# Pendulum dynamics: Newton’s vs Lagrange’s approach

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

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

## Definition of the mechanical systems

Define the body frame

> TP := rotate('Z',theta(t)) * translate(0,-L,0);

Define the body

> pendulum := make_BODY(TP,m,0,0,IG): show(pendulum);

Define acceleration due to gravity

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

Define the pin joint A

> A := origin(ground): show(A);

Reaction force (body coordinates)

> make_VECTOR(TP,TA,RA,0):
RF := make_FORCE(%, A, pendulum): show(RF);

Define the damping torque

> make_VECTOR(ground,0,0,-c*diff(theta(t),t)):
DT := make_TORQUE(%, pendulum): show(DT);

## Newton's equations

> Describe(newton_equations);

> eqnsN := newton_equations({ pendulum, RF, DT}): show(eqnsN);

# Compute the Newton's equations of the multibody system <MBS>, which may

# include bodies and forces.

newton_equations( MBS::{set, list}, verbose::boolean := false, \$ ) :: VECTOR

> eqnsN := project(eqnsN,TP): show(eqnsN);

Euler's equations with respect to the fixed point A, in moving frame

> Describe(euler_equations);

# Euler's equations of the multibody system <MBS> with respect the (optional)

# point <AA>.

euler_equations( MBS::{set, OBJECT, list}, AA::POINT := NULL,

verbose::boolean := false, \$ )

> eqnsE := euler_equations({ pendulum, RF, DT},CoM(pendulum)): show(eqnsE);

> eqnsE := euler_equations({ pendulum, RF, DT},A): show(eqnsE);

Equation(s) of motion for the system

> eqn := comp_Z(eqnsE);

## Numerical integration

Integration of the equation of motion

> data := [m=4, L=0.2, g=9.81, IG=0.12, c=0.5];

Define the initial conditions

> ic := theta(0)=2.8, D(theta)(0)=0.0;

Integrate the equation(s) of motion

> motion := dsolve( { subs(data,eqn) , ic }, type=numeric, range=0..5):

> plots[odeplot](motion, [[t,theta(t)],[t,D(theta)(t)]], color=[red,blue],numpoints=200);

## Reaction forces

Netwon's equations

> Describe(euler_equations);
eqnsN := newton_equations({ pendulum, RF, DT}): show(eqnsN);

# Euler's equations of the multibody system <MBS> with respect the (optional)

# point <AA>.

euler_equations( MBS::{set, OBJECT, list}, AA::POINT := NULL,

verbose::boolean := false, \$ )

Calculation of the reaction forces

> RA = solve(comp_Y(eqnsN),RA);

> TA = solve(comp_X(eqnsN),TA);

## Lagrange's equations

> KE := simplify(kinetic_energy(pendulum));

> PE := gravitational_energy(pendulum);

> eqnL := lagrange(KE-PE,theta(t),t) - generalized_force({DT},theta(t));

sintetically

> eqnL := lagrange_equations({pendulum,DT},[theta(t)]);

>

>