(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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These files must be placed in the same directory as the \ rest of the Mechanical Systems Pack files. 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Each generalized coordinate constraint is named like a standard \ constraint, takes approximately the same arguments as a standard constraint, \ returns a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["SysCon", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" constraint object that is passed to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["SetConstraints", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ", and generally appears in a mechanism model in exactly the same way as \ the standard ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " constraints. The difference between them is best illustrated by an \ example.\n\nConsider a standard ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech2D", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" model containing a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Translate2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " constraint that models a translational joint between body 2 and the \ ground. 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If the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Translate2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" constraint is replaced with a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["GenTranslate2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" constraint, ", Evaluatable->False, AspectRatioFixed->True], StyleBox["SetConstraints", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " does only one thing: one degree of freedom is added to the model by \ adding a single symbol to the dependent variable list. The name of the symbol \ is specified in the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["GenTranslate2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " constraint object.\n\nBody 2, which was represented by three symbols, is \ now represented by only one. All of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ "'s other constraint, load, output, and graphics functions can recognize \ and access body 2 just as if it were a standard body; the only difference is \ that the symbolic representation of all geometry on the body is a function of \ just one symbol, not three.\n\nNote that generalized coordinate constraints \ cannot be used within the multistage constraint objects, ", Evaluatable->False, AspectRatioFixed->True], StyleBox["TimeSwitch", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" or ", Evaluatable->False, AspectRatioFixed->True], StyleBox["StageSwitch", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ". This is simply because including them in a multistage constraint would \ have no effect on the model; the generalized coordinate constraints contain \ no constraint equations, so there is nothing to switch on and off.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["2D Constraints"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "There are only two 2D generalized coordinate constraints. The wide variety \ of standard 2D constraints cannot be meaningfully converted to generalized \ constraints; only constraints which attach exactly two bodies together such \ that only one degree of freedom remains between them are generally useful.\n\n\ The arguments given to the generalized coordinate constraints are of the same \ form as the arguments given to the standard constraints. However, each of \ these constraints requires an additional argument to specify the name of, and \ an initial guess for, the new symbol that represents the added degree of \ freedom."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[{ StyleBox["GenRevolute2[", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox["cnum, axis1, axis2, ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["{", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox["sym, guess", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["}]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " returns a constraint object that introduces a slave body into the model \ represented by a single generalized coordinate ", Evaluatable->False, AspectRatioFixed->True], StyleBox["sym", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[". 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Thus, the slave body cannot be the ground body. The \ master body may be the ground body, another body represented by standard \ coordinates, or another body represented by generalized coordinates that was \ created with another generalized constraint. The master-slave relationship \ cannot be circular; the master-slave chain must originate from the ground, or \ from a standard body. Also, a slave body cannot be a slave to more than one \ master body.\n\nAnother restriction on the arguments to generalized \ coordinate constraints is that the axis objects, ", Evaluatable->False, AspectRatioFixed->True], StyleBox["axis1", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["axis2", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ ", must each be defined entirely on a single body. 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Constraint 2, the revolute joint at the drive bar \ origin, is replaced first, and body 2 becomes a slave to the ground body, \ represented by the single coordinate ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Here is the new constraint."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "cs[2] = GenRevolute2[2, Axis[drivebar, 0, X],\n \ Axis[ground, 0, X], {alpha, 0.0} ];\nSetConstraints[ Array[cs, 5] ] "], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[{ StyleBox["The modified model is run again at ", Evaluatable->False, AspectRatioFixed->True], StyleBox["T", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" = 0.3.", Evaluatable->False, AspectRatioFixed->True] }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["SolveMech[0.3]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {T -> 0.3, X3 -> -2., Y3 -> 3., Th3 -> 0.2312912431860266, X4 -> -0.3090169943749474, Y4 -> 0.951056516295154, Th4 -> -0.4221363592795691, alpha -> 1.884955592153876}\ \>", "\<\ {T -> 0.3, X3 -> -2., Y3 -> 3., Th3 -> 0.231291, X4 -> -0.309017, Y4 -> 0.951057, Th4 -> -0.422136, alpha -> 1.88496}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[TextData[{ StyleBox["Note that the symbols ", Evaluatable->False, AspectRatioFixed->True], StyleBox["X2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Y2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Th2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" are no longer included in the solution rules returned by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["SolveMech", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". The location of body 2 is represented entirely by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", which has the same numerical value as ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Th2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" used to have. The numerical value of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" would be different from ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Th2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" if the direction vectors of the two axes in the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["GenRevolute2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" constraint were not identical (", Evaluatable->False, AspectRatioFixed->True], StyleBox["X", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["X", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[").\n\nTwo more of the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Revolute2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " constraints can also be replaced, but not the fourth one. This is because \ each generalized coordinate constraint creates a slave body, and a slave body \ can be a slave to only one master body. Thus, there is no way to use four \ generalized coordinate constraints in a model with only three bodies. Any \ three of the four revolute constraints can be replaced, but not all of them.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Here are two more new constraints."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "cs[3] = GenRevolute2[3, Axis[centerbar, 0, X], \n \ Axis[drivebar, 1, X], {beta, 0.0} ];\ncs[4] = GenRevolute2[4, \ Axis[drivenbar, 1, X],\n Axis[centerbar, 1, X], \ {gamma, 0.0} ];\nSetConstraints[ Array[cs, 5] ] "], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[{ StyleBox["The modified model is run again at ", Evaluatable->False, AspectRatioFixed->True], StyleBox["T", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" = 0.3.", Evaluatable->False, AspectRatioFixed->True] }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["SolveMech[0.3]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {T -> 0.3, alpha -> 1.884955592153876, beta -> -2.307091951433445, gamma -> 0.6534276024655956}\ \>", "\<\ {T -> 0.3, alpha -> 1.88496, beta -> -2.30709, gamma -> 0.653428}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[TextData[{ StyleBox[ "Now all of the Cartesian coordinates have been replaced with the \ generalized coordinates ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["beta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["gamma", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ". A quick degree of freedom count shows that this all makes sense: the \ three generalized coordinate constraints each add one degree of freedom to \ the model, while the two standard constraints ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Revolute2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["RotationLock1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " constrain a total of three degrees of freedom.\n\nThe physical meaning of \ the new coordinates is quite clear: ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" is the angle between the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["X", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" axis of the ground and the local ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" axis of the drive bar, ", Evaluatable->False, AspectRatioFixed->True], StyleBox["beta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" is the angle between the local ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" axes of the drive and center bars, and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["gamma", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" is the angle between the local ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" axes of the center and driven bars.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Output and Graphics Functions"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Now that we have a solution in terms of the generalized coordinates, all \ of the standard ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " output and graphics functions allow it to be used in the same manner as a \ solution in Cartesian coordinates. 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This is because of the master-slave relationship of the generalized \ coordinate bodies; the center bar is a slave to the drive bar, which is a \ slave to the ground. Thus, the location of any point on the driven bar is a \ function of all three generalized coordinates, and the location of any point \ on the drive bar is a function of only ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".\n\n", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " graphics functions also operate without regard to the use of generalized \ coordinates. 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Note the use of the \ zeroth point to avoid having to define {0, 0, 0}."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The ground (body 1) has one point defined in its body object.\nP1 is a point \ to locate the rotational axis of the crank."], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The crank (body 2) has one point defined in its body object. \nP1 is the \ attachment point of the connecting rod. The center of the crank is at the \ local origin."], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The connecting rod (body 3) has one point defined in its body object.\nP1 is \ the attachment point of the connecting rod to the slider, while the \ connecting rod is attached to the crank at P0."], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The slider (body 4) has no points in its body object, only an initial guess, \ because only the local origin P0 is used."], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Here are the body objects for the space-crank model."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "bd[1] = Body[ground = 1,\n PointList->{{ 0, 10, 12}}];\n\nbd[2] = \ Body[crank = 2,\n PointList->{{0, 8, 0}},\n InitialGuess->{{0, 10, 12}, {1, \ 0, 0, 0}} ];\n\nbd[3] = Body[conrod = 3,\n PointList->{{30, 0, 0}},\n \ InitialGuess->{{0, 18, 12}, {1, 0, 0, 0}} ];\n\nbd[4] = Body[slider = 4,\n \ InitialGuess->{{20, 0, 0}, {1, 0, 0, 0}} ];"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[ "The body properties are incorporated into the model."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "SetBodies[bd[ground],\n bd[crank ],\n bd[conrod],\n \ bd[slider]]"], "Input", AspectRatioFixed->True]}, Open]], Cell[TextData["Constraints"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Five constraints, one of which is a driving constraint, are used to model \ the space-crank mechanism. These constraints are simply standard ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " constraints which will be replaced with generalized coordinate \ constraints shortly.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["A ", Evaluatable->False, AspectRatioFixed->True], StyleBox["ProjectedAngle1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " constraint specifies the angular relationship between a line on the \ ground body and a line on the crank. This driving constraint is used to \ rotate the crank.", Evaluatable->False, AspectRatioFixed->True] }], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["A ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Revolute5", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " constraint forces the axis of the crank to be coincident with its pivot \ axis on the ground.", Evaluatable->False, AspectRatioFixed->True] }], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["A ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Spherical3", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " constraint between the crank and the connecting rod models a ball joint.", Evaluatable->False, AspectRatioFixed->True] }], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["An ", Evaluatable->False, AspectRatioFixed->True], StyleBox["OrthoRevolute4", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " constraint between the slider and the connecting rod models a universal \ joint. This is used instead of a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Spherical3", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " constraint so as to constrain the spin of the connecting rod about its \ own longitudinal axis.", Evaluatable->False, AspectRatioFixed->True] }], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["A ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Translate5", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " constraint between the slider and the ground allows translation of the \ slider.", Evaluatable->False, AspectRatioFixed->True] }], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Here are the constraint objects for the space-crank."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "cs[1] = ProjectedAngle1[1, Vector[crank, Y], Vector[ground, Y],\n \ Vector[ground, X], 2 N[Pi] T];\n\ncs[2] = Revolute5[2, \ Axis[crank, 0, X], Axis[ground, 1, X]];\n\ncs[3] = Spherical3[3, Point[crank, \ 1], Point[conrod, 0]];\n\ncs[4] = OrthoRevolute4[4, Axis[conrod, 1, Y], \ Axis[slider, 0, Z]];\n\ncs[5] = Translate5[5, Axis[slider, 0, X, Y], \ Axis[ground, 0, X, Y]];"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["The constraints are incorporated into the model."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["SetConstraints[ Array[cs, 5] ]"], "Input", AspectRatioFixed->True]}, Open]], Cell[TextData["Run-Time"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "The model can now be run through its range of motion by varying ", Evaluatable->False, AspectRatioFixed->True], StyleBox["T", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" with the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["SolveMech", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" command. The numerical value of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["T", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" specifies the angle of rotation of the crank in revolutions.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[{ StyleBox["The model is run at ", Evaluatable->False, AspectRatioFixed->True], StyleBox["T", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" = 0.05.", Evaluatable->False, AspectRatioFixed->True] }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["SolveMech[0.05]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {T -> 0.05, X2 -> 0., Y2 -> 10., Z2 -> 12., Eo2 -> 0.987688340595138, Ei2 -> 0.1564344650402309, Ej2 -> 0., Ek2 -> 0., X3 -> 0., Y3 -> 17.60845213036123, Z3 -> 14.47213595499958, Eo3 -> 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0.00323587}}]}, Open]], Cell[TextData["Generalized Coordinate Constraints"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Now one of the standard constraints is replaced with a generalized \ coordinate constraint. Constraint 2, the revolute joint that models the \ rotational axis of the crank, is replaced first, and body 2 becomes a slave \ to the ground body, represented by the single coordinate ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". Note that the arguments to the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["GenRevolute5", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" constraint are exactly the same as the arguments to the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Revolute5", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" constraint, with the addition of the new symbol ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and its initial guess.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Here is the new constraint."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "cs[2] = GenRevolute5[2, Axis[crank, 0, X], Axis[ground, 1, X],\n \ {alpha, 0.0}];\nSetConstraints[ Array[cs, 5] ] "], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[{ StyleBox["The modified model is run again at ", Evaluatable->False, AspectRatioFixed->True], StyleBox["T", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" = 0.05.", Evaluatable->False, AspectRatioFixed->True] }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["SolveMech[0.05]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {T -> 0.05, X3 -> 0., Y3 -> 17.60845213036123, Z3 -> 14.47213595499958, Eo3 -> 0.903943892415714, Ei3 -> 0.0893987074414874, Ej3 -> 0.2324512735198904, Ek3 -> -0.3476488399392477, X4 -> 19.50640137167271, Y4 -> 0., Z4 -> 0., Eo4 -> 1., Ei4 -> 0., Ej4 -> 0., Ek4 -> 0., alpha -> 0.3141592653589793}\ \>", "\<\ {T -> 0.05, X3 -> 0., Y3 -> 17.6085, Z3 -> 14.4721, Eo3 -> 0.903944, Ei3 -> 0.0893987, Ej3 -> 0.232451, Ek3 -> -0.347649, X4 -> 19.5064, Y4 -> 0., Z4 -> 0., Eo4 -> 1., Ei4 -> 0., Ej4 -> 0., Ek4 -> 0., alpha -> 0.314159}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[TextData[{ StyleBox[ "Note that all of the symbols associated with body 2 are no longer included \ in the solution rules returned by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["SolveMech", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". The location of body 2 is represented entirely by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ", which is the rotation angle of the crank in full revolutions. The \ definition (and hence the numerical value) of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " is dependent on the relative reference directions of the two axis objects \ passed to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["GenRevolute5", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ". Since the reference directions were not explicitly specified in the axis \ objects, default reference directions were assumed by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" (the local ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " axes of each body, in this case).\n\nOnly two more of the standard ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " constraints in this model can be replaced with generalized coordinate \ constraints. This is because each generalized coordinate constraint creates a \ slave body, and a slave body can be a slave to only one master body. Thus, \ there is no way to use more than three generalized coordinate constraints in \ a model with only three bodies.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Here are two more new constraints."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "cs[4] = GenOrthoRevolute4[4, Axis[conrod, 1, Y],\n \ Axis[slider, 0, Z],\n {beta, -1.0}, {gamma, 1.0} ];\ncs[5] = \ GenTranslate5[5, Axis[slider, 0, X, Y],\n \ Axis[ground, 0, X, Y], {delta, 20.0} ];\nSetConstraints[ Array[cs, 5] ] "], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[{ StyleBox["The modified model is run again at ", Evaluatable->False, AspectRatioFixed->True], StyleBox["T", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" = 0.05.", Evaluatable->False, AspectRatioFixed->True] }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["SolveMech[0.05]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {T -> 0.05, alpha -> 0.3141592653589793, beta -> -0.836490846423524, gamma -> 0.5033977045947692, delta -> 19.50640137167256}\ \>", "\<\ {T -> 0.05, alpha -> 0.314159, beta -> -0.836491, gamma -> 0.503398, delta -> 19.5064}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[TextData[{ StyleBox[ "Now the 21 Cartesian and Euler coordinates have been replaced with the \ four generalized coordinates ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["beta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["gamma", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["delta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ". A quick degree of freedom count shows that this all makes sense: the \ three generalized coordinate constraints add a total of four degrees of \ freedom to the model, while the two remaining standard constraints ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Spherical3", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["ProjectedAngle1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " constrain a total of four degrees of freedom.\n\nThe physical meaning of \ the new coordinates is somewhat confusing: ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" is simply the angle between the global ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" axis and the local ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" axis of the crank, and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["delta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" is the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["X", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " direction distance from the global origin to the origin of the slider. ", Evaluatable->False, AspectRatioFixed->True], StyleBox["beta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["gamma", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " are two angles that represent the rotation of the connecting rod with \ respect to the slider: ", Evaluatable->False, AspectRatioFixed->True], StyleBox["beta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " is the altitude angle, the angle between a horizontal plane and a vector \ down the axis of the connecting rod, and gamma is the azimuth angle, the \ rotation of the connecting rod axis about the global ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Z", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " axis.\n\nFinally, to reduce the mathematical model of the space-crank to \ absolute minimum size, constraint 1 (the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["ProjectedAngle1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " constraint) must be reexamined. This constraint uses a rather complicated \ expression relating the directions of three vectors to specify the rotation \ of the crank, but it is known that the rotation of the crank can be directly \ specified by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". While ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " attempts to simplify constraint expressions to some extent, the algebraic \ expression generated by constraint 1 is more complicated than it could be.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Here is the expression that results from constraint 1."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Constraints[1]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {-(Cos[6.283185307179587*T]*Sin[alpha]) + Cos[alpha]*Sin[6.283185307179587*T]}\ \>", "\<\ {-(Cos[6.28319 T] Sin[alpha]) + Cos[alpha] Sin[6.28319 T]}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[TextData[{ StyleBox["Constraint 1 really just sets ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" equal to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Pi", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" ", Evaluatable->False, AspectRatioFixed->True], StyleBox["T", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ", which can be done more directly with the all-purpose constraint function \ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Constraint", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Here is a simplified functional replacement for constraint 1."], "SmallText",\ Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Constraint[1, alpha == 2 N[Pi] T]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ SysCon[{alpha - 6.283185307179587*T}, {}, {}, 1, 1, {}, \"Cons\\ traint\"]\ \>", "\<\ SysCon[Constraint, <>]\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[TextData["Output and Graphics Functions"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Now that we have a solution in terms of generalized coordinates, all of \ the standard ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " output and graphics functions allow it to be used in the same manner as a \ solution in terms of Cartesian and Euler coordinates. For example, the \ location of a point is still found with the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Location", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" function and the spatial orientation of a body is found with ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Rotation", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ". The symbolic expressions returned by these functions reflect the new \ coordinate system being used, but the numerical results are equivalent. ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Here are the Cartesian coordinates of a point at the center of the \ connecting rod."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Location[conrod, {15, 0, 0}]\n%/.LastSolve[]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {delta + 15*Cos[gamma]*Sin[beta], 15*Cos[beta]*Cos[gamma], 15*Sin[gamma]}\ \>", "\<\ {delta + 15 Cos[gamma] Sin[beta], 15 Cos[beta] Cos[gamma], 15 Sin[gamma]}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {9.75320068583631, 8.80422606518059, 7.236067977499789}\ \>", "\<\ {9.7532, 8.80423, 7.23607}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[TextData[{ StyleBox[ "The location of the point on the connecting rod is a function of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["beta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["gamma", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["delta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", but not ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ". This is because of the master-slave relationship of the generalized \ coordinate bodies; the connecting rod is a slave to the slider, which is a \ slave to the ground. Thus, the location of any point on the connecting rod is \ a function of the generalized coordinates of the slider and the connecting \ rod, while the location of any point on the crank is a function of only ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Here is the global rotation angle of the crank."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["crrot = Rotation[crank]\ncrrot/.LastSolve[]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {2*ArcCos[(2 + 2*Cos[alpha])^(1/2)/2], {(2*Sin[alpha])/ ((2 - 2*Cos[alpha])*(2 + 2*Cos[alpha]))^(1/2), 0, 0}}\ \>", "\<\ Sqrt[2 + 2 Cos[alpha]] {2 ArcCos[----------------------], 2 2 Sin[alpha] {-------------------------------------------, 0, 0}} Sqrt[(2 - 2 Cos[alpha]) (2 + 2 Cos[alpha])]\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData["\<\ {0.3141592653589788, {1., 0, 0}}\ \>", "\<\ {0.314159, {1., 0, 0}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[TextData[{ StyleBox["The symbolic expression returned by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Rotation", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " is rather complicated, considering that the correct result is known to be \ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["{alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], StyleBox["{1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], StyleBox["0", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], StyleBox["0}}", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ". This is an unfortunate side effect of the general transformation that ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " uses to convert a rotation matrix to the single rotation angle and axis \ of rotation that it represents. The application of a couple of trig \ identities will reduce this expression back to the simpler form, but only in \ such trivial cases. In a more general case, such as the connecting rod, which \ has undergone two consecutive rotations relative to the ground, the symbolic \ expression for the angle and axis of rotation cannot be reduced \ significantly. \n\n", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " graphics functions also operate without regard to the use of generalized \ coordinates. The following graphics object will generate the same image with \ any of the three constraint formulations used in the preceding example.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Here is the graphics object representing the space-crank model."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "graph := Graphics3D[ {{Thickness[.008],\n Line[{{-1,-5,0}, {-1,18, 0}, \ {35,18, 0}, {35,-5,0},\n {-1,-5,0}, {-1,-5,22}, {-1,18,22}, \ {-1,18,0}}],\n Line[{{-1, 1.8,0}, {35, 1.8,0}}],\n Line[{{-1,-1.8,0}, \ {35,-1.8,0}}]},\n Bar[Line[crank, 0, {-1,0,0}], 10.0, 12],\n Box[ slider, \ {{-4,-1.5,-2}, {4,1.5,2}} ],\n Bar[ Line[conrod, 0, 1], 1.0, 8 ]} ];"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["This shows the graphic at T = 0.15."], "SmallText", Evaluatable->False, AspectRatioFixed->True], 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deals with issues related to applying loads to mechanisms \ modeled with generalized coordinates, and finding the reaction forces \ associated with those coordinates. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Applied Loads"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "The Generalized Coordinate Package has almost no effect on the modeling of \ applied Loads. The standard ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" load functions ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Force", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Moment", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Gravity", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["GyroMoment", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " are used to apply loads to a generalized coordinate slave body in exactly \ the same way as they are used with standard bodies. The one load function \ that is affected by the use of generalized coordinates is ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Load", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ", which is used to apply a load directly to a specified coordinate.\n\n\ Since ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Load", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " references the coordinates in a model directly by their symbolic name, it \ can be used to apply a load directly to a generalized coordinate. Conversely, \ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Load", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " cannot be used to apply a load directly to a Cartesian coordinate of a \ slave body because a slave body doesn't have any explicit Cartesian \ coordinates.\n\nFor example, suppose a model has two bodies, 2 and 3, and \ body 3 is a slave to body 2. The two bodies are joined by a revolute joint \ and the symbol ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " represents the rotation of the slave, relative to the master. The ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Force", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" function may be used to apply a force in the global ", Evaluatable->False, AspectRatioFixed->True], StyleBox["X", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" direction to the slave , but the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Load", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" function may not.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "This loads the Generalized Coordinate Package and defines names for bodies 2 \ and 3."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Needs[\"Mech`GenCrd2D`\"]\nmaster = 2;\nslave = 3;"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[ "Here is a load object that applies a 10 unit force to the slave body."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Force[slave, Axis[slave, 0, {1, 0}], 10]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ SysLoad[{Mech`Mech2D`Private`force[10/ (0 + Mech`Mech2D`Private`magvec[Axis[Point[3, {0, 0}], Vector[3, {1, 0}]]]), Axis[Point[3, {0, 0}], Vector[3, {1, 0}]]], Mech`Mech2D`Private`torque[10/ (0 + Mech`Mech2D`Private`magvec[Axis[Point[3, {0, 0}], Vector[3, {1, 0}]]]), Axis[Point[3, {0, 0}], Vector[3, {1, 0}]], 3]}, {Point[3, {0, 0}], Point[3, {1, 0}]}, 3, \"Force\"]\ \>", "\<\ SysLoad[Force, <>]\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData[{ StyleBox[ "This load object does not apply the 10 unit force to the slave body, \ because ", Evaluatable->False, AspectRatioFixed->True], StyleBox["X3", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" is not a coordinate in the model.", Evaluatable->False, AspectRatioFixed->True] }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Load[X3, 10]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ SysLoad[10, {}, X3, \"Load\"]\ \>", "\<\ SysLoad[Load, <>]\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[TextData[{ StyleBox["However, the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Load", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " function can be used to apply a moment to the slave body because the \ generalized coordinate ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" is an angular coordinate. A load applied directly to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " essentially applies a counterclockwise moment to the slave body, and a \ clockwise moment to the master.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Here is a load object that applies a 10 unit moment to the slave, and an \ equal and opposite moment to the master."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Moment[slave, master, 10] "], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {SysLoad[{{0, 0}, 10}, {}, 3, \"Moment\"], SysLoad[{{0, 0}, -10}, {}, 2, \"Moment\"]}\ \>", "\<\ {SysLoad[Moment, <>], SysLoad[Moment, <>]}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData[ "Here is a simpler load object that applies the same 10 unit couple between \ the slave and master."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Load[alpha, 10]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ SysLoad[10, {}, alpha, \"Load\"]\ \>", "\<\ SysLoad[Load, <>]\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[TextData[{ StyleBox["In general, the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Load", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" function should be used instead of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Force", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" or ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Moment", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " whenever possible because the resulting algebraic expressions that are \ added to the model can be much simpler. ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Constraint Reactions"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "While the use of generalized coordinates has little effect on the \ application of loads to a model, finding constraint reaction forces is \ another matter. When a constraint is eliminated from a model by using \ generalized coordinates, information regarding the reaction forces at the \ constraint is lost. This is one basic disadvantage of a reduced coordinate \ system; by algebraically eliminating coordinates from a model we lose \ information about those coordinates.\n\nThus, three restrictions are placed \ on models using generalized coordinates: the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Reaction", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " function cannot be used to find the reaction forces applied by a \ generalized coordinate constraint to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["any", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" body (because it is not really a constraint), the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Reaction", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " function cannot be used to find the reaction forces applied by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["any", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " constraint to a generalized coordinate slave body (because the body \ doesn't have Cartesian coordinates), and the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Loads", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " function cannot be used to find the loads applied to a slave body (for \ the same reason).\n\nBoth the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Reaction", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Loads", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " functions may still be used with generalized coordinates, but they must \ be used to access the coordinates directly.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[{ StyleBox["Reaction[", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox["cnum, sym", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" returns the generalized reaction applied by constraint ", Evaluatable->False, AspectRatioFixed->True], StyleBox["cnum", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" to generalized coordinate ", Evaluatable->False, AspectRatioFixed->True], StyleBox["sym", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ ". The physical meaning of the returned scalar load is dependent on the \ nature of the generalized coordinate ", Evaluatable->False, AspectRatioFixed->True], StyleBox["sym", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[".\n \n", Evaluatable->False, AspectRatioFixed->True], StyleBox["Loads[", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox["sym", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " returns the total external generalized load applied to generalized \ coordinate ", Evaluatable->False, AspectRatioFixed->True], StyleBox["sym", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[".\n\n", Evaluatable->False, AspectRatioFixed->True], StyleBox["Loads[", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox["sym, ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["Type->Reaction]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " returns the total generalized reaction load applied by all constraints to \ coordinate ", Evaluatable->False, AspectRatioFixed->True], StyleBox["sym", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[".\n\n", Evaluatable->False, AspectRatioFixed->True], StyleBox["Loads[", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox["sym, ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["Type->Dynamic]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " returns the total generalized inertial load applied to coordinate ", Evaluatable->False, AspectRatioFixed->True], StyleBox["sym", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Special3", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Alternate usages of ", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True], StyleBox["Reaction", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True], StyleBox["Loads", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True] }], "Special4", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox["The physical meaning of the quantities returned by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Reaction", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Loads", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " is quite clear, in the context of the simple generalized coordinate \ constraints provided by this package. If the generalized coordinate ", Evaluatable->False, AspectRatioFixed->True], StyleBox["sym", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " represents an angle in radians, then the loads on the coordinate are in \ consistent units of moment. If the coordinate ", Evaluatable->False, AspectRatioFixed->True], StyleBox["sym", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " represents a distance, then the loads on the coordinate are in consistent \ units of force.\n\nHowever, if abstract user-defined generalized coordinate \ constraints are used, the generalized loads may have little physical meaning, \ just like the generalized loads associated with each constraint (", Evaluatable->False, AspectRatioFixed->True], StyleBox["Lmb1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Lmb2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ", ...) sometimes have little physical meaning. The one meaning that always \ applies to a generalized load is this: a generalized load times a small \ variation in its associated generalized coordinate is equal to a small \ variation in work, in consistent energy units. ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Slider-Crank Example Mechanism"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["To demonstrate the use of the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Reaction", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Loads", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " functions with generalized coordinates a planar model of a simple \ slider-crank is developed. This model uses only two bodies, the slider and \ crank, and it is modeled with two generalized coordinates, the rotation angle \ of the crank and the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["X", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" displacement of the slider.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "This loads the Generalized Coordinate packages and defines some useful \ constants."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Needs[\"Mech`GenCrd2D`\"]\nOff[General::spell1]\nX = {1, 0};\nY = {0, 1};"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Here is the slider-crank mechanism. 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for the slider-crank model. Inertia properties \ are defined for each moving body so that the dynamic reaction forces can be \ calculated. No local point definitions are made in the body objects. All \ necessary point coordinates in this model are given explicitly in the \ constraint functions. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Here are the body objects for the slider-crank model."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "ground = 1;\nbd[2] = Body[crank = 2,\n Mass->200.0,\n Inertia->450.0];\n\ bd[3] = Body[slider = 3,\n Mass->25.0,\n Inertia->0.02];"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[ "The body properties are incorporated into the model."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["SetBodies[bd[crank], bd[slider]]"], "Input", AspectRatioFixed->True]}, Open]], Cell[TextData["Constraints"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Four constraints are used to model the slider-crank mechanism. Two \ generalized coordinate constraints are used along with two standard \ constraints."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["A ", Evaluatable->False, AspectRatioFixed->True], StyleBox["GenRevolute2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" constraint defines the symbol ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" as the angular coordinate of the crank.", Evaluatable->False, AspectRatioFixed->True] }], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["A ", Evaluatable->False, AspectRatioFixed->True], StyleBox["RotationLock1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" constraint controls the rotation of the crank.", Evaluatable->False, AspectRatioFixed->True] }], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["A ", Evaluatable->False, AspectRatioFixed->True], StyleBox["GenTranslate2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" constraint defines the symbol ", Evaluatable->False, AspectRatioFixed->True], StyleBox["beta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" as the displacement coordinate of the slider.", Evaluatable->False, AspectRatioFixed->True] }], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["A ", Evaluatable->False, AspectRatioFixed->True], StyleBox["RelativeDistance1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " constraint models the connecting rod between the crank and the slider.", Evaluatable->False, AspectRatioFixed->True] }], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Here are the constraint objects for the slider-crank."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "cs[1] = GenRevolute2[1, Axis[crank, 0, X],\n \ Axis[ground, 0, X], {alpha, 0.0}];\ncs[2] = RotationLock1[2, crank, 2 N[Pi] \ T];\ncs[3] = GenTranslate2[3, Axis[slider, 0, X],\n \ Axis[ground, 0, X], {beta, 5.0} ];\ncs[4] = RelativeDistance1[4, Point[crank, \ {2, 0}],\n Point[slider, 0], 3.5];"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[ "The constraints are incorporated into the current model."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["SetConstraints[ Array[cs, 4] ] "], "Input", AspectRatioFixed->True]}, Open]], Cell[TextData["Loads"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "A force that is a linear function of slider position is applied to the \ slider to simulate a simple spring. The spring has a spring constant of 10, \ and it is placed so that the spring force is zero when the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["X", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" displacement of the slider is three units. ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Here is a spring force applied to the slider."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["SetLoads[\n ld[1] = Load[beta, -10 (beta - 3)] ]"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[{ StyleBox["The model is run at ", Evaluatable->False, AspectRatioFixed->True], StyleBox["T", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" = 0.1 with the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Static", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" option to cause the Lagrange multipliers to be calculated.", Evaluatable->False, AspectRatioFixed->True] }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["SolveMech[0.1, Solution->Static]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {T -> 0.1, alpha -> 0.6283185307179587, beta -> 4.914704125228505, Lmb1 -> 33.55615853439749, Lmb2 -> -2.90399713341312}\ \>", "\<\ {T -> 0.1, alpha -> 0.628319, beta -> 4.9147, Lmb1 -> 33.5562, Lmb2 -> -2.904}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[TextData["Reactions"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Because generalized coordinates were used in this model, Mech's ability to \ find constraint reaction forces is limited. Normally, ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Reaction", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " could be used to find the reaction force vector that the revolute joint \ applies to the crank, but with this model only the moment applied to the \ crank by constraint 2 can be found. Note that ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Reaction", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " cannot directly find the moment applied to body 2, it can only find the \ generalized force associated with the symbol ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". Since ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" is an angular coordinate, its associated load is a moment. 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Dynamic motion synthesis is often where generalized \ coordinates have their greatest payoff. Using a coordinate system that is \ closer to being a minimal and independent set can drastically reduce the size \ of the mass matrix and centrifugal force vector, resulting in reduced \ run-times for the numerical integration routine.\n\nThe reduced mass matrix, \ in terms of generalized coordinates, is often useful in its own right for \ certain types of analysis. Examples of various mass matrix manipulations are \ presented in this section, including the conversion of an open-loop mass \ matrix to end-point coordinates, and the reduction of a closed loop mass \ matrix to minimal coordinates. 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Each body object is \ used only to define inertia properties for the body, no local point \ definitions are made. Thus the ground (body 1) needs no body object because \ its inertia is immaterial. Note that the lengths of the links are left as \ parameters ", Evaluatable->False, AspectRatioFixed->True], StyleBox["L1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["L2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", causing the resulting mass matrix to be completely general.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Here are the body objects for the manipulator model."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "ground = 1;\nlink1 = 2;\nlink2 = 3;\nbd[2] = Body[link1, Mass->1, \ Inertia->3, Centroid->{L1/2, 0}];\nbd[3] = Body[link2, Mass->1, Inertia->3, \ Centroid->{L2/2, 0}];"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[ "The inertia properties are incorporated into the model."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["SetBodies[ bd[link1], bd[link2] ]"], "Input", AspectRatioFixed->True]}, Open]], Cell[TextData["Constraints"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Four constraint objects are used to model the manipulator. Two constraints \ are generalized coordinate constraints which introduce the two degrees of \ freedom, ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["beta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ", and the other two constraints simply constrain these two degrees of \ freedom.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["A ", Evaluatable->False, AspectRatioFixed->True], StyleBox["GenRevolute2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " constraint sets up the relationship between link1 and the ground.", Evaluatable->False, AspectRatioFixed->True] }], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["A ", Evaluatable->False, AspectRatioFixed->True], StyleBox["GenRevolute2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" constraint sets up the relationship between link2 and link1.", Evaluatable->False, AspectRatioFixed->True] }], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Two ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Constraint", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" objects constrain ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["beta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" to be equal to the parameters ", Evaluatable->False, AspectRatioFixed->True], StyleBox["q1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["q2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Note that ", Evaluatable->False, AspectRatioFixed->True], StyleBox["L1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" appears explicitly in constraint 3 to locate the tip of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["link1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", where it is attached to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["link2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Here are the constraint objects for the manipulator."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "cs[1]=GenRevolute2[1, Axis[link1, 0, X],\n \ Axis[ground, 0, X], {alpha, 0}];\ncs[2]=Constraint[2, alpha == q1];\n\ cs[3]=GenRevolute2[3, Axis[link2, 0 , X],\n \ Axis[link1, {L1, 0}, X], {beta, 0}];\ncs[4]=Constraint[4, beta == q2];"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[ "The constraints are incorporated into the current model."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["SetConstraints[ Array[cs, 4] ] "], "Input", AspectRatioFixed->True]}, Open]], Cell[TextData["Mass Matrix"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "The kinematics of this model are truly trivial. The entire constraint set \ is nothing more than a pair of expressions setting ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["beta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" equal to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["q1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["q2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ". There is really no point in \"running\" the model at all, in the normal \ sense in which other ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " models are \"run\" because the configuration of the model can be \ specified directly by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["beta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ". The usefulness of the model lies in the other definitions that have been \ made by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" for inertia properties and coordinate transformations.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Here is the constraint vector."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Constraints[All]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {alpha - q1, beta - q2}\ \>", "\<\ {alpha - q1, beta - q2}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "Here is the global location of the tip of link 2."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["link2tip = Location[link2, {L2, 0}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {L1*Cos[alpha] + L2*Cos[alpha + beta], L1*Sin[alpha] + L2*Sin[alpha + beta]}\ \>", "\<\ {L1 Cos[alpha] + L2 Cos[alpha + beta], L1 Sin[alpha] + L2 Sin[alpha + beta]}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Here is the 2x2 system mass matrix."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["m = Simplify[ MassMatrix[All] ]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{6 + (5*L1^2)/4 + L2^2/4 + L1*L2*Cos[beta], 3 + L2^2/4 + (L1*L2*Cos[beta])/2}, {3 + L2^2/4 + (L1*L2*Cos[beta])/2, 3 + L2^2/4}}\ \>", "\<\ 2 2 5 L1 L2 {{6 + ----- + --- + L1 L2 Cos[beta], 4 4 2 L2 L1 L2 Cos[beta] 3 + --- + ---------------}, 4 2 2 2 L2 L1 L2 Cos[beta] L2 {3 + --- + ---------------, 3 + ---}} 4 2 4\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[TextData["End Point Mobility"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Now we wish to find the ellipse of mobility of the end point of link 2. \ The ellipse of mobility of an equivalent two-link manipulator model was found \ in Section 10.3 of the Mechanical Systems Pack manual. In that example, a one \ unit force was applied to the end point of link 2 while varying the direction \ of the force through 360 degrees, and the resulting free acceleration vector \ of the endpoint was found with the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["SetFree", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["SolveFree", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " functions.\n\nThis section presents a more elegant method of finding the \ ellipse of mobility directly from the mass matrix. The method presented in \ Section 10.3 is much more general than this one in that this method (as it is \ presented here) is restricted to finding the mobility of the end point at \ zero velocity, and is restricted to models that have the same total number of \ coordinates as the number of physical degrees of freedom of the end point.\n\n\ The basic trick here is to convert the mass matrix from the generalized \ (joint angle) coordinates to endpoint coordinates. The mass matrix ", Evaluatable->False, AspectRatioFixed->True], StyleBox["m", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", in the context of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f=m.a", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", transforms accelerations of the generalized coordinates ", Evaluatable->False, AspectRatioFixed->True], StyleBox["a", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ", which are the relative angular accelerations of each of the joints, into \ forces on the generalized coordinates ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ", which are moments applied at the joints. We need to have a matrix ", Evaluatable->False, AspectRatioFixed->True], StyleBox["P", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" in ", Evaluatable->False, AspectRatioFixed->True], StyleBox["P.F=A", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", which transforms a force vector ", Evaluatable->False, AspectRatioFixed->True], StyleBox["F", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" applied to the endpoint into the linear acceleration vector ", Evaluatable->False, AspectRatioFixed->True], StyleBox["A", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" of the end point. First, we need a transformation matrix ", Evaluatable->False, AspectRatioFixed->True], StyleBox["J", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " from generalized coordinates to endpoint coordinates such that ", Evaluatable->False, AspectRatioFixed->True], StyleBox["A = J.a", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". Then ", Evaluatable->False, AspectRatioFixed->True], StyleBox["P=J.Inverse[m].Transpose[J]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".\n\nThe matrix ", Evaluatable->False, AspectRatioFixed->True], StyleBox["J", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" is the Jacobian of the global coordinates of the endpoint.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Here is the transformation matrix ", Evaluatable->False, AspectRatioFixed->True], StyleBox["J", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["J = Outer[D, link2tip, {alpha, beta}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{-(L1*Sin[alpha]) - L2*Sin[alpha + beta], -(L2*Sin[alpha + beta])}, {L1*Cos[alpha] + L2*Cos[alpha + beta], L2*Cos[alpha + beta]}}\ \>", "\<\ {{-(L1 Sin[alpha]) - L2 Sin[alpha + beta], -(L2 Sin[alpha + beta])}, {L1 Cos[alpha] + L2 Cos[alpha + beta], L2 Cos[alpha + beta]}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox["Here is the inverse end point mass matrix ", Evaluatable->False, AspectRatioFixed->True], StyleBox["P", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", a rather large symbolic matrix.", Evaluatable->False, AspectRatioFixed->True] }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["P = J.Inverse[m].Transpose[J];"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "We can now find the acceleration of the endpoint that results from any \ force vector at any particular configuration simply by multiplying ", Evaluatable->False, AspectRatioFixed->True], StyleBox["P", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" times the force vector.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Here is the acceleration vector resulting from an applied force {0, 2}."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "config = {alpha->N[Pi]/6, beta->2 N[Pi]/3, L1->6, L2->6};\nP . {0, 2} /. \ config"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {2.267411966271985, 4.581818181818181}\ \>", "\<\ {2.26741, 4.58182}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "Now finding the ellipse of mobility is simply a matter of applying a unit \ force vector that revolves through a full circle and plotting the components \ of the resulting acceleration."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Here is the ellipse of mobility."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "ParametricPlot[Evaluate[P . 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While this can \ theoretically be done with or without generalized coordinates, it often \ becomes intractable when modeling in Cartesian coordinates because of the \ need to invert a large symbolic matrix.\n\nThe four-bar model that was \ developed previously in this notebook is used again here to demonstrate the \ formulation of a minimal mass matrix. In this case, the minimal mass matrix \ is a 1x1 matrix because the four-bar model has only one degree of freedom. \ Since the four-bar is modeled with three generalized coordinates ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["beta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["gamma", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", the default mass matrix generated by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " is a 3x3 matrix; the task at hand is to reduce it to 1x1.\n\nSince the \ Generalized Coordinate Package was loaded into Mathematica in the last \ section, it does not need to be loaded again here. Note, however, that \ inertia properties are included in the body objects of this version of the \ four-bar.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[{ StyleBox["This clears all data in the current ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mech", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" model.", Evaluatable->False, AspectRatioFixed->True] }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["ClearMech[]"], "Input", AspectRatioFixed->True]}, Open]], Cell[TextData["Bodies and Constraints"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The entire four-bar model is redefined without explanation here, because its \ kinematics are completely documented in a previous section."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Here are the body objects for the four-bar model."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "SetBodies[\n bd[1] = Body[ground = 1,\n PointList -> {{-2, 3}}],\n\n \ bd[2] = Body[drivebar = 2,\n PointList -> {{1, 0}},\n Mass -> 5,\n \ Inertia -> 1],\n\n bd[3] = Body[drivenbar = 3,\n PointList -> \ {{3, 0}},\n Mass -> 2,\n Inertia -> 4,\n InitialGuess \ -> {{-2, 3}, 0}],\n\n bd[4] = Body[centerbar = 4,\n PointList->{{0, 3}, \ {1, 2}},\n Mass -> 3,\n Inertia -> 12,\n \ InitialGuess->{{ 1, 0}, 0}] ]"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[ "Here are the generalized coordinate constraint objects."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "SetConstraints[\n cs[1] = RotationLock1[1, drivebar, 2 N[Pi] T ],\n cs[2] \ = GenRevolute2[2, Axis[drivebar, 0, X],\n \ Axis[ground, 0, X], {alpha, 0.0} ],\n cs[3] = GenRevolute2[3, \ Axis[centerbar, 0, X], \n Axis[drivebar, 1, X], \ {beta, 0.0} ],\n cs[4] = GenRevolute2[4, Axis[drivenbar, 1, X],\n \ Axis[centerbar, 1, X], {gamma, 0.0} ],\n cs[5] = Revolute2[5, \ Point[drivenbar, 0], Point[ground, 1] ] ]"], "Input", AspectRatioFixed->True]}, Open]], Cell[TextData["Mass Matrix"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["We intend to transform the 3x3 system mass matrix ", Evaluatable->False, AspectRatioFixed->True], StyleBox["M", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" into a 1x1 mass matrix ", Evaluatable->False, AspectRatioFixed->True], StyleBox["m", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" where ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f = m.alphadd", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" is the generalized force in the direction of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ", which is simply a counterclockwise moment applied to the drive bar. This \ transformation is of the form ", Evaluatable->False, AspectRatioFixed->True], StyleBox["m = Transpose[J] . M . J", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".\n\nThe necessary transformation matrix ", Evaluatable->False, AspectRatioFixed->True], StyleBox["J", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " converts motions in all three generalized coordinates into motions of the \ single coordinate ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". This matrix ", Evaluatable->False, AspectRatioFixed->True], StyleBox["J", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " is created by inverting the part of the model's Jacobian associated with \ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["beta", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["gamma", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and multiplying it by the part associated with ", Evaluatable->False, AspectRatioFixed->True], StyleBox["alpha", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". 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In this example, the drive bar is given an initial angular velocity \ of 2 pi radians/second, a condition that is inherent in the definition of \ constraint 1. Constraint 1 is dropped from the model, and the free motion of \ the underconstrained system is found. 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