# Describing Rigid Body Dynamics Using Newton-Euler Equations

### Key Takeaways

• Rigid bodies retain their shape irrespective of the force applied to them.

• Newton-Euler equations are the fundamental equations used in classical mechanics to describe the combined rotational and translational dynamics of a rigid body.

• Newton-Euler equations give 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. The Newton-Euler equation governs the motion of humanoid robots

Forces and motion are related. The fundamental law that correlates force and motion is Newton’s second law. According to this law, the net force acting on a body is equal to the product of mass and acceleration.

When applying Newton's second law to rigid bodies, usually the acceleration of the center of mass is considered for translational motion. The Euler equation relates moments with the centroidal mass moment of inertia and angular acceleration of the rigid body and is of great importance in rotational rigid body dynamics.

The two equations mentioned above for rigid body form the Newton-Euler equation. In engineering mechanics, the Newton-Euler equation is used to analyze rigid body dynamics.

## Rigid Body Dynamics

A rigid body can be defined as a body consisting of many mass particles. The mass particles are held inside the rigid body by massless bonds of length. The distance between the particles in a rigid body remains unchanged even under the application of force. Rigid bodies retain their shape irrespective of the force applied to them. 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. The concept of no internal degrees of freedom and constant distance between the particles helps in assessing the rigid body dynamics in the Newtonian approach.

## Mechanics of Rigid Body Systems

It is necessary to understand the mechanics of rigid bodies when designing systems with translational as well as rotational motion. There are two approaches:

1. Lagrange’s equation-based approach - Scalar quantities, such as potential energy and kinetic energy, are used. It also introduces the concepts of virtual displacements, generalized forces, and generalized coordinates.
2. Newtonian approach - Six scalar quantities are used to describe the motion of the rigid body in space. The Newton-Euler equation is utilized to describe rigid body motion.

## 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.

## The Newton-Euler Equation

The three laws for a hypothetical body (called a particle) were presented by Sir Isaac Newton. The equations formulated by Newton dealt with a body of mass with no extension. His successors extended Newton's laws of motion to various collections of bodies such as fluids and rigid bodies. One of the most famous contributions to the dynamics of rigid bodies was made by L. Euler.

Newton-Euler equations are the fundamental equations used in classical mechanics to describe the combined rotational and translational dynamics of a rigid body. The motion of a rigid body in a plane can be described using the Newton-Euler equation. It is the combination of Newton’s second law of motion and the Euler equation. Newton’s second law defines the relationship between forces and motion, whereas the Euler equation explains the rotational dynamics of rigid body motion.

Newton-Euler equations for a rigid body in plane motion in the x, y plane can be given by: 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. 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.

## Summarizing the Newton-Euler Equation

There are two summaries that can be deduced from the Newton-Euler equation:

1. The Newton-Euler equation groups Euler’s equations of motion of a rigid body into a single equation with six components, given below. • F is the total force acting on the center of mass

• τ is the total torque acting about the center of mass

• M is the mass of the body

• I3 is the identity matrix of order 3 x 3

• Icm is the moment of inertia about the center of mass

• αcm is the acceleration of the center of mass

• α is the angular acceleration of the body

• ω is the angular velocity of the body

1. 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.

The Newton-Euler equation governs the motion of humanoid robots and spacecraft. While designing robots and spacecraft, Cadence CFD tools can be utilized to understand the dynamics of motion.