Spring 2013 Pennsylvania State University

Joseph R. Felice

[DESIGN OF A COMPRESSOR

DRIVE TRAIN]

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Design of a Compressor Drive Train

By: Joseph R. Felice

Objective

The purpose of this case study is to supply a building contractor with a portable gasoline

powered air compressor for transportation to construction sites in order to drive hammers.

Elements of the compressor unit which relate to lecture topics include but are not limited to

the design of shafts, couplings, bearings, pinions, and gears.

Introduction

A building contractor desires a small air compressor unit used for driving hammers on

site. The compressor is to be powered by a single cylinder two-stroke gasoline engine

containing a flywheel. This engine operates at two horsepower regulated at 3,250 revolutions

per minute (rpm). A Schramm piston compressor containing one cylinder is operated by the

output shaft of the gearbox which is connected to the crankshaft. The input and output shafts

of the gearbox and engine respectively are connected by a clutch. Reduction of engine speed is

essential for the proper operation of driving hammers.

In order to reduce engine speed as well as increase torque, a clutch and gearbox system

is coupled to the engine. Inside of the housing for the gearbox unit a pinion and gear mesh,

thus transmitting rotational motion which allows for the reduction of engine speed. The gear

ratio for achieving this result will be determined during the course of this independent analysis.

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Shaft Analysis

Equations

00 < 924

792 2

Equation 1-1: The above relationship comes from the gas laws. This particular formula

demonstrates the pressure developed in the cylinder as a function of the crank angle.

= = 4 Equation 1-2: The pressure which was initially calculated generates a gas force. Featured here

is the relationship of the gas force on the piston and the cylinder head as a function of the

crank angle.

= !1 +$ %& Equation 1-3: The above expression relates the torque of the compressor crankshaft due to the

gas forces as a function of the crank angle.

' = 12() 2$ 2 +32$ 3 Equation 1-4: Shown here is the torque due to the inertia forces as a function of the crank

angle.

= +' Equation 1-5: Above the torques due to gas forces and inertia forces are simply added together

to yield the total torque.

All of the shaft analysis conducted for this case study was based on the selection of UNS No.

G10200/AISI No. 1020 (HR) steel (Shigley, 1040). The tensile strength of this type of steel is

55,000 psi.

Torque (foot-pounds) Crank Angle (degrees)

Minimum -202.56 143

Maximum 603.53 330

Table 1: Minimum and maximum torques necessary for driving the compressor crankshaft with

their respective crank angles.

= Equation 1-1

Equation 1-2

Equation 1-3

Equation 1-4

Equation 1-5

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Pressure (psi) Crank Angle (degrees)

Minimum 0 0-180

Maximum 132 360

Table 2: Minimum and maximum pressures developed in the cylinder with their respective

crank angles.

Force (pounds) Crank Angle (degrees)

Minimum 0 0-180

Maximum 932.56 360

Table 3: Minimum and maximum force on the piston and cylinder head due to the pressures

with their respective crank angles.

Figure 1: A plot of the Torque vs. Crankshaft Angle

-300

-200

-100

0

100

200

300

400

500

600

700

0 50 100 150 200 250 300 350 400To

rqu

e (

foo

t-p

ou

nd

s)

Angle (Degrees)

Torque

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Figure 2: A plot of the Pressure vs. Crankshaft Angle.

Figure 3: A plot of the Force vs. Crankshaft Angle.

0

20

40

60

80

100

120

140

0 50 100 150 200 250 300 350 400

Pre

ssu

re (

psi

)

Angle (Degrees)

Pressure

0

200

400

600

800

1000

0 50 100 150 200 250 300 350 400

Fo

rce

(p

ou

nd

s)

Angle (Degrees)

Force

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Force analysis:

Equations

+,-./ =-./

+,-'0 =-'0 Equations 2-1/2-2: Shown above are formulas for calculating the maximum and minimum

tangential force component acting on the gear tooth.

+-./ =+,-./%

+-'0 =+,-'0% Equations 2-3/2-4: These are formulas for calculating the maximum and minimum resultant

forces acting on the gears.

rg (inches) Tmax (foot-pounds) Tmin (foot-pounds) Wtmax (pounds) Wtmin (pounds)

2 603.5 -202.6 301.8 -101.3

Table 4: Shown here are the calculated values for the maximum and minimum torques of the

shaft as well as the corresponding maximum and minimum tangential components of the force.

(degrees) Wtmax (pounds) Wtmin (pounds) Wmax (pounds) Wmin (pounds) 20 301.8 -101.3 321.2 -107.8

Table 5: Featured above are the maximum and minimum tangential force components with

their respective maximum and minimum resultant forces.

Equation 2-1

Equation 2-2

Equation 2-3

Equation 2-4

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Shear-Force and Bending-Moment Diagram:

Figure 4: Demonstrated above is the shear-force and bending-moment diagram for the

maximum loading conditions experienced by the shaft.

Wtmax = 301.8 lbs

Ra = 150.9 lbs Rb = 150.9 lbs

Mmax = 301.8 ft-lbs

Vaxis

Maxis

2 in 2 in

4 in

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Shaft iteration:

Note: For the iterative process a safety factor of two was assumed for the calculations.

Equations

= 216 415 64789:.; + 3789 ? + 1@A, 64789:-; + 3789 ? BC> D? Equation 3-1: This formula is used in the iterative process for determining shaft diameter

(Shigley, 368).

E.F = E. + 3G.> ? = H!DIJKLMNO & + 3!>IJPQLMNO &R> ?

E-F = E- + 3G- > ? = H!DIJKSMNO & + 3!>IJPQSMNO &R> ?

Equations 3-2/3-3: Shown here are the expressions for von Mises stresses for a solid round

shaft in rotation (Shigley, 368).

89 = 1 + T8, 1 89< = 1 + T

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Iteration process

Torque (foot-pounds) Moment (foot-pounds)

Midrange 200.5 201.5

Amplitude 806.1 100.2

Table 6: Above are the midrange and amplitude torques and bending moments.

Iteration r/d D/d q qshear Kt Kts

1 - - - - 1 1

2 0.5 2 0.65 0.69 1.38 1.35

Table 7: Featured here are some factors that went into the computations for shaft iterations.

Iteration Kf Kfs se (kpsi) Diameter (inches) Safety Factor (n)

1 1 1 17.413 0.91 1.65

2 1.25 1.24 17.181 1.05 2.00

Table 8: The second iteration yielded the satisfactory safety factor of two.

The iteration process for shaft analysis is a repetitive technique which involves the use

of several design equations designated for properly sizing the diameter to meet project

specifications. Gear diameter and safety factor are a couple of key elements that establish the

design parameters for a shaft. In this specific case study a safety factor of two was selected

since it is adequately sufficient for satisfying the operational conditions associated with the air

compressor unit. An input (spur) pinion pitch diameter of four inches was given as a suggested

guideline for sizing the input shaft.

First iteration values for theoretical stress-concentration factors Kt and Kts were

estimated to be one since the shaft has a very well-rounded fillet. These estimates when

applied to Equations 3-4 and 3-5 also yielded values of one for fatigue stress-concentration

factors Kf and Kfs. Hence, the initial absence of values for the ratios r/d and D/d as well as for

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notch sensitivity values q and qshear, which necessitates these first iteration estimates. Also, a

value for the endurance limit was calculated using the Marin formula (Equation 3-7).

The first iteration calculation of the endurance limit was based on an assumed value for

the size factor, kb, of 0.9 as well as assumed values of one for both the temperature factor, kd

and the miscellaneous-effects factor, kf. When these values along with the midrange and

amplitude values for torque and bending moment were applied to Equation 3-1 a diameter of

0.91 inches along with a safety factor of 1.65 were acquired as featured in Table 6 above. Thus,

since the safety factor was below the desired value of two, a second iteration was performed.

For the second iteration, the original assumption for the kb value of 0.9 was discarded in

favor of being able to calculate an actual value based on the diameter acquired by the first

iteration. A value of 0.91 inches falls within the diameter range for the first formula featured in

Equation 6-20 of Shigley,

YZ = 0.3\].>] = 0.879\].>], which yields a value of 0.88, very close to the original assumed size factor (Shigley, 288). When

this new value for kb is applied to the Marin equation a more accurate endurance limit of

17.181 kspi is calculated (Table 5).

Values for q and qshear were obtained by taking a reading along an interpolated 55 kpsi

curve which corresponded to the appropriate radius in Figures 6-20 and 6-21 respectively

(Shigley, 295-296). The ratios r/d and D/d were used to acquire real values for Kt and Kts by

reading Figures A-15-8 and A-15-9 (Shigley, 1028). When the new values for endurance limit,

notch sensitivities and theoretical stress-concentration factors (Table 5) are applied once more

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to the iterative process a diameter of 1.05 inches was computed. This diameter when applied

to the modified-Goodman criteria (Equation 3-8) resulted in a satisfying the desired safety

factor of two for the shaft.

Gear and Pinion Design:

Equations

a = 2Y1 + 2( !( +b( + 1 + 2( sin & Equation 4-1: Featured above in the formula for determining the least amount of teeth a

pinion can contain in order to avoid interference (Shigley, 686).

af =gVa Equation 4-2: Shown here is the equation for determining the largest gear that can mesh with

the pinion in order to avoid interference (Shigley, 687).

E[h,f = 2+,8i8j8< 8- k9l C>/

Equation 4-3: This is the AGMA contact-stress equation both a pinion and gear (Shigley, 774).

@nh,f = !opqr IsItup & Equation 4-4: This is the formula for calculating the safety factors for both a pinion and gear

(Shigley, 774).

@, = 77.3vw + 12,800 Equation 4-5: Shown above is the formula for the allowable bending stress plot for Grade 1

through-hardened steel featured in Figure 14-2 of Shigley (Shigley, 747).

@[ = 349vw + 29,100 Equation 4-6: The formula for allowable contact stress for Grade 1 through-hardened steel

shown above is featured in Figure 14-5 of Shigley (Shigley, 750).

xh,f = /12 Equation 4-7: The formula for calculating the pitch line velocities for pinions and gears (Shigley,

707).

Equation 4-6

Equation 4-7

Equation 4-1

Equation 4-2

Equation 4-3

Equation 4-4

Equation 4-5

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Data Tables

dp (inches) gr Fp (inches) Np (teeth) Pd

(teeth/inch)

p

(inches)

(degrees)

4 2.6 2.7 14 3.5 0.89 20

2 2.6 1.5 14 7 0.45 20

Table 9: Pinion data for the iterative process related to acquiring the appropriate diameter.

dG (inches) gr Fp (inches) NG (teeth) Pd

(teeth/inch)

p

(inches)

(degrees)

6 2.6 1.6 36 6 0.52 20

5 2.6 1.5 36 7.2 0.44 20

Table 10: Gear data for the iterative process related to acquiring the appropriate diameter.

Note: Both Tables 9 and 10 show two iterations for the pinion and gear. The first row in each

table shows the first iteration with the give pitch diameter of four inches for the pinion. In the

second row of each table are the second and final iterations concluding with the correct values

for each gear. A detailed explanation of these numbers is featured in the Calculations

Appendix.

Gear Factors Pinion Gear

Speed (rpm) 3250 1250

Pitch-Line Velocity

(feet/minute)

1701.7 1701.7

Radial Load (pounds) 642.2 642.2

Compressive Load

(pounds)

567.1 567.1

Transmitted Load

(pounds)

603.5 603.5

Allowable Bending Stress

(psi)

32125 32125

Allowable Contact Stress

(psi)

109600 109600

Contact Stress (psi) 38471.4 24331.4

Safety Factor 2.08 3.28

Table 11: Featured above is information related to both the pinion and gear concerning

important speeds/velocities, loads, stresses and safety factors.

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Design Description

For the design of both the gear and pinion through-hardened Grade 1 steel was selected

for the casting material. A Brinell Hardness value of 250 was chosen for this steel. Therefore,

allowable bending stress calculations for both the gear and pinion were based on Equation 4-5

from Figure 14-2 in Shigley. Also, from Shigley the allowable contact stresses were calculated

by the application of the Brinell Hardness of the steel to Equation 14-6 from Figure 14-5. Given

the speeds for both the gear and pinion, the pitch line velocity was calculated by using Equation

4-7.

Once the overload, dynamic, load-distribution, surface condition and geometry factors

were determined as well as the elastic coefficient Equation 4-3 was applied to the pinion and

gear respectively, thus yielding the contact stresses. The earlier value of the transmitted load

of 301.8 pounds was recalculated to be 603.5 pounds in order to accommodate an interference

free design. Safety factors for the pinion and gear were calculated using Equation 4-4. The

safety factor for the pinion was determined to be 2.08, a perfectly acceptable value based on

the selection of a desired safety factor of two. Even though a safety factor of 3.28 for the gear

is slightly higher than expected it is still acceptable.

Bearing Design

Equations

k>] =y9z { |z|] + |]} 1 ~z> Z >/.

Equation 5-1: This formula is used for calculating the C10 catalog entry for both a pinion and

gear (Shigley, 578).

Equation 5-1

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Weibull Parameters

Manufacturer Rating Life, Rating Lives

Revolutions x0 b 1 90(10

6) 0 4.48 1.5

2 1(106) 0.02 4.459 1.483

Table 12: Above is a copy of a table shown in Shigley demonstrating the bearing data of two

different manufacturers experience with their life expectancy (Shigley, 608).

Bearing Design

k>],h'0'i0 = 1.80 301.8 56940.02 + 4.459 0.02} ! 1. 95&

>>.D>D = 11382.06}

Demonstrated above is the pinion application for the selection of Weibull parameters

shown in Table 12 (pg. 608 of Shigley) for the data findings regarding the industrial use of ball

bearings from Manufacturer two to Equation 5-1. As we can see the C10 catalog load rating in

this case yields a value of 11382.06 pounds and a similar application of the gear to Equation 5-

1, using the same application factor, gives a load rating of 8277.45 pounds (the life variate

value, xD, for the gear was 2190). For the calculation of the life variate for the deep-groove ball

bearing a Manufacturer twos catalog value of 106 was selected for the rating life. Calculations

based on the desired life expectancy of 10 years featured in the appendix will show an

operational duration of the ball bearings for the pinion was 5.694x109 revolutions and for the

gear 2.19x109 revolutions.

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Disk Plate Clutch

Equations

= 8-./ + Equation 6-1: This formula is rewritten from its original format as Equation 16-24 in Shigley. It

is used to solve for the outer diameter of a plate clutch (Shigley, 847).

= 4 + Equation 6-2: This formula solves for the actuating force in a plate clutch.

Disk Plate Clutch Design

Design Torque (foot-pounds) 348.17

Inside Diameter (inches) 2.45

Outside Diameter (inches) 4.25

D/d Ratio 1.73

Actuating Force (pounds) 2078.63

Application Factor 1.5

Number of Disks 1

Table 13: Above are the design parameters as well as torque and actuating force

measurements for the disk plate clutch.

Material Coefficient of Friction Maximum Pressure (psi)

Powdered Metal on Hard

Steel

0.1 300

Table 14: Shown here is the data for the material selected to craft the clutch plate.

Equation 6-1

Equation 6-2

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One plate should be sufficient since the pressure on the cylinder head is 132 psi and the

maximum pressure of the powdered metal on hard steel is 300 psi. Powdered metal on hard

steel was selected for the manufacturing of the clutch. The low coefficient of friction of this

material yields a high value for the actuating force value.

Conclusion

The purpose of this case study was to design a compressor unit for driving hammers at a

construction site. During the process all of the topics discussed in lecture this semester were

researched as a consequence of investigating how to best design this unit. The iteration

process of shaft design, pinion and gear design as well as several force analysis were conducted

in order to produce the optimal design. This case study was an effective means of better

understanding and appreciating course material covered during the semester.

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Bibliography

Budynas, Richard G. and J. Keith Nisbett. Shigleys Mechanical Engineering Design. 9th

ed.

New York, NY: McGraw Hill, 2008.