Ch. 9: Plane Kinematics of Rigid Bodies 473 9.0 Outline 473 Introduction 474 Rotation 478 Absolute Motion 486 Relative Velocity 508 Instantaneous Center of Zero Velocity 531 Relative Acceleration 544 Motion Relative to Rotating Axes 566 9.0 Outline Ch. 9: Plane Kinematics of Rigid Bodies 474 9.1 Introduction Kinematics of rigid bodies involves both linear and angular quantities. Usage 1. In designing the machines to perform the desired motion. 2. To determine the motion resulting from the applied force. A rigid body A system of particles for which the distance between the particles remain unchanged. Thus there will be no change in the position vector of any particle measured from the body-fixed coordinate system. 9.1 Introduction Ch. 9: Plane Kinematics of Rigid Bodies 475 9.1 Introduction Ch. 9: Plane Kinematics of Rigid Bodies 476 Plane motion of a rigid body all parts of the body move in parallel planes. The body then can be treated as a thin slab with motion confined to the plane of motion; plane that contains the mass center. Translation motion in which every line in the body remains parallel to its original position at all time. That is there is no rotation of any line in the body. The motion of the body is completely specified by the motion of any point in the body, since all points have same motion. Rectilinear translation all points in the body move in parallel straight lines of the same distance Curvilinear translation all points move on parallel curves of the same distance 9.1 Introduction Ch. 9: Plane Kinematics of Rigid Bodies 477 Rotation motion in which all particles move in circular paths about the axis of rotation. All lines in the body which are perpendicular to the axis of rotation rotate through the same angle in the same time. Circular motion of a point helps describe the rotating motion. General plane motion combination of translation and rotation. Principle of relative motion helps describe the general motion. Approaches 1. Direct calculation of the absolute displacement, velocity, and acceleration from the geometry 2. Use the principle of relative motion 9.1 Introduction Ch. 9: Plane Kinematics of Rigid Bodies 478 9.2 Rotation Rotation of a rigid body is described by its angular motion, which is dictated by the change in the angular position (specified by angle θ measured from any fixed line) of any line attached to the body. θθβ21= + ∆=∆θθ21 θθ21= θθ21= All lines on a rigid body in its plane of motion have the same angular displacement, the same angular velocity, and the same angular acceleration 9.2 Rotation Ch. 9: Plane Kinematics of Rigid Bodies 479 Angular motion relations Pick up any line in the plane of motion and associated it with the angular position coordinate θ. The angular velocity ω and accelerationα of a rigid body in plane rotation are defined. ωθ= αωθ= = ωωd= αθ d or θθd = θθ d Analogies between the linear and angular motion 9.2 Rotation Ch. 9: Plane Kinematics of Rigid Bodies 480 Rotation about a fixed axis All points other than those on the rotation axis move in concentric circles about the fixed axis. Curvilinear motion of a point A is related to the angular motion of the rigid body by the familiar n-t coordinate kinematic relationship. v= rω , r = constant at = vr = α v2 ar=ω 2 = n r 9.2 Rotation Ch. 9: Plane Kinematics of Rigid Bodies 481 Rotation about a fixed axis v = r =ω ×= r by vector differentiation, r constant ω = angular velocity of the vector r = angular velocity of the rigid body a=v= ωω××( r) + α × r arn =××ωω( ) art =α × 9.2 Rotation Ch. 9: Plane Kinematics of Rigid Bodies 482 P. 9/1 The two V-belt pulleys form an integral unit and rotate about the fixed axis at O. At a certain instant, point A on the belt of the smaller pulley has a velocity vA = 1.5 m/s, and point B 2 on the belt of the larger pulley has an acceleration aB = 45 m/s as shown. For this instant determine the magnitude of the acceleration aC of point C and sketch the vector in your solution. 9.2 Rotation Ch. 9: Plane Kinematics of Rigid Bodies 483 P. 9/1 find ωpulley from vA [v==×=ωωω r] 1.5 0.075, 20 rad/s CW find αpulley from a B 2 [at = rα] 45 =×= 0.4 αα , 112.5 rad/s CCW acceleration at C a= rω 2 a = 0.36 ×= 2022 144 m/s nCn a= rα a =×= 0.36 112.5 40.5 m/s2 [ tC] t a =a22 += a 149.6 m/s2 C CCnt aCn aCt 9.2 Rotation Ch. 9: Plane Kinematics of Rigid Bodies 484 P. 9/2 A V-belt speed-reduction drive is shown where pulley A drives the two integral pulleys B which in turn drive pulley C. If A starts from rest at time t = 0 and is given a constant angular acceleration α1, derive expressions for the angular velocity of C and the magnitude of the acceleration of a point P on the belt, both at time t. 9.2 Rotation Ch. 9: Plane Kinematics of Rigid Bodies 485 P. 9/2 2 dω rr α= ω = αω = 11 αω= α A1Bt, 1Ct, 1t dt r22 r 2 rr11 αA1B= αα, = αα 1C, = α 1 rr22 4 r 2 ar=ωα2 a = r1 ( t) n P2n 1 r2 2 r a= rαα a = r 1 [ t] P2t 1 r2 4 rr2 a= aa22 +=11αα 1 + 24t P PPnt 1 1 rr22 9.2 Rotation Ch. 9: Plane Kinematics of Rigid Bodies 486 9.3 Absolute Motion It is an approach to the kinematics analysis. It starts with the geometric relations that define the configuration involved. Then, the time derivatives of the relations are done to obtain velocities and accelerations. The +/- sense must be kept consistent throughout the analysis. If the geometric configuration is too complex, resort to the principle of relative motion is recommended. 9.3 Absolute Motion Ch. 9: Plane Kinematics of Rigid Bodies 487 P. 9/3 A wheel of radius r rolls on a flat surface w/o slipping. Determine the angular motion of the wheel in terms of the linear motion of its center O. Also determine the acceleration of a point on the rim of the wheel as the point comes into contact with the surface on which the wheel rolls. 9.3 Absolute Motion Ch. 9: Plane Kinematics of Rigid Bodies 488 P. 9/3 For ROLLING w/o SLIPPING only: displacement of the center O= arc length along the rim of the wheel distance s= arc C'A →= s rθ time derivatives: vOO= rωα and a= r vO = velocity of the center O of the wheel rolling w/o slipping ω = angular velocity of the wheel rolling w/o slipping DO NOT mix up the rolling motion with the rotation about a fixed axis Set up a fixed coordinate system originated at the point of contact between the rim and the ground. Trajectory of a point C on the rim is a cycloidal path. x-y coordinates of point C at arbitrary position C' is x=−=− s rsinθθ r( sin θ) y =−=− r rcos θ r( 1 cos θ) time derivatives: x =−=− rθ( 1 cos θ) vOO( 1 cos θ) y = r θθ sin = v sin θ 2 x=−+ v O( 1cosθ) v OO θθ sin =−+ a( 1cos θ) r ω sin θ 2 y=+=+ vsin OOθ v θ cos θ asin O θω r cos θ when point C comes to contact, θ = 0 x= 0, y = 0, x = 0, and y = rω 2 9.3 Absolute Motion Ch. 9: Plane Kinematics of Rigid Bodies 489 P. 9/4 The load L is being hoisted by the pulley and cable arrangement shown. Each cable is wrapped securely around its respective pulley so it does not slip. The two pulleys to which L is attached are fastened together to form a single rigid body. Calculate the velocity and acceleration of the load L and the corresponding angular velocity ω and angular acceleration α of the double pulley under the following conditions: Case (a) Pulley 1: ω1 = 0, α1 = 0 (pulley at rest) Pulley 2: ω2 = 2 rad/s 2 α2 = -3 rad/s Case (b) Pulley 1: ω1 = 1 rad/s 2 α1 = 4 rad/s Pulley 2: ω2 = 2 rad/s 2 α2 = -2 rad/s 9.3 Absolute Motion Ch. 9: Plane Kinematics of Rigid Bodies 490 P. 9/4 Tangential displacement, velocity, and acceleration of a point on the rim of the wheel equal the corresponding motion of the points on the wrapped no-slip, inextensible cable. (a) During time dt, line AB (on the integral pulley) moves to AB' through angle dθ . New (tangential) position of point A and B are determined from the motion induced by the cable. = =ωα = = = = − 2 vB v D22 r 0.2 m/s and (aB)t a D22 r 0.3 m/s dsA = 0 left cable does not move =θω = = α dsBBBAB d v AB (a )t AB AB' maintains straight and point O has absolute motion in vertical direction: dsO= AO dθωα v OO= AO a = AO ∴=ωα0.2 / 0.3 = 2 / 3 rad/s CCW, =−0.3/ 0.3 =− 1 rad/s2 CW 2 vOO= 0.1 × 2 / 3 = 0.0667 m/s and a= 0.1 ×− 1 =− 0.1 m/s 9.3 Absolute Motion Ch.