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chap_III_hwk - Chapter III: Moving Reference Frame...

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Unformatted text preview: Chapter III: Moving Reference Frame Kinematics Homework III-1 Chapter III Moving Reference Frame Kinematics Homework III-2 Chapter III: Moving Reference Frame Kinematics Homework Chapter III: Moving Reference Frame Kinematics Homework III-3 Problem III.1 Given: Particle P moves within a straight slot cut into a rotating disk with the x-position of P increasing at a constant rate of x. The disk is rotating with counter-clockwise sense at a speed of ˙ Ω, with changing at a rate of ω . The xyz coordinate system is attached to the disk with its origin ˙ at the center O of the disk. Frame Kinematics Homework Problems ME 274 Find: For this problem: • Determine the velocity and acceleration of particle P. Express your answers as vectors in terms of their x-y components. oves within a straight slot cut into a rotating disk with the x-position of P ! The disk is rotating with counter- acceleration vectors constant rate of x . Make a sketch of the velocity andclockwise sense at a found above. • ! ith ! changing at a rate of ! . The xyz coordinate system is attached to ts origin at the center O of the disk. y ine the velocity and acceleration of e P. Express your answers as vectors in f their x-y components. ! x sketch of the velocity and acceleration found in a) above. P h x O Use the analysis: x = 0.8 ft h = 0.6 ft , x = x f s , ˙ ! ing parameters in yourfollowing parameters ,in your analysis: 15=t1/.5f t, h = 0.6f t, x = −10f t/sec, Ω = 6rad/sec ˙2 = −4rad/sec2 . and Ω ! and ! = "5 rad / s . III-4 Chapter III: Moving Reference Frame Kinematics Homework Problem III.2 Given: A telescoping arm OP is pinned to a cart at end O. The cart moves along a horizontal ˙ surface with a constant speed of vO . The angle of the arm, θ, is increasing at a constant rate of θ ˙ . The xyz coordinate system is attached to the telescoping arm with and is extending at a rate of L its origin at end O of the arm. ing Reference Frame Kinematics Homework Problems ME 274 Find: For this problem: blem III-2 • Determine the velocity and acceleration of particle P. Express your answers as vectors in elescoping arm OP is pinned theircart at end O. The cart moves along a horizontal terms of to a x-y components. face with a constant speed of vO . The angle of the arm, !, is increasing at a constant • Make a sketch ! ! L The xyz coordinate system vectors found above. of ! and is extending at a rate of of .the velocity and accelerationis attached to the scoping arm with its origin at end O of the arm. a) Determine the velocity and acceleration of particle P. Express your answers as vectors in terms of their x-y components. b) Make a sketch of the velocity and acceleration vectors found in a) above. P x L y O ! vO ˙ Use the following parameters in your analysis: θ = !90◦ , vO = 6m/sec, θ = 5rad/sec, L = 2m, the following ˙parameters in your analysis: ! = 0 , vO = 10 m / s , ! = "3 rad / s , ¨ = 3m/sec2 . L = 0 and L !! ! 4 m , L = !3 m / s and L = 0 . Chapter III: Moving Reference Frame Kinematics Homework III-5 Problem III.3 Given: An L-shaped arm OBE is pinned to ground at O (section OB is perpendicular to section ME 274 is known to be rotating in the counterclockwise sense with a constant rotational speed of Ω. Rigid link AP is pinned to ground at A, and with P able to slide within a slot that is cut into section BE of arm OBE, as shown. The xyz coordinate system is attached to III-3 arm OBE with its origin at pin O. For the position shown, section OB is aligned with the direction ped arm OBE is pinned to P is at aat O (sectionyOB is perpendicular to section of OA, and ground position of P within the slot. rence Frame Kinematicsof OBE).Problems BE Homework Arm OBE E). Arm OBE is known to be rotating in the counterclockwise sense with a rotational speed of !.For this position: pinned to ground at A, and with P able to Find: Rigid link AP is in a slot that is cut into section BE of arm OBE, as shown. The xyz coordinate • Determine the angular velocity of link AP. Write your answer as a vector. attached to arm OBE with its origin at pin O. • Determine the angular acceleration of link AP. Write your answer as a vector. osition shown, section OB is aligned with the direction of OA, and P is at a f yP within the slot. For this E etermine the angular locity of link AP. Write P y ur answer as a vector. etermine the angular celeration of link AP. Write ur answer as a vector. L yP O B ! x A b Use the following parameters in your analysis: Ω = 4rad/sec, yP = 5f t, L = 2f t and b = 3f t. ollowing parameters in your analysis: ! = 2 rad / sec , yP = 2 ft , L = 2.5 ft ft . III-6 Chapter III: Moving Reference Frame Kinematics Homework Problem III.4 Given: Arm OB is pinned to ground at end O. Rigid link AP is pinned to ground at A, with end P able to slide within a straight slot that is cut into arm OB. Arm AP is rotating in the clockwise Reference Frame Kinematics Homework Problems of Ω. The xyz coordinate system is attached to arm OB with its origin ME 274 sense with a constant rate at pin O. For the position shown, link OB is perpendicular to the direction of OA, with arm AP at an angle of θ measured from OB. lem III-4 B is pinned to ground at this position: link AP is pinned to ground at A, with end P Find: For end O. Rigid o slide within a straight slot that is cut into arm OB. Arm AP is rotating in the wise sense with a constant rate of !. The xyz coordinate systemWrite your answer as a vector. • Determine the angular velocity of arm OB. is attached to arm ith its origin at pin O. • Determine the angular acceleration of arm OB. Write your answer as a vector. e position shown, • Determine the values ofto P and xP . of OA, with arm AP at link OB is perpendicular x the direction ˙ ¨ le of " measured from OB. For this position: Determine the angular velocity of arm OB. Write your answer as a vector. A ! Determine the angular acceleration of arm OB. Write your answer as a vector. Determine the values of ! xP and !!P . x L y " P O B x xP Use the following parameters in your analysis: Ω = 2rad/sec, L = 3m and θ = 36.87◦ . e following parameters in your analysis: ! = 8 rad / s , L = 2 m and ! = 53.13° . Chapter III: Moving Reference Frame Kinematics Homework III-7 Problem III.5 Given: A disk and shaft OC are mounted in a clevis that rotates about a fixed vertical axis at a rate of Ω. The shaft and disk rotate with respect to the clevis with a rate of ω0 in the direction shown below, with the angle θ held constant. The XYZ coordinate system is fixed with the Z-axis aligned with the fixed rotation axis of the clevis. The xyz coordinateEsystem is attached to the disk ng Reference Frame Kinematics Homework Problems M 274 with the y-axis aligned with OC for all time. For the position shown below, the x- and X-axes are aligned. blem III-5 Find: For the in a clevis that isk and shaft OC are mountedposition shown: rotates about a fixed vertical axis at a of !. The shaft and disk rotate with respect to the clevis with a rate of "0 in the • Determine the angular velocity of the disk. Write your answer as a vector in terms of its xyz ction shown below, with the angle # held constant. The XYZ coordinate system is components. d with the Z-axis aligned with the fixed rotation axis of the clevis. The xyz coordinate em is attached to theDetermine the y-axis aligned with OC the disk. Write your answer as a vector in terms of its • disk with angular acceleration of for all time. For the ition shown below, the x- and X-axes are aligned. xyz components. the position shown: z a) Determine the angular velocity of the disk. Write your answer as a vector in terms of its xyz components. Z y b) Determine the angular acceleration of the disk. Write your answer as a vector in terms of its xyz components. O C x Y X Use the following parameters in your analysis: θ = 36.87◦ , Ω = 4rad/sec = constant and ω0 = the following parameters in your analysis: ! = 53.13° , ! = 3 rad / s = constant and 6rad/sec = constant. = 5 rad / s = constant . III-8 Chapter III: Moving Reference Frame Kinematics Homework Problem III.6 Given: A disk and shaft OC are mounted in a clevis that is attached to a fixed vertical axis (Ω = 0). The shaft and disk rotate with respect to the clevis with a rate of ω0 in the direction ˙ shown below, with the angle θ increasing at a constant rate of θ. The XYZ coordinate system is fixed with the Z-axis aligned with the fixed vertical direction.ME 274 xyz coordinate system is The ng Reference Frame Kinematics Homework Problems attached to the disk with the y-axis aligned with OC for all time. For the position shown below, the x- and X-axes are aligned. blem III-6 Find: For the in a clevis that isk and shaft OC are mountedposition shown: is attached to a fixed vertical axis = 0 ). The shaft and disk rotate with respect to the clevis with a rate of !0 in the • Determine the angular velocity of the disk. Write! your answer as a vector in terms of its xyz ction shown below, with the angle " increasing at a constant rate of ! . The XYZ components. rdinate system is fixed with the Z-axis aligned with the fixed vertical direction. The coordinate system is attached to the disk with the y-axis aligned with OC for all time. as a vector in terms of its • Determine the angular acceleration of the disk. Write your answer xyz components. the position shown below, the x- and X-axes are aligned. the position shown: z a) Determine the angular velocity of the disk. Write your answer as a vector in terms of its xyz components. Z y b) Determine the angular acceleration of the disk. Write your answer as a vector in terms of its xyz components. O C x Y X ˙ Use the following parameters in your analysis: θ = 0, θ = 1.5rad/sec, ω0 = 2rad/sec and ω0 = ˙ ! the following 5rad/sec2 . in your analysis: ! = 0 , ! = 8 rad / s and ! 0 = 6 rad / s parameters ! ! 0 = 2 rad / s 2 . Chapter III: Moving Reference Frame Kinematics Homework III-9 Problem III.7 Given: Arm OB rotates about a fixed axis with a constant rate of Ω. A disk rotates about its central axis BC with a constant rate of ω0 relative to arm OB. The XYZ coordinate system is fixed with Homework Problems erence Frame Kinematics the Y-axis aligned with the fixed rotation axis of OB. The 274 coordinate system is attached ME xyz to the disk with the z-axis aligned with BC for all time. For the position shown below, the xyz axes are aligned with the XYZ axes. III-7 Find: For with a constant rate rotates about a fixed axis the position shown: of !. A disk rotates about its xis BC with a constant rate of "0 relative to arm OB. The XYZ coordinate • Determine the angular acceleration of the disk. Write your answer as a vector in terms of its fixed with the Y-axis aligned with the fixed rotation axis of OB. The xyz xyz components. te system is attached to the disk with the z-axis aligned with BC for all time. For ion shown below, • Determine are aligned with of point Aaxes. disk. Write your answer as a vector in terms of the xyz axes the acceleration the XYZ on the its xyz components. osition shown: etermine the angular elocity of the disk. Write our answer as a vector in rms of its xyz omponents. etermine the angular cceleration of the disk. rite your answer as a ector in terms of its xyz omponents. "0 d R h ! Use the following parameters in your analysis: Ω = −2rad/sec, ω0 = 3rad/sec, d = 10inches, h = 3inches and R = 5inches. ollowing parameters in your analysis: ! = "3 rad / s and ! 0 = 5 rad / s . III-10 Chapter III: Moving Reference Frame Kinematics Homework Problem III.8 Given: Yoke A rotates about a fixed axis with a constant rate of ω1 . A disk rotates about its center C with a constant rate of p relative to yoke A. The XYZ coordinate system is fixed with the X-axis aligned with the fixed rotation axis of yoke A. The xyz coordinate system is attached to the disk with the z-axis aligned with the rotation axis of the disk for all time. For the position shown below, the xyz axes are aligned with the XYZ axes. erence Frame Kinematics Homework Problems ME 274 Find: For the position shown: III-8 • Determine the angular velocity and angular acceleration of the disk. Write your answers as rotates about a fixed vectors in a constant rate xyz!1. A disk rotates about its axis with terms of their of components. with a constant rate of p relative to yoke A. The XYZ coordinate system is fixed • Determine the acceleration of A. The of the disk. Write your X-axis aligned with the fixed rotation axis of yokepoint B xyz coordinate system answer as a vector in terms of its xyz components. d to the disk with the z-axis aligned with the rotation axis of the disk for all the position shown below, the xyz axes are aligned with the XYZ axes. osition shown: etermine the angular velocity nd angular acceleration of the isk. Write your answers as ectors in terms of their xyz omponents. d C etermine the acceleration of oint B of the disk. Write your nswer as a vector in terms of s xyz components. R B b O Use the following parameters in your analysis: ω1 = 2rad/sec, p = 8rad/sec, d = 1.5m, b = 0.5m and R = your analysis: ! = 3 rad / s , p = 6 rad / s , d = 0.5 m , ollowing parameters in 0.25m. and R = 1 m . 1 Chapter III: Moving Reference Frame Kinematics Homework Moving Reference Frame Kinematics Homework Problems ME 274 III-11 Problem III.9 Problem Shaft AB rotates about a fixed axis with a constant rotational speed of Ω. A tube is hinged Given: III-9 ˙ Shaftshaftrotates about a fixed axis withtheconstant rotational speed of !.constantis on AB AB with the angle θ between a tube and shaft increasing at a A tube rate of θ. Particle P moves within the tube at a constant rate the tube and to the tube. The a constant ˙ hinged on shaft AB with the angle " between of xP relativeshaft increasing atXYZ coordinate system is fixed Particle X-axis aligned the the fixed rotation axis x the shaft AB. The xyz ! rate of ! . with theP moves within withtube at a constant rate ofof! P relative to the tube. coordinate system is attached to the tube with the x-axis aligned with the tube for all time. For the position The XYZ coordinate system is fixed with the X-axis aligned with the fixed rotation axis shown below, the z- and Z-axes are aligned. of the shaft AB. The xyz coordinate system is attached to the tube with the x-axis aligned with the tube for all time. For the position shown below, the z- and Z-axes are aligned. Find: For the position shown: For th• position shown: e Determine the angular velocity and angular acceleration of the tube. Write your answers as vectors in the a of their xyz components. a) Determine termsngular velocity and angular acceleration of the tube. Write your answers as vectors in terms of their xyz components. • Determine the acceleration of particle P. Write your answer as a vector in terms of its xyz b) Determine the acceleration of particle P. Write your answer as a vector in terms components. of its xyz components. x Y y xP " P X O A ! B U the following parameters in your analysis: ! = = ad / s , ! θ = , ◦! θ = r 4rad/sec Use se the following parameters in your analysis: Ω 4 r3rad/sec, = 30°60!, =˙ "2 −ad / s , , xP = 2m ! xPand3xP and 2m/sec. m / sec . = ˙m = − xP = !4 III-12 Chapter III: Moving Reference Frame Kinematics Homework Problem III.10 Given: The collar and clevis A are given a constant downward velocity of z < 0. This motion ˙ causes end B of bar AB to slide outward within a radial slot of the rotating disk. The disk has a constant rotational speed of Ω about a fixed axis passing through the center of the disk. The XYZ eference Frame Kinematics Homework Problems M rotation axis of the disk. The xyz coordinate system is fixed with the Z-axis aligned with the fixed E 274 coordinate system is attached to bar AB. For the position shown below, the xyz axes are aligned with the XYZ axes. m III-10 Find: For the position shown: ! llar and clevis A are given a constant downward velocity of z < 0 . This motion • Determine the angular velocity of bar AB. Write your answer end B of bar AB to slide outward within a radial slot of the rotating disk. The diskas a vector in terms of its xyz components. onstant rotational speed of ! about a fixed axis passing through the center of the he XYZ coordinate system is fixed angular acceleration of bar AB. Write your answer as a vector in terms of its with the Z-axis aligned with the fixed rotation • Determine the the disk. The xyz coordinate system is attached to bar AB. For the position shown xyz components. the xyz axes are aligned with the XYZ axes. z, Z position shown: Determine the angular velocity of bar AB. Write your answer as a vector in terms of its xyz components. Determine the angular acceleration of bar AB. Write your answer as a vector in terms of its xyz components. L y, Y x, X ! Use the in your parameters in your / s , L = Ω ˙ following parametersfollowinganalysis: ! = 3 rad analysis: 2.5=ft5,rad/sec, ft =ndf t, z = 1.5f t and z = −3m/sec. z = 1.5 L a 2 ft / sec . Chapter III: Moving Reference Frame Kinematics Homework III-13 Problem III.11 Given: Block A slides within a straight slot cut in a rotating disk with the position of A given by x = 2sin (4π t), where x and t are given in inches and seconds, respectively. The angular orientation of the disk is given by θ = 0.2sin (8π t), where and t are given in radians and seconds, respectively. Find: Determine the acceleration of block A when: • x = Problems ˙ e Frame Kinematics Homework 0 with x -11 (5/167) >0 • x = 2inches es within a straight slot cut in a rotating position of A given by x = 2 sin 4! t , t are given in inches and seconds, The angular orientation of the disk is 0.2 sin 8" t , where ! and t are given in econds, respectively. Determine the of block A when ! with x > 0 . inches. ME 274 III-14 Chapter III: Moving Reference Frame Kinematics Homework Problem III.12 Given: Link OA rotates with a constant angular speed of ωOA . Find: Determine: • the angular velocity of link BC. me Kinematics Homework Problems (5/183) ME 274 • the acceleration of pin A as seen by an observer on link BC. with a constant angular speed of ! OA . ar velocity of link BC. ration of pin A as seen by an observer on = 10rad/sec (CW) and θ = 30◦ . g parameters in your analysis: ! OA = 10 rad / sec ( CW ) and ! OA30° . = Use the following parameters in your analysis: ω Chapter III: Moving Reference Frame Kinematics Homework III-15 Problem III.13 Given: A motor is attached to a platform that is rotating with a constant rate of N about a fixed vertical axis. The body of the motor pivots about a moving horizontal axis at a constant rate of γ ˙ with the shaft of the motor rotating at a constant rate of ω0 . Find: Determine: • the angular acceleration of the disk attached to the shaft of the motor. rame Kinematics Homework Problems (7/12) ME 274 • the velocity of point A on the disk when A is at the top of the disk. hed to a platform that is rotating with f N about a fixed vertical axis. The or pivots about a moving horizontal nt rate of !! with the shaft of the motor stant rate of ! 0 . ne the angular acceleration of the disk to the shaft of the motor. ne the velocity of point A on the disk is at the top of the disk. Use the following parameters in your analysis: N = 0, γ = 30◦ , γ = 3π and ω0 = 120rpm. ˙ ng parameters in your analysis: N = 0 , ! = 30° , !! = 3" rad / sec and in . III-16 Chapter III: Moving Reference Frame Kinematics Homework Problem III.14 Given: System shown. rame Kinematics Homework Problems (7/17) ME 274 Find: Determine the angular velocity and angular acceleration of the jaws of arm O1 O2 . Assume all rotation rates to the constant. ngular velocity and angular the jaws of arm O1O2 . Assume all the constant. ng parameters inse theanalysis: ! 3 = ! 4 = ! 5 =your!1 = 2 rad ω3 e= ,ω4 = ω5 = 0, ω1 = 2rad/sec, θ = 60◦ and U your following parameters in 0 , analysis: / s c ˙ 1.5 rad / sec .θ = 1.5rad/sec. Chapter III: Moving Reference Frame Kinematics Homework III-17 Problem III.15 Given: System shown. Frame Kinematics Homework Problems 5 (7/25) ME 274 Find: Determine the angular velocity and angular acceleration of shaft B. Assume all rotation rates to the constant. angular velocity and angular f shaft B. Assume all rotation rates t. Use the following parameters in your analysis: Ω = 4rad/sec, ω0 = 3rad/sec, γ = 30◦ and ing parameters˙ in your analysis:. ! = 4 rad / sec , ! 0 = 3 rad / sec , γ = (π /4) rad/sec ! = " / 4 rad / sec . III-18 Chapter III: Moving Reference Frame Kinematics Homework Problem III.16 Given: System shown. K inematics Homework Problems /51) ME 274 Find: Determine the angular acceleration of the rotor of the gyroscope. Assume all rotation rates to the constant. ular acceleration of the rotor of the Assume all rotation rates to the constant. Use the following parameters in your analysis: N = 20rev/sec, γ = 30◦ and γ = 4rad/sec. ˙ parameters in your analysis: N = 20 rev / sec , ! = 30° and Chapter III: Moving Reference Frame Kinematics Homework III-19 Problem III.17 Moving Reference Frame Kinematics Homework Problems ME 274 ˙ Given: Person A is sitting on a merry-go-round that is rotating at a constant rate θ. Person B is running in a straight line at a constant speed vB along the sidewalk. Problem III-17 ! Person A is sitting this problem: -round that is rotating at a constant rate ! . Person B is Find: For on a merry-go running in a straight line at a constant speed vB along the sidewalk. ! • Determine the velocity and acceleration of person B as seen by the moving observer, person A, at the instant shown. Use vB = 10m/sec θ = 3rad/sec, θ = 30◦ , (a) Determine the velocity and acceleration of person ,B˙ as seen by the movingr = 1.5m, d = 8m, and h = 4m in A, at the instant ! observer, person your analysis. shown. Use v = 10 m/s, ! = 2 rad/s, ! = 60°, r B = 1 • ,Show5thatand h answerin your first part is independent of the location of observer A. That is, m d = m, your = 3 m in the analysis. (b) Show showythatanswer in the first part is independent of velocity and acceleration of the runner B, that our any observer on the disk sees the same the location of observer A. That is, show of the observer’s location.disk sees the same velocity and regardless that any observer on the acceleration of the runner B, regardless of the observer's location. ! ! ! III-20 Chapter III: Moving Reference Frame Kinematics Homework Problem III.18Frame Kinematics Homework Problems Moving Reference ME 274 Given: The mechanism shown below consists of a disk pinned at its center (point A), which rotates ! with a constant-18 Problem III angular velocity, ωAB , a slotted arm CD that is pinned at C, and a bar that can oscillate within the sguides below consists of a disk pinned at its peg at (point A), whichthe slotted The mechanism hown at E and F. As the disk rotates, the center B moves within arm, causing itato rock back and forth. As ! , a slotted arm CD thata slow advance and a quick rotates with constant angular velocity, the arm rocks, it provides is pinned at C, and a AB return to the reciprocating bar due to the change in distance between C and B. bar that can oscillate within the guides at E and F. As the disk rotates, the peg at B moves within the slotted arm, causing it to rock back and forth. As the arm rocks, it Find: For the position shown, determine: provides a slow advance and a quick return to the reciprocating bar due to the change in d• the angular velocity and For the position shown,the slotted armthe angular velocity istance between C and B. angular acceleration of determine: (i) CD. and angular acceleration of the slotted arm CD and (ii) the velocity and acceleration of • bar. Use ! = 30°, !AB = 40 rpm, bar. thethe velocity and acceleration of theR = 0.1 m, h = 0.2 m, and d = 0.24 m in your analysis. ! ! Use the following parameters in your analysis: θ = 36.87◦ , ωAB = 30rpm, R = 0.1m, h = 0.3m and d = 0.24m. Moving Reference Frame Kinematics Homework Problems ME 274 Chapter III: Moving Reference Frame Kinematics Homework ! III-21 Problem III-19 At the III.19 Probleminstant shown, the shaft rotates with a constant angular velocity !P . At the same instant, the disk spins about its axle with a constant angular velocity !S . Determine the Given: angular instant shown, the shaft the velocity of constantat this instant. total At the velocity of the disk and rotates with a point C angular velocity ωP . At the same instant, the disk spins about its axle with a constant angular velocity ωS . Use ! = 10 rad/s and ! = -6 rad/s in your analysis. P S Find: Determine the total angular velocity of the disk and the velocity of point C at this instant. ! ! Use the following parameters in your analysis: ωP = 15rad/sec and ωS = 5rad/sec. Moving Reference Frame Kinematics Homework Problems ! III-22 ME 274 Chapter III: Moving Reference Frame Kinematics Homework Problem III-20! At the instant shown, the shaft rotates with an angular velocity !P and has an angular Problem III.20 ! acceleration !P . At the same instant, the disk spins about its axle with an angular ! velocity !S and has an angular acceleration !S . an angular velocity angular acceleration Given: At the instant shown, the shaft rotates with Determine the total ωP . At the same instant, theodisk spins and the accelerationan angular velocity ωS . f the disk about its axle with of point C at this instant. Find: Determine the! = 1angular ! = -3 rad/s2, and disk and the ivelocityanalysis. C at this !P ! Use !P = 8 rad/s, total 2 rad/s, acceleration of the !S = 2 rad/s2 n your of point S instant. ! ! Use the following parameters in your analysis: ωP = 10rad/sec, ωS = 8rad/sec, ωP = −2rad/sec2 ˙ and ωS = 3rad/sec2 . ˙ Moving Reference Frame Kinematics Homework Problems ME 274 ! Chapter III: Moving Reference Frame Kinematics Homework Problem III-21! III-23 At the instant shown, the helicopter is moving upwards with speed vH and acceleration aH . At the same instant, the frame H, not the horizontal blade, rotates about a vertical Problem III.21 axis At a constant shown, the helicopter . moving upwards rotates with a and acceleration Given:with the instant angular velocity of !H is If the tail blade Bwith speed vH constant aH angular velocity !B / H ,the frame H, (not the H, determine the velocityabout a vertical axis with . At the same instant, measured relative to horizontal blade) rotates and acceleration of a constant angular velocity of ωH . The tail blade B rotates with a constant angular velocity ωB/H , point P, located on the end of the blade, at the instant the blade is in the vertical position. measured relative to H. Use v = 4 m/s, a = 2 m/s2, ! = 0.6 rad/s, and ! = 200 rad/s in your analysis. H H B Find: Determine the velocity andHacceleration of point/ HP, located on the end of the blade, at the instant the blade is in the vertical position. Use the following parameters in your analysis: vH = 6m/sec, aH = 3m/sec2 , ωH = 0.5rad/sec and ωB/H = 250rad/sec. Moving Reference Frame Kinematics Homework Problems III-24 ME 274 Chapter III: Moving Reference Frame Kinematics Homework Problem III-22! A particle P (having a mass of m) slides along bar OA with a constant speed of vrel Problem III.22 relative to OA. OA is pinned to a vertical shaft that is rotating about a vertical shaft with Given: Particle Pof !. In addition,OA with a constant speed thevvertical shaft with a is pinned a constant rate slides along bar OA is rotating relative to of rel relative to OA. OA to a vertical shaft that is rotating about a vertical shaft with a constant rate of Ω. In addition, OA constant rate of " In vertical shaft with a you are rate of θ . In moving observer who is rotating relative to .the solving this problem,constant asked to˙ use asolving this problem, you are is to use a moving observer who is attached to bar OA, as shown below. origin at pin O. askedattached to bar OA, as shown below. Let xyz be attached to OA with its Let xyz be attached to OA with its origin = 3 rad/sec, " = 5 rad/sec, R = 2 meters, v = 10 m/sec and " = 0. Use m = 20 kg, ! at pin O. ˙ ˙ rel i Find:) For Determine the angular velocity and angular acceleration of bar OA in terms of the position shown, determine: their xyz components. • the angular velocity and angular acceleration of bar OA in terms of their xyz components. ii) Determine the acceleration of particle P in terms of its xyz components. • ithe acceleration theparticle P acting onof its xyzin terms of its xyz components. ii) Determine of net force in terms particle components. observer A P y x R " vrel O ! ˙ Use the following parameters in your analysis: Ω = 3rad/sec, θ = 5rad/sec, R = 2m, vrel = 10m/sec and θ = 0. Chapter III: Moving Reference Frame Kinematics Homework ference Frame Kinematics Homework Problems m III-23 III-25 ME 274 Problem III.23 ˙ ˙ A particle P travels in a tube P travels 6 ft /tube with R = The tube.isThe tube is being raised at a constant Given: Particle with R = in a sec = constant . constant being ˙rate of ! = 3 rad / sec . In addition, the a vertical shaft which is rotating about the fixed Y ˙ raised at a crate of θ. In addition, the tube is attached to tube is attached to a onstant axis withis rotating aboutof ω .fixed observer is attached to the tube with the xyz axes also attached a constant rate the An Y axis with a constant rate of vertical shaft which to . An observer is origin at point O. ! = 4 rad / secthe tube with its attached to the tube with the xyz axes also attached to the tube with its origin at point O. Find: For the position shown, determine: You are asked • the angular velocity vector of the observer.= 5 feet and ! = to find the acceleration of particle P when R 36.87°. In doing so, answer the following questions. a) b) c) d) e) Find the • ngular velocity vector of vector of the. observer. a the angular acceleration the observer Find the • ngular acceleration vectorseen byobserver. a the velocity of point P as of the the observer. What is the velocity of point P as seenas seen observer? • the acceleration of point P by the by the observer. What is the acceleration of point P as seen by the observer? • the acceleration of point P using the above results. Using the results from a)-d) above, find the acceleration of point P. Y x P y R ! O X " ˙ ˙ Use the following parameters in your analysis: R = 5f t, θ = 36.87◦ , R = 6f t/sec, θ = 3rad/sec and ω = 4rad/sec. Moving Reference Frame Kinematics Homework Problems ME 274 III-26 Chapter III: Moving Reference Frame Kinematics Homework Problem III-24 Given: Arm AB is pinned to ground at point A. A block that is pinned to AB Problem III.24 at B is constrained to slide within a slot on a disk. The disk is rotating clockwise about a shaft at O a constant rate of 10 rad/sec. At the instant Given: shown, theArm ABhorizontal, arm ABat point A. A fblock5that iswith the AB at B is constrained slot is is pinned to ground is at angle o ! = 3.13° pinned to to slide horizontalwithin a slot ft. a disk. The disk is rotating about a shaft at O with a constant rate of and d = 0.5 on ωdisk . At the instant shown, the slot is horizontal. Find: At Find: At this position, determine the angular velocity angular this position, determine the angular velocity and and angular acceleration of arm AB. Write acceleration of arm AB. Write your answers as vectors. your answers as vectors. " B O d L ! A Use the following parameters in your analysis: ωdisk = 10rad/sec (CW), θ = 53.13◦ , L = 3f t and d = 0.5f t. Moving Reference Frame Kinematics Homework Problems ME 274 Chapter III: Moving Reference Frame Kinematics Homework III-27 Problem III-25 Given: The mechanism shown below is driven by a motor attached at B. End A of Problem III.25 link AB is free to slide within a straight slot within link OC. The angular elocity of !AB i a constant 2 is driven by a motor attached Given: vThe mechanism sshown below rad/s in the direction shown. at B. End A of link AB is free to slide within a straight slot within link OC. The angular velocity of ωAB is in the direction shown. For the position shown below, determine Find: Find: For a) tposition shown, determine: the he angular velocity of OC. b) the velocity of A relative to an observer positioned on OC and rotating with OC. • the angular velocity of OC. • the velocity of A relative to an observer positioned on OC and rotating with OC. c) the angular acceleration of OC. • the angular acceleration of OC. elative to an observer positioned on OC and d) the acceleration of A r rotating with OC. • the acceleration of A relative to an observer positioned on OC and rotating with OC. Express your answers as vectors in terms of either their xyz or their XYZ components. Y B !AB 30 ° y # L 0.17 m x C " 60 ° O A 0.1 m X Use the following parameters in your analysis: ωAB = 2rad/sec, L = 0.17m, θ = 30◦ and φ = 60◦ . Moving Reference Frame Kinematics Homework Problems ME 274 Problem III-26 Chapter III: Moving Reference Frame Kinematics Homework Given: A mechanism is made up of links AB, BD and DE. This mechanism is pinned to a shaft that is rotating at a constant rate of ! = 4 rad/sec about a fixed Problem III.26 horizontal axis. All pins in the mechanism allow rotation about axes that are perpendicular to the plane containing links AB, BD and DE. When " = 90°, it Given: A mechanism is made up of links AB, BD and DE. This mechanism is pinned to a shaft ˙ is known that ! = 5 rad / Ω about a fixed horizontal axis. All pins in the mechanism that is rotating at a constant rate of sec = cons tan t . Shown in the figure below are the X about that that are to ground and the xyz axes containing links to link allow rotationYZ axesaxes are fixed perpendicular to the plane that are attached AB, BD and DE. ˙ A, When θ = 90◦B.it is known that θ = constant. Shown in the figure below are the XYZ axes that III-28 are fixed to ground and the xyz axes that are attached to link AB. For the position of " = 90°, determine the Find: Find: For the position of θ = 90◦ , determine: a) angular velocity of link AB. • the angular )velocity of acceleration of link AB. b angular AB. c acceleration AB. • the angular)acceleration of of pin B. • the acceleration in vector Express ALL answersof pin B. form. In using the rotating reference frame kinematics equations, please state to which body your observer is attached. x B Y y d D b A " e h E X ! ˙ Use the following parameters in your analysis: Ω = 4rad/sec, θ = 5rad/sec, b = 2f t, d = 2.5f t, e = 1f t and h = 0.5f t. A shaft rotates about a fixed vertical axis at a constant rate of ! = 5 rad/sec, as shown below. A straight bar OA, having a length of L = 2 meters, is pinned to point O on the shaft, with O being on the rotation axis of the shaft. At the Chapter III: Moving Reference Frame Kinematics Homework instant when " = 0, bar OA is being raised at a rate of 4 rad/sec from the horizontal plane. In addition, the rate at which the bar is being raised is Problem decreasing at a rate 3 rad / sec 2 . III.27 Given: III-29 A set of xyz coordinate axes is attached to bar OA with its origin at O. A Given: A shaft rotates about a fixed vertical axis at a constant rate of Ω, as shown below. A second having a length axes, is pinned to point O on At shaft, with O being straight bar OA, set of coordinate of L, XYZ, are fixed to ground. the the instant when " on the = , the yz and XYZ a instant when θ with each other. ˙ rotation axis0ofthe xshaft. At the xes are aligned = 0, bar OA is being raised at a rate of θ from the ¨ horizontal plane, with this rate changing at a rate of θ. A set of xyz coordinate axes is attached Fto bar OAFwith its originwhen " = 0: ind: or the instant at O. A second set of coordinate axes, XYZ, are fixed to ground. At the instant when θ = 0, the xyz and XYZ axes are aligned with each other. a) find the angular velocity and angular acceleration of bar OA. Find: For the ) find the acceleration of point A on the bar. b instant when θ = 0, determine: • the angular be expressed in vector form. You may OA. All answers shouldvelocity and angular acceleration of bar choose to write these vectors in terms ofacceleration xyzpointYZ on the bar. • the either their of or X A components. Y A y x L " vertical shaft bar O ! bearing X ˙ ¨ Use the following parameters in your analysis: Ω = 5rad/sec, θ = 4rad/sec, θ = −3rad/sec2 and L = 2m. Problem III-28 Given: A robotic manipulator is made up of two links OA and ADB as shown in the figure below left. Link OA has a fixed length of L = 4 ft, and the length link III-30 ADB is increasing at a cChapter rIII: Moving Reference Frame Kinematics links onstant ate of 6 ft/sec. The orientation angles of Homework OA and ADB are increasing at constant rates of d!/dt = 2 rad/sec and d"/dt = 3 rad/sec, respectively. Problem III.28 Given: A robotic manipulator is made up of two links OA and ADBaas shown0°, the figure below Find: For the position shown below right with b = 3 ft, ! = 0 nd " = 9 in ˙ left. Link OA etermine the acceleration ofthe lengthnlink ADB is changing at a constant rate of b. d has a fixed length of L, and point B o the manipulator. Find: For In solving this problem, use an observer that is attached ,to AD and with acceleration of HINT: the position shown below right with θ = 0 and φ = 90◦ determine the a set of point B on coordinate axes also attached to AD. the manipulator. x x B B b y D " L O ! D A O 90° y A ˙ ˙ Use the following parameters in your analysis: b = 3f t, b = 6f t/sec = constant, θ = 2rad/sec = ˙ constant, φ = 3rad/sec = constant and L = 4f t. a) find the angular velocity vector of the boom OP. b) find the angular acceleration vector of the boom OP. ( ) ( ) c) evaluate the Frame aO , v P / O , a P/O Chapter III: Moving Reference vectorsKinematics Homework rel and r P / O in the rel III-31 following equation for the acceleration of point P: a P = aO + a P / O + ! " r P / O + 2# " v P / O + # " # " r P/O Problem III.29 rel rel ( ) ( ) ( ) Note that you are NOT asked to evaluate the right hand side of this Given: A crane is moving to the right with a constant speed of vC and is rotating about a vertical equation for a P . ˙ ˙ axis with a constant rate of Ω. The boom is being raised at a rate of θ with θ changing at a rate ¨.this problem, use an observer attached to the boom and with a moving set of xyz of θ For coordinate axes also attached to the boom, as shown in the figure below. XYZ are fixed Find: axes. Determine the acceleration of end P of the boom. Y ! P x d vC L y " O X Use the following parameters in your analysis: vC = 0.5f t/sec, Ω = 1.5rad/sec, θ = 36.87◦ , ˙ ¨ θ = 0.6rad/sec, θ = 1.2rad/sec2 , L = 20f t and d = 4f t. ...
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