Dynamics: Newton-Euler Equations of Motion 0Contents7 Direct DynamicsNewton-Euler Equations of Motion 17.1 Compound Pendulum . . . . . . . . . . . . . . . . . . . . . . . 17.2 Double Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . 107.3 One-Link Planar Robot Arm . . . . . . . . . . . . . . . . . . . 197.4 Two-Link Planar Robot Arm . . . . . . . . . . . . . . . . . . 22Dynamics: Newton-Euler Equations of Motion 17 Direct DynamicsNewton-Euler Equations of MotionThe Newton-Euler equations of motion for a rigid body in plane motion arem¨rC=XF and ICzzα =XMC,or using the cartesian componentsm¨xC=XFx, m¨yC=XFy, and ICzz¨θ =XMC.The forces and moments are known and the differential equations are solvedfor the motion of the rigid body (direct dynamics).7.1 Compound PendulumFigure 7.1(a) depicts a compound pendulum of mass m and length L. Thependulum is connnected to the ground by a pin joint and is free to swingin a vertical plane. The link is moving and makes an instant angle θ(t) withthe horizontal. The local acceleration of gravity is g. Numerical application:L = 3 ft, g=32.2 ft/s2, G = m g=12 lb. Find and solve the Newton-Eulerequations of motion.SolutionThe system of interest is the link during the interval of its motion. Thelink in rotational motion is constrained to move in a vertical plane. First areference frame will be introduced. The plane of motion will be designatedthe xy plane. The y-axis is vertical, with the positive sense directed verticallyupward. 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 inertialreference frame are ı, , and k. The angle between the x and the link axis isdenoted by θ. The link is moving and hence the angle is changing with timeat the instant of interest. In the static equilibrium position of the link, theangle, θ, is equal to −π/2. The system has one degree of freedom. The angle,θ, is an appropriate generalized coordinate describing this degree of freedom.The system has a single moving body. The only motion permitted that bodyis rotation about a fixed horizontal axis (z-axis). The body is connected tothe ground with the rotating pin joint (R) at O. The mass center of the linkDynamics: Newton-Euler Equations of Motion 2is at the point C. As the link is uniform, its mass center is coincident withits geometric center.KinematicsThe mass center, C, is at a distance L/2 from the pivot point O and theposition vector isrOC= rC= xCı + yC, (7.1)where xCand yCare the coordinates of CxC=L2cos θ and yC=L2sin θ. (7.2)The link is constrained to move in a vertical plane, with its pinned location,O, serving as a pivot point. The motion of the link is planar, consisting ofpure rotation about the pivot point. The directions of the angular velocityand angular acceleration vectors will be perpendicular to this plane, in the zdirection.The angular velocity of the link can be expressed asω = ωk =dθdtk =˙θk, (7.3)ω is the rate of rotation of the link. The positive sense is clockwise (consistentwith the x and y directions defined above). This problem involves only a singlemoving rigid body and the angular velocity vector refers to that body. For thisreason, no explicit indication of the body, 1, is included in the specificationof the angular velocity vector ω = ω1. The angular acceleration of the linkcan be expressed asα =˙ω = αk =d2θdt2k =¨θk, (7.4)α is the angular acceleration of the link. The positive sense is clockwise.The velocity of the mass center can be related to the velocity of the pivotpoint using the relationship between the velocities of two points attached tothe same rigid bodyvC= vO+ ω × rOC=ı  k0 0 ωxCyC0= ω(−yCı + xC) =Lω2(− sin θı + cos θ) =L˙θ2(− sin θı + cos θ). (7.5)The velocity of the pivot point, O, is zero.The acceleration of the mass center can be related to the acceleration of theDynamics: Newton-Euler Equations of Motion 3pivot point (aO= 0) using the relationship between the accelerations of twopoints attached to the same rigid bodyaC= aO+ α × rOC+ ω × (ω × rOC) = aO+ α × rOC− ω2rOC=ı  k0 0 αxCyC0− ω2(xCı + yC) = α(−yCı + xC) − ω2(xCı + yC) =−(αyC+ ω2xC)ı + (αxC− ω2yC) =−L2(α sin θ + ω2cos θ)ı +L2(α cos θ − ω2sin θ) =−L2(¨θ sin θ +˙θ2cos θ)ı +L2(¨θ cos θ −˙θ2sin θ). (7.6)It is also useful to define a set of body-fixed coordinate axes. These are axesthat move with the link (body-fixed axe). The n-axis is along the length of thelink, the positive direction running from the origin O toward the mass centerC. The unit vector of the n-axis is n. The t-axis will be perpendicular tothe link and be contained in the plane of motion as shown in Fig. 7.1(a).Theunit vector of the t-axis is t and n × t = k. The velocity of the mass centerC in body-fixed reference frame isvC= vO+ ω × rOC=n t k0 0 ωL20 0=Lω2t =L˙θ2t, (7.7)where rOC= (L/2)n. The acceleration of the mass center C in body-fixedreference frame isaC= aO+ α × rOC− ω2rOC=Lα2t − ω2L2n =L¨θ2t −˙θ2L2n, (7.8)oraC= atC+ anC,with the componentsatC=L¨θ2t and anC= −L˙θ22n.Dynamics: Newton-Euler Equations of Motion 4Newton-Euler equation of motionThe link is rotating about a fixed axis. The mass moment of inertia of thelink about the fixed pivot point O can be evaluated from the mass momentof inertia about the mass center C using the transfer theorem. ThusIO= IC+ mL22=mL212+mL24=mL23. (7.9)The pin is frictionless and is capable of exerting horizontal and vertical forceson the link at OF01= F01xı + F01y, (7.10)where F01xand F01yare the components of the pin force on the link in thefixed axis system.The force driving the motion of the link is gravity. The weight of the linkis acting through its mass center will cause a moment about the pivot point.This moment will give the link a tendency to rotate about the pivot point.This moment will be given by the cross product of the vector from the pivotpoint, O, to the mass center, C, crossed into the weight force G = −mg.As the pivot point, O, of the link is fixed, the appropriate moment sum-mation point will be about that pivot point. The sum of the moments aboutthis