Table of Contents
Abstract………………………… Abstract………………………………………………… …………………………………………… ……………………………………….… ………………….……………….2 …………….2 Introduction…………………………………………………………………………………………………....2 Experiment ……………………………………………………………………………………………………..4 ……………………………………………………………………………………………………..4 Procedure…………………………………………………………………………………………………….…7 Results, Analysis, and Discussion…………………………………………………………………….9 Discussion…………………………………………………………………….9 Conclusion and Recommendations…………………………………………………………….....12 Bibliography……………………………………………………………………………………………..…..12 Appendix………………………………… Appendix………………………………………………………… …………………………………………… …………………………………….. ………………..……..13 ……..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.
√ () Equation 1
Where N cr 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, F so 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
- Figure 1
Figure 1 roller and follower-Figure 2
Figure 2
5
Figure 3
Figure 3 - Figure 4
6 Figure 4 - 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/dθ and d2Z/dθ2 verses θ, obtaining the maximum
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-θ plot. 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, N cr , is obtained by differentiating the Z in figure 6 below, twice. The Z -
curve, shown
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 2π/39.
Z - Ɵ Curve Z= -0.4033x6 + 3.1385x5 - 8.6954x4 + 9.9149x3 - 4.1064x2 + 0.9772x + 0.0102 R² = 0.9985
1.2 1 0.8 ) n i ( 0.6 Z
0.4 0.2 0 0
0.5
1
1.5
2
2.5
3
Ɵ (rad)
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
d2Z/dƟ2 vs Ɵ 4 2
)
2
0
d a r / n i (
-2
2
Ɵ
d / Z 2 d
0
0.5
1
1.5
2
2.5
3
-4 -6 -8
-10
Ɵ (rad)
Figure 7: d2Z/dƟ2 vs Ɵ
The realization of this maximum negative displacement value mathematically closes equation 3 (lab manual) and N cr 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 N cr values are depicted in Table 1 below.
Table 1: Theoretical and Actual N CR Values Theoretical Red Spring Added Weight NCR (g)
0 400 800 1200 1600 2000
113.43 103.20 95.49 89.44 84.52 80.43
Actual
White Spring Added NCR Weight (g)
0 400 800 1200 1600 2000
95.54 87.41 81.25 76.39 72.44 69.14
Red Spring Added Weight NCR (g)
0 400 800 1200 1600 2000
299.00 280.67 266.67 249.67 238.67 228.00
White Spring Added NCR Weight (g)
0 400 800 1200 1600 2000
265.67 244.67 227.67 215.67 210.67 202.00
11
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.
Actual and Theoretical N CR Values 350.00 300.00 ) M250.00 P R ( d200.00 e e p S150.00 l a c i t i r 100.00 C
Red Spring Theoretical White Spring Theoretical Red Spring Actual White Spring Actual
50.00 0.00 0
400
800
1200
1600
2000
2400
Added Weight (g)
Figure 8: Actual and Theoretical N CR Values
12
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.
Bibliography
[1]
ME 4201 Lab Manual – “Cam Experiment ”. Pages 17-22. Fall 2014
13 Appendix
Average Critical Speed (RPM) for Red Spring Extra Weight (g)
Average RPM
Standard Dev.
0 400 800 1200 1600 2000
299.00 280.67 266.67 249.67 238.67 228.00
1.63 0.94 1.70 0.47 1.70 1.41
Table 2
Average Critical Speed (RPM) for White Spring Extra Weight (g)
Average RPM
Standard Dev.
0 400 800 1200 1600 2000
265.67 244.67 227.67 215.67 210.67 202.00
1.70 1.89 0.47 0.47 0.47 0.00
Table 3
Cam-Follower Experiment Spring Dimensions Dimensions Length
Spring Weight (g)
Nominal Stiffness (lb/in)
Retainer Weight (lb)
1/8
2.99
0.138
31.4
0.156
1/8
3.02
0.294
22.5
0.300
Mean Diameter (in)
Diameter
Red
1.12
White
1.85
Color
Table 4
14
Miscellaneous Weight of roller follower and attachment
3.78 lb
Diameter of roller follower
1-1/8"
Diameter of paper recording drum
3.673"
Table 5
15
Table 6
16 Sample Calculation – Red Spring with No Extra Weight Theoretical
√ ( ) √
