RBDyn problem

Introduction

So, Rigid body dynamics problems are problems that are described by the following requirements:

• There is a inertial frame of reference, static, absolute, that we call R0, in the origin
• There are some referential frames, which are relatively connected to the frame R0 using translation and rotation vectors
• There are some bodies in the space
  • with a Reference Frame called baseframe, which moves and rotates with the object.
  • with a mass m
  • with a center of mass CM, relative to the baseframe
  • with a moment of inertia II, relative to the baseframe
• Initial configuration of the system: position and velocity at t=0 for each object

Exemple Simple Pendulum

On the page Exemple: Simple Pendulum (Angular parameterization) we have the mathematical description of the problem.

The code to describe the problem is:

import rbdyn

# Definition of constant values
a = 1 # radius
m = 1 # mass

# Definition of the variables
theta = rbdyn.variable("theta")

# Definition of the frames of reference
R0 = rbdyn.FrameReference()  # Inertial frame of reference
R1 = rbdyn.FrameReference(R0, rotation=(theta, "z"))
R2 = rbdyn.FrameReference(R1, translation=(a, 0, 0))

ball = rbdyn.Object(R2, name="ball")
ball.mass = m

E = ball.KineticEnergy(R0)
print("Kinetic Energy = ")
print(E)

gives us the result

Kinetic Energy = 
0.5 * m * dtheta**2

Exemple rotating bar

The second example, we have a bar of mass m, length l, its center at the position (x, y, 0) and it's rotationed at an angle theta relative to the vector z. So, we describe the problem as

import rbdyn

# Definition of constant values
m = 1  # The mass
l = 1  # The bar's length

# Definition of the variables
x = rbdyn.variable("x")
y = rbdyn.variable("y")
theta = rbdyn.variable("theta")

# Definition of the frames of reference
R0 = rbdyn.FrameReference()  # Inertial frame of reference
R1 = rbdyn.FrameReference(base=R0, translation=(x, y, 0))
R2 = rbdyn.FrameReference(base=R1, rotation=(theta, "z"))

# Definition of the object
CM = (0, 0, 0)  # Center of mass
II = (l**2 / 12) * np.array([[0, 0, 0],
                             [0, 1, 0],
                             [0, 0, 1]])
bar = Objet(base=R2, name="bar")
bar.mass = m  # Put the mass into the object
bar.CM = CM  # Put the CM into the object
bar.II = II  # Put the kinematic moment of inertia tensor

# Calculus of Kinetic Energy relative of the frame R0
E = bar.KineticEnergy(R0)
print("Kinetic Energy = ")
print(E)

And it gives the result

Kinetic Energy = 
m * (0.0833333333333333 * l**2 * dtheta**2 + dx**2 + dy**2) / 2