dyna_newton.html

2. Newtonian Mechanics

Newtonian Mechanics has a VECTORIAL approach to the problems

restart:

libname := `C:/libs/MBSymba.lib`, libname: with(MBSymba_r4):

PDEtools[declare](prime=t,quiet):

2.1 Components of a Mechanical System

Bodies

Figure depicting the rigid body

Creating a rigid body

command structure: body1 := make_BODY (point of Center of Mass, mass, Ix comp, Iy comp, Iz comp, prod Cyz, prod Cxz, prod Cxy):

1 - Define the body reference frame

> TM := translate(x(t),0,z(t)) * rotate('Y',mu(t)) * rotate('Z',psi(t));

2 - Define the point of depicting the center of gravity

>G := make_POINT(TM,xg,yg,zg): show(G);

3 - Define the rigid body

If the body has only principal moments of inertia

> body1 := make_BODY (G , m , Ix,Iy,Iz): show(body1);

If the body has principal moments and products of inertia

body1 := make_BODY (G , m , Ix,Iy,Iz, Cyz,Cxz,Cxy): show(body1);

Kinetic energy of the body

> kinetic_energy(body1);

Define the acceleration due to the gravity as a vector

`>`	_gravity := make_VECTOR(ground, 0,0,g): show(_gravity);

Gravitational (potential) energy of the body

`>`	gravitational_energy(body1);

Check the expression of gravitational energy (result should equal zero)

`>`	- Zcomp(project(G,ground)) * g * m; % - gravity_energy(body1);

Forces

Define a force acting on a body

command structure: make_FORCE(ref frame, x-comp, y-comp, z-comp , pt of application, body acted on, body reacted against):

1 - The reference frame

`>`	TF := rotate('X',phi(t));

2 - The application point of the force

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

3 - The body on which the force acts

`>`	show(body1);

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

`>`	Fext := make_FORCE(TF,FX,FY,FZ, A , body1): show(Fext);

4b - Define a force acting between two bodies

`>`	body2 := make_BODY(TF,m): show(body2);

`>`	F12 := make_FORCE(TF,FX12,FY12,FZ12, A , body1,body2): show(F12);

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

`>`	project(F12,ground): show(%);

Torques

Torques can be defined in a similar fashion to forces

command structure: make_TORQUE(ref frame, Mx-comp, My-comp, Mz-comp , body acted on, body reacted against)

Define a torque acting on a single body

`>`	M := make_TORQUE(TF, MX,MY,MZ, body1): show(M);

Define a torque acting between two bodies

`>`	M12 := make_TORQUE(TF, MX12,MY12,MZ12, body1,body2): show(M12);

2.2 Linear Momentum

The linear momentum of a body is the product of its mass multiplied by the velocity of the CoM (Center of Mass)

`>`	LM1 := linear_momentum(body1): show(LM1); ground coordinates

`>`	LM1 := project(LM1,TM): show(LM1); body coordinates

2.3 Newton's Equations

command structure: eqnsN := newton_equations({body under consideration}, reference coordinate frame):

`>`	eqnsN := newton_equations({body1},ground): show(eqnsN); ground frame

2.4 Angular Momentum

1 - The angular momentum of a point mass G with respect to a point P is equal to the cross product of PG (the distance from the axis of rotation to that point) multiplied by the linear momentum of the mass.

`>`	MF := translate(x(t),y(t),z(t));

`>`	point_mass := make_BODY( MF, m): #show(point_mass);

`>`	G := CoM(point_mass): #show(G);

Axis or point about which the mass is rotating

`>`	P := make_POINT(ground,px,py,pz): #show(P);

Angular momentum

command structure: AM := cross_prod( perp dist btwn axis and pnt, linear momentum of the selected body):

`>`	PG := make_VECTOR(P,G): #show(PG);

`>`	AM := cross_prod(PG,linear_momentum(point_mass)): show(AM);

2 - The angular momentum of a rigid body with respect its center of mass is equal to the product of its inertia tensor by its angular velocity

command structure: AM1 := angular_momentum( selected body, CoM(selected body)):

`>`	show(body1);

`>`	show( angular_velocity(body1[frame]) );

`>`	AM1 := angular_momentum( body1, CoM(body1)): show(AM1);

3 - The angular momentum of a rigid body with respect to a point P is the sum of the angular moment with respect the CoM plus the cross product of PG and the linear momentum

command structure AMP: 0 angular_momentum (specified body, axis of point of reference):

`>`	AMP := angular_momentum(body1, P ): show(AMP);

2.5 Euler Equations

The time derivative of the angular momentum must be equal to the angular momentum of all external forces (active and reactive)

`>`	{body1}: The system

`>`	P: Axis or point about which the angular momentum is being calculated

`>`	ground: The reference frame

Euler eqns for the system

command structure: eqnsE := euler_equations({specified body} , axis of point of rotation , reference frame):

eqnsE := euler_equations({body1} , P , ground ):

`>`	show(eqnsE);