Design of a Cam with Reciprocating Roller Follower Charles Hunter AME40423 Mechanisms and Machines October 11, 2010 Abstract The purpose of this project was to design a cam with a reciprocating roller follower. The follower had specific cycloidal motion characteristics as constraints which were used to develop a cycloidal displacement diagram. The cam was constrained to have an offset of 0.25 inches and the pressure angle, δ, of the cam could not exceed 30o. The roller follower was constrained to have a 0.375 inch diameter. A flowchart of the equations used to plot the cam profile is presented. After analysis using the equations, the minimum diameter of the base circle which fit the constraints mentioned earlier was 0.625 inches. This base circle diameter leads to a minimum follower position, yo, of 0.433 inches. The maximum pressure angle for the designed cam was 30o. A plot of the cam profile with a scale of 1’’ = 1’’ is presented in the results section. The appendix includes all of the equations and MATLAB code used to develop the cam profile.
1 Engineering Analysis A flowchart of the steps used to calculate the position values, ̅ , of the cam profile is presented below as Figure 1. The flowchart was used to develop the MATLAB code presented in Appendix C. A value of the base circle radius was input into the code which then presented a
plot of the cam profile and the maximum pressure angle of the cam as the output. This step was repeated until the minimum value of the base circle was found that fit into the constraints. A balance needed to be found between the pressure angle and the diameter of the base circle. When the base circle diameter was decreased, the pressure angle increased. Once the value for the base circle diameter was obtained, the minimum follower distance, yo, was calculated. The equations to calculate the values for each step in the flowchart are presented in Appendix A.
Figure 1: Flowchart for Cam Design
2 Results The roller follower’s cycloidal motion displacement diagram is presented in Figure 2. Cycloidal motion was used to describe the motion of the roller follower. Cycloidal motion was chosen so the motion has continuous and smooth accelerations, and therefore no infinite jerk. The equations used to calculate the cycloidal motion are found in Appendix B. The roller follower has a lift of 0.6 inches which is accomplished in 150o of cam rotation. The follower maintains the lift for 30o and then returns to the lowest position in 180o of cam rotation.
Figure 2: Cycloidal Motion Displacement Diagram
The equations used to calculate the displacement of the roller follower were differentiated with respect to Ď• to see its kinematic properties. The velocity of the cycloidal motion is described
when the displacement equations are differentiated once. A diagram of the cycloidal motion velocity is presented in Figure 3.
Figure 3: Cycloidal Motion Velocity Diagram
The minimum base circle diameter was 0.625 inches. The cam profile with a scale of 1” = 1” is presented in Figure 4. The maximum pressure angle, δ, of a cam with a reciprocating roller follower and base circle diameter of 0.625 inches was calculated to be 30o.
Figure 4: Cam Profile 1” = 1” scale
Appendix A: Equations for Cam with Reciprocating Roller Follower1 √
(1) (2)
√
(3)
(4)
(5)
(6)
(7)
̅
1
(
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(
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Equations from AME 40423 Notes; Stanisic, Michael. Introduction to the Synthesis and Analysis of Planar Mechanisms and Machines. Fall 2010.
(8)
Appendix B: Equations for Cycloidal Motion2 Rise Zone: (9)
(10)
Top Zone: (11)
(12)
Return Zone:
(13)
(14)
Bottom Zone: (15)
(16)
2
Equations from AME 40423 Notes; Stanisic, Michael. Introduction to the Synthesis and Analysis of Planar Mechanisms and Machines. Fall 2010.
Appendix C: MATLAB Code for Cam Profile %Charles Hunter %10/08/10 %Mechanism & Machines %Design a cam with a reciprocating roller follower where the follower has %the following motion characteristics: %1) A lift of .6 inches %2) The lift is accomplished in 150 degrees of cam rotation %3) The follower maintains the lift of 30 degrees. %4) The follower returns to its lowest position in 180 degrees of cam %rotation. %Cam should have an offset of 0.25 inches. Use a roller follower which has %a 0.375 inch diameter. %Design the smallest possible cam which has a pressure angle whose %magnitude never exceeds 30 degrees. close all clear all %Input of rf, m, rb, and y0. m = 0.25; rf = 0.1875; Rb = input('Enter Radius of Base Circle:'); y0 = sqrt(((Rb+rf)^2) - m^2); %Set Variables DMax = 0; Phi = 0; L = 0.6; Brise = ((5*pi)/6); Breturn = pi; Btop = (pi/6); PhiZ = (Brise + Btop); t = 1;
for Phi = (0:360) Phi = ((Phi*pi)/180); %Compute y % dy/d(phi) if Phi < ((5*pi)/6) y(t) = (((L*Phi)/Brise)-((L/(2*pi))*(sin((2*pi*Phi)/Brise)))); dy(t) = (L/Brise)*(1-cos((2*pi*Phi)/Brise)); elseif Phi >= ((5*pi)/6) && Phi <= pi y(t) = L; dy(t) = 0; elseif Phi > pi y(t) = (L - ((L*(Phi-PhiZ))/Breturn) + ((L/(2*pi))*sin((2*pi*(PhiPhiZ))/Breturn))); dy(t) = (L/Breturn)*(-1+cos((2*pi*Phi)/Breturn)); elseif Phi >= 2*pi
y(t) = 0; dy(t) = 0; end
%Compute D D = y+y0; %Compute Rf Rf = sqrt(D.^2+m^2); %Compute Psi Psi = atan(m./D); %Compute a a = dy; %Compute Alpha Alpha = atan((a.*cos(Psi))./(Rf-a.*sin(Psi))); %Compute Delta Delta = Alpha - Psi; %Check on Pressure Angle if abs(Delta) > DMax DMax = abs(Delta); end %Compute R Rx = (D.*cos(Phi) + m.*(cos(Phi + (pi/2))) + rf.*(cos(Phi+pi-Delta))); Ry = (D.*sin(Phi) + m.*(sin(Phi + (pi/2))) + rf.*(sin(Phi+pi-Delta))); %Increment t = t+1; end %Output Max pressure angle in Degrees. DMax = (DMax*180)/pi %Output of various plots plot(y) xlabel('Beta Rotation [Degrees]') ylabel('Follower Displacement [y]') title('Cycloidal Motion Displacement Diagram') %% figure; plot(dy) xlabel('Beta Rotation [Degrees]') ylabel('dy/d\phi') title('Cycloidal Motion Velocity Diagram') %% figure; plot(Rx,Ry) xlabel('X Position') ylabel('Y Position') title('Cam Profile') axis equal