I.8 Packages for Kinematic Chains 0Contents8 Packages for Kinematic Chains 18.1 Driver Link . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Position Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 58.3 Velocity and Acceleration Analysis . . . . . . . . . . . . . . . 158.4 Force Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48I.8 Packages for Kinematic Chains 18 Packages for Kinematic Chains8.1 Driver LinkPackages can be used a to calculate the position, velocity and accelerationof a driver link in rotational motion (Fig. 8.1.(a)). For the p osition analysisthe input data are the coordinates (xA, yA) of the start joint A with respectto the reference frame xOyz, the length of the link AB, and the angle φ withthe horizontal. For the velocity and the acceleration analysis, the angularvelocity ω =˙φ, and the angular acceleration α =¨φ are considered. Theoutput data are the position, velocity and acceleration of the end point B.The position equations for the driver link arexB= xA+ AB cos φ,yB= yA+ AB sin φ, (8.1)where xBand yBare the coordinates of the point B.The velocity equations for the driver link arevBx= −ABω sin φ,vBy= ABω cos φ, (8.2)where vBxand vByare the velocity components of the point B on the x andy axes.The acceleration equations for the driver link areaBx= −ABω2cos φ − ABα sin φ,aBy= −ABω2sin φ + ABα cos φ, (8.3)where aBxand aByare the acceleration components of the point B on the xand y axes.In order to compute the position, velocity and acceleration of the jointB, using MathematicaT M, the necessary commands can be can collected ina function. The name of the function is Driver.Driver[xA ,yA ,AB ,phi ,omega ,alpha ]:=Block[{ xB, yB, vBx, vBy, aBx, aBy },xB = xA + AB Cos[phi] ;yB = yA + AB Sin[phi] ;I.8 Packages for Kinematic Chains 2vBx = - AB omega Sin[phi] ;vBy = AB omega Cos[phi] ;aBx = - AB omegaˆ2 Cos[phi] - AB alpha Sin[phi] ;aBy = - AB omegaˆ2 Sin[phi] + AB alpha Cos[phi] ;Return[{ xB, yB, vBx, vBy, aBx, aBy } ] ;] ;The input data, the variable parts of the computation, and the outputdata are defined as parameters to this function.All the variables local to Driver[] are declared in the Block[] state-ment to isolate them from any values they might have globally. The Math-ematicaT Mcommand Block[{x, y, ...}, expr] specifies that expris to be evaluated with local values for the symbols x, y, ... . In ourcase, the local variables are xB, yB, vBx, vBy, aBx, aBy, and expris the body of the function.The MathematicaT Mcommand Return[expr] returns the value exprfrom a function. For the driver, expr represents the output data and it is avector which contains the elements xB, yB, vBx, vBy, aBx, and aBy.The mechanism that MathematicaT Mprovides for keeping the variablesused in a package different from those used in the main session is calledcontext. As each symbol is read from the terminal or from a file, Mathemat-icaT Mchecks to see whether this symbol has already been used be fore. If ithas been encountered before, the new instance is made to refer to that pre-viously read symbol. If the symbol has not been encountered be fore, a newentry in the symb ol table is created. Each symbol belongs to a certain con-text. Within one context the names of the symbols are unique, but the samename can occur in two different contexts. For the driver the proper context isDriver::usage = "Driver[xA,yA,AB,phi,omega,alpha]computes the driver link position,velocity and acceleration vectors."Begin["Private‘"]Driver[xA ,yA ,AB ,phi ,omega ,alpha ]:=Block[ { xB, yB, vBx, vBy, aBx, aBy },xB = xA + AB Cos[phi] ;yB = yA + AB Sin[phi] ;vBx = - AB omega Sin[phi] ;vBy = AB omega Cos[phi] ;I.8 Packages for Kinematic Chains 3aBx = - AB omegaˆ2 Cos[phi] - AB alpha Sin[phi] ;aBy = - AB omegaˆ2 Sin[phi] + AB alpha Cos[phi] ;Return[ { xB, yB, vBx, vBy, aBx, aBy } ] ;]End[ ]The local variables xB, yB, vBx, vBy, aBx, and aBy are now cre-ated in the context Private‘ which is not searched when one types a vari-able name later on.The usage message defined for the symbol Driver is there to providedocumentation for the function and to make sure that Driver is defined inthe current context. If it had not been defined before entering the contextPrivate‘, it would not be found later on.The MathematicaT Mcommand End[] returns the present context andreverts to the previous one.The functions that the package provides are put into a separate contextwhich must be visible to be able to use the functions later on. This can bedone using the pair of MathematicaT Mcommands BeginPackage[] andEndPackage[]. Thus, the following MathematicaT Mpackage is introducedBeginPackage["Driver‘"]Driver::usage = "Driver[xA ,yA ,AB ,phi ,omega ,alpha ]computes the driver position, velocity and accelerationvectors."Begin["‘Private‘"]Driver[xA ,yA ,AB ,phi ,omega ,alpha ]:=Block[ { xB, yB, vBx, vBy, aBx, aBy },xB = xA + AB Cos[phi] ;yB = yA + AB Sin[phi] ;vBx = - AB omega Sin[phi] ;vBy = AB omega Cos[phi] ;aBx = - AB omegaˆ2 Cos[phi] - AB alpha Sin[phi] ;aBy = - AB omegaˆ2 Sin[phi] + AB alpha Cos[phi] ;Return[ { xB, yB, vBx, vBy, aBx, aBy } ] ;]End[ ]EndPackage[ ]I.8 Packages for Kinematic Chains 4The command BeginPackage["Driver‘"] sets Driver‘ to be thecurrent context, and the command EndPackage[] ends the package, prepe nd-ing Driver‘ to the context search path.Note the initial backquote in the context name inside the commandBegin["‘Private‘"]. This establishes ‘Private‘ as a subcontext ofthe context Driver‘ (so its full name is Driver‘Private‘).The name of the source file for the MathematicaT Mpackage Driver isDriver.m (see Program 8.1).ExampleA driver link is shown in Fig. 8.1(b). The input data are AB = 0.20m, the angle between the driver link AB and the horizontal axis, φ = 30◦,and the angular velocity, ω = 5 rad/s. Calculate the position, velocity andacceleration components of the joint B. The cartesian reference frame xOyzis chosen with A ≡ O.The MathematicaT Mpackage Driver is loaded in the main Mathemat-icaT Msession using the command<<Driver.m ;To compute the numerical values of the position, velocity and accelera-tion components for the joint B, the MathematicaT Mfunction Driver isused.Position analysisSince the joint A is the origin of the reference frame xAyz, the coordinatesof the joint A arexA= yA= 0.The coordinates of the joint B arexB=