DYNAMICS. Kinematics of Rigid Bodies VECTOR MECHANICS FOR ENGINEERS: Tenth Edition CHAPTER

Transcription

1 Tenth E CHAPTER 15 VECTOR MECHANICS FOR ENGINEERS: DYNAMICS Ferdinand P. Beer E. Russell Johnston, Jr. Phillip J. Cornwell Lecture Notes: Brian P. Self California Polytechnic State University Kinematics of Rigid Bodies

2 Quiz Date 17 August Time: 6:15 p.m. -

3 Contents Introduction Translation Rotation About a Fixed Axis: Velocity Rotation About a Fixed Axis: Acceleration Rotation About a Fixed Axis: Representative Slab Equations Defining the Rotation of a Rigid Body About a Fixed Axis Sample Problem 5.1 General Plane Motion Absolute and Relative Velocity in Plane Motion Sample Problem 15. Sample Problem 15.3 Instantaneous Center of Rotation in Plane Motion Sample Problem 15.4 Sample Problem 15.5 Absolute and Relative Acceleration in Plane Motion Analysis of Plane Motion in Terms of a Parameter Sample Problem 15.6 Sample Problem 15.7 Sample Problem 15.8 Rate of Change With Respect to a Rotating Frame Coriolis Acceleration Sample Problem 15.9 Sample Problem Motion About a Fixed Point General Motion Sample Problem Three Dimensional Motion. Coriolis Acceleration Frame of Reference in General Motion Sample Problem

4 Applications How can we determine the velocity of the tip of a turbine blade? 15-4

5 Applications Planetary gear systems are used to get high reduction ratios with minimum weight and space. How can we design the correct gear ratios? - 5

6 Applications Biomedical engineers must determine the velocities and accelerations of the leg in order to design prostheses. - 6

7 Introduction A rigid body is made of infinite points and distance between any two points does not change during motion. Kinematics of rigid bodies: relations between time the positions, velocities, and accelerations of the particles forming a rigid body. 15-7

8 Introduction Classification of rigid body motions: - translation: rectilinear translation curvilinear translation - rotation about a fixed axis - general plane motion - motion about a fixed point - general motion 15-8

9 enth Scalar The rate of change of a scalar A is independent of frame of reference. If F is a fixed frame of reference and m is a moving coordinate system Ȧǀ F = Ȧǀ m

10 Topics 1. Rate of change of vector in translating frame..rate of change of vector in a rotating frame 3. Relation between velocities and accelerations of points on a rigid body. 10

11 Translating frames Let F & T be the two reference frames such that frame T translates w.r.t F i.e no rotations are involved. During translatory motion, the coordinate axes in the two frames retain same orientation w.r.t each other.

12 xyz translate relative to XYZ. i,j,k as seen from XYZ remain constant in magnitude and direction.

13 enth Rate of change of vector in a translating frame

14 enth Velocity of a point moving in a translating frame

15 Velocity of a point moving in a translating enth frame

16 Acceleration of a point in translating frame

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18 Rate of change of vector in rotating frame enth

19 XYZ fixed frame F X Y Z fixed frame F with Z oriented in direction of w xyz moving frame with angular velocity w. z is also oriented in direction of w

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23 Rate of change of vector in fixed frame = Rate of change of vector in moving frame + w x A

24 For a vector fixed in m, i.e.,

25 enth nd derivative

26 Relation between velocity of two points on a rigid body in fixed frame of reference.

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30 enth Relation between acceleration of two points on a rigid body in fixed frame of reference.

31 While Solving problems se will use vectors. We will not use graphical method, i.e. velocity or acceleration diagrams - 31

32 Concept Quiz What is the direction of the velocity of point A on the turbine blade? w a) A b) c) d) y x L va w r v wkˆ Li A v Lw ˆj A ˆ 15-3

33 Concept Quiz What is the direction of the normal acceleration of point A on the turbine blade? a) A w b) c) d) an an w r ˆ w ( Li ) ˆ a Lw i n y x L 15-33

34 Group Problem Solving A series of small machine components being moved by a conveyor belt pass over a 150 mm-radius idler pulley. At the instant shown, the velocity of point A is 375 mm/s to the left and its acceleration is 5 mm/s to the right. Determine (a) the angular velocity and angular acceleration of the idler pulley, (b) the total acceleration of the machine component at B

35 Group Problem Solving SOLUTION: Using the linear velocity and accelerations, calculate the angular velocity and acceleration. Using the angular velocity, determine the normal acceleration. Determine the total acceleration using the tangential and normal acceleration components of B

36 Group Problem Solving Find the angular velocity of the idler pulley using the linear velocity at B. v= 375 mm/s a t = 5 mm/s B v r 375 mm/s (150 mm) w.50 rad/s 15-36

37 Group Problem Solving Find the angular acceleration of the idler pulley. v= 375 mm/s a t = 5 mm/s B a r 5 mm/s (150 mm) rad/s 15-37

38 Group Problem Solving Find the normal acceleration of point B. v= 375 mm/s a t = 5 mm/s B an r (150 mm)(.5 rad/s) a n mm/s What is the direction of the normal acceleration of point B? Downwards, towards the center 15-38

39 Group Problem Solving - 39

40 Video - 40

41 Knee joint video - 41

42 Group Problem Solving In the position shown, bar AB has an angular velocity of 4 rad/s clockwise. Determine the angular velocity of bars BD and DE. Which of the following is true? a) The direction of v B is b) The direction of v D is c) Both a) and b) are correct - 4

43 Group Problem Solving In the position shown, bar AB has an angular velocity of 4 rad/s clockwise. Determine the angular velocity of bars BD and DE. - 43

44 Group Problem Solving y x Determine the angular velocity of bars BD and DE. How should you proceed? w AB = 4 rad/s Determine v B with respect to A, then work your way along the linkage to point E. Write v B in terms of point A, calculate v B. v v w r B A AB B/A w AB (4 rad/s) k r (175 mm) i v r ( 4 k) ( 175 i) B/ A B AB B/ A v B (700 mm/s) j Does it make sense that v B is in the +j direction? - 44

45 y x Write v B in terms of point A, calculate v B. w AB = 4 rad/s v v w r B A AB B/A w (4 rad/s) AB k - 45

46 y x w AB = 4 rad/s r (175 mm) i / BA v w r ( 4 k) ( 175 i) B AB B/ A v B (700 mm/s) j Does it make sense that v B is in the +j direction? - 46

47 v D, y x w AB = 4 rad/s Determine v D with respect to E, then equate it to equation above. w DE w DE k r (75 mm) i (75 mm) j DE / - 47

48 v D, y x w AB = 4 rad/s Determine v D with respect to E, then equate it to equation above. v w r D DE D/ E ( w k ) ( 75i 75 j) DE v 75w j 75w i D DE DE - 48

49 v D, y x w AB = 4 rad/s Determine v D with respect to B. w BD w BD k r (00 mm) j / DB - 49

50 v D, y x w AB = 4 rad/s Determine v D with respect to B. v v w r D B BD D/ B 700 j ( w k) ( 00 j) BD v 700j 00w i D BD - 50

51 v D, Equating components of the two expressions for v D j: w w.5455 rad/s DE DE 3 i: 00w 75w w w BD DE BD DE 8 w DE.55 rad/s w BD rad/s - 51

52 Group Problem Solving Knowing that at the instant shown bar AB has a constant angular velocity of 4 rad/s clockwise, determine the angular acceleration of bars BD and DE. Which of the following is true? a) The direction of a D is b) The angular acceleration of BD must also be constant c) The direction of the linear acceleration of B is - 5

53 Group Problem Solving SOLUTION: The angular velocities were determined in a previous problem by simultaneously solving the component equations for v v v D B D B Knowing that at the instant shown bar AB has a constant angular velocity of 4 rad/s clockwise, determine the angular acceleration of bars BD and DE. The angular accelerations are now determined by simultaneously solving the component equations for the relative acceleration equation. - 53

54 w AB = 4 rad/s From our previous problem, we used the relative velocity equations to find that: w.55 rad/s DE w BD rad/s AB We can now apply the relative acceleration equation with - 54

55 w AB = 4 rad/s Analyze Bar AB w B A AB B/A AB B/A a a r r w (4) ( ).8 m/s AB B/ A a r i i B - 55

56 w AB = 4 rad/s Analyze Bar BD w D B BD D/ B BD D/ B a a r r.8 i k ( 0. j) ( ) ( 0. j) BD - 56

57 a (.8 0. ) i j D BD - 57

58 w AB = 4 rad/s Analyze Bar DE a r w r D DE D/ E DE D/ E k ( 0.75i j) DE (.5455) ( 0.75i j) - 58

59 a D ( ) i DE ( ) j DE - 59

60 a D ( ) i DE ( ) j DE a (.8 0. ) i j D BD - 60

61 w AB = 4 rad/s Equate like components of a D j: (0.75 DE 0.486) i:.8 0. BD [ (0.075)(.43) ] DE BD.43rad/s 4.18rad/s - 61

62 Planetary gears Video - 6

63 General Plane Motion General plane motion is neither a translation nor a rotation. General plane motion can be considered as the sum of a translation and rotation

64 General Plane Motion General plane motion can be considered as the sum of a translation and rotation

65 General Plane Motion General plane motion can be considered as the sum of a translation and rotation. Displacement of particles A and B to A and B can be divided into two parts: - translation to A and - rotation of B about A to B 1 B

66 Instantaneous Center of Rotation in Plane Motion The same translational and rotational velocities at A are obtained by allowing the slab to rotate with the same angular velocity about the point C on a perpendicular to the velocity at A

67 Instantaneous Center of Rotation in Plane Motion Plane motion of all particles in a slab can always be replaced by the translation of an arbitrary point A and a rotation about A with an angular velocity that is independent of the choice of A

68 Instantaneous Center of Rotation in Plane Motion The velocity of all other particles in the slab are the same as originally defined since the angular velocity and translational velocity at A are equivalent. As far as the velocities are concerned, the slab seems to rotate about the instantaneous center of rotation C

69 Instantaneous Center of Rotation in Plane Motion If the velocity at two points A and B are known, the instantaneous center of rotation lies at the intersection of the perpendiculars to the velocity vectors through A and B

70 Instantaneous Center of Rotation in Plane Motion If the velocity vectors at A and B are perpendicular to the line AB, the instantaneous center of rotation lies at the intersection of the line AB with the line joining the extremities of the velocity vectors at A and B

71 Instantaneous Center of Rotation in Plane Motion If the velocity vectors are parallel and equal, the instantaneous center of rotation is at infinity and the angular velocity is zero

72 Instantaneous Center of Rotation in Plane Motion The instantaneous center of rotation lies at the intersection of the perpendiculars to the velocity vectors through A and B. w v B v A AC va l cos BC w l sin v A tan va l cos The velocities of all particles on the rod are as if they were rotated about C. The particle at the center of rotation has zero velocity. 15-7

73 Instantaneous Center of Rotation in Plane Motion The particle coinciding with the center of rotation changes with time and the acceleration of the particle at the instantaneous center of rotation is not zero. The acceleration of the particles in the slab cannot be determined as if the slab were simply rotating about C

74 Instantaneous Center of Rotation in Plane Motion At the instant shown, what is the approximate direction of the velocity of point G, the center of bar AB? G a) b) c) d) 15-74

75 Group Problem Solving What is the velocity of B? What direction is the velocity of B? What direction is the velocity of D? v B ( AB) w (0.5 m)(4 rad/s) 1m/s AB w AB = 4 rad/s v B Find m tan m v D - 75

76 Group Problem Solving Locate instantaneous center C at intersection of lines drawn perpendicular to v B and v D. Calculate w BD C v B B 100 mm v B ( BC) w BD D 1m/s (0.5 m) w BD v D w BD 4 rad/s - 76

77 Group Problem Solving C B Find w DE v B 100 mm v D D v D 0.5 m ( DC) wbd (4 rad/s) cos 1 m/s 0.15 m ( ) ; ; cos cos v DE w w D DE DE w DE 6.67 rad/s - 77

78 enth Absolute and Relative Velocity in Plane Motion Any plane motion can be replaced by a translation of an arbitrary reference point A and a simultaneous rotation about A. A B A B v v v w rw v r k v A B A B A B A B A B r k v v w

79 Sample Problem 15. The double gear rolls on the stationary lower rack: the velocity of its center is 1. m/s. Determine (a) the angular velocity of the gear, and (b) the velocities of the upper rack R and point D of the gear

80 Sample Problem

81 Sample Problem 15. The double gear rolls on the stationary lower rack: the velocity of its center is 1. m/s. Determine (a) the angular velocity of the gear, and (b) the velocities of the upper rack R and point D of the gear

82 Sample Problem 15. SOLUTION: The displacement of the gear center in one revolution is equal to the outer circumference. Relate the translational and angular displacements. Differentiate to relate the translational and angular velocities. 15-8

83 Sample Problem 15. The velocity for any point P on the gear may be written as v v v P A P A Evaluate the velocities of points B and D. v A w k r P A 15-83

84 Sample Problem 15. SOLUTION: y x w < 0 (rotates clockwise). w wk 8rad sk v w A r v r 1 A 1 w 1.m s 0.150m 15-84

85 Sample Problem 15. For any point P on the gear, v P v A v P A v A w k r P A 15-85

86 Sample Problem 15. Velocity of the upper rack is equal to velocity of point B: v R v B v A k r B 1.m si 8rad sk 0.10 m w A 1.m si 0.8m si j v R m si 15-86

87 Sample Problem 15. v D Velocity of the point D: v A wk r D A 1.m si 8rad sk mi v v D D 1.m s i 1.697m s 1.m s j 15-87

88 Sample Problem

89 Sample Problem

90 Sample Problem 15.6 The center of the double gear has a velocity and acceleration to the right of 1. m/s and 3 m/s, respectively. The lower rack is stationary. Determine (a) the angular acceleration of the gear, and (b) the acceleration of points B, C, and D

91 Sample Problem 15.6 SOLUTION: The expression of the gear position as a function of is differentiated twice to define the relationship between the translational and angular accelerations. The acceleration of each point on the gear is obtained by adding the acceleration of the gear center and the relative accelerations with respect to the center. The latter includes normal and tangential acceleration components

92 Sample Problem 15.6 a A r r1 1 a A r k 1 3m s m 0rad s k 15-9

93 Sample Problem 15.6 The acceleration of each point is obtained by adding the acceleration of the gear center and the relative accelerations with respect to the center. The latter includes normal and tangential acceleration components

94 enth Sample Problem j i i j j k i r r k a a a a a a a A B A B A n A B t A B A A B A B s 6.40m s m s 3m m s 8rad m s 0rad s 3m w s 8.1m s 6.40m m 5 B B a j i s a

95 enth Sample Problem j i i j j k i r r k a a a a A C A C A A C A C s 9.60m s 3m s 3m m s 8rad m s 0rad s 3m w j a c s 9.60m i j i i i k i r r k a a a a A D A D A A D A D s 9.60m s 3m s 3m 0.150m s 8rad m s 0rad s 3m w s 1.95m s 3m.6m 1 D D a j i s a

96 Figure shows a cyclist on a bicycle. Assume that he is pedaling at 90 rpm and chain runs between the 1 cm radius front chain ring and the 3 cm radius rear cog. The wheels have a radius of 5 cm. Determine how fast the cyclist is travelling? Assume no slipping between the wheel and the road. 1-96

97 1-97

98 Since, O only moves horizontally, b 1 component must be

99 To keep in shape during the winter, cyclists often put their bikes on rollers which allows them to pedal as hard as they wish without going anywhere. Assume that the cyclist is pedaling at 4 rad/s and then begins to accelerate so that the angular acceleration of the main crank assembly is equal to 1.rad/s. After accelerating at this rate for s. what is the acceleration of the pedal spindle P? The distance from O, the center of the bottom bracket, to the pedal spindle is 17 cm, and the bottom bracket center O is fixed in space. Find the acceleration of a bike's pedal spindle. 1-99

100 First we will need to determine the rotation rate after seconds by integrating with respect to time

101 Example General Plane Motion The knee has linear velocity and acceleration from both translation (the runner moving forward) as well as rotation (the leg rotating about the hip)

102 Sample Problem 15.4 SOLUTION: The point C is in contact with the stationary lower rack and, instantaneously, has zero velocity. It must be the location of the instantaneous center of rotation. Determine the angular velocity about C based on the given velocity at A. The double gear rolls on the stationary lower rack: the velocity of its center is 1. m/s. Evaluate the velocities at B and D based on their rotation about C. Determine (a) the angular velocity of the gear, and (b) the velocities of the upper rack R and point D of the gear

103 Sample Problem 15.4 SOLUTION: The point C is in contact with the stationary lower rack and, instantaneously, has zero velocity. It must be the location of the instantaneous center of rotation. Determine the angular velocity about C based on the given velocity at A. v 1.m s v A A raw w 8rad s r 0.15 m A Evaluate the velocities at B and D based on their rotation about C. v R v B r w B 0.5 m8rad s v R m si r v D D 0.15 m rdw 0.11 m 0.11 m8rad s v v D D 1.697m s 1. i 1. j m s

104 Golf Robot Not ours maybe Tom Mase has pic? A golf robot is used to test new equipment. If the angular velocity of the arm is doubled, what happens to the normal acceleration of the club head? If the arm is shortened to ¾ of its original length, what happens to the tangential acceleration of the club head? w, If the speed of the club head is constant, does the club head have any linear accelertion? - 104

105 Instantaneous Center of Zero Velocity What happens to the location of the instantaneous center of velocity if the crankshaft angular velocity increases from 000 rpm in the previous problem to 3000 rpm? What happens to the location of the instantaneous center of velocity if the angle is 0? - 105

106 ighth 007 The McGraw-Hill Companies, Inc. All rights reserved

107 ighth A disc C is mounted on a shaft AB in Figure. The shaft and disc rotate with a constant angular speed ω of 10 rad/sec relative to the platform to which bearings A and B are attached. Meanwhile, the platform rotates at a constant angular speed ω 1 of 5 rad/sec relative to the ground in a direction parallel to the Z axis of the ground reference XYZ. What is the angular velocity vector ω for the disc C relative to XYZ? What are (dω/dt) xyz and (d ω /dt ) xyz? 007 The McGraw-Hill Companies, Inc. All rights reserved

108 Consider the vector w. Note that this vector is constrained in direction to be always collinear with the axis AB of the bearings of the shaft. This clearly is a physical requirement. Also, since w is of constant value, we may think of the vector w as fixed to the platform along AB.

109 w w w 1 1 w w w 1 1 w w w 1 1 1

110 Example 15.1: sec / 50 ) 10 (5 5 ) ( 0 sec / jrad j k k irad j k w w w w w w w w w w w w

111 Consider a position vector ρ between two points on the rotating disc. The length of ρ is 100 mm and at the instant of interest is the vertical direction. What are the first and second time derivatives of ρ at this instant as seen from ground reference?

112 r is fixed in a frame rotating at w 1 + w 1 r w w r ) ( 1 1 / sec ) 10 5 ( mm i k j k r r w w r w w r ) ( ) ( r w w r w w r ) ( ) 0 (

113 r w w r w w r ) ( ) ( r w w r w w r ) ( ) 0 ( sec / sec / 1000 ) 10 (5 100 ) 10 (5 mm k j mm i j k k j k r r

114 ighth A tank is maneuvering its gun into position. At the instant of interest, the turrent A, is rotating at an angular speed of rad/sec relative to the tank and is in position. Also at this instant, the gun is rotating at an angular speed turret and forms an angle horizontal plane. What are gun relative to the ground? of 1 rad/sec relative to the =30 0 with the w w, w and of the 007 The McGraw-Hill Companies, Inc. All rights reserved

115 ighth 007 The McGraw-Hill Companies, Inc. All rights reserved

116 007 The McGraw-Hill Companies, Inc. All rights reserved. ighth w (1)cos 0 i (1)sin 0 j i 0.34 j Consider w 1 Fixed in turret. w w w 1 1 w w w 0 w w

117 w k (0.940 i 0.34 j) j irad / sec w w 1 w 1 w 1 0 ( w ) ( w w ) 1 1 1

118 At instant of interest: w k [(k ) (0.940 i 0.34 j)] 3.76i j rad / 3 sec

119 Concept Question The figure depicts a model of a coaster wheel. If both w 1 and w are constant, what is true about the angular acceleration of the wheel? a)it is zero. b)it is in the +x direction c)it is in the +z direction d)it is in the -x direction e)it is in the -z direction

120 Sample Problem

121 The crane rotates with a constant angular velocity w 1 = 0.30 rad/s and the boom is being raised with a constant angular velocity w = 0.50 rad/s. The length of the boom is l = 1 m. - 11

122 Determine: angular velocity of the boom, angular acceleration of the boom, velocity of the boom tip, and acceleration of the boom tip. 15-1

123 SOLUTION: With w 1 r 0.30 j w 1cos30i 10.39i 6 j 0.50k sin 30 j Angular velocity of the boom, w w w

124 Angular acceleration of the boom, w1 w 1 w w w w Oxyz w v w Velocity of boom tip, r 15-14

125 enth 15 Acceleration of boom tip, v r r r a w w w

126 Sample Problem j w r 10.39i 6 j w 0.50k 15-16

127 Sample Problem SOLUTION: Angular velocity of the boom, w w w 1 w 0.30rad s j 0.50rad sk 15-17

128 w Angular acceleration of the boom, w 1 1 w w w w Oxyz w 0.30rad s j 0.50rad sk 0.15 rad s i - 18

129 Sample Problem Velocity of boom tip, i j k v w r v 3.54m si 5.0m s j 3.1m sk 15-19

130 a a a Acceleration of boom tip, r w w r r w v i j k i j k i i 1.50 j 0.90k k m s i 1.50 m s j 1.80 m s k

131 Absolute and Relative Acceleration in Plane Motion As the bicycle accelerates, a point on the top of the wheel will have acceleration due to the acceleration from the axle (the overall linear acceleration of the bike), the tangential acceleration of the wheel from the angular acceleration, and the normal acceleration due to the angular velocity

132 Concept Question You have made it to the kickball championship game. As you try to kick home the winning run, your mind naturally drifts towards dynamics. Which of your following thoughts is TRUE, and causes you to shank the ball horribly straight to the pitcher? A) Energy will not be conserved when I kick this ball B) In general, the linear acceleration of my knee is equal to the linear acceleration of my foot C) Throughout the kick, my foot will only have tangential acceleration. D) In general, the angular velocity of the upper leg (thigh) will be the same as the angular velocity of the lower leg - 13

133 Applications Rotating coordinate systems are often used to analyze mechanisms (such as amusement park rides) as well as weather patterns

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139 Vector moving in moving frame of reference 139

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147 Coriolis Acceleration Consider a collar P which is made to slide at constant relative velocity u along rod OB. The rod is rotating at a constant angular velocity w. The point A on the rod corresponds to the instantaneous position of P. Absolute acceleration of the collar is a P a A a P F a c

148 Coriolis Acceleration a a P A a P F a a A a P r c where F r 0 Oxy a v a P F c r a rw c w The absolute acceleration consists of the radial and tangential vectors shown A u

149 Coriolis Acceleration at t, at t t, v v v v A A u u

150 Coriolis Acceleration at t, at t t, v v v v A A Change in velocity over t is represented by the sum of three vectors v RR TT T T TT is due to change in direction of the velocity of point A on the rod, u u lim t0 TT t lim t0 v A t rww rw a A a r A recall, r a rw A

151 enth Coriolis Acceleration u v v t t u v v t A A, at, at result from combined effects of relative motion of P and rotation of the rod T T R R and u u u t r t u t T T t RR t t w w w w lim lim 0 0 u a v a c P c w F recall,

152 Concept Question You are walking with a constant velocity with respect to the platform, which rotates with a constant angular velocity w. At the instant shown, in which direction(s) will you experience an acceleration (choose all that apply)? a P y x v r w r r Oxy r Oxy - 15

153 Concept Question a) +x b) -x c) +y d) -y e) Acceleration = 0 y x v w a P r r r Oxy r Oxy - 153

154 Determine the acceleration of the virtual point D. Calculate the Coriolis acceleration. Add the different components to get the overall acceleration of point D

155 155

156 Group Problem Solving The sleeve BC is welded to an arm that rotates about stationary point A with a constant angular velocity w = (3 rad/s) j. In the position shown rod DF is being moved to the left at a constant speed u = 400 mm/s relative to the sleeve. Determine the acceleration of Point D

157 SOLUTION: The absolute acceleration of point D may be expressed as a a a a D D' D BC c - 157

158 Given: u= 400 mm/s, w = (3 rad/s) j. Find: a D Write overall expression for a D a r r r r D Oxy Oxy - 158

159 Do any of the terms go to zero? a r r D r Oxy r Oxy 159

160 Determine the normal acceleration term of the virtual point D a D r where r is from A to D (3 rad/s) j (3 rad/s) j [ (100 mm) j (300 mm) k] (700 mm/s ) k - 160

161 D Determine the Coriolis acceleration of point D a r r r r Oxy Oxy a C w v D/ F (3 rad/s) j(400 mm/s) k (400 mm/s ) i - 161

162 Add the different components to obtain the total acceleration of point D a a a a D D D/ F C (700 mm/s ) k 0 (400 mm/s ) i a (400 mm/s ) i (700 mm/s ) k D - 16

163 In the previous problem, u and w were both constant. w What would happen if u was increasing? a)the x-component of a D would increase b)the y-component of a D would increase c)the z-component of a D would increase d)the acceleration of a D would stay the same

164 What would happen if w was increasing? w a)the x-component of a D would increase b)the y-component of a D would increase c)the z-component of a D would increase d)the acceleration of a D would stay the same

165 Geneva mechanism - 165

166 Sample Problem 15.9 Disk D of the Geneva mechanism rotates with constant counterclockwise angular velocity w D = 10 rad/s. At the instant when = 150 o, determine (a) the angular velocity of disk S, and (b) the velocity of pin P relative to disk S

167 Sample Problem 15.9 SOLUTION: The absolute velocity of the point P may be written as v v v P P P s Magnitude and direction of velocity v P of pin P are calculated from the radius and angular velocity of disk D. Direction of velocity vp of point P on S coinciding with P is perpendicular to radius OP. Direction of velocity v P s of P with respect to S is parallel to the slot. Solve the vector triangle for the angular velocity of S and relative velocity of P

168 Sample Problem 15.9 SOLUTION: The absolute velocity of the point P may be written as v v v P P P s Magnitude and direction of absolute velocity of pin P are calculated from radius and angular velocity of disk D. v P Rw D 50 mm 10 rad s 500mm s

169 Sample Problem 15.9 Direction of velocity of P with respect to S is parallel to slot. From the law of cosines, r R l Rl cos R r 37.1mm sin sin30 sin30 sin 4. 4 R r 0.74 The interior angle of the vector triangle is

170 Sample Problem 15.9 Direction of velocity of point P on S coinciding with P is perpendicular to radius OP. v P 500 mm s v P vp sin 500mm s sin mm s 151.mm s rws ws ws 4.08rad sk 37.1mm v P v P s vp cos 500m s cos m s cos 4.4i sin 4. 4 j s

171 Sample Problem In the Geneva mechanism, disk D rotates with a constant counter-clockwise angular velocity of 10 rad/s. At the instant when j = 150 o, determine angular acceleration of disk S

172 Sample Problem SOLUTION: The absolute acceleration of the pin P may be expressed as a a a a P P P s The instantaneous angular velocity of Disk S is determined as in Sample Problem The only unknown involved in the acceleration equation is the instantaneous angular acceleration of Disk S. Resolve each acceleration term into the component parallel to the slot. Solve for the angular acceleration of Disk S. c 15-17

173 Sample Problem SOLUTION: Absolute acceleration of the pin P may be expressed as a a a a P P From Sample Problem v P s 4.4 w S 477mm s cos 4.4i sin 4.4 j P s 4.08rad s k c

174 - 174

175 Sample Problem Considering each term in the acceleration equation, a a P P Rw D 500mm 10rad 5000 mm s cos30i sin30 j s 5000 mm s

176 Sample Problem a P a a a rw cos4.4i sin 4.4 j P a r sin 4.4i cos4.4 j P a 37.1mm sin 4.4i cos4.4 j P n t t P n S S S P t note: S may be positive or negative

177 Sample Problem The direction of the Coriolis acceleration is obtained by rotating the direction of the relative velocity v P by 90 o in the sense of w S. s a c w SvP s sin 4.4i cos 4.4 j 4.08rad s477mm s sin 4.4i cos 4.4 j 3890 mm s sin 4.4i cos 4.4 j

178 Sample Problem The relative acceleration must be parallel to the slot. a P s Equating components of the acceleration terms perpendicular to the slot, 37.1 S cos rad s S S 33rad sk

179 Sample Problem For the disk mounted on the arm, the indicated angular rotation rates are constant. Determine: the velocity of the point P, the acceleration of P, and angular velocity and angular acceleration of the disk

180 Sample Problem

181 Sample Problem SOLUTION: Define a fixed reference frame OXYZ at O and a moving reference frame Axyz or F attached to the arm at A. r Li w 1 j Rj rp A Rj wd F w k

182 Sample Problem With P of the moving reference frame coinciding with P, the velocity of the point P is found from vp vp vp F v r j Li Rj P w1 w1l k v w r w k Rj w R i P F D F P A v P w R i w 1 L k 15-18

183 Sample Problem

184 enth Sample Problem The acceleration of P is found from c P P P a a a a F i L Lk j r a P w w w j R Ri k r a A P D D P w w w w w F F F k R R i j v a P c 1 1 w w w w F Rk Rj Li a P 1 1 w w w w

185 Sample Problem Angular velocity and acceleration of the disk, w w D F w j w 1 w k w F w j w1 w1 j w k w 1 w i

186 The crane shown rotates at the constant rate w 1 = 0.5 rad/s; simultaneously, the telescoping boom is being lowered at the constant rate w = 0.40 rad/s. Knowing that at the instant shown the length of the boom is 6 m and is increasing at the constant rate u= 0.5 m/s determine the acceleration of Point B

187 Group Problem Solving SOLUTION: Define a moving reference frame Axyz or F attached to the arm at A. The acceleration of P is found from a a a a B B' B F c The angular velocity and angular acceleration of the disk are w w B F w w F

188 Group Problem Solving Given: w 1 = 0.5 rad/s, w = rad/s. L= 6 m, u= 0.5 m/s. Find: a B. Equation of overall acceleration of B D a r r r r Do any of the terms go to zero? Oxy Oxy a r r r r D Oxy Oxy

189 Group Problem Solving Let the unextending portion of the boom AB be a rotating frame of reference. What are and? w i w j 1 (0.40 rad/s) i(0.5 rad/s) j. w jw i 1 ww k 1 (0.10 rad/s ) k

190 D a r r r r Oxy Oxy Determine the position vector r B/A r B/ A r B (6 m)(sin 30j cos30 k) (3 m) j(3 3 m) k Find r i j k r (0.3 m/s ) i B

191 Find r r B (0.40i 0.5 j) (0.40i 0.5 j) (3j 3 3 k) (0.3 m/s ) i (0.48 m/s ) j ( m/s ) k

192 D a r r r r Determine the Coriolis acceleration first define the relative velocity term Oxy Oxy vbf / u(sin 30j cos30 k) (0.5 m/s)sin30 j (0.5 m/s)cos30k 15-19

193 Calculate the Coriolis acceleration Ω v ()(0.40i 0.5 j) (0.5sin 30j 0.5cos30 k) BF / (0.165 m/s ) i ( m/s ) j (0. m/s ) k Add the terms together ab (0.817 m/s ) i (0.86 m/s ) j (0.956 m/s ) k