By Mark D. Ardema

Not like different books in this topic, which are likely to pay attention to 2-D dynamics, this article makes a speciality of the applying of Newton-Euler the right way to advanced, real-life 3-D dynamics difficulties. it truly is hence excellent for non-compulsory classes in intermediate dynamics.

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I y* i Dynamics zk / J / ^ yj Fig. 3-1 As an example, suppose a point moves with constant speed v along a curve given by a; = e'', y = sinr, and z = r^ where r is a known function of time. Then the position vector of the point at any time is Z = xi + yj + zk = eJ'i + sin rj + r k and from Eq. 2) the velocity is dr v_ = ieJ'i + cosrj + IrkA The speed of the point is therefore — V e'^'" + cos^ r + 4r2 dr ~dt so t h a t we may write V = -v/e^'" + cos^ r + Ar^ (fl + cos rj + 2rfc) In a similar way, the acceleration may be found from Eqn.

22) This very important relation is called the basic kinematic equation (BKE). We will see later that it is the same for 3-D motion as it is for 2-D. 10 Relative Velocity and Acceleration Let frame {i, i } have angular velocity a; and its origin have position ZB relative to frame {J, J } , and let point A move with respect to both frames (Fig. 2-13). It is our goal to relate the velocity and acceleration of point A as measured in one frame to the velocity and acceleration of A as measured in the other.

This frame is needed because we are given the problem's data relative to this frame (Fig. 2-16). The second frame, {i,j}, is attached to car B with origin at point B; this frame is needed because the answers to the problem are requested relative to this frame. Note that at the instant of interest, both frames are in identical positions; this does not mean they jReview of Planar Kinematics 25 I, i, -en Fig. 2-16 are identical frames, however, because an instant later, frame {i,j} will have moved relative to the ground (and thus to frame {I, J}).