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where [N(f2)](I) = -i[N(f2)]/fi2 is the ith order partial derivative With respect to (f2, [M(s2)])(j) = j[M(s2)], sj2 is the jth order partial derivative, and with respect to s2, w 2 = df2/dt is the angular velocity of the cam, and v2= ds2/dt is the linear velocity of the cam Note that the elements in the matrices above for the derivatives of B-splines in the rotating and translating directions are found in Eqs (543) to (546) Example Application The following numerical examples demonstrate the feasibility of synthesizing the follower motions In these two cases, a cam with a translating spherical follower is considered The dimensional parameters used for the cam-follower mechanism in both examples are: a = 22 mm l = 50 mm r = 5 mm

3 6 10 15

.

EXAMPLE 9: Synthesis of a Three-Dimensional Surface A rectangular grid data set of 7 7 displacement constraints for the follower is assumed as in Table 58 To guarantee continuity of the fourth derivative of the motion function (ie, to ensure that derivative of jerk is continuous), the orders of the B-spline surface are chosen to be six in both parametric directions of rotation and translation A uniform knot sequence in the f2 direction is adopted as

t/100

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As the output shows, both threads compute the proper sum of 15 Let s examine this program in detail The program creates three classes The first is SumArray It defines the method SumIt( ), which sums an integer array The second class is MyThread, which uses a static object called sa that is of type SumArray Thus, only one object of SumArray is shared by all objects of type MyThread This object is used to obtain the sum of an integer array Notice that SumArray stores the running total in a field called sum Thus, if two threads use SumIt( ) concurrently, both will be attempting to use sum to hold the running total Because this will cause errors, access to SumIt( ) must be synchronized Finally, the class Sync creates two threads and has them compute the sum of an integer array Inside SumIt( ), the lock statement prevents simultaneous use of the method by different threads Notice that lock uses lockOn as the object being synchronized This is a private object that is used solely for synchronization Sleep( ) is called to purposely allow a task-switch to occur, if one can but it can t in this case Because the code within SumIt( ) is locked, it can be used by only one thread at a time Thus, when the second child thread begins execution, it does not enter SumIt( ) until after the first child thread is done with it This ensures the correct result is produced To understand the effects of lock fully, try removing it from the body of SumIt( ) After doing this, SumIt( ) is no longer synchronized, and any number of threads can use it concurrently on the same object The problem with this is that the running total is stored in sum, which will be changed by each thread that calls SumIt( ) Thus, when two threads call SumIt( ) at the same time on the same object, incorrect results are produced because sum reflects the summation of both threads, mixed together For example, here is sample output from the program after lock has been removed from SumIt( ):.

Child #1 starting Running total for Child Child #2 starting Running total for Child Running total for Child Running total for Child Running total for Child Running total for Child #1 is 1 #2 #1 #2 #1 #2 is is is is is 1 3 5 8 11

[0, 2p 7, 4p 7, 6p 7, 8p 7, 10p 7, 12p 7, 2p ]

23:

Running total for Child Running total for Child Running total for Child Running total for Child Sum for Child #1 is 29 Child #1 terminating Sum for Child #2 is 29 Child #2 terminating

t/100

[0, 0, 0, 0, 0, 0, 25, 50, 50, 50, 50, 50, 50]

#1 #2 #1 #2

15 19 24 29

As the output shows, both child threads are using SumIt( ) at the same time on the same object, and the value of sum is corrupted The effects of lock are summarized here: For any given object, once a lock has been acquired, the object is locked and no other thread can acquire the lock Other threads trying to acquire the lock on the same object will enter a wait state until the code is unlocked When a thread leaves the locked block, the object is unlocked

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