![]() Representations for the general motionĪ general motion of a rigid body is one in which the body may not have a fixed point. Invoking Euler’s theorem, the rotation axis of the tensor is the axis of rotation for the motion of the rigid body. Thus, the motion of the body at the end of the time interval is characterized by the rotation tensor that, in general, has an axis and angle of rotation that differ from those associated with. To do this, we consider the motion of the body with a fixed point during the time interval, where we emphasize that, for the motion at hand, the body’s configuration changes from the initial state to the current configuration. We can arrive at an alternative, and more common, interpretation of Euler’s theorem that does not feature the reference configuration. Because the motion of the body in question is from the reference configuration to the current configuration, this axis depends on the choice of reference configuration. The axis referred to here is the rotation axis of the tensor. Thus, from ( 4), we can then infer Euler’s theorem on the motion of a rigid body:Įvery motion of a rigid body about a fixed point is a rotation about an axis through the fixed point. If we assume that one point of the body is fixed, then we can simplify ( 3) by choosing the fixed point to be the origin, in which case Recall that has an associated axis and angle of rotation. ![]() Specifically, in 1775, Euler showed that the motion of a body that satisfies ( 2) is such that Second, the rigid body’s motion preserves orientations. First, the distance between any two material points, say, and, of a rigid body remains constant for all motions. Euler’s theoremįor a rigid body, the nature of the motion function ( 1) can be simplified dramatically. Notice that the motion of a material point of depends on the particular material point and instant of time. As a result, using the reference configuration, we can then define the motion of the body as a function of and : This configuration is defined by the invertible function, and thus we may use the position vector of in the reference configuration to uniquely define the material point. We also establish a fixed reference configuration of the body. It is important to note that defines the state of the body at time. We use a subscript to indicate that the function depends on time because the location of the particle varies with time. ![]() This function maps a material point of to a point in three-dimensional Euclidean space. īorrowing from developments in continuum mechanics, we define the current configuration of the body,, as a smooth, one-to-one, onto function that has a continuous inverse. In both configurations, three material points of the body are denoted:, , and. Reference configuration and current configuration of a body. Referring to Figure 1, we denote a material point of by, say,, and the vector locates the material point, relative to a fixed origin, at time. 1.2 Representations for the general motionĪ body is considered to be a collection of material points, i.e., mass particles.
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