EGR 280 Mechanics 14 Kinematics of Rigid Bodies

EGR 280 Mechanics 14 – Kinematics of Rigid Bodies

Kinematics of rigid bodies We will group the various types of rigid-body motion into three categories: Translation – rectilinear or curvilinear motion where all of the particles that make up the body move along parallel paths Rotation about a fixed axis – the particles of the rigid body move in parallel planes along circles centered on the axis of rotation General plane motion – plane motion that is a combination of translation and rotation

Translation Consider a rigid body in translation. Let A and B be points on that body. A Y r. B/A r. A B r. B = r. A + r. B/A Z X Note that r. B/A does not change magnitude or direction. Therefore v. A = v. B and a. A = a. B If a rigid body is in translation, all of the points of the body have the same velocity and the same acceleration at any given instant.

Rotation about a fixed axis Consider a rigid body rotating about a fixed axis. Let P be a point of the body and r be its position with respect to a fixed point O. Angle θ is the angular coordinate. The velocity of P is tangent to the circle that P makes around the axis of rotation. Graphics and problem statements © 2004 R. C. Hibbeler. Published by Pearson Education, Inc. , Upper Saddle River, NJ. Let ω = dθ/dt be the angular velocity of the body, acting in the direction (by the RHR) of the axis of rotation. The velocity of P can be written as v = dr/dt = ω × r

and the acceleration is, by definition a = dv/dt = d(ω × r)/dt = (dω/dt × r) + (ω × dr/dt) = α × r + ω × (ω × r) where α = dω/dt is the angular acceleration. Two-dimensional fixed-point rotation If the motion takes place in two dimensions, then the angular velocity and angular acceleration will always be perpendicular to the plane of the motion. v = ω × r = ωk × r (v = rω) v Y a = α × r + ω × (ω × r) = αk × r + ωk × (ωk × r) r = αk × r – ω2 r ωk X

Note that the acceleration has two components, one directed toward O with magnitude equal to rω2, and the other tangent to the circular path of P with magnitude equal α (compare to a = (dv/dt)et + (v 2/ρ)en). We have: ω = dθ/dt α = dω/dt = d 2θ/dt 2 α = dω/dt = (dω/dθ)(dθ/dt) = ω(dω/dθ) Special cases: Uniform rotation: α = 0 θ = θ 0 + ωt Uniformly accelerated rotation: α constant θ = θ 0 + ω0 t + ½ αt 2 ω = ω0 + αt ω2 = ω02 + 2αθ

General plane motion A Any plane motion can be considered as the sum of a translation and a rotation: r. A B Y v. B = v. A + v. B/A where v. B/A is simply a rotation about A: v. B/A = ωk × r. B/A v. B = v. A + ωk × r. B/A Z r. B X v. B/A = ωr. B/A • Velocity is a property of a point; angular velocity is a property of a body • The sense of the relative velocity may change with a change in reference point

and the absolute acceleration of any point B on the rigid body can be written as: a. B = a. A + a. B/A = a. A + α × r. B/A + ω × (ω × r. B/A) = a. A + αk × r. B/A + ωk × (ωk × r. B/A) = a. A + αk × r. B/A – ω2 r. B/A

Special cases: Rolling If a circular cylinder rolls without slipping on a surface: v. B = v. A + ωk × r. B/A 0 = -vi + ωk × (-rj) v v = ωr ω A r B If two pulleys are joined by a belt: B v. B = v. A + ω1 k × r. B/A ω1 = ω1 k × r. B/A A v. C = v. D + ω2 k × r. C/D = ω2 k × r. C/D If the belt doesn’t slip or stretch, v. B=v. C and ω1 r 1= ω2 r 2 C D ω2

Special cases: Rolling When the teeth of two gears mesh, their pitch circles roll on each other without slipping: The velocity of point C on gear A must be the same as the velocity of point C on gear B: v. C = v. A + ω1 k × r. C/A = -ω1 k × r. C/A v. C = v. B + ω2 k × r. C/B = ω2 k × r. C/B ω1 r 1= -ω2 r 2 ω2 ω1 A B C v. C The number of teeth on each gear, N, must be an integer value, and is proportional to the pitch diameter: ω1 N 1= -ω2 N 2