Table of Contents

Abstract………………………………………………………………………………………….……………….2

Introduction…………………………………………………………………………………………………....2

Experiment……………………………………………………………………………………………………..4

Procedure…………………………………………………………………………………………………….…7

Results, Analysis, and Discussion…………………………………………………………………….9

Conclusion and Recommendations…………………………………………………………….....12

Bibliography……………………………………………………………………………………………..…..12

Appendix………………………………………………………………………………………………..……..13

2

Abstract

This experiment was performed to determine the effect different weights had

on the follower and witness the behavior of the follower as the cam shaft reached

critical speed. In addition to the weight varying, the springs along the follower,

holding the weight, were changed throughout the experiment. The revolutions per

minute of the cam shaft were recorded each time the springs and weights were

changed. From these measurements the effect of spring force on the critical speed of

the cam shaft could be analyzed.

Introduction

The purpose of a cam is to convert rotational motion to linear motion. This

occurs when the follower, in this case a roller follower, follows the shape of the cam.

When the follower goes around the peak of the cam, it is forced in an upward, then

downward motion. This linear motion is increased as the speed of the motor

spinning the cam increases. The follower will remain tracing the outside of the cam

until it reaches the critical speed which occurs when the revolutions per minute are

so high that the follower cannot keep up with the cam and then separates. This

separation can be heard due to the large impact force produced by the follower

regaining contact with the cam. The importance of this critical speed is that the

force created during critical speed can eventually cause fatigue failure of the

surfaces of the follower and the cam.

This experiment shows the change in critical speed when two different

springs, each with added weights from no weight to 2000 grams in increments of

3

400 grams, were added to the follower. The two springs used were the red spring,

with a stiffness of 31.4 lb/in and the white spring, with a stiffness of 22.5 lb/in. The

resulting critical speed for each change in weight, for each spring was recorded.

This critical speed was then calculated using the following equation.

N cr=602π √−(W cos β+F so+kz)

Wg ( d2 zd θ2 )

Equation 1

Where Ncr is the critical speed, W is the weight of the follower, β is the angle

between the line of motion of the follower and the vertical, Fso is the spring force

behind the follower, k is the spring constant, z is the vertical displacement of the

follower, and θ is the angular displacement of the cam.

Then, the critical speed measured and the critical speed calculated were

compared. These results can be seen in Table 1.

4

Equipment

Experimental apparatus frame and assembly- Figure 1

Figure 1

A roller and follower-Figure 2

Figure 2

5

Springs with different stiffness values (Red and White spring were used)- Figure 3

Figure 3

Variable speed drive motor- Figure 4

6

Figure 4 Necessary instrumentation and recording devices- Figure 5 (Omega Non-Contact Pocket Optical Tachometer)

Figure 5

7

Procedure

1) While the variable speed drive motor is turned off, select the cam rotation

direction by moving the switch to the right or left. Do not change the

direction of rotation for the cam while the motor is running.

2) Select a cam, follower and spring and connect these parts to create the entire

cam follower assembly.

3) Wrap a piece of Teledeltos paper around the recording drum and secure it

with scotch tape.

4) Attach the rubber belt to the cam and then put the cam follower through one

full rotation manually, with the motor off. This will provide a sketch of the

follower displacement. Once the sketch has been recorded, remove the paper

from the recording drum.

5) Turn on the motor, and slowly increase the speed until you reach the critical

point. The critical point is distinguished by a loud tapping sound, indicating

the separation between the cam and follower. Decrease and increase the

speed a few times and listen to be sure that you have found the real critical

point. Using the Omega Non-Contact Pocket Tachometer, read the speed of

the camshaft and record it. Each person in the group should take their own

readings since the data collection is based on each individuals’ own hearing

sensitivity.

6) Repeat step 5 five times, adding a 400 gram weight to the follower each time.

8

7) To change the spring force, Fs, replace the spring and then repeat steps 5 and

6 with the new spring.

8) Calculate the critical speed (Ncr) analytically by differentiating twice, the Z- θ

curve and plotting dZ/d and dθ 2Z/dθ2 verses , obtaining the maximum θ

negative value into Eq.(3) from the lab manual. Because a curve-fit is

required to approximate the cam surface in this step, 95% confidence

intervals should also be shown on the Z- plot.θ

9) Plot the variation of Ncr versus the weight of the follower as well as the

spring force (either Fso or k if the spring was replaced). Be sure to indicate

the mean and standard deviation of each data point on graphs containing

measured values

10)Compare the theoretical Ncr with the experimental Ncr values obtained and

comment on the results.

9

Results, Analysis, and Discussion

The critical speed, Ncr , is obtained by differentiating the Z - θ curve, shown

in figure 6 below, twice. The Z - θ curve was plotted after analyzing the analog cam

profile recorded on Teledeltos paper attached to the recording drum. The plotted

profile was divided into thirty-nine points corresponding to a time interval of

2 /39.π

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

f(x) = − 0.403322 x + 3.13855 x − 8.69544 x + 9.91486 x³ − 4.10645 x²⁶ ⁵ ⁴ + 0.977227 x + 0.0102115R² = 0.9984931110376

Z - CurveƟ

Ɵ (rad)

Z (i

n)

Figure 6: Cam Profile

A polynomial curve fit was incorporated over the relevant portion of the cam

profile using Microsoft Excel. This curve fit displayed a high correlation coefficient,

near unity, reassuring the linearity between both the vertical displacement of the

follower and the angular displacement of the cam. Upon differentiating figure 6

twice, the maximum negative vertical displacement of the follower is found to be

-3.87 inches as shown in figure 7 below, as well as tabulated in table 6 found in the

appendix.

10

0 0.5 1 1.5 2 2.5 3

-10

-8

-6

-4

-2

0

2

4

d2Z/d 2 vs Ɵ Ɵ

Ɵ (rad)

d2Z/

d2

(in/

rad2

)Ɵ

Figure 7: d2Z/dƟ2 vs Ɵ

The realization of this maximum negative displacement value mathematically

closes equation 3 (lab manual) and Ncr is calculated for both the red and white

springs having nominal stiffness values of 31.4 lb/in and 22.5 lb/in respectively. The

critical speed is then calculated while increasing weight in increments of 400 grams,

added to the top of the follower for both springs. The theoretical and calculated Ncr

values are depicted in Table 1 below.

Table 1: Theoretical and Actual NCR Values

Theoretical ActualRed Spring White Spring Red Spring White Spring

Added Weight (g)

NCRAdded

Weight (g)NCR

Added Weight (g)

NCRAdded

Weight (g)NCR

0113.4

30

95.54

0299.0

00

265.67

400103.2

0400

87.41

400280.6

7400

244.67

800 95.49 80081.2

5800

266.67

800227.6

7

1200 89.44 120076.3

91200

249.67

1200215.6

71600 84.52 1600 72.4 1600 238.6 1600 210.6

11

4 7 7

2000 80.43 200069.1

42000

228.00

2000202.0

0

As can be seen from table 1, the actual and theoretical NCR values of both

springs decrease in a semi linear fashion with an increase in follower weight,

depicted in in figure 8. A 95% confidence interval is incorporated into the

polynomial curve fit consisting of error bars indicating the standard deviation of

each data point containing measured values. The variables of interest in regard to

NCR are spring constant and mass of the follower. The results show that as the spring

constant increases, the critical speed of separation will also increase. Furthermore,

as the mass of the follower increases, the critical speed decreases. It is important to

note that the theoretical NCR values are roughly three times less than that of actual

values calculated. This relationship defies logic and a calculation error is inherently

present in regards to theoretical values.

12

0 400 800 1200 1600 2000 24000.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

Actual and Theoretical NCR Values

Red Spring Theoretical

White Spring Theoretical

Red Spring Actual

White Spring Actual

Added Weight (g)

Crit

ical

Spe

ed (

RPM

)

Figure 8: Actual and Theoretical NCR Values

Conclusion and Recommendations

It can be concluded that the additional weight to each of the springs changes

the critical speed dramatically. Due to a calculation error, it is impossible to note

the exact difference between theoretical and calculated critical speeds. They do,

however, have the same linear pattern (as seen in Table 8). One recommendation to

this experiment is to have a more accurate way of measuring the theoretical critical

speed. The factor of human error could be eliminated if there was another way to

measure when the follower lost contact with the cam other than just listening for

the sound of the impact.

13

Bibliography

[1] ME 4201 Lab Manual – “Cam Experiment”. Pages 17-22. Fall 2014

14

Appendix

Average Critical Speed (RPM) for Red SpringExtra Weight (g) Average RPM Standard Dev.

0 299.00 1.63400 280.67 0.94800 266.67 1.70

1200 249.67 0.471600 238.67 1.702000 228.00 1.41

Table 2

Average Critical Speed (RPM) for White Spring

Extra Weight (g) Average RPM Standard Dev.0 265.67 1.70

400 244.67 1.89800 227.67 0.47

1200 215.67 0.471600 210.67 0.472000 202.00 0.00

Table 3

Cam-Follower Experiment Spring Dimensions

Color

DimensionsSpring Weight

(g)

Nominal Stiffness (lb/in)

Retainer Weight

(lb)

Mean Diameter

(in)Diameter Length

Red 1.12 1/8 2.99 0.138 31.4 0.156White 1.85 1/8 3.02 0.294 22.5 0.300

Table 4

15

MiscellaneousWeight of roller follower

and attachment 3.78 lb

Diameter of roller follower 1-1/8"

Diameter of paper recording drum 3.673"

Table 5

16

Table 6

17

Sample Calculation – Red Spring with No Extra Weight Theoretical

N cr=602π √−(W cos β+F so+kz)

Wg ( d2 zd θ2 )

N cr=602π

√−¿¿¿