Newton-Euler's Law of Rigid body motion.pptx

1. Newton Euler’s Law of Rigid
Body Motion

2. Rigid Body
• In physics, a rigid body, also known as a rigid object, is a solid body in
which deformation is zero or negligible.
• The distance between any two given points on a rigid body remains constant in time
regardless of external forces or moments exerted on it.
• A rigid body is usually considered as a continuous distribution of mass.
• In the study of special relativity, a perfectly rigid body does not exist; and objects can
only be assumed to be rigid if they are not moving near the speed of light.
• In quantum mechanics, a rigid body is usually thought of as a collection of point
masses. For instance, molecules (consisting of the point masses: electrons and nuclei)
are often seen as rigid bodies.
• The rigid body particles are not affected by stress, strain, or vibrations.
• There are no internal degrees of freedom for the particles in a rigid body.

3. Rigid Body Motion
• Rigid motion can be translational, rotational, or both.
• Rigid body structures and dynamics can get complicated; most rigid body
motion is three-dimensional. The motion of robots and the motion
of aerospace vehicles are two examples.
• When rigid body motion is confined to two-dimensional space, it is easy to
visualize and solve problems mathematically. Such rigid body dynamic
problem-solving requires limited mathematical tools and manipulations.
• However, if rigid bodies are involved with the three-dimensional motion, the
solving process becomes more tedious. The Newtonian approach to mechanics
is helpful in such rigid body motion problems. The Newton-Euler equation is
employed in most rigid body dynamics problem-solving.

4. Introduction
• In classical mechanics, the Newton–Euler equations describe the combined
translational and rotational dynamics of a rigid body.
• Traditionally the Newton–Euler equations is the grouping together of Euler's
two laws of motion for a rigid body into a single equation with 6 components,
using column vectors and matrices.
• These laws relate the motion of the center of gravity of a rigid body with the
sum of forces and torques (or moments) acting on the rigid body.

5. Newton’s Law of Motion
Euler’s Law of Motion
The equations (1) and (2) correspond to Newton’s second law and equation (3) is the Euler equation. IcZZ is the
central moment of inertia. Θ is the angle of rotation. The double dots above the rotations represent the second
derivative. The displacement in the x and y direction are given by xc and yc . Mc is the total torque acting about the
center of mass.

6. Newton Euler’s Equation
The Newton-Euler equation gives the relationship between the motion of the center of gravity
of a rigid body with the sum of forces and moments acting on the rigid body.
With respect to a coordinate frame whose origin coincides with
the body's center of mass for τ(torque) and an inertial frame of
reference for F(force), they can be expressed in matrix form as:

7. Application
• The Newton-Euler equation governs the motion of humanoid robots and
spacecraft. While designing robots and spacecraft.