ws 40 - C15 MEEN 221 Summer 2009 - Worksheet 15 Dr. A....

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Unformatted text preview: C15 MEEN 221 Summer 2009 - Worksheet 15 Dr. A. Palaggolo (1) HWSlS Ch 13 (105, 108, 111, 112) (— ‘D we Tues. 3‘2, {8 (2) Attendance Mandatory for 10:00 am — 12:39pm, unless excused by instructor. Multiple Quizzes may be given anytime during this period. (3) Today’s Material: Kinematics Curvilinear Motion Section 13.5 FINAL EXAM WILL BE HELD ONT” ESOAV, «11 AUGUST 10:30 am—12z30 pm ZACH 104B (I) Plane 'Curvilinear Motion Consider a particle P' that travels along the path P in -the x-y "lane as shown in Figure 1. Y) may :1“ “oP ‘ y - REC-mu in km . QOQO\HA-T€S Rectangular: Since x(t) and y(t) locate particle P the rectangular coordinate motion quantities are: POSITION: R = x(t)z° '+ y(t)}' (1) VELOCITY: I7 =' Vj + Vyj' = xi + y} ‘ (2) ACCELERATION: ii = aj + ay} = fo + 17,} = + (3) Polar: The polar coordinate unit vectors vary in direction with time as they follow the particle, although their magnitudes remain equal to one, as shown in Figure. 2 I Figure 2. Polar Coordinate Unit Vec‘lbrs Note. that Aé, = (1A 60!?a (4) A so ‘ é, = EEAA‘géa = 9 ég (5) Similarly a, = —9' (E, ' (6) So that if ‘ Rp(t)=ré, (7) _. d V t=" + — é 2V ér + V9 éa ' V 5P =§;(I7P)=a'rér +a9 ée +fér + féfifl +réé9 + :> ar=i"—r92 (10) as flaw-a (11) Normal - Tangential The tangential and normal unit vectors vary in direction with time as ' they follow the path of particle P, although their magnitudes remain equal to one, as shown in Fig. 3. .. a; 42.97111.-. 1_+__6_3}ae:2_;;: ‘ “a . a7. , : V 0 = L Then smce 0&— a 14 . 17P=Vé, / - JO ( ) . - V2 (15) (16) 59/3 1723'1‘3'”: 5 Particle is f°11°wing a .Spiral path given by r(t) agate 9(t) is in radians and r is in millimeters. Given that 6 = ID/t 'Ead/s and that e = 0 When t = 1 S,.determine the velocity and acceleration l . r . o 16f the particle when 6 = 240 o 13,110* A collar that slides around a circular wine has-a pin that isfi sconstrained to move in the slot of arm AB. The arm rotates oounter— 'élockwise at a constant angular speed I of w = 2 rad/s. When_the arn; is 30° ,L :above the horizontal, :a. Determine the radial distanoe r ' ' from the pivot A to the pin 3. Sb. Determine the velocity compdnents VI and v9 o§,the‘tollar. io.- Determine the-aGCeleration ‘ components a? and as of the collar. “d. Verify that the velocity vettor ' V is directed along the wire. 13—118 Arm AC of the‘cam follower mechanism shown is rotating at a constant angular speed of 0 s 150 _ r rev/min. A spring holds the pin 3 against the cam lobes. If the equation that describes the shape of the cam lobes is . R = 125 f 50 cos 39 phere R is in millimeters, -a. Calculate and plot the 'magnitude of the velocity VB and the acceleration a8 ( bf the pin 8 as functions 0 '0 of 9 for-O < 9 < 180 . r . b. Will the shape of the ' curves change if the angular speed 0 is doubled? % Note Sheet 0: ENGR 221 _ . Instructor Dr. Alan Palazzolo clear .=linspace(0,0.l,500); thetadot=31.4; theta_radians=thetadot*t; R=125+50*cos(3*theta_radians); Rdot=—150*thetadot*sin(3.*theta_radians); Rdotdot=-450*thetadot“2*cos(3*theta_radians); magnitude_VB=sqrt(Rdot.*Rdot+thetadot“2*R.*R); magnitude_aB=sqrt( (Rdotdot—thetadot“2*R).*(Rdotdot—thetadot“2*R) + 4*thetadot“2*Rdot.*Rdot); % convert theta to degrees, V to m/s and a to m/s“2 , then plot subplot(2,l,l) plot(theta_radians*57,magnitude_VB/lOOO,'k—'); title('Magnitudes of Velocity and Acceleration for thetadot =3l.4 rad/sec') xlabel('theta in degrees'); ylabel(‘Velocity in m/s'); grid on subplot(2,l,2) plot(theta_radians*57,magnitude_aB/lOOO,'k—'); xlabel('theta in degrees'); ylabel('Acceleration in m/SAZ'); grid on Velocity in m/s Acceleration in m/s2 3.5 160 150 140 _|. 03 O Magnitudes of Velocity and Acceleration for thetadot =15.71 rad/sec l l l . . . 4 . . _ . _ I _ _ _ _ _ a _ . . _ _ I . . _ _ _ _ 60 80 100 120 140 1 60 180 theta in degrees 20 40 _ _ _ _ . _ . . _ _ _ h . _ . _ . _ i _ _ _ _ _ _ . _ _ _ . _ _ 4 _ _ _ . _ _ . _ _ _ _ _ . a _ . . . _ _ . _ _ _ _ _ _ _ _ _ _ _ r . _ _ _ _ _ h . . . . . . I . _ . . . _ _ _ _ _ _ . A 100 120 140 160 180 a theta in degrees Magnitudes of Velocity and Acceleration for thetadot =31.4 rad/sec P m .b Velocity in m/s .03 m 0 20 40 60 80 100 120 140 160 180 theta in degrees 650 600 A U1 ()1 01 O 01 O O O A O O Acceleration in m/s2 350 300 250 0 20 40 60 80 1 00 120 140 160 180 theta in degrees ...
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ws 40 - C15 MEEN 221 Summer 2009 - Worksheet 15 Dr. A....

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