Thomas Industrial Library

8.12 MODELING AN INDUSTRIAL CAM-FOLLOWER SYSTEM

We will now put together the concepts of this chapter in an example that is representative of a common cam-follower system as used in industrial machinery. We will use a follower train similar to the one shown on the left side of Figure 8-10 (p. 198).

EXAMPLE 8-3

Creating a One-DOF Equivalent System Model of an Industrial Cam-Follower Train.

Given: A cam-follower system as is shown in Figure 8-16

Problem: Create a suitable, approximate, one- DOF , lumped-parameter model of the system as shown in Figure 8-4 (p. 186). Define its effective mass and effective spring rate at the roller follower in terms of the individual elements’ parameters.

Assumptions: The air cylinder is connected to a large accumulator that gives it an effective spring rate that is close to zero. The preload on the cam follower due to air pressure is sufficient to prevent follower jump. All parts are steel.

Solution:

1 For this example, only the linkage shown on the left side of Figure 8-10 will be modeled, as can be seen in Figure 8-16. The approach would be the same for the companion linkage on the right side of Figure 8-10, or for any other similar linkage.

2 The first step is to lump the masses of the links as shown in Figure 8-17a. The tooling mass m 5 is lumped at the right end of link 4, at point D . It has a value of 0.9 kg in this linkage.

3 The bellcrank, link 4, is in rotation about O 4 and so must be converted to an equivalent mass at point C using equations 8.11 with radius r 4 = 0.173 m. The mass moment of inertia of link 4 about pivot O 4 was determined from a CAD solid model of the link to be 0.0087 kg-m 2 . The effective mass of link 4 placed at point C is then

4 The mass of the connecting rod, link 3, is found from the CAD solid model to be 0.8338 kg.

5 The follower arm, link 2, is in rotation about O 2 and so must be converted to an equivalent mass at point B using equations 8.11 with radius r 2 = 0.283 m. The mass moment of inertia of link 2 about pivot O 2 was determined from a CAD solid model of the link to be 0.0325 kg-m 2 . The effective mass of link 2 placed at point B is then

6 The mass m 1 of the roller follower is 0.196 kg obtained from the manufacturer’s catalog information.

7 The next step is to bring all these lumped masses back to the roller follower location at point A where we wish to place the effective mass of the system. This will require the application of any link ratios that may be present.

8 First, bring mass m 5 across the bellcrank from point D to C using the radii of link 4 in equation 8.18b.

9 Add the effective masses of links 4 and 5 at point C .

10 Bring the mass m C and the mass of the connecting rod m 3 down to point B and add them to the effective mass of link 2 at that point.

11 Bring the total mass lumped at point B back to the follower at point A with equation 8.18b and add it to the mass of the roller that is there.

This is the effective mass to be used in the 1-DOF model of the system.

12 Next, the compliances of each element must be combined to find the overall effective spring constant of the system as felt at the cam follower, point A . Figure 8-18 shows a schematic of the various compliant elements that comprise this system. Note that the air cylinder does not contribute to this compliance. It essentially provides a near constant force as shown in Figure 8-12c, due to the accumulator. As will be seen in Chapter 10, as long as the spring or air cylinder used to close the cam follower joint in this type of “industrial” cam-follower system has sufficient force to prevent follower jump, then its spring constant will not affect the overall compliance of the system.

13 The bellcrank, link 4, can best be modeled as a double-cantilever beam. Standard beam equations available from references such as [4] can be used in combination with the beam’s cross-section geometry to determine the deflection and thus the spring rate of each half of the double-cantilever beam. However, in this case, the cross section of the beam is

not uniform along its length, causing variation in its area moment of inertia that complicates a classic computation of its deflection. The best approach is to use a CAD solids modeler and Finite Element Analysis (FEA) software to calculate the deflection at the locations where the loads are applied. Figure 8-19a shows the result of such an FEA analysis of link 4. Two calculations were done, one for each half of the beam. In each case the elements at the section containing the pivot axis were fixed and a 1000 N load applied at one pin joint. The calculated deflections were then divided into the applied load to get a spring rate. This analysis gives k 4 = 2.474E6 N/m and k 5 = 2.242E6 N/m.

14 The follower arm, link 2, is a simply-supported, overhung beam. Its deflection was analyzed by fixing the pivot locations and applying a 1000 N load to its pin center at point B of Figure 8-18. The resulting deflection shown in Figure 8-19b indicates a spring rate of k 2 = 7.813E6 N/m.

15 The connecting rod is in axial compression, and assuming no buckling, its spring rate can be found from the equation for axial deflection of a uniform rod. In this case, the rod is a steel tube of 25.5 mm OD and 22.1 mm ID, giving a cross-sectional area of 1.231E-4 m2. The length l 3 is 0.576 m. Its spring rate is then:

16 Now the compliances can be combined to form an effective spring rate at the roller follower. First bring the right hand side of link 4, k 5, across to point C .

17 All the compliances are in series because they pass a common force and each has a different deflection. Combine springs k 2, k 3, k 4, and k 5@C according to equation 8.15c and bring them to point B .

18 Now bring the combined compliance at B back to point A with equation 8.19b.

This is the effective spring rate to be used in the 1-DOF model of the system.

19 The air cylinder force in this example is applied at radius ra on the follower arm as shown in Figure 8-16 (p. 207). This force must be brought back to the roller follower by the ratio of the radii of follower and cylinder to the first power.

20 An approach for estimating the damping in the system is described in the next chapter.

8.13 REFERENCES

1 Koster, M. P. (1974). Vibrations of Cam Mechanisms . Phillips Technical Library Series, Macmillan Press Ltd.: London.

2 Hundal, M. S. (1963). “Aid of Digital Computer in the Analysis of Rigid Spring-Loaded Valve Mechanisms.” SAE Progress in Technology , 5 , pp. 4-9, 57.

3 O’Brien, C and P. Duperry . (2001). “Modeling a Cam-Driven Linkage with an Air-Cylinder Spring”, Major Qualifying Project, Worcester Polytechnic Institute.

4 Norton, R. L. (2000). Machine Design: An Integrated Approach , 2ed. Prentice-Hall, Upper Saddle River, NJ.

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