Forces and torques

by Roberto Lot

> restart: libname := libname, "C:/MBSymba": with(MBSymba_r6):

> PDEtools[declare](x(t),y(t),z(t),alpha(t),beta(t),r(t),prime=t, quiet):

> T1 := translate(x(t),y(t),0); #set_frame_name(T1,'T1'):
body1 := make_BODY (T1 , m1 , Ix1,Iy2,Iz2): show(body1);

> T2 := T1 . rotate('Z',alpha(t)) . translate(r(t),0,0); #set_frame_name(T2,'T2'):

> body2 := make_BODY (T2 , m2 , Ix2,Iy2,Iz2): show(body2);

Forces

> Describe(make_FORCE);

# Make a FORCE from a VECTOR <FF> applied in <PP> that acts on the body <BA>

# and (optionally) reacts on the body <BR>.

make_FORCE( FF::VECTOR, PP::POINT, BA::{BODY, mframe},

BR::{BODY, mframe} := master_frame, $ )

The application point of the force

> A := make_POINT(T1,a,0,0): show(A);

4a - Define a force acting on a single body (i.e. reacting on the ground)

> make_VECTOR(T2,FX,FY,0):

> FA := make_FORCE(%, A , body1): show(FA);

> project(FA,ground): show(%);

4b - Define a force acting between two bodies

> make_VECTOR(T1,FX12,FY12,0):
F12 := make_FORCE(%, A , body1,body2): show(F12);

Vector operations such as projection can be performed on forces since they are vector entities

> project(F12,T2): show(%);

Torques

> Describe(make_TORQUE);

# Make a TORQUE from a VECTOR <FF> that acts on the body <BA> and (optionally)

# reacts on the body <BR>.

make_TORQUE( FF::VECTOR, BA::{BODY, mframe},

BR::{BODY, mframe} := master_frame, $ )

Define a torque acting on a single body

> M := make_TORQUE( make_VECTOR(ground, MX,MY,MZ), body2): show(M);

Define a torque acting between two bodies

> M12 := make_TORQUE( make_VECTOR(T2,MX12,MY12,MZ12), body1,body2): show(M12);

> project(M12,ground): show(%);

> O2A := join_points(origin(T2),A): show(O2A);

> make_VECTOR(ground,FX12,FY12,0):
F2 := make_FORCE(%, A , body2): show(F2);

> MO2 := cross_prod(O2A,F2): show(MO2);

Power and generalized forces

> show(FA);

> power(FA);

> velocity(CoM(body1)): show(%);

> generalized_force(FA, [x(t),y(t)]): Vector(%);

> simplify(power(F12));

> generalized_force(F12, r(t));

> generalized_force(F12, alpha(t));

> show(MO2);

> power(MO2);

> generalized_force(MO2, [r(t),alpha(t)]): Vector(%);

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