizing shaft stress equations shown below the stress can be determined (400 MPa)

▪ When comparing this to the shaft’s yield strength, a factor of safety of 1.33 is calculated

Direct pulley impact at max speed

▪ Utilizing this same force and finding the stress on the shaft due to bending. [pic]=8510 MPa

▪ This means the shaft will permanently bend due to the moment applied on it

▪ The way to avoid this catastrophic failure is to ensure the chassis protects these open gears by extending past its edges or enclosing it completely. While this may not completely ensure the module’s safety, it will fix nearly every probable scenario.

Shaft Stress Calculations

Shaft 1 (Diameter=3/8”)

• Material: 1045 Steel, Yield Strength (Sy)= 530 MPa, Ultimate Strength= 625MPa

• Max Stress

o The shaft is keyed for a 3/32” key, thus a close approximation for the actual yield strength is ¾ the materials yield strength (Keyed Yield Strength=398 MPa)

o Loading is comprised of three components

▪ Moment-Based on cantilevered distance from bearing and radial load exerted on shaft from the miter gear (2.1 N-m)

▪ Force- Based on axial load exerted on shaft from miter gear (156.12 N)

▪ Torque- Exerted by the stall torque of the motor, through a gear ratio of 2:1 (9.64 N-m)

o Stress Calculation-

▪ [pic]=102 MPa

▪ [pic]= 58.4 MPa

o Factors of Safety-

▪ [pic]= 3.9

▪ [pic]= 3.4

• Fatigue Life

o Infinite Life- 2000RPM (Average operating speed)=33.3 cycles/second

▪ 5 year life @ 1 hour operating time (2 hr per week)-approximately 1,908,000 seconds of use

▪ 33.3*1,908,000=6.4E7 cycles to failure for infinite life

o The endurance strength can be calculated using the stress concentration factors from the keyway (197 MPa)

o σ’F=Sut+345MPa= 970 MPa

o [pic]=-0.109915548

o [pic]=.673

o [pic]=900 MPa

o Loads are based on typical operating conditions, not max conditions

▪ Moment-Based on cantilevered distance from bearing and radial load exerted on shaft from the miter gear (2.1 N-m)

▪ Force- Based on axial load exerted on shaft from miter gear (156.12 N)

▪ Torque- Exerted by the operating torque of the motor, through a gear ratio of 2:1 (2.82 N-m)

o [pic]= 39.4 MPa

o [pic]= 2.25E12 cycles to failure

Shaft 2 (Diameter=1/2”)

• Material: 1045 Steel, Yield Strength= 530 MPa, Ultimate Strength= 625MPa

• Max Stress

o The shaft is keyed for a 1/8” key, thus the actual yield strength can be equated to ¾ the materials yield strength (Keyed Yield Strength=398 MPa)

o Loading is comprised of three components

▪ Moment-Based on the axle length between bearings and radial load exerted on shaft from the miter gear (4.28 N-m)

▪ Force- Based on axial load exerted on shaft from miter gear (156.12 N)

▪ Torque- Exerted by the stall torque of the motor, through a gear ratio of 2:1 (9.64 N-m)

o Stress Calculation-

▪ [pic]= 47.2 MPa

▪ [pic]= 26.5 MPa

o Factors of Safety-

▪ [pic]= 8.4

▪ [pic]= 7.5

• Fatigue Life

o Infinite Life- 1000RPM=16.67 cycles/second

▪ 5 year life @ 1 hour operating time (2 hr per week)-approximately 1,908,000 seconds of use

▪ 16.67*1,908,000=3.2E7 cycles to failure for infinite life

o The endurance strength can be calculated using the stress concentration factors from the keyway (197 MPa)

o σ’F=Sut+345MPa= 970 MPa

o [pic]=-0.109915548

o [pic]=.673

o [pic]=900 MPa

o Loads are based on typical operating conditions, not max conditions

▪ Moment-Based on the axle length between bearings and radial load exerted on shaft from the miter gear (4.28 N-m)

▪ Force- Based on axial load exerted on shaft from miter gear (156.12 N)

▪ Torque- Exerted by the operating torque of the motor, through a gear ratio of 2:1 (2.82 N-m)

o [pic]= 25.6 MPa

o [pic]= 1.15E14 cycles to failure

Shaft 3 (Diameter=3/4”)

• Material: 1045 Steel, Yield Strength= 530 MPa, Ultimate Strength= 625MPa

• Max Stress

o The shaft is keyed for a 3/16” key, thus the actual yield strength can be equated to ¾ the materials yield strength (Keyed Yield Strength=398 MPa)

o Loading is comprised of three components

▪ Moment-Based on the axle length between bearings and the force exerted by the weight of the system (21.53 N-m)

▪ Force- Based on axial load exerted on the shaft from turning forces (235.44 N)

▪ Torque- Exerted by the stall torque of the motor, through a gear ratio of 8:1 (38.56 N-m)

o Stress Calculation-

▪ [pic]= 59.0 MPa

▪ [pic]= 32.7 MPa

o Factors of Safety-

▪ [pic]= 6.7

▪ [pic]= 6.1

• Fatigue Life

o Infinite Life- 500RPM=8.34 cycles/second

▪ 5 year life @ 1 hour operating time (2 hr per week)-apprx 1,908,000 seconds of use

▪ 8.34*1,908,000=1.6E7 cycles to failure for infinite life

o The endurance strength can be calculated using the stress concentration factors from the keyway (197 MPa)

o σ’F=Sut+345MPa= 970 MPa

o [pic]=-0.109915548

o [pic]=.673

o [pic]=900 MPa

o Loads are based on typical operating conditions, not max conditions

▪ Moment-Based on the axle length between bearings and the force exerted by the weight of the system (21.53 N-m)

▪ Force- Based on axial load exerted on the shaft from turning forces (235.4 N)

▪ Torque- Exerted by the operating torque of the motor, through a gear ratio of 8:1 (11.28 N-m)

o [pic]= 35.6 MPa

o [pic]= 5.7E12 cycles to failure

Steering Shaft (Diameter=1/4”)

• Material: 303 Stainless Steel, Yield Strength= 240 MPa, Ultimate Strength= 620 MPa

• Max Stress

o Loading is based on torque alone (0.745 N-m)

o Stress Calculation-

▪ [pic]= 25.7 MPa

▪ [pic]= 14.8 MPa

o Factors of Safety-

▪ [pic]= 9.4

▪ [pic]= 8.1

• Fatigue Life

o σ’F=Sut+345MPa=965E6 MPa

o [pic]=-0.07772

o [pic]=.862

o [pic]=914 MPa

o Load is comprised of torque alone (.745 N-m)

o [pic]= 25.7 MPa

o [pic]= 9.3E19 cycles to failure

Spur Gears (Calculated using ANSI standards)

Driving Spur

▪ Material- Carbon Steel, Yield Strength=76900 psi, Modulus of Elasticity=30E6 psi, Poisson’s Ratio=.29, Brunell Hardness 179

▪ Max Bending Stress

o [pic]= 306.8 lbf

o Ko= 1.25 - Overload Factor, based on light shocks encountered

o Kv= 1.15 - Dynamic Factor, based on quality and velocity of gears

o Ks= 1 - Size Factor

o Pd= .833” – Pitch diameter

o F= .25” – face width

o Km= 1.20 – Load-Distribution factor, based on geometry

o KB= 1 – Rim Thickness factor, based on geometry

o J= .325- Geometry factor, based on number of teeth of gears

o [pic]=5357.1 psi

o [pic]= 9.0

▪ Endurance Stress

o [pic]=2284.7 lbf/in2

o Cf=1

o I=0.08- Geometry Factor

o [pic]=56972.4 psi

o Sc= 180000 psi- Repeatedly applied contact strength @ 107 cycles, material property

o Zn=.59 - Stress cycle life factor, based on hardness and number of cycles

o CH=1 -Hardness ratio factor

o KT= 1- Temperature factor

o KR= 1 – Reliability factor

o [pic]=1.9

o Comparable factor of safety= SH2=3.5

Driven Spur

▪ Material- Carbon Steel, Yield Strength=76900 psi, Modulus of Elasticity=30E6 psi, Poisson’s Ratio=.29, Brunell Hardness 179

▪ Max Bending Stress

o [pic]= 306.7 lbf

o Ko= 1.25 - Overload Factor, based on light shocks encountered

o Kv= 1.15 - Dynamic Factor, based on quality and velocity of gears

o Ks= 1 - Size Factor

o Pd= 1.667” – Pitch diameter

o F= .25” – face width

o Km= 1.19 – Load-Distribution factor, based on geometry

o KB= 1 – Rim Thickness factor, based on geometry

o J= .389- Geometry factor, based on number of teeth of gears

o [pic]=9011.0 psi

o [pic]= 5.4

▪ Endurance Stress

o [pic]=2284.7 lbf/in2

o Cf=1

o I=0.08- Geometry Factor

o [pic]= 40010.7 psi

o Sc= 180000 psi- Repeatedly applied contact strength @ 107 cycles, material property

o Zn=.60 - Stress cycle life factor, based on hardness and number of cycles

o CH=1 -Hardness ratio factor

o KT= 1- Temperature factor

o KR= 1 – Reliability factor

o [pic]= 2.70

o Comparable factor of safety= SH2=7.2

Ring/Pinion Gears (Calculated using ANSI standards)

Steering Spur

▪ Material- 2024-T4 Aluminum, Yield Strength=47000 psi, Modulus of Elasticity=10.4E6 psi, Poisson’s Ratio=.333

▪ Max Bending Stress

o [pic]= 39.3 lbf

o Ko= 1.25 - Overload Factor, based on light shocks encountered

o Kv= 1.10 - Dynamic Factor, based on quality and velocity of gears

o Ks= 1 - Size Factor

o Pd= .4375” – Pitch diameter

o F= .125” – face width

o Km= 1.20 – Load-Distribution factor, based on geometry

o KB= 1 – Rim Thickness factor, based on geometry

o J= .24- Geometry factor, based on number of teeth of gears

o [pic]=951.8 psi

o [pic]= 44.1

Steering Ring

▪ Material- 2024-T4 Aluminum, Yield Strength=47000 psi, Modulus of Elasticity=10.4E6 psi, Poisson’s Ratio=.333

▪ Max Bending Stress

o [pic]= 39.3 lbf

o Ko= 1.25 - Overload Factor, based on light shocks encountered

o Kv= 1.10 - Dynamic Factor, based on quality and velocity of gears

o Ks= 1 - Size Factor

o Pd= 3.125” – Pitch diameter

o F= .125” – face width

o Km= 1.8 – Load-Distribution factor, based on geometry

o KB= 1 – Rim Thickness factor, based on geometry

o J= .4- Geometry factor, based on number of teeth of gears

o [pic]= 3996.0 psi

o [pic]= 10.5

Miter Gears (Calculated using ANSI standards)

Both Miters (At max torque)

▪ Material- Medium Carbon Steel, Yield Strength=76900 psi

▪ Max Bending Stress

o Pd= 1.25” – Pitch diameter

o

o [pic]= 84.7 lbf

o Ko= 1.25 - Overload Factor, based on light shocks encountered

o Kv= 1 - Dynamic Factor, based on quality and velocity of gears

o Ks= .5 - Size Factor

o F= .27” – face width

o Km= 1.10 – Load-Distribution factor, based on geometry

o J= 0.175- Geometry factor, based on number of teeth of gears

o Kx= 1, Lengthwise curvature factor

o [pic]=14922.2 psi

o [pic]= 5.15

Both Miters (At max speed)

▪ Material- Medium Carbon Steel, Yield Strength=76900 psi

▪ Max Bending Stress

o Pd= 1.25” – Pitch diameter

o

o [pic]= 4 lbf

o Ko= 1.25 - Overload Factor, based on light shocks encountered

o Kv= 1.28 - Dynamic Factor, based on quality and velocity of gears

o Ks= .5 - Size Factor

o F= .27” – face width

o Km= 1.10 – Load-Distribution factor, based on geometry

o J= 0.175- Geometry factor, based on number of teeth of gears

o Kx= 1, Lengthwise curvature factor

o [pic]=901.6 psi

o [pic]= 85.3

▪ Forces

o Knowing max torque on miter (9.63 N-m), we can find the max tangential force by dividing by half the pitch diameter=> Ftan=606.6 N

o α= 20 degress -pressure angle

o d= 45 degrees

o [pic]=645.5 N

o [pic]=220.8 N

o [pic]=156.1 N

Retaining Rings

▪ On 3/8” shaft

o Ring can withstand 542.7 N of axial force

o Miter gear provides axial load= 156.1 N

o Factor of safety= 3.48

▪ On 1/2” shaft

o Ring can withstand 542.7 N of axial force

o Miter gear provides axial load= 156.1 N

o Factor of safety= 3.48

▪ On 3/4” shaft

o Ring can withstand 631.6 N of axial force

o Axial load is from turning

▪ Assume wheel instantaneously turns 90 degrees, the max force that can be applied axially would be equivalent to the frictional force

▪ [pic]=235.4 N (assuming μ=.6)

o Factor of safety= 2.68

Mechanical Brake

Max Temperature

▪ Assuming all kinetic energy is converted directly into heat energy, [pic]

▪ Assume emergency brake will not be used continuously, but rather for one cycle during the emergency

▪ Assume initial temperature of 23 ° Celcius

▪ Solving the above equation for Tfinal we find it to be 38.9 °Celcius

Heat Dissipation

▪ Assuming Free Convection, the time required for heat dissipation can be calculated

▪ Utilizing the properties of air at room temperature, the Rayleigh number, Nusselt number, and convection heat transfer coefficient can be calculated

▪ Using this information the heat transfer rate is determined

▪ [pic] gives the time to dissipate the heat (4.8 minutes)

▪ This resultant was later verified by the manufacturer of the brake

Keys

On 3/8” Shaft

▪ Key is High carbon steel, Yield Strength 427 MPa, 3/32” square

▪ Knowing the diameter of and torque on the shaft, the shear force on the key can be calculated (2024.1 N)

▪ Assuming a factor of safety of 4, the required length of the key is calculated (.63”)

On 1/2” Shaft

▪ Key is High carbon steel, Yield Strength 427 MPa, 1/8” square

▪ Knowing the diameter of and torque on the shaft, the shear force on the key can be calculated (1518.1 N)

▪ Assuming a factor of safety of 4, the required length of the key is calculated (.35”)

On 3/4” Shaft

▪ Key is High carbon steel, Yield Strength 427 MPa, 3/16” square

▪ Knowing the diameter of and torque on the shaft, the shear force on the key can be calculated (4048.3 N)

▪ Assuming a factor of safety of 4, the required length of the key is calculated (.63”)

Keyways

• Keyway analysis was done using Cosmos FEA software

• By utilizing shaft diameters and torques, forces on keyway surfaces were calculated and input into the program

• Factor of Safety

o Driving Miter=16

o Driven Miter=15

o Driving Pulley=3.4

o Driven Pulley=8.4

o Driven Spur=8.1

o Wheel=1.8, but in reality, failure would result in the slip of a pressed insert, rather than physical failure of the wheel

Set Screws

To connect spur gear to 5/16” drive motor shaft

▪ By choosing a screw size and quantity (2- #8’s), the maximum force at the shaft surface can be calculated (3425.1 N)

▪ The torque and diameter of the shaft is used to determine the actual force seen at this shaft surface (1214.5 N)

▪ By comparing these two values the factor of safety is determined (2.82)

To connect spur gear to 8mm steering motor shaft

▪ By choosing a screw size and quantity (2-#6’s), the maximum force at the shaft surface can be calculated (2224.1 N)

▪ The torque and diameter of the shaft is used to determine the actual force seen at this shaft surface (234.6 N)

▪ By comparing these two values the factor of safety is determined (9.5)

Timing Belt and Pulleys

• Utilizing MITCalc simulation software and inputting various parameters including distance between centers, power applied to belt, operating speeds, and other operating conditions a belt type and specific model was selected

• From this the 5M Powergrip GT2 belt was chosen and matched with pulleys of 18 and 72 teeth

• The selection of these parts was also verified with an engineer at the supplier sdp-

Bearings

C10 = Catalog Load Rating (lbf)

LR = Rating Life (hrs)

nR = Rating Speed (RPM)

FD = Desired Radial Load (lbf)

LD = Desired Life (hrs)

nD = Desired Speed (RPM)

FR = Radial Force (lbf)

FA = Axial Force (lbf)

Fe = Equivalent Radial Load (lbf)

C0 = Static Load Rating (lbf)

X2 = Factor dependent on bearing geometry

Y2 = Factor dependent on bearing geometry

V = Rotation Factor

a = 3 (ball bearing)

e = abscissa

|Lower Drive Bearing | | | | | | |

|C10 (lbs) |1171 | |Upper Drive Bearing | |Center Bearing |

|LR * nR |1.0E+06 | |C10 (lbs) |1187 | |C10 (lbs) |691 |

|FD (lbs) |44.125 | |LR * nR |1.0E+06 | |LR * nR |1.0E+06 |

|nD (RPM) |5.0E+02 | |Fe (lbs) |84.6 | |Fe (lbs) |70.6 |

|a |3 | |nD (RPM) |2000 | |nD (RPM) |4000 |

|LD (hrs) |3.74E+07 | |FA (lbs) |35.1 | |FA (lbs) |35.1 |

|Steering Bearing | |V |1 | |V |1 |

|C10 (lbs) |300 | |e |0.24 | |e |0.3 |

|LR * nR |1.0E+06 | |X2 |0.56 | |X2 |0.56 |

|FD (lbs) |10 | |Y2 |1.85 | |Y2 |1.45 |

nD (RPM) |340 | |LD (hrs) |1.38E+06 | |LD (hrs) |2.34E+05 | |a |3 | |LD (years) |158 | |LD (years) |27 | |LD (hrs) |7.94E+07 | | | | | | | |LD (years) |9065 | | | | | | | |

Screws

Bolts connecting Yoke to Turntable

Bolt type: 4 * 10-32 (SAE)

Torque applied to the turntable

T= 2.38 N-m

Converted ASTM

T = 21 lb-in

Resultant load on each bolt

[pic]

[pic]

Primary Shear Load per Bolt is

[pic]

Since the secondary shear Forces are equal we have

[pic]

The resultant force is

[pic]

Fr = Fa = Fb = Fc = Fd = 2.867lb

Maximum Shear Stress

As = 0.155

[pic]

Bolts connecting Brake to Brake plate

Bolt type: 4 * 8-32 (SAE)

Torque applied to the turntable

T = 15 lb-in

Resultant load on each bolt

[pic]

[pic]

Primary Shear Load per Bolt is

[pic]

Since the secondary shear Forces are equal we have

[pic]

The resultant force is

[pic]

Fr = Fa = Fb = Fc = Fd = 4.714 lb

Maximum Shear Stress

As = .0992

[pic]

Yoke

[pic]

• Yoke stress analysis was done using Cosmos FEA software

• Loading

o Weight vertically loads lower bearing holes (196.2 N each)

o Turning force loads inside wall (158.3 N)

o Driven Miter axial force loads inside wall (156 N)

o Driven Miter radial force loads upper bearing holes

o Driving Miter axial force loads upward on top plate

• Minimum factor of safety= 20

Brake Plate

[pic]

• Brake plate stress analysis was done using Cosmos FEA software

• Loading

o Outside edge was fixed, as it is welded to the motor mount assembly

o Each brake mounting hole was loaded with a force corresponding to the brake’s torque output and the holes distance from center

• Minimum factor of safety= 200

Turntable

• Capable of withstanding 750 lbs, or 340 kg

• Actual weight is about 40 kg per module

• Factor of Safety= 8.5

[pic]

Figure X: Shaft Stress and Fatigue Strength Calculations

[pic]

Figure X: Drive Motor Spur Gear Stress Calculation

[pic]

Figure X: Miter Gear Stress and Force Calculation

[pic] [pic]

Figure X: Retaining Ring Calculations Figure X: Impact Calculations

Figure X: Brake Temperature and Heat Dissipation Calculations

[pic]

Figure X: Key and Set Screw Analysis

[pic]

[pic]

Figure X: Timing Belt and Pulley Analysis

[pic]

Figure X: Driven Miter Gear Keyway Analysis

[pic]

Figure X: Driving Miter Gear Keyway Analysis

[pic]

Figure X: Driven Pulley Keyway Analysis

[pic]

Figure X: Driving Pulley Keyway Analysis (Representation)

[pic]

Figure X: Driven Spur Gear Keyway Analysis

[pic]

Figure X: Wheel Keyway Analysis (Representation)

Note: Actual wheel utilizes press fit keyway insert, thus failure during stall will result in slip of this insert, rather than the physical failure of a mechanical part

[pic]

Figure X: Yoke Displacement

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[pic]

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