MIT2_017JF09_p14

# MIT2_017JF09_p14 - 14 PENDULUM DYNAMICS AND LINEARIZATION...

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± 14 PENDULUM DYNAMICS AND LINEARIZATION 35 14 Pendulum Dynamics and Linearization Consider a single-link arm, with length l and all the mass m concentrated at the end. A motor at the ﬁxed pivot point supplies a controllable torque τ . As drawn, a positive torque drives the arm counter-clockwise, so as to drive the arm angle θ positive. The arm is free to swing around the full 360 degrees; gravity pulls the arm downward. There is no damping. T l m W g 1. Derive and state the equation of motion for this system. Solution: It is a basic rotary moment of inertia with a gravity eﬀect and input torque. We will get ml 2 θ ¨ = τ mgl cos θ. 2. For each of the linearization angles [-90,0,90] degrees, answer the following: (a) What is the static torque needed to support the arm in this conﬁguration? Solution: the three angles given, the static torque is the amount needed to just balance the gravity torque. These values are [0, mgl , 0]; note that the upward and downward conﬁgurations do not take any static torque to maintain.

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MIT2_017JF09_p14 - 14 PENDULUM DYNAMICS AND LINEARIZATION...

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