MIT16_07F09_Lec06

# MIT16_07F09_Lec06 - S Widnall J Peraire 16.07 Dynamics Fall...

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Unformatted text preview: S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6- Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed cartesian coordinate system . Then we showed how they could be expressed in polar coordinates. While it is clear that the choice of coordinate system does not affect the final answer, we shall see that, in practical problems, the choice of a specific system may simplify the calculations and/or improve the understanding considerably. In many problems, it is useful to use a coordinate system that aligns with both the velocity vector of the particle—which is of course tangent to the particle’s trajectory— and the normal to this trajectory, forming a pair of orthogonal unit vectors. The unit vectors aligned with these two directions also define a third direction, call the binormal which is normal to both the velocity vector and the normal vector. This can be obtained from these two unit vectors by taking their cross product. Such a coordinate system is called an Intrinsic Coordinate System . Intrinsic coordinates: Tangential, Normal and Binormal compo- nents. We follow the motion of a point using a vector r ( t ) whose position along a known curved path is given by the scalar function s ( t ) where s ( t ) is the arc length along the curve. We obtain the velocity v from the time rate of change of the vector r ( t ) following the particle d r d r ds d r v = = = s ˙ (1) dt ds dt ds We identify the scalar s ˙ as the magnitude of the velocity v and d ds r as the unit vector tangent to the curve at the point s ( t ). Therefore we have v = v e t , (2) where r ( t ) is the position vector, v = s ˙ is the speed, e t is the unit tangent vector to the trajectory, and s is the path coordinate along the trajectory. 1 The unit tangent vector can be written as, d r e t = . (3) ds The acceleration vector is the derivative of the velocity vector with respect to time. Since d ds r depends only on s , using the chain rule we can write d v d r d d r d r 2 d 2 r d r 2 d 2 r a = = ¨ s + ˙ s = ¨ s + ˙ s = v ˙ + v . (4) dt ds dt ds ds ds 2 ds ds 2 The second derivative d 2 r /ds 2 is another property of the path. We shall see that it is relate to the radius of curvature. Taking the time derivative of Equation (2), an alternate expression can be written in terms of the unit vector e t as a = dv e t + v d e t . (5) dt dt The vector e t is the local unit tangent vector to the curve which changes from point to point. Consequently, the time derivative of e t will, in general, be nonzero. The time derivative of e t can be written as, d e t d e t ds d e t = = v . (6) dt ds dt ds In order to calculate the derivative of e t , we note that, since the magnitude of e t is constant and equal to one, the only changes that e t can have are due to rotation, or swinging ....
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MIT16_07F09_Lec06 - S Widnall J Peraire 16.07 Dynamics Fall...

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