Direct Dynamics: Newton–Euler Equations of Motion

Abstract

The Newton-Euler equations of motion for a rigid body in plane motion are m¨rm¨r C = ∑ F and I Czz α = ∑ M C , or using the Cartesian components m ¨ x C = ∑ F x , m ¨ y C = ∑ F y , and I Czz¨θCzz¨ Czz¨θ = ∑ M C. The forces and moments are known and the differential equations are solved for the motion of the rigid body (direct dynamics). 5.1 Compound Pendulum Exercise Figure 5.1a depicts a compound pendulum of mass m and length L. The pendulum is connnected to the ground by a pin joint and is free to swing in a vertical plane. The link is moving and makes an instant angle θ (t) with the horizontal. The local acceleration of gravity is g. Numerical application: L = 3 ft, g = 32.2 ft/s 2 , G = mg = 12 lb. Find and solve the Newton-Euler equations of motion. Solution The system of interest is the link during the interval of its motion. The link in rota-tional motion is constrained to move in a vertical plane. First, a reference frame will be introduced. The plane of motion will be designated the (x, y) plane. The y-axis is vertical, with the positive sense directed vertically upward. The x-axis is horizontal and is contained in the plane of motion. The z-axis is also horizontal and is perpendicular to the plane of motion. These axes define an inertial reference frame. The unit vectors for the inertial reference frame are ı, j, and k. The angle between the x and the link axis is denoted by θ. The link is moving and hence the angle is changing with time at the instant of interest. In the static equilibrium position of the link, 183