MACHINE ELEMENTS (Phase 2) PART 1 – INTRODUCTION TO CAM DESIGN DEFINITION OF TERMS CAM – A rotating mechanism of irregular shape, as on a wheel on a machine: used to change the direction of the motion of another part moving against it, as rotary into reciprocating or variable motion. COMMON CLASSIFICATION OF CAM ACCORDING TO SHAPE: DISK CAM – Cam profile curve is worked in a flat plate as shown in Figure 1. CYLINDRICAL CAM - Cam profile curve is worked around a blank cylinder. Figure 2: Cylindrical Cam Figure 1: Common types of Disk Cam CAM FOLLOWER – A part with a sliding contact with the cam. Two (2) types and most common used are Reciprocating and Oscillating follower. CLASSIFICATION ACCORDING TO TYPES OF FOLLOWER: IN-LINE/RADIAL FOLLOWER – The follower is considered to be inline or radial when its centerline passes through the center of cam rotation. Seldom used in actual practice since it produces high contact stresses. OFFSET FOLLOWER – The centerline of the follower has an offset distance with respect to the center of cam rotation. OSCILLATING FOLLOWER – Special type cam mechanism with a follower oscillating at a certain center having a defined distance with respect to center of cam rotation. 1 CLASSIFICATION BASED ON THE TYPE OF MOTION: SIMPLE MOTION – The cam profile is generated in reference with the predefined motion of the follower. PARABOLIC MOTION – Analytically, the cam profile is generated by following the motion curve of an equation of a Parabola. Graphical method will be discussed in the proceeding topics. HARMONIC MOTION – Cam profile is generated by following the motion curve of a harmonic motion. CYCLOIDAL MOTION – Cam profile is generated by following the motion curve of a cycloid. Figure 3: Cam Nomenclature (Radial Cam with Roller follower) CAM NOMENCLATURE: BASE CIRCLE – The smallest circle tangent to the cam surface about the center of cam rotation. CAM PROFILE – The actual contacting surface of a disk type cam. Its curvature is generated using either one of the following considerations: (1) Assumption of the required motion for the follower then design the cam profile to give this motion and, (2) Assumption of the shape of the cam then determine what characteristics of displacement, velocity and acceleration this contour will give. Dilemma: After the cam is designed, some contour will be difficult to manufacture. Solution: Follow the second method by making the cam symmetrical and by using shape for the cam contours that can be generated through machining and other metal working process. Result: Manufacturing of cams for automotive application that must be produced cheaply and accurately has been solved. STROKE – The distance measured between two extreme position of the follower. TRACE POINT – Is a point on the follower that corresponds to the contact point of fictitious knife edge follower. PITCH CURVE – Is the path of trace point relative to the cam profile. PRESSURE ANGLE – The angle between the direction of motion of the trace point and the common normal (line of action) to the contacting surfaces. It is also a measure of the instantaneous force transmission properties of the mechanism. The larger is the pressure angle, more power is required to operate the cam mechanism. REMEMBER! Line of action is the axis where the force from rotating cam profile is ACTUALLY transmitted on the follower. 2 1.0 PROCEDURES IN GENERATING CAM CONTOUR There are two distinct methods of how Cam contours or cam profiles are generated. The methods are (1) Graphical Method and (2) Analytical Method. The first method is of relatively the simplest method of generating cam contours which uses only graphical techniques while the second method uses mathematical equations to determine the curvature of the cam profile. thus, we need to determine the plot of motion curve as shown in Figure 4. With the corresponding intersection of the motion curve and the angle corresponding to this displacement, the procedures above will then follow to generate the required cam contour. GRAPHICAL METHOD FOR SIMPLE MOTION To determine the cam contours graphically, it will be necessary to invert the mechanism and to hold the cam stationary while the follower moves around it. This will not affect the relative motion between the cam and the follower, and the procedure is as follows: 1. Rotate the follower about the center of the cam in a direction opposite to that of the proposed cam rotation. 2. Move the follower radially outward the correct amount for each division of rotation. 3. Draw the cam outline tangent to the polygon that is formed by the various positions of the follower surface. Unfortunately, in the last step, there is no graphical way of determining the contact point between the cam and the follower, and it must be determined by eye with the use of a French curve. The methods summarized above are applied in all types of disk cam as follows: 1. DISK CAM WITH RADIAL ROLLER FOLLOWER By following the construction method of cam contour as summarized above, we may arrive in the desirable contour provided that we have the data of follower displacement per degree of cam rotation, Figure 4. Disk Cam with Radial Roller Follower Design Plate Problems: 1. A disk cam with a radial roller follower is rotating clockwise through a total displacement of 1½ in. with the following lift figures: Cam Angle, θ 30 60 90 120 150 180 Lift, in. 0.10 0.37 0.75 1.13 1.40 1.50 Cam Angle, θ 210 240 270 300 330 360 Lift, in. 1.40 1.13 0.75 0.37 0.10 0.00 Layout the cam contour and determine by trial the maximum pressure angle and the position (angle) to where it will 3 occur provided that the base diameter of the cam is 2in. and a diameter of roller follower as 5/8in. 2. A radial roller follower disk cam is rotating clockwise through a total displacement of 30mm. with the following lift figures: Cam Angle, θ 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Lift, mm 0.00 0.23 0.90 2.01 3.51 5.36 7.50 9.87 12.40 15.00 17.60 20.13 22.50 24.64 26.49 27.99 29.10 29.77 30.00 Cam Angle, θ 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 Lift, mm 29.77 29.10 27.99 26.49 24.64 22.50 20.13 17.60 15.00 12.40 9.87 7.50 5.36 3.51 2.01 0.90 0.23 0.00 Using CAD Software, layout the cam details and contour. Determine the maximum pressure angle and the position (angle) to where it will occur provided that the base circle diameter of the cam is 20mm. and a diameter of roller follower as 8mm. with a follower rod diameter of 4mm. Provide square (1.5mmx1.5mm) keyway in the layout. Assume a cam thickness of 10mm, hub diameter of 15mm, hub thickness of 3mm and a bore diameter of 8mm. Generate the 3D model of the cam. 2. DISK CAM WITH OFFSET ROLLER FOLLOWER Consider the same type of cam assembly but this time, the follower axis has an offset distance from the vertical axis passing through the cam rotation. The method of construction will be almost the same in an exception that roller follower is being offset with the specified distance from the reference axis given. The development of this type of cam has been integrated to minimize the pressure angle of the contacting surface, thus, reducing the power requirement during outward rotation (lifting motion) although there is a large pressure angle in the return motion. Figure 5. Disk Cam with Radial Roller Follower Design Plate Problems: 1. A disk cam with an offset roller follower is rotating clockwise through a total displacement of 1½ in. with the following lift figures: 4 Cam Angle, θ 30 60 90 120 150 180 Lift, in. 0.10 0.37 0.75 1.13 1.40 1.50 Cam Angle, θ 210 240 270 300 330 360 Lift, in. 1.40 1.13 0.75 0.37 0.10 0.00 Layout the cam contour and determine by trial the maximum pressure angle during outward and return motion provided that the base diameter of the cam is 2in. and a diameter of roller follower as 5/8in. The centerline of the follower has an offset distance of ½in. from the vertical axis passing through the center of cam rotation. 2. An offset roller follower disk cam is rotating clockwise through a total displacement of 33mm. with the following lift figures: Cam Angle, θ 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Lift, mm 0.00 0.25 1.00 2.21 3.86 5.89 8.25 10.86 13.63 16.50 19.37 22.14 24.75 27.11 29.14 30.79 32.00 32.75 33.00 Cam Angle, θ 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 Lift, mm 32.75 32.00 30.79 29.14 27.11 24.75 22.14 19.37 16.50 13.63 10.86 8.25 5.89 3.86 2.21 1.00 0.25 0.00 Using CAD Software, layout the cam details and contour. Determine the maximum pressure angle during outward and return motion provided that the base circle diameter of the cam is 22mm and a diameter of roller follower as 8.80mm with a follower rod diameter of 3.3mm. Provide square (1.7mmx1.7mm) keyway in the layout. Assume a cam thickness of 11mm, hub diameter of 15.5mm, hub thickness of 3.30mm and a bore diameter of 8.80mm. The offset distance of axis of the follower from the vertical axis passing through the cam rotation is measured as 6mm. Generate the 3D model of the cam. 3. DISK CAM WITH FLAT-FACED FOLLOWER Figure 6 shows a typical layout of a Cam with flat-faced follower. The principal construction of this type of cam are the same as in the radial roller follower with the exception that the cam profile is drawn tangent to the various position of the flat face follower. Figure 6. Disk Cam with Flat-Faced Follower 5 In this case, there is no need to distinguish between radial and offset followers because they are kinematically equivalent. That is any follower shaft parallel to the one shown will produce the same output motion. However, it may be necessary to change the length of the follower face when the follower is offset. One important factor of the design of this type of cam is the determination of the minimum length of follower face to ensure a smooth contact of the mating surfaces during the operation. This minimum length can be defined graphically by measuring the maximum parallel distance between the axis passing through the center of cam rotation and the normal axis to the point of contact corresponding to the angle of cam rotation. Figure 7. Determination of the length of follower face (Radial symmetrical construction) From Figure 7, we can determine graphically the length of the follower face using the equation: 𝑳 𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒇𝒐𝒍𝒍𝒐𝒘𝒆𝒓 𝒇𝒂𝒄𝒆 = 𝟐 [ + 𝒂𝒍𝒍𝒐𝒘𝒂𝒏𝒄𝒆] 𝟐 In case of an offset follower, for symmetrical construction, the length of the follower face will be determined using the equation: 𝑳 𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒇𝒐𝒍𝒍𝒐𝒘𝒆𝒓 𝒇𝒂𝒄𝒆 = 𝟐 [ + 𝒂𝒍𝒍𝒐𝒘𝒂𝒏𝒄𝒆 + 𝒐𝒇𝒇𝒔𝒆𝒕 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆] 𝟐 Figure 8. Determination of the length of follower face (Offset symmetrical construction) 6 Design Plate Problems: 1. A disk cam rotating clockwise drive a radial flat-faced follower through a total displacement of 1.50in. with the following lift figures: Cam Angle, θ 30 60 90 120 150 180 Lift, in. 0.10 0.37 0.75 1.13 1.40 1.50 Cam Angle, θ 210 240 270 300 330 360 Lift, in. 1.40 1.13 0.75 0.37 0.10 0.00 Lay out the cam using the following conditions: a. Base circle diameter of 2in. Add 0.125in (symmetrical construction) after determining the minimum length of the follower face. b. Base circle diameter of 2.50in and a follower offset distance of 0.25in. Add 1/8 in. (symmetrical construction) after determining the minimum length of the follower face. 2. A Flat-faced follower disk cam is rotating clockwise through a total displacement of 33mm. with the following lift figures: Cam Angle, θ 0 10 20 30 40 50 60 70 80 90 100 110 120 130 Lift, mm 0.00 0.04 0.29 1.02 2.16 4.00 6.45 9.46 12.87 16.50 20.13 23.54 26.55 29.00 Cam Angle, θ 140 150 160 170 180 190 200 210 220 230 240 250 260 270 Lift, mm 30.84 32.05 32.71 32.96 33.00 32.96 32.71 32.05 30.84 29.00 26.55 23.54 20.13 16.50 Cam Angle, θ 280 290 300 310 320 Lift, mm 12.87 9.46 6.45 4.00 2.16 Cam Angle, θ 330 340 350 360 Lift, mm 1.02 0.29 0.04 0.00 Using CAD Software, layout the cam details and contour. Determine the minimum length of the follower face with a symmetrical construction allowance of 2mm on both ends. The base circle diameter of the cam is 25mm and a diameter of roller follower is 8.80mm with a follower rod diameter of 2.25mm. Provide square (1.5mmx1.5mm) keyway in the layout. Assume a cam thickness of 11mm, hub diameter of 20mm, hub thickness of 3.30mm and a bore diameter of 8mm. Generate a 3D model of the cam. 4. DISK CAM WITH OSCILLATING ROLLER FOLLOWER Figure 9. Disk Cam with an Oscillating Roller Follower 7 Figure 9 shows a typical layout of a disk cam with an oscillating roller follower. In this type of cam, an oscillating follower move at a certain degree as the cam rotates and touches the mating surface of the follower. The principal construction of this type of cam is the same as in the reciprocating roller follower, only, various condition shall be established first before doing the layout of the cam contour. These are: 1. Distance between the center of cam rotation and center of oscillation of the follower, 2. Position of center of oscillation relative to the center of cam rotation and, 3. Radius of oscillation of the follower. Design Plate Problems: 1. A disk, cam rotating counterclockwise drives an oscillating roller follower through a total angle of 200. Lay out the cam using a minimum diameter of 2in. The center of the hub of the follower is to be 3in to the right of the center and on the horizontal centerline of the cam. The diameter of the roller is to be 0.75in and the distance from the center of the hub to the center of the roller is to be 2.75in. Using a bore of 0.75in, hub diameter of 1in., and a 0.10in x 0.10 in key in both cam and follower hub, draw the cam assembly layout and determine graphically the maximum pressure angle. Use the following lift figures: Cam Angle 30 60 90 120 150 180 Follower Angle, degree 1.50 5.50 10.00 14.50 18.50 20.00 Cam Angle 210 240 270 300 330 360 Follower Angle, degree 18.50 14.50 10.00 5.50 1.50 0.00 2. A disk, cam rotating counterclockwise drives an oscillating roller follower through a total angle of 300. Lay out the cam using a CAD software providing a minimum cam diameter of 33mm. The center of the hub of the follower is to be 75mm to the right and 15mm above the center of rotation of the cam. The center of the roller is to be 67.50mm from the center of oscillation. Using a bore of 15mm, hub diameter of 22.50mm, and a 2.25mm x 2.25mm key in both cam and follower hub with cam and hub thickness of 10mm and 3mm respectively, draw the cam contour and generate its 3D model using the below lift figures. Determine also the maximum pressure angle. Cam Angle 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Follower Angle, degree 0.00 0.22 0.86 1.92 3.35 5.12 7.18 9.47 11.92 14.48 17.06 19.60 22.02 24.25 26.20 27.81 29.01 29.75 30.00 Cam Angle 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 Follower Angle, degree 29.75 29.01 27.81 26.20 24.25 22.02 19.60 17.06 14.48 11.92 9.47 7.18 5.12 3.35 1.92 0.86 0.22 0.00 8 5. DISK CAM WITH OSCILLATING FLAT-FACED FOLLOWER Graphically, the construction method in generating the cam contour of a disk cam with an oscillating flat-faced follower is a combination of knowledge in a reciprocating and oscillating follower. Review that cam contour in a flat faced follower is always drawn tangent to the enclosing polygon generated by following the displacement of the follower with respect to cam rotation. The concept of finding the minimum length as in the reciprocating flat face follower is also applied in this type of cam. The construction is almost similar with the oscillating roller follower in an exception that cam contour is drawn normal to a flat surface. Design Plate Problems: 1. A disk, cam rotating counterclockwise drives a flat-faced follower through a total angle of 200. Lay out the cam using a minimum diameter of 2in. The center of the hub of the follower is to be 3in to the right of the center and on the horizontal centerline of the cam. The radius of oscillation is 2.75in. Using a bore of 0.75in, hub diameter of 1in., and a 0.10in x 0.10 in key in both cam and follower hub, draw the cam assembly layout and determine the minimum length of follower faced (symmetrical construction) by adding a 0.10in allowance in both ends to ensure smooth contact. Use the following lift figures: Cam Angle 30 60 90 120 150 180 Figure 10. Disk Cam with an Oscillating Flat-faced Follower Follower Angle, degree 1.50 5.50 10.00 14.50 18.50 20.00 Cam Angle 210 240 270 300 330 360 Follower Angle, degree 18.50 14.50 10.00 5.50 1.50 0.00 2. A disk, cam rotating counterclockwise drives an oscillating flatfaced follower through a total angle of 300. Lay out the cam using a CAD software providing a minimum cam diameter of 33mm. The center of the hub of the follower is to be 70mm to the left and 10mm above the center of rotation of the cam. The radius of oscillation is to be 65mm. Using a bore of 15mm, hub diameter of 22.50mm, and a 2.25mm x 2.25mm key in both cam and follower hub with cam and hub thickness of 10mm and 3mm respectively, draw the cam contour and generate its 3D model using below lift figures. Upon constructing the cam contour, determine the minimum length of 9 follower faced (symmetrical construction) by adding a 1mm allowance in both ends to ensure smooth contact Cam Angle 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Follower Angle, degree 0.00 0.22 0.86 1.92 3.35 5.12 7.18 9.47 11.92 14.48 17.06 19.60 22.02 24.25 26.20 27.81 29.01 29.75 30.00 Cam Angle 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 Follower Angle, degree 29.75 29.01 27.81 26.20 24.25 22.02 19.60 17.06 14.48 11.92 9.47 7.18 5.12 3.35 1.92 0.86 0.22 0.00 1. PARABOLIC MOTION Parabolic motion has the lowest theoretical acceleration for a given rise and cam speed for the motions listed, and for this reason it has been used for many cam profiles. However, in slow-speed work, this has little significance. Parabolic motion may or may not have equal intervals of acceleration and deceleration, depending on requirements. Parabolic motion may also be modified to include an interval of constant velocity between the acceleration and deceleration; this is often spoken of as modified constant velocity. Although, analytically, the parabolic motion can be constructed using the displacement formula as S = ½ At2, it can be graphically constructed as follow: GRAPHICAL METHOD FOR CAM DISPLACEMENT CURVE The above discussion introduced us the method of generating the cam contour but following the real engineering process, before a cam contour can be determined, it is necessary to select the motion with which the follower will move in accordance with the requirements of the system. If operation is to be at slow speed, the motion may be any one of several common motions, for example, parabolic (constant acceleration and deceleration), parabolic with constant velocity, simple harmonic, or cycloidal. Figure 11. Parabolic Motion Procedures: 1. Determine the distance of follower displacement and construct a displacement graph. 2. Divide the right side of the displacement curve by the number of interval as the cam rotation. (In case of Figure 11, there are nine (9) intervals) 10 3. Project a construction line (dotted) from the starting motion (zero (0) degrees) extending to the point of intersection of the right side of the displacement graph with the interval line. 4. Trace the curve corresponding to the point of intersection of each construction line generated in procedure 3 with the degree of cam rotation. In plotting the displacement-time graph, it is necessary to determine the inflection point if the motion is parabolic or a modification thereof. The inflection point for parabolic motion will be at the midpoint of the displacement scale and of the time scale if the intervals are equal. The slope of this point of inflection can be used as a guide in constructing the section of the displacement having a constant velocity. Figure 12. Finding the slope in the point of inflection (Method 1) Figure 13. Finding the slope in the point of inflection (Method 2) After the inflection points have been determined as, for example, in Figure 12 and Figure 13, the constantly accelerated portion OA of the displacement curve is constructed as shown in Figure 11. The deceleration portion BC of the curve in Figure 12 and 13 is constructed in a similar manner for its particular displacement and corresponding time interval. Design Plate Problems: 1. A symmetrical contour disk cam of flat-faced follower having a minimum diameter of 25mm follows a constantly accelerated parabolic motion during the first 600 of cam rotation, a constant velocity motion during the second 600 rotations and a constantly decelerated parabolic motion during the third 600 of cam rotation. If the total follower displacement is 50mm, draw the cam contour and generate its 3D model. Upon constructing the cam contour, determine the minimum length of follower face (symmetrical construction) by adding a 1.50mm allowance in 11 both ends to ensure smooth contact. Use a bore of 15mm, hub diameter of 22.50mm, and a 2.25mm x 2.25mm key with cam and hub thickness of 10mm and 3mm respectively. Note: Analyze the cam contour in every 100 of cam rotation. 2. An offset roller follower disk cam is rotating clockwise through a total displacement of 33mm with the following motion curves: Cam Angle of Rotation 0-60 61-120 121-180 181-240 241-300 301-360 overhung in its construction. Less power will also be needed to operate the cam. Although, analytically, the harmonic motion can be constructed using the displacement formula S = r(1-cos(θ)), where ‘r’ is the radius of the construction circle (r = S/2) and ‘θ’ being the angle of cam rotation, it can be simply presented graphically as follow: Motion Curve Constant Accelerated Parabolic motion Constant Velocity motion Constant Decelerated Parabolic motion Constant Accelerated Parabolic motion Constant Velocity motion Constant Decelerated Parabolic motion Using CAD Software, layout the cam details and contour. Determine the maximum pressure angle during outward and return motion provided that the base circle diameter of the cam is 20mm and a diameter of roller follower as 8.85mm with a follower rod diameter of 3.3mm. Provide square (1.5mmx1.5mm) keyway in the layout. Assume a cam thickness of 11mm, hub diameter of 15.5mm, hub thickness of 3.30mm and a bore diameter of 8.80mm. The offset distance of axis of the follower from the vertical axis passing through the cam rotation is measured as 6mm. Generate the 3D model of the cam. Note: Analyze the cam contour in every 100 of cam rotation. 2. HARMONIC MOTION Generally, harmonic motion has the advantage that, with a radial roller follower, the maximum pressure angle will be smaller than with parabolic motion with equal time intervals or with cycloidal motion. This will allow the follower to be less rigidly supported and more Figure 14. Harmonic Motion Procedures: 1. Determine the distance of follower displacement and construct a displacement graph. 2. Draw the construction circle (semicircle, r = S/2) with the center at the midpoint of zero (0) degree ordinate. 3. Divide the circle radially with the numbers of division to be used in the displacement graph. 4. Project a horizontal line extending from the intersection of construction circle and the line generated in procedure 3 up to the corresponding angle of rotation in displacement graph. 5. Trace the intersection of ordinates in displacement graph and the line generated in procedure 4. 12 Note: A dwell can also be integrated with the cam design. Dwell in cam design is defined as the interval of cam rotation (in degrees) where the follower has no considerable motion or the cam is at rest position. Design Plate Problems: 1. Design the contour of an offset roller follower disk cam rotating clockwise through a total displacement of 33mm following a simple harmonic motion. Using CAD Software, layout the cam details and contour. Determine the maximum pressure angle during outward and return motion and compare the obtained result in problem 2 under parabolic motion provided that the base circle diameter of the cam is 20mm and a diameter of roller follower as 8.85mm with a follower rod diameter of 3.3mm. Provide square (1.5mmx1.5mm) keyway in the layout. Assume a cam thickness of 11mm, hub diameter of 15.5mm, hub thickness of 3.30mm and a bore diameter of 8.80mm. The offset distance of axis of the follower from the vertical axis passing through the cam rotation is measured as 6mm. Generate the 3D model of the cam. Note: Analyze the cam contour in every 100 of cam rotation. 2. A disk cam of an offset flat-faced follower having a minimum diameter of 25mm and a follower offset distance of 3mm follows the following motion curves during its operation: Cam Angle of Rotation 0-60 61-120 121-180 181-240 241-300 301-360 Motion Curve Dwell Constant Accelerated Parabolic motion Decelerating Harmonic Motion Accelerating Harmonic Motion Constant Decelerated Parabolic motion Dwell If the total follower displacement is 55mm, draw the cam contour and generate its 3D model. Upon constructing the cam contour, determine the minimum length of follower face (symmetrical construction) by adding a 2.00mm allowance in both ends to ensure smooth contact. Use a bore of 15mm, hub diameter of 22.50mm, and a 2.25mm x 2.25mm key with cam and hub thickness of 10mm and 3mm respectively. Note: Analyze the cam contour in every 100 of cam rotation. Each half of follower displacement is divided equally into types of motion curves given. 3. CYCLOIDAL MOTION Figure 15 shows the construction for cycloidal motion. Although, analytically, the type of cycloidal motion presented below (other types will be presented in the preceding topics) can be constructed using the displacement formula: 𝜃 1 𝜃 𝑠 = 𝑆[ − 𝑠𝑖𝑛2𝜋 ] 𝛽 2𝜋 𝛽 where ‘S’ is the total displacement of the follower, ‘s’ is the specific lift with respect to cam rotation, ‘θ’ being the angle of cam rotation and ‘β’ is the total angle in consideration. (In the case of Figure 15, β = 1800) The radius of the construction circle is r = S/2π. The circumference of this circle is divided into the same number of parts as in the time scale, in the case of figure 15, eighteen (18) parts. The eighteen (18) marks on the circumference are projected horizontally onto the vertical diameter of the circle. The marks on the vertical diameter are then projected parallel to OA to the corresponding line on the time axis. Procedures: 1. Determine the distance of follower displacement and construct a displacement graph. 13 Figure 15. Cycloidal Motion 2. Draw the construction circle (r = S/2π) with the center at point ‘O’. 3. Divide the circumference of the construction circle with intervals equal to the number of division in the displacement graph. 4. Project horizontally the intersection of each constructed line in procedure 3 up to the zero (0) degree ordinate of the displacement graph. 5. Construct the line O-A. (From origin and zero (0) degree ordinate up to displacement ‘S’ and 180 degrees’ ordinate intersection. 6. Project lines parallel with O-A starting from the intersection of lines constructed in procedure 4 and trace the curve from the intersection of ordinates and the line constructed in procedure 5 corresponding to the angle division under consideration. The significance of constructing the types of motion discussed in this section will be tackled in the preceding topics of cams. This is due to the fact that different types of motion curve produce dynamic effect that tends to separate the cam with the follower and produces significant level of machine vibration that should be avoided. Design Plate Problems: 1. Design the contour of an offset flat faced follower disk cam rotating clockwise through a total displacement of 50mm following a cycloidal motion. Using CAD Software, layout the cam details and contour. Determine the length of follower face permitting a 1.5mm allowance on both sides (symmetrical construction) provided that the base circle diameter of the cam is 25mm. Provide square (1.5mmx1.5mm) keyway in the layout. Assume a cam thickness of 11mm, hub diameter of 15.5mm, hub thickness of 3.30mm and a bore diameter of 8.80mm. The offset distance of axis of the follower from the vertical axis passing through the cam rotation is measured as 6mm. Generate the 3D model of the cam. Note: Analyze the cam contour in every 100 of cam rotation. 2. A disk cam of a roller follower having a minimum diameter of 25mm and a roller diameter of 10mm follows the following motion curves during its operation: Cam Angle of Rotation 0-60 61-120 121-180 181-240 241-300 301-360 Motion Curve Dwell Accelerating Cycloidal motion Decelerating Harmonic Motion Accelerating Harmonic Motion Decelerating Cycloidal motion Dwell If the total follower displacement is 45mm, draw the cam contour and generate its 3D model. Upon constructing the cam contour, determine the maximum pressure angle between the cam and the follower. Use a bore of 17.50mm, hub diameter of 20mm, and a 2.25mm x 2.25mm key with cam and hub thickness of 15mm 14 and 5mm respectively. Note: Analyze the cam contour in every 100 of cam rotation. Each half of follower displacement is divided equally into types of motion curves given. 15