Homework No. 6 (40 points). Due on Blackboard before 8:00 am on Friday, October 13th.

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1 ME 35 - Machine Design I Fall Semester 017 Name of Student: Lab Section Number: Homework No. 6 (40 points). Due on Blackboard before 8:00 am on Friday, October 13th. The important notes for this homework assignment are as printed on Homework 1. Part I. (0 points). Draw the lift curve the cam profile of Problem 6.8 on page 365. Templates to draw the lift curve the cam profile are provided on pages 3 of this homework. Part II. (0 points). At the cam angle 5 o, determine numerical values for: (i) The first, second, third-order kinematic coefficients of the displacement diagram. (ii) The radius of the curvature of the cam surface. (iii) The unit tangent normal vectors to the cam at the point of contact with the follower. (iv) The coordinates of the point of contact between the cam the follower. Express your answers in the moving Cartesian coordinate reference frame attached to the cam. (v) The pressure angle of the cam. Is your answer acceptable for this cam-follower system? - 1 -

2 Solution to Homework Set 6. Part I. 0 Points. (i) 8 Points. The displacement diagram (or lift curve) is plotted in Figure 1. Figure 1. The Displacement Diagram (or Lift Curve). - -

3 (ii) 8 Points. The cam profile is plotted in Figure. Figure. The Cam Profile

4 The lift curve can be drawn by following the procedure outlined on Figures , see page 303. The cam profile can be drawn by following the procedure outlined on Figures 6.8, 6.9, 6.10, 6.11, see pages 304, 305, 306, 307. (iii) 4 Points. The pressure angle of the cam was discussed in lecture can be written from Equation (6.3), see page 333 in the text book, as 1 tan y RO y The plot of the pressure angle against the cam position is shown in Figure Pressure Angle - [deg] Cam - Cam Angle - [deg] Figure 3. The pressure angle against the cam position. The radius of curvature of the pitch curve the radius of curvature of the cam profile were discussed in lecture. The radius of curvature of the pitch curve can be written in terms of the first-order the second-order kinematic coefficients of the follower center as PC 3 R x y x y Also, see Equation (6.37), see page 344 in the text book. A plot of the radius of curvature of the cam profile against the cam position is shown in Figure

5 10 8 Cam - Cam Radius of Curvature - [in] Cam - Cam Angle - [deg] Figure 4. A plot of the radius of curvature of the cam profile against the cam position. Comments on the curves, see Figures 1,, 3, 4, the answers to the design issues: (i) Position discontinuities never occur. Discontinuities in the derivatives will only occur at transitions between dwell segments lifting/returning segments of motion. Discontinuities in the derivatives are undesirable especially for a high speed cam follower system. There is an acceleration discontinuity at the beginning end of the simple harmonic motions, both rise return. There is a jerk discontinuity at the beginning end of the cycloidal motions, both rise return. (See the notes on page 34.) (ii) The recommended value of the pressure angle for a cam-follower system, see page 33 in the text book, is less that 30 or 35. The pressure angle defines the steepness of the cam profile is a measure of the efficiency of the cam. In this design, the pressure angle is more than the accepted value at the cam angles 1664, , 1656, Therefore, from this point of view, this cam profile is not a good cam design. The high values of the pressure angle may be due to the selection of the segments for the displacement diagram, /or the dimensions of the cam the diameter of the follower. (iii) The radius of curvature of a cam profile should always be negative for a good cam design. A positive radius of curvature means that the cam has a concave surface there is the possibility that the follower may lose contact with the cam. Note that if the radius of curvature of the cam profile is positive then the designer must ensure that the radius of curvature of the cam is greater than the radius of the follower. In the proposed design, the positive values of the radius of curvature of the cam are always greater than 0.5 (i.e., the radius of the follower)

6 The radius of curvature of the cam is positive for the following ranges of the cam angle: 35, 17019, 085, The radius of curvature of the cam is positive, smaller than the radius of the roller follower, for the following ranges of the cam angle: 913, , 1516 Note that the radius of curvature of the cam is zero between the cam angles degrees meaning that pointing has occurred. Also, it could imply that undercutting has occurred. Also, with the exception of where the radius of curvature of the cam goes to zero, there is an inflection point at the boundary of each range of angles for which the radius of curvature is positive (goes to infinity). (iv) Possible design changes to the cam-follower system include: (a) Increasing the radius of the prime circle (with the same lift curve) in general would reduce the pressure angle. (b) Change the profiles to match acceleration at the transition (blend or meshing) points to eliminate acceleration discontinuities. (c) Change the two SHM profiles to cycloidal would make accelerations continuous but would also increase accelerations (, therefore, the values of the pressure angle) in the middle parts of the rise the return profiles. (d) Could try to increase the diameter of the roller follower if the contact stresses are too large. (e) Change the eccentricity (or the offset). The effects may not be obvious from observation, or intuition, so a spreadsheet may be required to explore this problem numerically. Part II (0 Points). The base circle diameter of the cam is D = 3 inches the diameter of the reciprocating roller follower is d = 1 inch. Therefore, the radius of the prime circle of the cam is R 0 D d 3 1 inches (1) Since the problem states that there is a radial follower then the follower offset (or eccentricity) is 0. Therefore, the coordinates of the follower center on the prime circle (that is, when the roller follower is touching the base circle of the cam) in the fixed reference frame are X 0 0 (a) Y R (b) inches The displacement of the follower center for full-rise simple harmonic motion, see Equation (6.1a), page 314 in the text book, can be written as L y ybase 1cos * (3) where y base is the base lift (that is, the lift at the beginning of the full-rise simple harmonic motion). Therefore, from Table P6.8, see page 365, in the problem statement, the base lift is y 0 (4) base

7 Also, the total lift during the full-rise simple harmonic motion is The range of the full-rise simple harmonic motion is L inches (5) o o o rad (6a) * (6b) 0 where 0 is the starting cam angle for the full-rise simple harmonic motion. The cam angle is 5 o, the starting cam angle for the full-rise simple harmonic motion is 0 10 o, therefore * o o o rad (6c) * rad (7) Substituting Equations (4), (5), (7) into Equation (3), the displacement of the follower center is or in y 0in 1cos 4 (8a) y 1in ( ) 0.99 in (8b) o This result implies that the follower center has risen from 0 in to 0.99 in after 15 of the full-rise * o simple harmonic motion (i.e., when the angle 15 or the cam angle θ 5 ). o * o For example, note that after 60 of full-rise simple harmonic motion (i.e., when the angle 60 or the cam angle θ 70) then the displacement of the follower center, from Equation (8a), is in y 0in 1cos in (8c) (i) 3 Points. The first, second, third-order kinematic coefficients of the displacement diagram (or the lift curve), see Equations (6.1b), (6.1c), (6.1d), page 314 in the text book, can be written as * L y sin (9a) y * L cos (9b) 3 * L y sin 3 (9c) - 7 -

8 Substituting Equations (4), (5), (6), (7) into Equations (9), the first-order, second-order, thirdorder kinematic coefficients of the displacement diagram (or the lift curve) are y 3 sin.11in/rad (10a) 4 y y 9 cos in/rad 4 (10b) 7 sin in/rad 4 3 (10c) Partial Check: Note that the slope of the displacement diagram (i.e., the first-order kinematic coefficient or velocity), see Equation (10a), is positive. From intuition (for a simple harmonic curve with rise) this is known to be correct, see the template in Figure 1. (ii) 8 Points. At the cam angle 5 o, the coordinates of the follower center (in the fixed Cartesian reference frame) are X X0 0 (11a) Y Y0 y in 0.93 in.93 in (11b) The coordinates of the follower center in the moving reference frame can be written as x X cos Y sin (1a) y X sin Y cos (1b) Substituting Equations (11a) (11b), the cam angle 5 o, into Equations (1), the coordinates of the follower center in the moving reference frame are x 0 cos 5.93 sin in (13a) y 0 sin 5.93 cos in (13b) Differentiating Equations (1) with respect to the cam angle, the first-order kinematic coefficients of the follower center in the moving reference frame are x X sin Y cos ysin (14a) y X cos Y sin ycos (14b) Therefore, the first-order kinematic coefficients of the follower center in the moving reference frame are x 0 sin 5.93 cos 5 (.11) sin in/rad (15a) y 0 cos 5.93 sin 5 (.11) cos in/rad (15b) Differentiating Equations (14) with respect to the cam angle, the second-order kinematic coefficients of the follower center in the moving reference frame are - 8 -

9 x X cos Y sin y cos y sin (16a) y X sin Y cos y sin y cos (16b) Therefore, the second-order kinematic coefficients of the follower center in the moving reference frame are x 0 cos 5.93 sin 5.11 cos 5 ( 6.364)sin in/rad (17a) y 0 sin 5.93cos 5.11 sin 5 ( 6.364) cos in/rad (17b) The coordinates of the point of contact between the cam the follower can be written as where x y cam cam d y x (18a) R d x y (18b) R R x y (19a) The positive sign is used here because the input cam angle can always be chosen as positive, that is, counterclockwise. Substituting Equations (15a) (15b) into Equation (19a) gives R in (19b) CHECK: Squaring adding Equations (14) substituting into Equation (19a) gives the symbolic equation that was presented in lecture, that is R ( yx ) Y (19c) Substituting Equations (10a), (11a) (11b) into Equation (19c) gives R in (19d) Substituting Equation (19b) the known values into Equations (18a) (18b), the coordinates of the point of contact between the cam the follower are x cam y cam in in 3.13 (0a) (0b) - 9 -

10 (iii) 3 Points. The radius of curvature of the pitch curve can be written in terms of the first-order the second-order kinematic coefficients of the follower center, as PC 3 R x y x y (1a) Substituting the known numerical values into this equation, the radius of curvature of the pitch curve is PC in (1b) The positive sign implies that the center of curvature is along the positive unit normal direction. This means that the cam profile is concave may not be a good cam profile. Check: Substituting Equations (14), (16), (19c) into Equation (1a), the radius of curvature of the pitch curve can be written in symbolic form as PC 3/ [( y X ) Y ] Y ( y Y ) ( y X )( y X ) (a) Then substituting the known numerical values into Equation (a), the radius of curvature of the pitch curve is 3/ [(.11 0).93 ] PC (b).93 ( ) (.11 0) [(.11) 0] Therefore, the radius of curvature of the pitch curve is PC in (c) Note that Equation (c) is in good agreement with Equation (1b). There is a rounding error on the order of one-half of one percent of the full value. Since the radius of curvature of the pitch curve is a positive value then the radius of curvature of the of the cam profile can be written as d cam PC (3a) Substituting Equation (1b) the diameter of the roller follower into this equation, the radius of curvature of the cam profile is 1 cam in (3b) Note that the radius of curvature of the cam profile is a larger positive value than the radius of curvature of the pitch curve (which agrees with fact that the cam profile is concave). (iv) 4 Points. The unit tangent vector to the pitch curve can be written as x ˆ y u ˆ t i j R R (4a) Substituting the known values into Equation (4a), the unit tangent vector to the pitch curve is

11 that is 3.11 ˆ 0.1 u ˆ t i j (4b) u iˆ ˆj (4c) t The unit normal vector to the pitch curve can be written as y ˆ x u ˆ N i j R R (5a) Substituting the known values into this equation, the unit normal vector to the pitch curve is that is 0.1 ˆ 3.11 u ˆ N i j (v) Points. The pressure angle of the cam can be written as (5b) u iˆ ˆj (5c) N cos x y cos sin R R (6a) Substituting Equations (15) (19b), the specified cam angle θ 5, into Equation (6a) gives cos cos sin (6b) Therefore, the pressure angle of the cam (for the given cam angle) is (7) Check: The pressure angle of the cam (for the given cam angle) can also be written from Equation (6.3), see page 333 in the text book, as 1 tan y R O y (8a) Substituting Equations (15) (19b) into Equation (8a) gives tan 1.11in 0 in 0 in 0.93 in (8b) Therefore, the pressure angle of the cam can be written as.11 in.93 in 1 1 tan tan 0.95 (8c) The pressure angle of the cam is

12 o (9) Note that Equation (9) is in very good agreement with Equation (7). The value of the pressure angle given by Equations. (7) or (9) is not an acceptable value for a cam-follower system. Recall that a good cam design should have a pressure angle in the range 0 30, see page 33 in the text book. The conclusion is that this cam design is not satisfactory for machine design applications (at least not for this cam angle of 5 o )