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کانیکمیسدمهـندادکشنه

Chapter 3 : Relative Motion

Advanced Dynamics

Lecturer: Hossein Nejat

• A moving body, such as an automobile, frequently provides a useful reference frame for our observations of motion.
• It is apparent from above figure that the absolute position is related to the relative position :
• The task of adding right side terms of above equation is complicated, because the axes used to represent the vectors are not parallel.

• Let us consider a general situation in which two coordinate systems, xyz and x'y'z', are employed to represent the components of a vector.
• The components of i' are the projections of the vector onto the axes of xyz, which, in turn, are determined from the direction angles according to Chapter 3 : Relative Motion, 3.1 Rotation Transformations

• Define lp'q = lqp' to be the cosine of the angle between axis p' and axis q, with p and q representing x, y, or z. Extending to other directions leads :
• It is convenient to rewrite these equations in matrix form as
• Where
• The matrix [R] is the rotation transformation matrix. Chapter 3 : Relative Motion, 3.1 Rotation Transformations (I)

• Matrices satisfying are said to be orthonormal. On the other hand. Thus, the elements of the product are
• where denotes the Kronecker delta; = 1 if p = q and = 0 otherwise.
• The equality gives rise to an important property, leading to: which is a useful check for computations. Although the rotation matrix have 9 components, due to orthonormality, it must satisfy 6 relations, which means that there are only three independent direction angles. For example, the values of in last figure are not independent because

• The importance of the transformation matrix stems from the fact that it relates the components of arbitrary vectors with respect to two coordinate systems, not just the unit vectors.
• We have:

• Example: A force F may be described in terms of its components with respect to either the XYZ or xyz reference frames shown in the sketch. a) If determine the components of the force relative to the xyz coordinate system. b) If determine the components of the force relative to the XYZ coordinate system.

• The corresponding equations derived from each element of the product are

• Now that [R] is known, we may transform the vectors. In case (a), we know the XYZ components, so
• In case (b), we use the inverse transformation because we know the xyz components. Specifically,

• The general task of evaluating the rotation transformation [R] between two coordinate systems often is more readily achieved by picturing one coordinate system as having moved away from the other in a sequence of simple rotations.
• The ultimate orientation of a reference frame that undergoes a spatial rotation clearly will depend upon both the orientation of each axis of rotation and the amount of rotation about each axis.
• In a body-fixed rotation sequence, each rotation is about one of the axes of the coordinate system at the preceding step in the sequence.
• In a space-fixed rotation sequence, each rotation is about one of the axes of the fixed coordinate system. Chapter 3 : Relative Motion, 3.2 Finite Rotations

3.2.1 BODY-FIXED ROTATIONS

• As shown in last figure, we choose the fixed XYZ system such that it coincides with the initial orientation of xyz.
• The result of the first rotation is
• The result of the second rotation is

Concluding Remark:

• Let xyz be a reference frame that undergoes a sequence of rotations about its own axes, and let XYZ mark the initial orientation of xyz. The transformation from XYZ to the final xyz components is obtained by premultiplying (from right to left) the sequence of transformation matrices for the individual single-axis rotations. For n rotations,