MGBeamOnTwoCablesStaticsViaKaneEmbedConstraints.html (MotionGenesis input/output).
(1) % File: MGBeamOnTwoCablesStaticsViaKaneEmbedConstraints.txt
(2) % Problem: Static configuration of a beam supported by two cables.
(3) %--------------------------------------------------------------------
(4) NewtonianFrame N % Ceiling with Nx> horizontally right, Ny> vertically down
(5) RigidBody B % Beam with Bx> pointing from Bo to Bc and Bz> = Nz>.
(6) Point Nc( N ) % Point of N attached to cable C.
(7) Point Bc( B ) % Point of B attached to cable C.
(8) %--------------------------------------------------------------------
(9) Constant LN = 6 m % Distance between No and NC.
(10) Constant LB = 4 m % Distance between Bo and BC.
(11) Constant LA = 2.7 m % Length of cable A (connects Bo to No).
(12) Constant LC = 3.7 m % Length of cable C (connects BC to NC).
(13) Constant g = 9.8 m/s^2 % Earth's gravitational acceleration.
(14) B.SetMass( m = 120 kg )
(15) %--------------------------------------------------------------------
(16) Variable x', y' % Nx> and Ny> measures of Bo's position vector from No.
(17) Variable q' % Angle from Nx> to Bx> with positive sense +Nz>.
(18) %--------------------------------------------------------------------
(19) % Rotational kinematics relating Bx>, By>, Bz> to Nx>, Ny>, Nz>.
(20) B.RotateZ( N, q )
-> (21) B_N = [cos(q), sin(q), 0; -sin(q), cos(q), 0; 0, 0, 1]
-> (22) w_B_N> = q'*Bz>
(23) %--------------------------------------------------------------------
(24) % Position and velocity vectors (for partial velocities).
(25) Bo.SetPositionVelocity( No, x*Nx> + y*Ny> )
-> (26) p_No_Bo> = x*Nx> + y*Ny>
-> (27) v_Bo_N> = x'*Nx> + y'*Ny>
(28) Bcm.SetPositionVelocity( Bo, 0.5*LB*Bx> )
-> (29) p_Bo_Bcm> = 0.5*LB*Bx>
-> (30) v_Bcm_N> = 0.5*LB*q'*By> + x'*Nx> + y'*Ny>
(31) BC.SetPositionVelocity( Bo, LB*Bx> )
-> (32) p_Bo_Bc> = LB*Bx>
-> (33) v_Bc_N> = LB*q'*By> + x'*Nx> + y'*Ny>
(34) NC.SetPosition( No, LN*Nx> )
-> (35) p_No_Nc> = LN*Nx>
(36) %--------------------------------------------------------------------
(37) % Constraint relating cable lengths to magnitudes of position vectors.
(38) LengthConstraint[1] = LA^2 - Bo.GetDistanceSquared( No )
-> (39) LengthConstraint[1] = LA^2 - x^2 - y^2
(40) LengthConstraint[2] = LC^2 - Bc.GetDistanceSquared( Nc )
-> (41) LengthConstraint[2] = LC^2 + 2*LB*cos(q)*(LN-x) - LB^2 - y^2 - 2*LB*y*sin(q)
- (LN-x)^2
(42) %--------------------------------------------------------------------
(43) % Relevant forces on beam (to calculate generalized forces).
(44) % Note: Cable tensions TA and TC do not contributes to generalized force
(45) % because both LengthConstraint[1] and LengthConstraint[2] are embedded.
(46) Bcm.AddForce( m*g*Ny> )
-> (47) Force_Bcm> = m*g*Ny>
(48) %--------------------------------------------------------------------
(49) % Differentiate length constraints to solve for y' and q' in terms of x'.
(50) Solve( Dt(LengthConstraint) = 0, y', q' )
-> (51) y' = -x*x'/y
-> (52) q' = -(LB*cos(q)-LN-LB*x*sin(q)/y)*x'/(LB*(y*cos(q)+sin(q)*(LN-x)))
(53) %--------------------------------------------------------------------
(54) % Statics via Kane's embedded method (y' and q' are in terms of x').
(55) % This method embeds both LengthConstraint[1] and LengthConstraint[2].
(56) SetGeneralizedSpeed( x' )
(57) Statics = System.GetStaticsKane()
-> (58) Statics[1] = -0.5*m*g*(2*x/y+cos(q)*(LB*cos(q)-LN-LB*x*sin(q)/y)/(y*cos
(q)+sin(q)*(LN-x)))
(59) %--------------------------------------------------------------------
(60) % Solve set of 3 nonlinear algebraic equations (with guess).
(61) Solve( [Statics; LengthConstraint] = 0, x = 1 m, y = 3 m, q = 15 deg )
-> (62) x = 0.8153669
-> (63) y = 2.573942
-> (64) q = 0.2257464 % or q = 12.93432 deg.
(65) %--------------------------------------------------------------------
(66) % Post-process: Calculate distance between No and Bcm.
(67) distanceNoToBcm = EvaluateToNumber( Bcm.GetDistance(No) )
-> (68) distanceNoToBcm = 4.095517
(69) %--------------------------------------------------------------------
(70) % Save input and output responses.
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