DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.

Transcript

1 DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec Choose suitable programs for rise and fall to mimize accelerations Plot the SVAJ diagrams Given: RISE DWELL FALL DWELL h 1 2 h 2 h 2 h 4 Cycle time: τ 4 sec Solution: See Mathcad file P7 1 The camshaft turns 2π rad durg the time for one cycle Thus, its speed is ω 2 rad τ ω = 1571 rad sec 2 From Table 9-2, the motion program with lowest acceleration that does not have fite jerk is the modified trapezoidal The modified trapezoidal motion is defed local coordates by equations 1 The numerical constants these equations are first defed below constants C 1 through C 11 c c c 4124 c c c c c c c c The equations for the rise or fall terval () are divided to 5 subtervals These are: for <= <= / where, for these equations, is a local coordate that ranges from to, s 1 (,, h) h c 1 c 2 s 4 v 1 (,, h) c h 1 1 cos 4 for / <= <= / a 1 (,, h) c h s 4 j 1 (,, h) c h 4 cos s 2 (,, h) h c 5 c 6 c 7 v 2 (,, h) h c c 6 a 2 (,, h) c h j 2 (,, h) for / <= <= 5/ s (,, h) h c c 2 s 4 π c 9 2 v (,, h) h c c 1 cos 4 π

2 DESIGN OF MACHINERY SOLUTION MANUAL -7-2 a (,, h) c h s 4 π j (,, h) c h 4 cos 4 π 2 for 5/ <= <= 7/ 2 s 4 (,, h) h c 5 c 1 c 11 v 4 (,, h) h c c 1 a 4 (,, h) c h j 4 (,, h) 2 for 7/ <= <= s 5 (,, h) h c 1 c 2 s 4 π c 1 v 5 (,, h) c h 1 1 cos 4 π a 5 (,, h) c h s 4 π j 5 (,, h) c h 4 cos 4 π 2 4 The above equations can be used for a rise or fall by sertg the proper values of,, and h To plot the SVAJ curves, first defe a range function that has a value of one between the values of a and b and zero elsewhere R(, a, b) if( ( > a) ( b), 1, ) 5 The global SVAJ equations are composed of four tervals (rise, dwell, fall, and dwell) The local equations above must be assembled to a sgle equation each for S, V, A, and J that applies over the range <= <= 6 6 Write the global SVAJ equations for the first terval, <= <= 1 For this terval, the local and global frames are cocident so the local equations can be used as written, substitutg only for h 1 for h and 1 for Note that each subterval function is multiplied by the range function so that it will have nonzero values only over its subterval For <= <= 1 S 1 ( ) R, 1, 1 s 1, 1, h 1 R,, 1 s 2, 1, h 1 R, 5 1 s, 1, h 1 R, 7 1 s 4, 1, h 1 R, 7 1, 1 s 5, 1, h 1 V 1 ( ) R, 1, 1 v 1, 1, h 1 R,, 1 v 2, 1, h 1 R, 5 1 v, 1, h 1 R, 7 1 v 4, 1, h 1 R, 7 1, 1 v 5, 1, h 1

3 DESIGN OF MACHINERY SOLUTION MANUAL -7- A 1 ( ) R, 1, 1 a 1, 1, h 1 R,, 1 a 2, 1, h 1 R, 5 1 a, 1, h 1 R, 7 1 a 4, 1, h 1 R, 7 1, 1 a 5, 1, h 1 J 1 ( ) R, 1, 1 j 1, 1, h 1 R,, 1 j 2, 1, h 1 R, 5 1 j, 1, h 1 R, 7 1 j 4, 1, h 1 R, 7 1, 1 j 5, 1, h 1 7 Write the global SVAJ equations for the second terval, 1 <= <= 1 2 For this terval, the value of S is the value of S at the end of the previous terval and the values of V, A, and J are zero because of the dwell For 1 <= <= 1 2 S 2 ( ) R, 1, 1 2 S 1 1 V 2 ( ) R, 1, 1 2 A 2 ( ) R, 1, 1 2 J 2 ( ) R, 1, 1 2 Write the global SVAJ equations for the third terval, 1 2 <= <= 1 2 For this terval, the local and global frames are not cocident so we must transform the local to the global by subtractg the first two tervals from it Note that, for a fall, the local equation for s is subtracted from the value of S at the end of the previous terval while V, A, and J are simply negated For 1 2 <= <= 1 2 Let a 1 2 S ( ) R, a, a s 1 a,, h R, a, a s 2 a,, h R, a 5 s a,, h R, a 7 s 4 a,, h R, a 7, a s 5 a,, h R, a, a S 2 ( a)

4 DESIGN OF MACHINERY SOLUTION MANUAL -7-4 V ( ) R, a, a v 1 a,, h R, a, a v 2 a,, h R, a 5 v a,, h R, a 7 v 4 a,, h R, a 7, a v 5 a,, h A ( ) R, a, a a 1 a,, h R, a, a a 2 a,, h R, a 5 a a,, h R, a 7 a 4 a,, h R, a 7, a a 5 a,, h J ( ) R, a, a j 1 a,, h R, a, a j 2 a,, h R, a 5 j a,, h R, a 7 j 4 a,, h R, a 7, a j 5 a,, h 9 Write the global SVAJ equations for the fourth terval, 1 2 <= <= For this terval, the values of S, V, A, and J are zero because of the dwell For 1 2 <= <= Let a 1 2 S 4 ( ) R, a, a 4 V 4 ( ) R, a, a 4 A 4 ( ) R, a, a 4 J 4 ( ) R, a, a 4 1 Write the complete global equation for the displacement and plot it over one rotation of the cam, which is the sum of the four tervals defed above S( ) S 1 ( ) S 2 ( ) S ( ) S 4 ( ), 1 6

5 DESIGN OF MACHINERY SOLUTION MANUAL -7-5 DISPLACEMENT, S Displacement, S( ) Cam Rotation Angle, 11 Write the complete global equation for the velocity and plot it over one rotation of the cam, which is the sum of the four tervals defed above V( ) V 1 ( ) V 2 ( ) V ( ) V 4 ( ) 5 VELOCITY, V Velocity, V( ) Cam Rotation Angle, 12 Write the complete global equation for the acceleration and plot it over one rotation of the cam, which is the sum of the four tervals defed above A( ) A 1 ( ) A 2 ( ) A ( ) A 4 ( )

6 DESIGN OF MACHINERY SOLUTION MANUAL ACCELERATION, A 25 Acceleration, A( ) Cam Rotation Angle, 1 Write the complete global equation for the jerk and plot it over one rotation of the cam, which is the sum of the four tervals defed above J( ) J 1 ( ) J 2 ( ) J ( ) J 4 ( ) 15 JERK, J 1 Jerk, J( ) Cam Rotation Angle,