Rigid body


In physics, a rigid body, also known as a rigid object, is a solid body in which deformation is zero or negligible, when a deforming pressure or deforming force is applied on it. 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. Mechanics of rigid bodies is a field within mechanics where motions and forces of objects are studied without considering effects that can cause deformation.
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, where the mass is infinitely large. In quantum mechanics, a rigid body is usually thought of as a collection of point masses. For instance, molecules are often seen as rigid bodies.

Principles

Linear and angular position

The position of a rigid body is the position of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, namely that all its particles maintain the same distance relative to each other. If the body is rigid, it is sufficient to describe the position of at least three non-collinear particles. This makes it possible to reconstruct the position of all the other particles, provided that their time-invariant position relative to the three selected particles is known. However, typically a different, mathematically more convenient, but equivalent approach is used. The position of the whole body is represented by:
  1. the linear position or position of the body, namely the position of one of the particles of the body, specifically chosen as a reference point, together with
  2. the angular position of the body.
Thus, the position of a rigid body has two components: linear and angular, respectively. The same is true for other kinematic and kinetic quantities describing the motion of a rigid body, such as linear and angular velocity, acceleration, momentum, impulse, and kinetic energy.
The linear position can be represented by a vector with its tail at an arbitrary reference point in space and its tip at an arbitrary point of interest on the rigid body, typically coinciding with its center of mass or centroid. This reference point may define the origin of a coordinate system fixed to the body.
There are several ways to numerically describe the orientation of a rigid body, including a set of three Euler angles, a quaternion, or a direction cosine matrix. All these methods actually define the orientation of a basis set which has a fixed orientation relative to the body, relative to another basis set, from which the motion of the rigid body is observed. For instance, a basis set with fixed orientation relative to an airplane can be defined as a set of three orthogonal unit vectors b1, b2, b3, such that b1 is parallel to the chord line of the wing and directed forward, b2 is normal to the plane of symmetry and directed rightward, and b3 is given by the cross product.
In general, when a rigid body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as translation and rotation, respectively. Indeed, the position of a rigid body can be viewed as a hypothetic translation and rotation of the body starting from a hypothetic reference position.

Linear and angular velocity

Velocity and angular velocity are measured with respect to a frame of reference.
The linear velocity of a rigid body is a vector quantity, equal to the time rate of change of its linear position. Thus, it is the velocity of a reference point fixed to the body. During purely translational motion, all points on a rigid body move with the same velocity. However, when motion involves rotation, the instantaneous velocity of any two points on the body will generally not be the same. Two points of a rotating body will have the same instantaneous velocity only if they happen to lie on an axis parallel to the instantaneous axis of rotation.
Angular velocity is a vector quantity that describes the angular speed at which the orientation of the rigid body is changing and the instantaneous axis about which it is rotating. All points on a rigid body experience the same angular velocity at all times. During purely rotational motion, all points on the body change position except for those lying on the instantaneous axis of rotation. The relationship between orientation and angular velocity is not directly analogous to the relationship between position and velocity. Angular velocity is not the time rate of change of orientation, because there is no such concept as an orientation vector that can be differentiated to obtain the angular velocity.

Kinematical equations

Addition theorem for angular velocity

The angular velocity of a rigid body B in a reference frame N is equal to the sum of the angular velocity of a rigid body D in N and the angular velocity of B with respect to D:
In this case, rigid bodies and reference frames are indistinguishable and completely interchangeable.

Addition theorem for position

For any set of three points P, Q, and R, the position vector from P to R is the sum of the position vector from P to Q and the position vector from Q to R:
The norm of a position vector is the spatial distance.
Here the coordinates of all three vectors must be expressed in coordinate frames with the same orientation.

Mathematical definition of velocity

The velocity of point P in reference frame N is defined as the time derivative in N of the position vector from O to P:
where O is any arbitrary point fixed in reference frame N, and the N to the left of the d/dt operator indicates that the derivative is taken in reference frame N. The result is independent of the selection of O so long as O is fixed in N.

Mathematical definition of acceleration

The acceleration of point P in reference frame N is defined as the time derivative in N of its velocity:

Velocity of two points fixed on a rigid body

For two points P and Q that are fixed on a rigid body B, where B has an angular velocity in the reference frame N, the velocity of Q in N can be expressed as a function of the velocity of P in N:
where is the position vector from P to Q., with coordinates expressed in N
This relation can be derived from the temporal invariance of the norm distance between P and Q.

Acceleration of two points fixed on a rigid body

By differentiating the equation for the Velocity of two points fixed on a rigid body in N with respect to time, the acceleration in reference frame N of a point Q fixed on a rigid body B can be expressed as
where is the angular acceleration of B in the reference frame N.

Angular velocity and acceleration of two points fixed on a rigid body

As mentioned above, all points on a rigid body B have the same angular velocity in a fixed reference frame N, and thus the same angular acceleration

Velocity of one point moving on a rigid body

If the point R is moving in the rigid body B while B moves in reference frame N, then the velocity of R in N is
where Q is the point fixed in B that is instantaneously coincident with R at the instant of interest. This relation is often combined with the relation for the Velocity of two points fixed on a rigid body.

Acceleration of one point moving on a rigid body

The acceleration in reference frame N of the point R moving in body B while B is moving in frame N is given by
where Q is the point fixed in B that instantaneously coincident with R at the instant of interest. This equation is often combined with Acceleration of two points fixed on a rigid body.

Other quantities

If C is the origin of a local coordinate system L, attached to the body,
the spatial or twist 'acceleration' of a rigid body is defined as the spatial acceleration of C :
where
  • represents the position of the point/particle with respect to the reference point of the body in terms of the local coordinate system L
  • is the orientation matrix, an orthogonal matrix with determinant 1, representing the orientation of the local coordinate system L, with respect to the arbitrary reference orientation of another coordinate system G. Think of this matrix as three orthogonal unit vectors, one in each column, which define the orientation of the axes of L with respect to G.
  • represents the angular velocity of the rigid body
  • represents the total velocity of the point/particle
  • represents the total acceleration of the point/particle
  • represents the angular acceleration of the rigid body
  • represents the spatial acceleration of the point/particle
  • represents the spatial acceleration of the rigid body.
In 2D, the angular velocity is a scalar, and matrix A simply represents a rotation in the xy-plane by an angle which is the integral of the angular velocity over time.
Vehicles, walking people, etc., usually rotate according to changes in the direction of the velocity: they move forward with respect to their own orientation. Then, if the body follows a closed orbit in a plane, the angular velocity integrated over a time interval in which the orbit is completed once, is an integer times 360°. This integer is the winding number with respect to the origin of the velocity. Compare the amount of rotation associated with the vertices of a polygon.

Instantaneous rotation axis formulae

Assume that is a smooth 3-d vector field and is a point in,
with.
Denote the ball of radius centered at, and
.
We examine the expression
Linearizing the velocity field at gives
where is the Jacobian matrix at.
Decompose it into symmetric and antisymmetric parts:,
with antisymmetric.
By linear algebra, there exists a vector
such that.
In fact, direct computation shows that.
The symmetric part does not contribute to the integral, hence
Using the triple product identity, there holds
Integrating over the ball and using spherical symmetry,
so that
Incidentally, this formula provides an integral formulation of the curl of the vector field at :