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

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. I cZZ  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 x c  and y c  . M c  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.