MGFourBarDynamicsKaneEmbedded.html  (MotionGenesis input/output).
```   (1) % MotionGenesis file:  MGFourBarDynamicsKaneEmbedded.txt
(3) %--------------------------------------------------------------------
(4) NewtonianFrame  N                 % Ground link.
(5) RigidBody       A, B, C           % Crank, coupler, rocker links.
(6) Point           BC( B )           % Point of B connected to C.
(7) Point           CB( C )           % Point of C connected to B.
(8) %--------------------------------------------------------------------
(9) Constant   LN = 1 m,  LA = 1 m    % Length of ground link, crank link.
(10) Constant   LB = 2 m,  LC = 2 m    % Length of coupler link, rocker link.
(11) Constant   g = 9.81 m/s^2         % Earth's gravitational acceleration.
(12) Specified  H = 200                % Horizontal force at point CB.
-> (13) H = 200

(14) Specified  TA = 0                 % Motor torque on crank A from ground N.
-> (15) TA = 0

(16) Variable   qA'',  qB'',  qC''     % Link angles (relative to ground).
(17) SetGeneralizedSpeed( qA' )
(18) %--------------------------------------------------------------------
(19) A.SetMassInertia( mA = 10 kg,  0,  IA = mA*LA^2/12,  IA  )
-> (20) IA = 0.08333333*mA*LA^2

(21) B.SetMassInertia( mB = 20 kg,  0,  IB = mB*LB^2/12,  IB  )
-> (22) IB = 0.08333333*mB*LB^2

(23) C.SetMassInertia( mC = 20 kg,  0,  IC = mC*LC^2/12,  IC  )
-> (24) IC = 0.08333333*mC*LC^2

(25) %--------------------------------------------------------------------
(26) %   Rotational kinematics.
(27) A.RotateZ( N,  qA )
-> (28) A_N = [cos(qA), sin(qA), 0;  -sin(qA), cos(qA), 0;  0, 0, 1]
-> (29) w_A_N> = qA'*Az>
-> (30) alf_A_N> = qA''*Az>

(31) B.RotateZ( N,  qB )
-> (32) B_N = [cos(qB), sin(qB), 0;  -sin(qB), cos(qB), 0;  0, 0, 1]
-> (33) w_B_N> = qB'*Bz>
-> (34) alf_B_N> = qB''*Bz>

(35) C.RotateZ( N,  qC )
-> (36) C_N = [cos(qC), sin(qC), 0;  -sin(qC), cos(qC), 0;  0, 0, 1]
-> (37) w_C_N> = qC'*Cz>
-> (38) alf_C_N> = qC''*Cz>

(39) %--------------------------------------------------------------------
(40) %   Translational kinematics.
(41) Ao.Translate(   No,          0> )
-> (42) p_No_Ao> = 0>
-> (43) v_Ao_N> = 0>
-> (44) a_Ao_N> = 0>

(45) Acm.Translate(  Ao,  0.5*LA*Ax> )
-> (46) p_Ao_Acm> = 0.5*LA*Ax>
-> (47) v_Acm_N> = 0.5*LA*qA'*Ay>
-> (48) a_Acm_N> = -0.5*LA*qA'^2*Ax> + 0.5*LA*qA''*Ay>

(49) Bo.Translate(   Ao,      LA*Ax> )
-> (50) p_Ao_Bo> = LA*Ax>
-> (51) v_Bo_N> = LA*qA'*Ay>
-> (52) a_Bo_N> = -LA*qA'^2*Ax> + LA*qA''*Ay>

(53) Bcm.Translate(  Bo,  0.5*LB*Bx> )
-> (54) p_Bo_Bcm> = 0.5*LB*Bx>
-> (55) v_Bcm_N> = LA*qA'*Ay> + 0.5*LB*qB'*By>
-> (56) a_Bcm_N> = -LA*qA'^2*Ax> + LA*qA''*Ay> - 0.5*LB*qB'^2*Bx> + 0.5*LB*qB''*By>

(57) BC.Translate(   Bo,      LB*Bx> )
-> (58) p_Bo_BC> = LB*Bx>
-> (59) v_BC_N> = LA*qA'*Ay> + LB*qB'*By>
-> (60) a_BC_N> = -LA*qA'^2*Ax> + LA*qA''*Ay> - LB*qB'^2*Bx> + LB*qB''*By>

(61) Co.Translate(   No,      LN*Ny> )
-> (62) p_No_Co> = LN*Ny>
-> (63) v_Co_N> = 0>
-> (64) a_Co_N> = 0>

(65) Ccm.Translate(  Co,  0.5*LC*Cx> )
-> (66) p_Co_Ccm> = 0.5*LC*Cx>
-> (67) v_Ccm_N> = 0.5*LC*qC'*Cy>
-> (68) a_Ccm_N> = -0.5*LC*qC'^2*Cx> + 0.5*LC*qC''*Cy>

(69) CB.Translate(   Co,      LC*Cx> )
-> (70) p_Co_CB> = LC*Cx>
-> (71) v_CB_N> = LC*qC'*Cy>
-> (72) a_CB_N> = -LC*qC'^2*Cx> + LC*qC''*Cy>

(73) %--------------------------------------------------------------------
(74) %   Add relevant forces and torques (replace gravity forces with equivalent set).
(75) Bo.AddForce( 0.5*(mA+mB)*g*Nx> )
-> (76) Force_Bo> = 0.5*(mA+mB)*g*Nx>

(77) CB.AddForce( 0.5*(mB+mC)*g*Nx> + H*Ny> )
-> (78) Force_CB> = 0.5*(mB+mC)*g*Nx> + H*Ny>

(79) A.AddTorque( N,  TA * Az> )
-> (80) Torque_A_N> = TA*Az>

(81) %--------------------------------------------------------------------
(82) %   Configuration (loop) constraints and their time-derivatives.
(83) Loop> = LA*Ax> + LB*Bx> - LC*Cx> - LN*Ny>
-> (84) Loop> = LA*Ax> + LB*Bx> - LC*Cx> - LN*Ny>

(85) Loop[1] = Dot( Loop>, Nx> )
-> (86) Loop[1] = LA*cos(qA) + LB*cos(qB) - LC*cos(qC)

(87) Loop[2] = Dot( Loop>, Ny> )
-> (88) Loop[2] = LA*sin(qA) + LB*sin(qB) - LN - LC*sin(qC)

(89) %--------------------------------------------------------------------
(90) %   Use the loop constraints to solve for initial values of qB, qC and qB',qC'
(91) %   (results depend on constants and initial values of qA and qA').
(92) Input  qA = 30 deg,  qA' = 0 rad/sec
(93) SolveSetInputDt( Loop = 0,   qB = 60 deg,  qC = 20 deg )

->   %  INPUT has been assigned as follows:
->   %   qB                        74.47751218592991       deg
->   %   qC                        45.52248781407007       deg

-> (94) qB' = -LA*sin(qA-qC)*qA'/(LB*sin(qB-qC))
-> (95) qC' = -LA*sin(qA-qB)*qA'/(LC*sin(qB-qC))
-> (96) qB'' = (LC*qC'^2-cos(qC)*(LB*cos(qB)*qB'^2+LA*(cos(qA)*qA'^2+sin(qA)*
qA''))-sin(qC)*(LB*sin(qB)*qB'^2+LA*(sin(qA)*qA'^2-cos(qA)*qA'')))/(LB*
sin(qB-qC))

-> (97) qC'' = -(LB*qB'^2-cos(qB)*(LC*cos(qC)*qC'^2-LA*(cos(qA)*qA'^2+sin(qA)*
qA''))-sin(qB)*(LC*sin(qC)*qC'^2-LA*(sin(qA)*qA'^2-cos(qA)*qA'')))/(LC*
sin(qB-qC))

(98) %--------------------------------------------------------------------
(99) %   Equations of motion with Kane's method (Optional: Solve for qA'').
(100) Dynamics = System.GetDynamicsKane()
-> (101) Dynamics[1] = 0.5*(mA+mB)*g*LA*sin(qA) + 0.5*LA*sin(qA-qB)*(2*H*cos(
qC)-(mB+mC)*g*sin(qC))/sin(qB-qC) + 0.25*(4*IA+mA*LA^2+4*IB*LA^2*sin(
qA-qC)^2/(LB^2*sin(qB-qC)^2)+LA^2*(mC+4*IC/LC^2)*sin(qA-qB)^2/sin(qB-
qC)^2+mB*LA^2*(4+sin(qA-qC)^2/sin(qB-qC)^2-4*sin(qA-qC)*cos(qA-qB)/sin
(qB-qC)))*qA'' - TA - 0.25*LA*(4*IB*sin(qA-qC)*(LC*qC'^2-sin(qC)*(LA*
sin(qA)*qA'^2+LB*sin(qB)*qB'^2)-cos(qC)*(LA*cos(qA)*qA'^2+LB*cos(qB)*qB'^2))
/(LB^2*sin(qB-qC)^2)-mC*sin(qA-qB)*(LB*qB'^2+sin(qB)*(LA*sin(qA)*qA'^2
-LC*sin(qC)*qC'^2)+cos(qB)*(LA*cos(qA)*qA'^2-LC*cos(qC)*qC'^2))/sin(
qB-qC)^2-4*IC*sin(qA-qB)*(LB*qB'^2+sin(qB)*(LA*sin(qA)*qA'^2-LC*sin(
qC)*qC'^2)+cos(qB)*(LA*cos(qA)*qA'^2-LC*cos(qC)*qC'^2))/(LC^2*sin(qB-
qC)^2)-mB*(2*LB*sin(qA-qB)*qB'^2+(2*cos(qA-qB)*(LC*qC'^2-sin(qC)*(LA*
sin(qA)*qA'^2+LB*sin(qB)*qB'^2)-cos(qC)*(LA*cos(qA)*qA'^2+LB*cos(qB)*qB'^2))
+sin(qA-qC)*(2*LA*sin(qA-qB)*qA'^2-(LC*qC'^2-sin(qC)*(LA*sin(qA)*qA'^2
+LB*sin(qB)*qB'^2)-cos(qC)*(LA*cos(qA)*qA'^2+LB*cos(qB)*qB'^2))/sin(
qB-qC)))/sin(qB-qC)))

(102) Solve( Dynamics = 0,   qA'' )
-> (103) qA'' = -(2*(mA+mB)*g*LA*sin(qA)+2*LA*sin(qA-qB)*(2*H*cos(qC)-(mB+mC)*g
*sin(qC))/sin(qB-qC)-4*TA-LA*(4*IB*sin(qA-qC)*(LC*qC'^2-sin(qC)*(LA*
sin(qA)*qA'^2+LB*sin(qB)*qB'^2)-cos(qC)*(LA*cos(qA)*qA'^2+LB*cos(qB)*qB'^2))
/(LB^2*sin(qB-qC)^2)-mC*sin(qA-qB)*(LB*qB'^2+sin(qB)*(LA*sin(qA)*qA'^2
-LC*sin(qC)*qC'^2)+cos(qB)*(LA*cos(qA)*qA'^2-LC*cos(qC)*qC'^2))/sin(
qB-qC)^2-4*IC*sin(qA-qB)*(LB*qB'^2+sin(qB)*(LA*sin(qA)*qA'^2-LC*sin(
qC)*qC'^2)+cos(qB)*(LA*cos(qA)*qA'^2-LC*cos(qC)*qC'^2))/(LC^2*sin(qB-
qC)^2)-mB*(2*LB*sin(qA-qB)*qB'^2+(2*cos(qA-qB)*(LC*qC'^2-sin(qC)*(LA*
sin(qA)*qA'^2+LB*sin(qB)*qB'^2)-cos(qC)*(LA*cos(qA)*qA'^2+LB*cos(qB)*qB'^2))
+sin(qA-qC)*(2*LA*sin(qA-qB)*qA'^2-(LC*qC'^2-sin(qC)*(LA*sin(qA)*qA'^2
+LB*sin(qB)*qB'^2)-cos(qC)*(LA*cos(qA)*qA'^2+LB*cos(qB)*qB'^2))/sin(
qB-qC)))/sin(qB-qC))))/(4*IA+mA*LA^2+4*IB*LA^2*sin(qA-qC)^2/(LB^2*sin(
qB-qC)^2)+LA^2*(mC+4*IC/LC^2)*sin(qA-qB)^2/sin(qB-qC)^2+mB*LA^2*(4+sin
(qA-qC)^2/sin(qB-qC)^2-4*sin(qA-qC)*cos(qA-qB)/sin(qB-qC)))

(104) %--------------------------------------------------------------------
(105) %   Numerical integration parameters.
(106) Input  tFinal = 7 sec,  tStep = 0.02 sec,  absError = 1.0E-07
(107) %--------------------------------------------------------------------
(108) %   List quantities to be output from ODE.
(109) Output  t sec,  qA deg,  qB deg,  qC deg
(110) %--------------------------------------------------------------------
(111) %   Solve ODEs (plot results).
(112) ODE()  MGFourBarDynamicsKaneEmbedded

(113) % Plot  MGFourBarDynamicsKaneEmbedded.1 [1, 2, 3, 4]
(114) %
(115) %
(116) %
(117) %********************************************************************
(118) %   Design 4-bar linkage to draw Valentines heart.
(119) Input LN := 2,  LA := 4 m,  LB := 4 m,  LC := 4 m,  mA := 20 kg
(120) %--------------------------------------------------------------------
(121) %   Control motor torque so qA' is nearly 60 degrees/sec.
(122) Constant  wMotor = 60 deg/sec     % Desired motor angular speed.
(123) Constant  kw = 9600 N*m/rad       % Control constant.
(124) TA := kw * (wMotor - qA')         % Motor torque on crank A from ground N.
-> (125) TA = kw*(wMotor-qA')

(126) H := 0                            % Remove horizontal force at point CB.
-> (127) H = 0

(128) %--------------------------------------------------------------------
(129) %   Output coupler point position for Valentines heart.
(130) Point     CouplerPoint( B )
(131) Constant  CouplerDistance = sqrt(8) m     % For conventional heart, use  2.66
(132) Constant  CouplerAngle = 45 degrees       % For conventional heart, use 41.20
(133) couplerPointBx = CouplerDistance * cos(CouplerAngle)
-> (134) couplerPointBx = CouplerDistance*cos(CouplerAngle)

(135) couplerPointBy = CouplerDistance * sin(CouplerAngle)
-> (136) couplerPointBy = CouplerDistance*sin(CouplerAngle)

(137) CouplerPoint.SetPosition( Bo,  couplerPointBx * Bx> + couplerPointBy * By> )
-> (138) p_Bo_CouplerPoint> = couplerPointBx*Bx> + couplerPointBy*By>

(139) couplerPointHorizontal = Dot( CouplerPoint.GetPosition(No),  Ny> )
-> (140) couplerPointHorizontal = LA*sin(qA) + couplerPointBx*sin(qB) + couple
rPointBy*cos(qB)

(141) CouplerPointVertical  = -Dot( CouplerPoint.GetPosition(No),  Nx> )
-> (142) CouplerPointVertical = couplerPointBy*sin(qB) - LA*cos(qA) - couplerP
ointBx*cos(qB)

(143) Output  couplerPointHorizontal m,  CouplerPointVertical m,  qA' deg/sec, TA N*m
(144) %--------------------------------------------------------------------
(145) %   Initial configuration is coupler link B horizontal (qB = 90 degrees).
(146) %   Use the loop constraints to solve for initial values of qA, qB.
(147) %   Use loopDt to solve for initial values of qB', qC'.
(148) Input  qB := 90 deg,  qA' := Input(wMotor, noUnitSystem) deg/sec
(149) SolveSetInput( Loop = 0,   qA = -15 deg,   qC  = 15 deg )

->    %  INPUT has been assigned as follows:
->    %   qA                       -14.47751218592985       deg
->    %   qC                        14.47751218592986       deg

(150) %--------------------------------------------------------------------
(151) %   Solve ODEs (plot results).
(152) ODE() MGFourBarDynamicsValentinesHeart

(153) Plot MGFourBarDynamicsValentinesHeart.2 [1, 2]
(154) %--------------------------------------------------------------------
(155) %   Save input together with program responses.
```
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