Chapter 21 - Engineering Mechanics Dynamics Chapter 21 a...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Engineering Mechanics - Dynamics Chapter 21 a Iyz 0 2m a h 2 h( a y) h a y dy 2 2 1 mah 6 Iyz 1 mah 6 Problem 21-7 Determine by direct integration the product of inertia Ixy for the homogeneous prism. The density of the material is . Express the result in terms of the mass m of the prism. Solution: 2m a h 2 717 Engineering Mechanics - Dynamics Chapter 21 a Ixy 0 2m a h 1 2 a m 12 2 h( a y) y a 2 y dy 1 4 m a 2 12 a Ixy *Problem 21-8 Determine the radii of gyration kx and ky for the solid formed by revolving the shaded area about the y axis. The density of the material is Given: a b 4 ft 0.25 ft slug ft Solution: b a 2 2 3 12 M 0 a dy b b 2 a b y 2 dy M 292.17 slug a Ix 0 b a 4 2 y 2 a dy b a 2 a b 4y 2 2 2 y 2 a b y 2 2 2 dy Ix 948.71 slug ft 2 Iy 0 a 2 2 a dy b 2 a b 2y 2 2 2 a b y 2 2 2 dy Iy 1608.40 slug ft 2 kx Ix M kx 1.80 ft 718 Engineering Mechanics - Dynamics Chapter 21 ky Iy M ky 2.35 ft Problem 21-9 Determine the mass moment of inertia of the homogeneous block with respect to its centroidal x' axis. The mass of the block is m. Solution: m abh h m abh Ix' m abh 2 1 2 a 12 h z a b dz 2 m 1 3 a bh a b h 12 1 3 abh 12 2 Ix' m 2 a 12 h 2 Problem 21-10 Determine the elements of the inertia tensor for the cube with respect to the x, y, z coordinate system. The mass of the cube is m. 719 Engineering Mechanics - Dynamics Chapter 21 Solution: Ixx Iyy Izz a m 2 2 2 2 ma 3 2 Ixy Ixz a m 4 2 Iyz a m 2 a m 4 2 I ma 12 2 8 3 8 3 3 3 8 3 3 Remember to change the signs of the products of inertia to put them in the inertia tensor Problem 21-11 Compute the moment of inertia of the rod-and-thin-ring assembly about the z axis. The rods and ring have a mass density . Given: 2 l h kg m 500 mm 400 mm 120 deg Solution: r l 2 h h l l 2 acos Iz 3 l sin 3 2 2 2 r r 2 Iz 0.43 kg m 2 720 Engineering Mechanics - Dynamics Chapter 21 *Problem 21-12 Determine the moment of inertia of the cone about the z' axis. The weight of the cone is W, the height is h, and the radius is r. Given: W h r 15 lb 1.5 ft 0.5 ft ft s Solution: r h 2 2 g 32.2 atan Ix Iy Iz' 3 2 W 4r 80 Ix Ix sin Iz 2 h 3h W 4 3 2 Wr 10 2 Iz cos 2 Iz' 0.0962 slug ft 2 Problem 21-13 The bent rod has weight density Locate the center of gravity G(x', y') and determine the principal moments of inertia Ix', Iy', and Iz' of the rod with respect to the x', y', z' axes. Given: 1.5 a b g 1 ft 1 ft 32.2 ft s 2 lb ft 721 Engineering Mechanics - Dynamics Chapter 21 Solution: 2a x' a ba b b b y' 0.50 ft 2 2 2a x' 0.667 ft ab y' 2a 2 b Ix' a y' 2 a (b y' ) 2 1 2 bb 12 2 b b 2 2 y' Ix' 0.0272 slug ft 2 Iy' 2 2 aa 12 Ix' Iy' a 2 a 2 x' b (a x' ) Iy' 0.0155 slug ft 2 Iz' Iz' 0.0427 slug ft 2 Problem 21-14 The assembly consists of two square plates A and B which have a mass MA each and a rectangular plate C which has a mass MC. Determine the moments of inertia Ix, Iy and Iz. Given: MA MC 3 kg 4.5 kg 60 deg 1 2 90 deg 30 deg 0.3 m 0.2 m 0.4 m a b c Solution: MA A c( 2b) 722 Engineering Mechanics - Dynamics Chapter 21 Ix 1 2 MC ( 2a) 12 c 2 0 b A( 2b) a cos 2 sin 2 d Iy 1 2 MC ( 2b) 12 c 2 Ax b 0 b c 2 Ax b 0 2 2 sin 2 d dx Iz 1 2 MC ( 2b) 12 ( 2a) 2 2 a cos 2 d dx Ix 1.36 kg m 2 Iy 0.380 kg m 2 Iz 1.25 kg m 2 Problem 21-15 Determine the moment of inertia Ix of the composite plate assembly. The plates have a specific weight . Given: 6 lb ft a b c g 2 0.5 ft 0.5 ft 0.25 ft 32.2 ft s 2 Solution: a b 2 2 2 c atan I1 c2 a b ( 2a) 2 ( 2b) 12 2 3 723 Engineering Mechanics - Dynamics Chapter 21 I2 c2 a 2 b 2 2 2c 3 I2 cos 2 Ix 2 I1 sin ( 2a) ( 2a) ( 2b) 12 2 Ix 0.0293 slug ft 2 *Problem 21-16 Determine the product of inertia Iyz of the composite plate assembly. The plates have a specific weight Solution: Due to symmetry, Iyz 0 Problem 21-17 Determine the moment of inertia of the composite body about the aa axis. The cylinder has weight Wc and each hemisphere has weight Wh. Given: Wc Wh b c g Solution: atan c b 2 20 lb 10 lb 2 ft 2 ft 32.2 ft s 2 Iz c 2 2 Wh 2 5 1 c Wc 2 2 2 Iz 724 0.56 slug ft 2 Engineering Mechanics - Dynamics Chapter 21 Iy Iy Iaa 2 c 83 Wh 2 320 2 2 2Wh b 2 3 c 8 2 2 Wc b 12 2 c 2 2 1 4 1.70 slug ft Iz cos 2 Iy sin 2 Iaa 1.13 slug ft 2 Problem 21-18 Determine the moment of inertia about the z axis of the assembly which consists of the rod CD of mass MR and disk of mass MD. Given: MR MD r l 1.5 kg 7 kg 100 mm 200 mm Solution: atan r l 1 2 MD r 4 MD l 2 I1 I2 I3 1 2 MR l 3 I1 1 2 MD r 2 1 0 0 sin cos Iz I1 0 0 1 0 0 sin cos Imat 0 cos 0 sin 0 I2 0 0 0 2 0 cos 0 sin I3 Iz Imat 2 2 0.0915 kg m 725 Engineering Mechanics - Dynamics Chapter 21 Problem 21-19 The assembly consists of a plate A of weight WA, plate B of weight WB, and four rods each of weight Wr. Determine the moments of inertia of the assembly with respect to the principal x, y, z axes. Given: WA WB Wr rA rB h Solution: L rB h L 2 15 lb 40 lb 7 lb 1 ft 4 ft 4 ft rA 2 h 2 L 5.00 ft asin 53.13 deg Ix L 2Wr 3 WA sin 2 2 L 2 Wr 12 2 2 h Wr 2 2 rA 2 rB 2 WB rB 4 2 rA 4 WA h Iy Ix by symmetry 2 2 2 2 Iz Ix Iy Iz L 4 Wr cos 12 20.2 2 Wr rB 2 rA WA rA 2 WB rB 2 20.2 slug ft 16.3 2 *Problem 21-20 The thin plate has a weight Wp and each of the four rods has weight Wr. Determine the moment of 726 Engineering Mechanics - Dynamics Chapter 21 inertia of the assembly about the z axis. Given: Wp Wr h a 5 lb 3 lb 1.5 ft 0.5 ft Solution: L h 2 a h L 2 a 2 acos Iz 1 2 4 Wr L sin 3 2 1 12 Wp ( 2a) 2 ( 2a) 2 Iz 0.0881 slug ft 2 Problem 21-21 If a body contains no planes of symmetry, the principal moments of inertia can be determined mathematically. To show how this is done, consider the rigid body which is spinning with an angular velocity directed along one of its principal axes of inertia. If the principal moment of inertia about this axis is I, the angular momentum can be expressed as H = I = I xi + I yj + I zk. The components of H may also be expressed by Eqs. 21-10, where the inertia tensor is assumed to be known. Equate the i, j, and k components of both expressions for H and consider x, y, and z to be unknown. The solution of these three equations is obtained provided the determinant of the coefficients is zero. Show that this determinant, when expanded, yields the cubic equation I3 -(Ixx+ Iyy + Izz)I2 + (IxxIyy + IyyIzz + IzzIxx- I2xy - I2yz-I2zx)I -(IxxIyyIzz - 2IxyIyzIzx - IxxI2yz - IyyI2zx - IzzI2xy) = 0. The three positive roots of I, obtained from the solution of this equation, represent the principal moments of inertia Ix, Iy, and Iz. Solution: H I I xi I yj I zk Equating the i, j, and k components to the scalar (Eq. 21 - 10) yields Ixx I x Ixy y Iyy I y Ixz z Iyz z 0 0 Iyx x 727 Engineering Mechanics - Dynamics Chapter 21 Izx x Izy y Izz I x, z y, 0 requires Solution for nontrivial and z Ixx Iyx Izx I Ixy Iyy Izy I Ixz Iyz Izz I QED 0 Expanding the determinant produces the required equation Problem 21-22 Show that if the angular momentum of a body is determined with respect to an arbitrary point A, then HA can be expressed by Eq. 21-9. This requires substituting A = G + GA into Eq. 21-6 and expanding, noting that G dm = 0 by definition of the mass center and vG = vA+ GA. Solution: HA A dm vA A A dm HA G GA dm vA G GA G GA dm HA G dm vA GA m vA G G dm G dm GA GA G dm GA GA m 728 Engineering Mechanics - Dynamics Chapter 21 Since G dm 0 and HG G G dm HA HA GA m vA m vG HG HG GA GA m GA m vA GA HG G Q.E.D Problem 21-23 The thin plate of mass M is suspended at O using a ball-and-socket joint. It is rotating with a constant angular velocity = k when the corner A strikes the hook at S, which provides a permanent connection. Determine the angular velocity of the plate immediately after impact. Given: M 1 5 kg 2 rad s a b 300 mm 400 mm Solution: Angular Momentum is conserved about the line OA. 0 OA a b 1 3 2 oa OA OA I2 Mb I3 0 1 12 M ( 2a) 2 I1 I2 I3 I1 0 Imat 0 I2 0 0 0 I3 Ioa Guess oa Imat oa 1 rad s 729 T 2 Engineering Mechanics - Dynamics Chapter 21 0 Given Imat 0 1 oa Ioa 2 0.00 2 Find 2 2 oa 0.75 1.00 rad s *Problem 21-24 Rod AB has weight W and is attached to two smooth collars at its end points by ball-and-socket joints. If collar A is moving downward at speed vA, determine the kinetic energy of the rod at the instant shown. Assume that at this instant the angular velocity of the rod is directed perpendicular to the rod's axis. Given: W vA a b c 6 lb 8 3 ft 6 ft 2 ft L a 2 ft s Solution: Guesses vB 1 ft s b 2 c 2 x 1 rad s rad s c b a vB 0 0 y 1 rad s 0 z x y z 1 Given 0 vA x y z c b a 730 0 ft s Engineering Mechanics - Dynamics Chapter 21 vB x x y z z 0.98 1.06 1.47 rad s vB 12.00 ft s Find vB x y z y 0 vG 0 vA c 2 b 2 a 2 T 1 W 2 g vG vG 1 W L 2 g 2 12 T 6.46 lb ft Problem 21-25 At the instant shown the collar at A on rod AB of weight W has velocity vA. Determine the kinetic energy of the rod after the collar has descended a distance d. Neglect friction and the thickness of the rod. Neglect the mass of the collar and the collar is attached to the rod using ball-and-socket joints. Given: W vA a b c d Solution: L Guesses vB 1 ft s rad s z x 6 lb 8 3 ft 6 ft 2 ft 3 ft ft s a 2 b 2 c 2 1 rad s rad s 731 y 1 1 Engineering Mechanics - Dynamics Chapter 21 0 Given 0 vA vB x y z x y z x y z c b a vB 0 0 x y z c b a 0 ft s Find vB x y z 0.98 1.06 1.47 c 2 b 2 a 2 1 W 2 g L 2 rad s vB 12.00 ft s 0 vG 0 vA T1 1 W 2 g vG vG 12 T1 6.46 lb ft In position 2 the center of mass has fallen a distance d/2 T1 T2 0 T2 W W d 2 d 2 T2 15.5 lb ft T1 Problem 21-26 The rod AB of mass MAB is attached to the collar of mass MA at A and a link BC of mass MBC using ball-and-socket joints. If the rod is released from rest in the position shown, determine the angular velocity of the link after it has rotated 180 . Given: MAB MA 4 kg 1 kg 732 Engineering Mechanics - Dynamics Chapter 21 MBC a b c Solution: 2 kg 1.2 m 0.5 m a 2 b 2 atan 1 3 b a 2 2 I MAB c sin rad s 1 3 MBC b 2 Guess 1 Given MAB MBC g b 1 2 I 2 Find 10.85 rad s Problem 21-27 The rod has weight density and is suspended from parallel cords at A and B. If the rod has angular velocity about the z axis at the instant shown, determine how high the center of the rod rises at the instant the rod momentarily stops swinging. Given: 3 lb ft rad s 2 a g 3 ft 32.2 ft s 2 Solution: T1 V1 T2 V2 733 Engineering Mechanics - Dynamics Chapter 21 1 1 2 12 ( 2a) g ( 2a) 2 2 2a h h 1 2 a 6 2 g h 2.24 in *Problem 21-28 The assembly consists of a rod AB of mass mAB which is connected to link OA and the collar at B by ball-and-socket joints. When = 0 and y = y1, the system is at rest, the spring is unstretched, and a couple moment M, is applied to the link at O. Determine the angular velocity of the link at the instant = 90. Neglect the mass of the link. Units Used: kN Given: mAB M a y1 k 2 4 kg 7Nm 200 mm 600 mm kN m 10 N 3 Solution: L 1 3 L a 2 a 2 y1 2 I mAB L y1 2 Guess 1 a 2 rad s M( 90 deg) 1 2 I 2 Given mAB g 1 2 k 2 734 Engineering Mechanics - Dynamics Chapter 21 Find L OA 6.10 rad s 20.2 rad s a OA Problem 21-29 The assembly consists of a rod AB of mass mAB which is connected to link OA and the collar at B by ball-and-socket joints. When 0 and y = y1, the system is at rest, the spring is unstretched, and a couple moment M = M0(b + c), is applied to the link at O. Determine the angular velocity of the link at the instant 90. Neglect the mass of the link. Units Used: Given: mAB M0 y1 a b c k 4 kg 1Nm 600 mm 200 mm 4 2 2 kN m kN 10 N 3 Solution: L 1 3 L a 2 a 2 y1 2 I mAB L y1 2 Guess 1 a 2 rad s 90 deg Given mAB g M0 b 0 c d 1 2 I 2 1 2 k 2 Find 735 Engineering Mechanics - Dynamics Chapter 21 5.22 rad s OA L a OA 17.3 rad s Problem 21-30 The circular plate has weight W and diameter d. If it is released from rest and falls horizontally a distance h onto the hook at S, which provides a permanent connection, determine the velocity of the mass center of the plate just after the connection with the hook is made. Given: W d h g Solution: vG1 W g 2g h d 2 vG1 5 W 4 g 12.69 d 2 13.53 2 2 19 lb 1.5 ft 2.5 ft 32.2 ft s 2 ft s vG1 8vG1 2 rad s ft s 5d d 2 2 vG2 2 vG2 10.2 Problem 21-31 A thin plate, having mass M, is suspended from one of its corners by a ball-and-socket joint O. If a stone strikes the plate perpendicular to its surface at an adjacent corner A with an impulse Is, determine the instantaneous axis of rotation for the plate and the impulse created at O. Given: M a 4 kg 200 mm 736 Engineering Mechanics - Dynamics Chapter 21 45 deg 60 Is Solution: I1 2 3 Ma 2 0 0 Ns I2 1 3 Ma a 2 2 I3 I2 1 I23 0 M 4 0 Cmat 0 cos 0 sin I1 0 I2 sin cos 0 I23 Cmat I3 vx Ns vy vz 1 1 1 m s x y z T Imat Cmat 0 0 IOx 1 1 1 I23 1 1 1 rad s Guesses IOy IOz Given IOx Is IOy IOz 0 a 2 1 1 0 0 y z vx M vy vz x Is Imat y z x vx vy vz a 2 737 Engineering Mechanics - Dynamics Chapter 21 IOx IOy IOz x y z Find IOx IOy IOz x y z vx vy vz vx vy vz IOx Impulse IOy IOz x y z 8.57 axis Impulse 0.00 N s 0.00 0.00 axis 0.14 0.99 *Problem 21-32 Rod AB has weight W and is attached to two smooth collars at its ends by ball-and-socket joints. If collar A is moving downward with speed vA when z = a, determine the speed of A at the instant z = 0. The spring has unstretched length c. Neglect the mass of the collars. Assume the angular velocity of rod AB is perpendicular to its axis. Given: W vA a b c k 6 lb 8 3 ft 6 ft 2 ft 4 lb ft ft s 2 ft 738 Engineering Mechanics - Dynamics Chapter 21 g 32.2 ft s 2 Solution: First Position Guesses vB1 1 L a 2 b 2 c 2 ft s rad s 0 z1 x1 1 rad s rad s c b a vB1 0 0 y1 1 1 x1 y1 z1 Given 0 vA x1 y1 z1 c b a 0 ft s x1 x1 y1 z1 0.98 1 Find x1 y1 z1 vB1 1 y1 z1 1.06 1.47 rad s vB1 0 vG1 0 vA 6.00 vG1 0.00 4.00 T1 1 W 2 g vG1 vG1 T1 V1 1 W L 2 1 1 c 1 1 2 b a ft s 2 g 12 T2 V2 T1 6.46 lb ft Work - Energy 739 Engineering Mechanics - Dynamics Chapter 21 T2 T1 W a 2 1 2 k L 2 b 2 2 c T2 10.30 lb ft Second Position Guesses vA2 Note that B becomes the instantaneous center 1 ft s rad s rad s x2 y2 z2 2 2 y2 vB2 1 ft s rad s x2 1 1 z2 1 0 Given 0 vA2 x2 y2 z2 L 2 b 2 vB2 0 0 b 0 L b b 0 2 0 ft s T2 vA2 vB2 x2 y2 z2 1 W L 2 g 3 2 x2 2 y2 2 z2 x2 2.23 1.34 0.00 vA2 18.2 ft s rad s vB2 0.00 ft s Find vA2 vB2 x2 y2 z2 y2 z2 Problem 21-33 The circular disk has weight W and is mounted on the shaft AB at angle with the horizontal. Determine the angular velocity of the shaft when t = t1 if a constant torque M is applied to the shaft. The shaft is originally spinning with angular velocity 1 when the torque is applied. 740 Engineering Mechanics - Dynamics Chapter 21 Given: W 15 lb 45 deg t1 M 1 3s 2 lb ft 8 rad s r g Solution: 0.8ft 32.2 ft s 2 IAB W g M IAB r 2 4 cos 2 W g r 2 2 sin 2 IAB 0.11 lb ft s 2 2 1 t1 2 61.7 rad s Problem 21-34 The circular disk has weight W and is mounted on the shaft AB at angle of with the horizontal. Determine the angular velocity of the shaft when t = t1 if a torque M = M0 ebt applied to the shaft. The shaft is originally spinning at Given: W 15 lb 45 deg t1 M0 1 1 when the torque is applied. 2s 4 lb ft 8 rad s r g 0.8 ft 32.2 ft s 2 741 Engineering Mechanics - Dynamics Chapter 21 b 0.1 s 1 Solution: W g r cos 4 t1 0 2 2 IAB W g r 2 2 sin 2 IAB 0.11 lb ft s 2 1 2 1 IAB M0 e bt dt 2 87.2 rad s Problem 21-35 The rectangular plate of mass mp is free to rotate about the y axis because of the bearing supports at A and B. When the plate is balanced in the vertical plane, a bullet of mass mb is fired into it, perpendicular to its surface, with a velocity v. Compute the angular velocity of the plate at the instant it has rotated 180. If the bullet strikes corner D with the same velocity v, instead of at C, does the angular velocity remain the same? Why or why not? Given: mp mb v a b g 15 kg 0.003 kg 2000 m s 150 mm 150 mm 9.81 m s 2 Solution: Given Guesses mb v a 1 a mp 2 3 2 2 2 1 rad s 3 1 rad s 1 2 mp a 2 3 2 mp g a 2 a 1 mp 3 2 2 3 2 mp g a 2 742 Engineering Mechanics - Dynamics Chapter 21 2 3 Find 2 3 2 8.00 rad s 3 21.4 rad s If the bullet strikes at D, the result will be the same. *Problem 21-36 The rod assembly has a mass density and is rotating with a constant angular velocity = when the loop end at C encounters a hook at S, which provides a permanent connection. Determine the angular velocity of the assembly immediately after impact. Given: 2.5 a 1 1k kg m 0.5 m 2 rad s h 0.5 m Solution: 0 OC a h 1 3 1 3 3 oc OC OC I1 h 1 12 ( 2a) 3 2a h 2 I2 h 3 2a h 2 I3 I1 I2 I1 0 Imat 0 IOC oc Imat oc T 0 I2 0 0 0 1 I3 rad s Guess 2 743 Engineering Mechanics - Dynamics Chapter 21 0 Given Imat 0 1 oc IOC 2 2 Find 2 2 0.63 rad s rad s 0.00 2 oc 0.44 0.44 Problem 21-37 The plate of weight W is subjected to force F which is always directed perpendicular to the face of the plate. If the plate is originally at rest, determine its angular velocity after it has rotated one revolution (360). The plate is supported by ball-and-socket joints at A and B. Given: W F a b g 15 lb 8 lb 0.4 ft 1.2 ft 32.2 ft s Solution: atan a b a 2 2 18.43 deg 2 IAB IAB Guess W g 12 cos 2 2 W g b 12 sin 2 0.0112 lb ft s 1 rad s 2 1 2 IAB 2 Given F a cos Find 58.4 rad s 744 Engineering Mechanics - Dynamics Chapter 21 Problem 21-38 The space capsule has mass mc and the radii of gyration are kx = kz and ky. If it is traveling with a velocity vG, compute its angular velocity just after it is struck by a meteoroid having mass mm and a velocity vm = ( vxi +vyj +vzk ). Assume that the meteoroid embeds itself into the capsule at point A and that the capsule initially has no angular velocity. Units Used: Mg Given: mc kx ky vG a Solution: Guesses x 1000 kg 3.5 Mg 0.8 m mm vx vy vz 1m c 0.60 kg 200 m s m s 0.5 m m s 400 600 200 m s 1m b 3m 1 rad s y 1 rad s z 1 rad s 2 a Given c b mm vy vx vG vz mc kx 0 ky 2 0 0 kx 2 x y z 0 0 0 0.107 0.000 0.107 x y z x Find x y z y z rad s Problem 21-39 Derive the scalar form of the rotational equation of motion along the x axis when moments and products of inertia of the body are not constant with respect to time. 745 and the Engineering Mechanics - Dynamics Chapter 21 Solution: In general M d dt M Hxi Hyj Hzk H'x i H'y j xi z Hy H'z k yj y Hz i x Hy k zk Hxi Hyj Hzk and expanding the cross product yields Substitute M H'x H'z H'y x Hz z Hx j y Hx Substitute Hx, Hy, and Hz using Eq. 21 - 10. For the i component Mx d dt Ix x Ixy y Ixz z z Iy y Iz z Izx x Izy y y Iyz z Iyx x One can obtain y and z components in a similar manner. *Problem 21-40 Derive the scalar form of the rotational equation of motion along the x axis when moments and products of inertia of the body are constant with respect to time. Solution: In general M d dt M Hxi Hyj H'y j xi z Hy y Hx and the Hzk H'z k yj y Hz i x Hy k zk H'x i Hxi Hyj Hzk and expanding the cross product yields Substitute M H'x H'z H'y x Hz z Hx j Substitute Hx, Hy, and Hz using Eq. 21 - 10. For the i component Mx d dt Ix x Ixy y Ixz z z Iy y y Iz z Izx x Izy y 746 Iyz z Iyx x Engineering Mechanics - Dynamics Chapter 21 For constant inertia, expanding the time derivative of the above equation yields Mx I x 'x Ixy 'y Ixz 'z z Iy y Iz z Izx x Izy y y Iyz z Iyx x One can obtain y and z components in a similar manner. Problem 21-41 Derive the Euler equations of motion for Solution: In general M d dt M Substitute M H'x H'z Hxi Hyj Hzk i.e., Eqs. 21-26. H'x i H'y j xi z Hy H'z k yj y Hz i x Hy k zk Hxi Hyj Hzk and expanding the cross product yields H'y x Hz z Hx j y Hx Substitute Hx, Hy, and Hz using Eq. 21 - 10. For the i component Mx d dt Ix x Ixy y Ixz z z Iy y y Iz z Izx x Izy y Iyz Izx 0 and require Iyz z Iyx x Set Mx Ixy Ix 'x Ix Iy Iz to be constant. This yields Iy z y Iz y z One can obtain y and z components in a similar manner. Problem 21-42 The flywheel (disk of mass M) is mounted a distance d off its true center at G. If the shaft is rotating at constant speed , determine the maximum reactions exerted on the journal bearings at A and B. 747 Engineering Mechanics - Dynamics Chapter 21 Given: M d 40 kg 20 mm rad s a b 0.75 m 1.25 m m s 2 8 g 9.81 Solution: Guesses Check both up and down positions Aup Aup 1N B up Mg 1N 2 Given Bup Md Aup a A up Bup B up b 0 A up Bup 213.25 127.95 Find A up B up N Guesses Given Adown Adown 1N B down B down Mg Md 1N 2 Adown a A down Bdown Bdown b 0 A down Bdown 277.25 166.35 Find Adown B down N Thus Amax B max max A up Adown max B up B down Amax B max 277 N 166 N Problem 21-43 The flywheel (disk of mass M) is mounted a distance d off its true center at G. If the shaft is rotating at constant speed , determine the minimum reactions exerted on the journal bearings at A and B during the motion. 748 Engineering Mechanics - Dynamics Chapter 21 Given: M d 40 kg 20 mm rad s a b 0.75 m 1.25 m m s 2 8 Solution: Guesses g 9.81 Check both up and down positions Aup Aup 1N B up Mg 1N 2 Given Bup Md Aup a A up Bup B up b 0 A up Bup 213.25 127.95 Find A up B up N Guesses Adown Adown 1N B down Mg 0 Md 1N 2 Given B down Adown a A down Bdown Bdown b Find Adown B down A down Bdown 277.25 166.35 N Thus Amin B min min A up Adown min B up B down Amin B min 213 N 128 N 749 Engineering Mechanics - Dynamics Chapter 21 *Problem 21-44 The bar of weight W rests along the smooth corners of an open box. At the instant shown, the box has a velocity v = v1k and an acceleration a = a1k. Determine the x, y, z components of force which the corners exert on the bar. Given: W v1 a1 4 lb 5 ft s ft s Solution: Guesses Ax Ay 1 lb 1 lb Bx By Bz Given Ax Ay Bz Bx By W 0 0 W a1 g Bx By Bz 1 2 c b a Ax Ay 0 0 1 lb 1 lb 1 lb 2 a b c 2 ft 1 ft 2 ft 2 c 1 2 b a Ax Ay Bx By Bz Find Ax Ay B x By B z Ax Ay Bx 2.12 1.06 lb By Bz 2.12 1.06 lb 4.25 750 Engineering Mechanics - Dynamics Chapter 21 Problem 21-45 The bar of weight W rests along the smooth corners of an open box. At the instant shown, the box has a velocity v = v1j and an acceleration a = a1j. Determine the x, y, z components of force which the corners exert on the bar. Given: W v1 a1 4 lb 3 ft s 6 2 a b ft c s 2 ft 1 ft 2 ft Solution: Guesses Ax Ay 1 lb 1 lb Bx By Bz Given Ax Ay Bz Bx By W 0 Bx By Bz 1 2 Ax Ay 0 0 0 W a1 g 1 lb 1 lb 1 lb c 1 2 b a Ax Ay Bx By Bz c b a Find Ax Ay B x By B z Ax Ay Bx 2.00 0.63 lb By Bz 2.00 1.37 lb 4.00 751 Engineering Mechanics - Dynamics Chapter 21 Problem 21-46 The conical pendulum consists of a bar of mass m and length L that is supported by the pin at its end A. If the pin is subjected to a rotation , determine the angle that the bar makes with the vertical as it rotates. Solution: Ix Iy y Iz 0 1 2 mL 3 x 0 cos z sin 0 'y Ix 'x L sin 2 1 2 L cos 3 Iy 0 'z 0 'x Mx Iz y x mg 0 0 1 2 mL 3 cos sin g 2 acos 3g 2L 2 Problem 21-47 The plate of weight W is mounted on the shaft AB so that the plane of the plate makes an angle with the vertical. If the shaft is turning in the direction shown with angular velocity , determine the vertical reactions at the bearing supports A and B when the plate is in the position shown. 752 Engineering Mechanics - Dynamics Chapter 21 Given: W 20 lb 30 deg 25 a b c rad s 18 in 18 in 6 in Solution: W g Ix 2 c 6 Iy sin 0 rad s FA FA 0 0 FB b FA FB FA a 1 lb FB W x y z 2 Ix Iz x z Iz y cos Guesses Given FB 0 1 lb Ix 0 0 0 Iy 0 0 FA FB 0 Iz 8.83 x y z Find FA F B 11.17 lb *Problem 21-48 The car is traveling around the curved road of radius such that its mass center has a constant speed vG. Write the equations of rotational motion with respect to the x, y, z axes. Assume that the car's six moments and products of inertia with respect to these axes are known. 753 Engineering Mechanics - Dynamics Chapter 21 Solution: Applying Eq. 21-24 with vG z x 0 y 0 'x 'y 'z 0 Mx Iyz vG 2 My Izx vG 2 Mz 0 Note: This result indicates the normal reactions of the tires on the ground are not all necessarily equal. Instead, they depend upon the speed of the car, radius of curvature, and the products of inertia, Iyz and Izx. (See Example 13-6.) Problem 21-49 The rod assembly is supported by journal bearings at A and B, which develops only x and z force reactions on the shaft. If the shaft AB is rotating in the direction shown with angular velocity , determine the reactions at the bearings when the assembly is in the position shown. Also, what is the shaft's angular acceleration? The mass density of each rod is Given: rad s kg 1.5 m 5 a b c d e g 500 mm 300 mm 500 mm 400 mm 300 mm 9.81 m s 2 754 Engineering Mechanics - Dynamics Chapter 21 Solution: Ixx (a b c) (a b c) 3 b c) 3 2 da 2 e e 12 2 2 e (a 2 b) 2 e 2 2 Izz (a 2 b c) (a 2 d d 12 d 2 2 da d 2 e (a b) 2 Iyy d d 3 e e 3 Ixy Iyy Iyz 1N 2 Ixy da Iyz e( a b) e 2 Ixx Imat Ixy 0 Guesses Given Ax 0 Az Bx 0 Bz Ax 0 Iyz Izz Az 1 N Bx Imat 1.5500 0.0600 0.0000 0.0600 0.0455 0.0540 0.0000 0.0540 kg m 1.5685 rad s 2 2 1 N Bz 1N 'y 1 0 0 (a b c d e)g d d 2 2 e e 2 'y 0 d d 'y 2 e e 2 2 0 a b 2 0 0 a e 2 Ax Az Bx Bz 'y b c (a 0 0 eg 0 0 b c)g 0 a b 0 c d 2 a 0 Bx 0 Bz 0 0 dg Imat 0 'y 0 0 Imat 0 0 0 Find A x A z B x Bz 'y Ax Az 1.17 12.33 N Bx Bz 0.0791 12.3126 rad s 2 N 'y 25.9 755 Engineering Mechanics - Dynamics Chapter 21 Problem 21-50 The rod assembly is supported by journal bearings at A and B, which develops only x and z force reactions on the shaft. If the shaft AB is subjected to a couple moment M0 j and at the instant shown the shaft has an angular velocity j, determine the reactions at the bearings when the assembly is in the position shown. Also, what is the shaft's angular acceleration? The mass density of each rod is Given: rad s kg 1.5 m 5 a 500 mm c d e 500 mm 400 mm 300 mm m s M0 8Nm 2 b 300 mm g 9.81 Solution: (a b c) 3 2 2 Ixx (a b c) da e e 12 2 e (a b) 2 e 2 2 Izz (a 2 b c) (a b c) 3 2 d d 12 d 2 2 da 2 d 2 2 e (a b) 2 Iyy d d 3 e e 3 Ixy Iyy Iyz 1N 2 Ixy da Iyz e( a b) e 2 Ixx Imat Ixy 0 Guesses Ax 0 Iyz Izz Az 1 N Bx Imat 1.5500 0.0600 0.0000 0.0600 0.0455 0.0540 0.0000 0.0540 kg m 1.5685 rad s 2 2 1 N Bz 1N 'y 1 756 Engineering Mechanics - Dynamics Chapter 21 Given Ax 0 Az Bx 0 Bz (a b 0 0 c d e)g d d 'y 2 d d 2 0 e e 2 2 0 Imat 'y 0 0 0 Imat 0 0 2 e e 2 'y 0 a b 2 0 0 a e 2 b c (a 0 0 eg 0 0 b c)g 0 a b 0 c d 2 a 0 Bx 0 Bz 0 0 dg 0 M0 0 Ax Az Bx Bz 'y Find A x A z B x Bz 'y Ax Az 3.39 0.66 Bx Bz 7.2243 4.1977 rad s 2 N N 'y 201.7 Problem 21-51 The rod assembly has a weight density r. It is supported at B by a smooth journal bearing, which develops x and y force reactions, and at A by a smooth thrust bearing, which develops x, y, and z force reactions. If torque M is applied along rod AB, determine the components of reaction at the bearings when the assembly has angular velocity at the instant shown. 757 Engineering Mechanics - Dynamics Chapter 21 Given: 5 M lb ft a b c 4 ft 2 ft 2 ft d g 2 ft 32.2 ft s 2 50 lb ft 10 rad s Solution: g Iyz cb c c 2 2 dc b d 2 Izz c 3 dc 2 Ixx (a b) (a 3 (a 3 0 Iyy Iyz b) 2 c 2 c 2 12 2 cb 2 c 2 2 d 2 d 2 12 dc 2 b d 2 2 Iyy (a Ixx b) b) cb d d 2 12 d b d 2 0 Iyz Izz 1 lb Az 1 lb Imat 16.98 0.00 0.00 0.00 15.32 2.48 0.00 2.48 lb ft s 1.66 rad s 2 2 Imat 0 0 Guesses Given Ax Ay Az Ax 1 lb Ay Bx 1 lb By 1 lb 1 Bx By 0 (a b 0 0 c d)g c c c 2 c 2 2 dc dc 2 0 758 Engineering Mechanics - Dynamics Chapter 21 0 0 a b 0 c b Ax Ay Az Bx By d 2 Bx By 0 0 0 dg 0 c 2 b 0 0 M 0 0 cg 0 Imat 0 0 0 0 Imat 0 Ax Find Ax Ay Az Bx B y Ay Az 15.6 46.8 lb 50.0 Bx By 12.5 46.4 rad s 2 lb 30.19 *Problem 21-52 The rod AB supports the sphere of weight W. If the rod is pinned at A to the vertical shaft which is rotating at a constant rate k, determine the angle of the rod during the motion. Neglect the mass of the rod in the calculation. Given: W 10 lb s 0.5 ft 2 ft 32.2 ft s Solution: I3 2 W 5 g I3 d 2 W 2 l g 2 2 7 rad d l g I1 759 Engineering Mechanics - Dynamics Chapter 21 Guess Given 50 deg W l sin Find I3 I1 cos sin 70.8 deg Problem 21-53 The rod AB supports the sphere of weight W. If the rod is pinned at A to the vertical shaft which is rotating with angular acceleration k, and at the instant shown the shaft has an angular velocity k, determine the angle of the rod during the motion. Neglect the mass of the rod in the calculation. Given: W 10 lb 2 rad s 7 d l g 2 rad s 0.5 ft 2 ft 32.2 ft s 2 Solution: 2 W 5 g I3 d 2 W 2 l g 50 deg W l sin Find I3 I1 cos sin 2 I3 I1 Guess Given 70.8 deg 760 Engineering Mechanics - Dynamics Chapter 21 Problem 21-54 The thin rod has mass mrod and total length L. Only half of the rod is visible in the figure. It is rotating about its midpoint at a constant rate ', while the table to which its axle A is fastened is rotating at angular velocity . Determine the x, y, z moment components which the axle exerts on the rod when the rod is in position . Given: mrod L ' 0.8 kg 150 mm 6 rad s rad s 2 Solution: L mrod 12 sin v 2 IA cos ' sin cos 0 v ' cos ' sin 0 v M Mx My Mz Mx My Mz Imat 0 0 Imat v 0 0 ' cos ' sin 0 sin cos ' 0 2IA ' 1 2 IA 2 0 0 0 IA 0 0 0 0 IA sin cos ' 0 IA 0 0 IA 0 2IA ' sin kx ky kz 1 2 IA sin 2 2 761 Engineering Mechanics - Dynamics Chapter 21 Mx My Mz 0 ky sin kz sin 2 ky kz 0.036 N m 0.0030 N m Problem 21-55 The cylinder has mass mc and is mounted on an axle that is supported by bearings at A and B. If the axle is turning at j, determine the vertical components of force acting at the bearings at this instant. Units Used: kN Given: mc a 30 kg 1m 40 d L g Solution: atan d L 2 10 N 3 rad s 0.5 m 1.5 m 9.81 m s 2 Ix' mc L 12 mc d 2 4 2 Iz' Ix' Iy' mc d 2 2 2 1 IG 0 0 sin cos Ix' 0 0 0 Iy' 0 0 0 Iz' 1 0 0 sin cos 0 cos 0 sin 0 cos 0 sin 762 Engineering Mechanics - Dynamics Chapter 21 Guesses Given Ax 0 Az Ax Az Bx Bz Ax 1N Az 1N Bx 1N Bz 1N Bx 0 Bz 0 0 mc g 0 0 a 0 Bx 0 Bz 0 a 0 Ax 0 Az 0 IG 0 0 0 Find Ax Az Bx B z Ax Bx 0.00 0.00 N Az Bz 1.38 1.09 kN *Problem 21-56 The cylinder has mass mc and is mounted on an axle that is supported by bearings at A and B. If the axle is subjected to a couple moment M j and at the instant shown has an angular velocity j, determine the vertical components of force acting at the bearings at this instant. Units Used: Given: mc a 30 kg 1m 40 rad s d L g 0.5 m 1.5 m 9.81 m s 2 kN 10 N 3 M Solution: 30 N m atan d L 2 Ix' L mc 12 Ix' mc d 2 4 2 Iy' mc d 2 2 2 763 Iz' Engineering Mechanics - Dynamics Chapter 21 1 IG 0 0 sin cos 1N Az Ix' 0 0 1N 0 Iy' 0 Bx 0 0 Iz' 1 0 0 sin cos 1N 1 rad s 2 0 cos 0 sin 0 cos 0 1N sin Bz Guesses Given Ax 0 Az 0 M 0 Ax Az Bx Bz Ax Bx 0 Bz 0 a 0 0 0 mc g Bx 0 Bz 0 a 0 Ax 0 Az IG 0 0 0 0 IG 0 0 0 Find Ax Az Bx B z Ax Bx 15.97 15.97 N Az Bz 1.38 1.09 kN 20.65 rad s 2 Problem 21-57 The uniform hatch door, having mass M and mass center G, is supported in the horizontal plane by bearings at A and B. If a vertical force F is applied to the door as shown, determine the components of reaction at the bearings and the angular acceleration of the door. The bearing at A will resist a component of force in the y direction, whereas the bearing at B will not. For the calculation, assume the door to be a thin plate and neglect the size of each bearing. The door is originally at rest. Given: M F a b 15 kg 300 N c d 100 mm 30 mm 30 mm 9.81 m s 2 764 200 mm e 150 mm g Engineering Mechanics - Dynamics Chapter 21 Solution: Ax Given 1N Guesses Ay Ax Ay Az 1N Bx 0 Bz F Az 1 N Bx 0 0 Mg M 1 N Bz 0 0 'y a 1N 'y 1 rad s 2 a b 0 Ax Ay Az Bx Bz 'y Bx 0 Bz a b 0 Ax Ay Az b 0 a c 0 e d 0 0 F M ( 2a) 12 0 2 'y Ax Find A x A y A z B x Bz 'y Ay Az 0 0 297 N Bx Bz 0 143 rad s 2 N 'y 102 Problem 21-58 The man sits on a swivel chair which is rotating with constant angular velocity He holds the uniform rod AB of weight W horizontal. He suddenly gives it an angular acceleration measured relative to him, as shown. Determine the required force and moment components at the grip, A, necessary to do this. Establish axes at the rod's center of mass G, with +z upward, and +y directed along the axis of the rod towards A. Given: 3 W L rad s 5 lb 3 ft 765 Engineering Mechanics - Dynamics Chapter 21 a 2 ft 2 rad s 2 g 32.2 ft s 2 Solution: Guesses Ax Mx Given IG W L g 12 2 1 lb 1 lb ft Ay My 1 lb 1 lb ft Az Mz 1 lb 1 lb ft 0 Ax Ay Az 0 L 2 0 Ax Ay Az Mx My Mz Find Ax Ay Az Mx My Mz Ax Ay Az 0.00 4.89 lb 5.47 Mx My Mz 8.43 0.00 0.00 lb ft 0 0 W W g a L 2 Mx My Mz IG 0 0 0 0 0 0 0 0 0 0 IG 0 0 0 0 0 0 0 0 L 2 2 Ax Ay Az 0 IG 0 IG Problem 21-59 Four spheres are connected to shaft AB. If you know mC and mE, determine the mass of D and F and the angles of the rods, D and F so that the shaft is dynamically balanced, that is, so that the bearings at A and B exert only vertical reactions on the shaft as it rotates. Neglect the mass of the rods. 766 Engineering Mechanics - Dynamics Chapter 21 Given: mC mE E 1 kg 2 kg 30 deg 0.1 m a Solution: We need to put the center of mass along AB and to make the product of inetia go to zero. mD 1 kg mF 1 kg D Guesses Given mE a cos mC a mC a a 40 deg F 10 deg E mD( 2a) sin D D mF a sin F 0 F mD( 2a) cos mE a sin E D mF a cos 0 F mD( 2a) ( 2a) cos mE( 3a)a sin E E mF( 4a)a cos 0 0 mD( 2a) ( 2a) sin D mD mF Find mD mF D F mE( 3a)a cos mF( 4a)a sin F mD D F 0.661 1.323 mF kg D F 139.1 40.9 deg *Problem 21-60 The bent uniform rod ACD has a weight density , and is supported at A by a pin and at B by a cord. If the vertical shaft rotates with a constant angular velocity , determine the x, y, z components of force and moment developed at A and the tension of the cord. Given: 5 a b lb ft 1 ft 1 ft 767 Engineering Mechanics - Dynamics Chapter 21 c 0.5 ft 20 rad s ft s 2 g 32.2 Solution: a a 3 b b 3 2 g b b 12 2 Ixx ba 2 b 2 2 2 Iyy Izz Iyz a a 3 2 ba b 2 0 Iyy Iyz 0 2 ba Ixx IA 0 0 Iyz Izz 1 lb ft 1 lb ft Ax Ay 1 lb 1 lb Az T 1 lb 1 lb Guesses My Mz Given Ax Ay Az T 0 0 (a b) a 0 a 2 2 0 ba 2 a a 2 ba My Mz Tc 0 0 0 IA 0 768 Engineering Mechanics - Dynamics Chapter 21 Ax Ay Az T My Mz Find Ax Ay Az T My Mz Ax Ay Az T 0.0 93.2 57.1 47.1 lb My Mz 0.00 0.00 lb ft Problem 21-61 Show that the angular velocity of a body, in terms of Euler angles , and may be expressed as ' sin ) j + ( ' cos + ') k, where i, j, and k = ( ' sin sin + ' cos ) i + ( ' sin cos are directed along the x, y, z axes as shown in Fig. 21-15d. Solution: From Fig. 21 - 15b, due to rotation , the x, y, z components of ' are simply ' along z axis From Fig. 21 - 15c, due to rotation , the x, y, z components of ' and ' are ' sin in the y direction, ' cos in the z direction, and ' in the x direction. Lastly, rotation , Fig 21 - 15d, produces the final components which yields ' sin sin ' cos i ' sin cos ' sin j ' cos ' k Q.E.D Problem 21-62 A thin rod is initially coincident with the Z axis when it is given three rotations defined by the Euler angles , , and If these rotations are given in the order stated, determine the coordinate direction angles , , of the axis of the rod with respect to the X, Y, and Z axes. Are these directions the same for any order of the rotations? Why? Given: 30 deg 45 deg 60 deg Solution: cos u sin 0 sin cos 0 0 0 1 1 0 0 sin cos 769 cos sin 0 sin cos 0 0 0 1 0 0 1 0 cos 0 sin Engineering Mechanics - Dynamics Chapter 21 69.3 acos u 127.8 deg 45.0 The last rotation ( ) does not affect the result because the rod just spins around its own axis. The order of application of the rotations does affect the final result since rotational position is not a vector quantity. Problem 21-63 The turbine on a ship has mass M and is mounted on bearings A and B as shown. Its center of mass is at G, its radius of gyration is kz, and kx = ky. If it is spinning at angular velocity determine the vertical reactions at the bearings when the ship undergoes each of the following motions: (a) rolling 1, (b) turning 2, (c) pitching 3. Units Used: kN Given: M 400 kg 200 rad s rad s rad s rad s kx kz a 0.5 m 0.3 m 0.8 m 1000 N 1 0.2 b 1.3 m 2 0.8 3 1.4 Solution: kx IG M 2 0 kx 2 0 0 kz 2 0 0 0 Ay By Guesses Ax Bx 1N 1N 1N 1N 770 Engineering Mechanics - Dynamics Chapter 21 (a) Rolling Ax Ay 0 0 0 b Ax Ay Bx By (b) Turning Ax Ay 0 0 0 b Ax Ay Bx By (c) Pitching Ax Ay 0 Bx By 0 Bx By 0 Ax Ay 0 0 0 a 0 Mg 0 Bx By 0 Ax Ay 0 0 0 a 0 Mg 0 Given 0 0 0 Bx By 0 0 IG 0 0 rad s 2 N 0 0 0 IG 0 Find Ax Ay B x By Ax Bx 0.00 0.00 kN Ay By 1.50 2.43 kN Given 0 0 0 Bx By 0 IG 0 0 2 N 0 2 0 IG 2 Find Ax Ay B x By Ax Bx 0.00 0.00 kN Ay By 1.25 5.17 kN Given 0 Mg 0 0 0 0 N 771 Engineering Mechanics - Dynamics Chapter 21 0 0 b Ax Ay Bx By Ax Ay 0 0 0 a Bx By 0 IG 0 3 3 3 0 IG 0 0 Find Ax Ay B x By Ax Bx 4.80 4.80 kN Ay By 1.50 2.43 kN *Problem 21-64 An airplane descends at a steep angle and then levels off horizontally to land. If the propeller is turning clockwise when observed from the rear of the plane, determine the direction in which the plane tends to turn as caused by the gyroscopic effect as it levels off. Solution: As noted on the diagram Mx represents the effect of the plane on the propeller. The opposite effect occurs on the plane. Hence, the plane tends to turn to the right when viewed from above. Problem 21-65 The propeller on a single-engine airplane has a mass M and a centroidal radius of gyration kG computed about the axis of spin. When viewed from the front of the airplane, the propeller is turning clockwise at s about the spin axis. If the airplane enters a vertical curve having a radius and is traveling at speed v, determine the gyroscopic bending moment which the propeller exerts on the bearings of the engine when the airplane is in its lowest position. 772 Engineering Mechanics - Dynamics Chapter 21 Given: M kG v 15 kg 0.3 m km hr rad s 200 s 350 80 m Solution: v y Mz Mz M kG 2 y s 328 N m Problem 21-66 The rotor assembly on the engine of a jet airplane consists of the turbine, drive shaft, and compressor. The total mass is mr, the radius of gyration about the shaft axis is kAB, and the mass center is at G. If the rotor has an angular velocity AB, and the plane is pulling out of a vertical curve while traveling at speed v, determine the components of reaction at the bearings A and B due to the gyroscopic effect. Units Used: kN Given: mr kAB AB 10 N 3 700 kg 0.35 m 1000 rad s 773 Engineering Mechanics - Dynamics Chapter 21 1.30 km a b v 0.8 m 0.4 m 250 m s M A Aa mr kAB 1N Bb B M 2 Solution: Guesses Given v AB 1N A B 0 A B Find ( A B) A B 13.7 13.7 kN Problem 21-67 A motor has weight W and has radius of gyration kz about the z axis. The shaft of the motor is supported by bearings at A and B, and is turning at a constant rate s = zk, while the frame has an angular velocity of y = yj. Determine the moment which the bearing forces at A and B exert on the shaft due to this motion. Given: W kz 50 lb 0.2 ft rad s z 100 y 2 rad s a g Solution: 0.5 ft 32.2 ft s 2 0 M y 0 0 W 2 kz z g M 12.4 0.0 0.0 lb ft 0 774 Engineering Mechanics - Dynamics Chapter 21 *Problem 21-68 The conical top has mass M, and the moments of inertia are Ix = Iy and Iz. If it spins freely in the ball-and-socket joint at A with angular velocity s compute the precession of the top about the axis of the shaft AB. Given: M 0.8 kg 3 2 a 100 mm Ix Iz 3.5 10 kg m 3 30 deg 2 0.8 10 rad s kg m g 9.81 m s 2 s 750 Solution: Using Eq. 21-30. Mx Guess Ix ' sin ' 1 rad s a 2 cos Iz ' sin ' cos ' Given M g sin Ix ' sin rad s 2 cos Iz ' sin ' cos s ' Find ' ' rad s a 1.31 low precession Guess ' 200 Given M g sin Ix ' sin 255 rad s 2 cos Iz ' sin ' cos s ' Find ' ' high precession Problem 21-69 A wheel of mass m and radius r rolls with constant spin about a circular path having a radius a. If the angle of inclination is , determine the rate of precession. Treat the wheel as a thin ring. No slipping occurs. 775 Engineering Mechanics - Dynamics Chapter 21 Solution: Since no sipping occurs, r ' a r cos 2 ' ' a r cos r ' Also, ' ' mr 2 2 F m a ' N mg 0 Ix Iy ' sin Iz ' y mr 2 j ' cos k z Thus, x 0 ' ' ' sin ' ' sin 'y ' ' cos ' 'x ' ' sin Applying Mx F r sin Ix 'x 'z 0 Iz Iy z y N r cos mr 2 2 ' ' sin mr 2 mr 2 2 ' ' cos ' sin Solving we find m a ' r sin 2 m g r cos 2 2 mr 2 r ' sin 2 ' sin 2 a r cos r ' mr 2 2 a r ' sin 2 2g cos a ' sin cos 2g cot a r cos Problem 21-70 The top consists of a thin disk that has weight W and radius r. The rod has a negligible mass and length L. If the top is spinning with an angular velocity s, determine the steady-state precessional angular velocity p. 776 Engineering Mechanics - Dynamics Chapter 21 Given: W r 8 lb 0.3 ft g 40 deg 32.2 ft 2 L 0.5 ft s s rad 300 s Solution: Mx I ' sin 2 cos Iz ' sin ' cos ' Guess p 1 rad s r 4 2 Given 2 p sin p cos s W L sin W g W 2 L g rad s p sin 2 cos W g r 2 p Find p p 1.21 low precession Guess p 70 rad s r 4 2 Given 2 p sin p cos s W L sin W g W 2 L g rad s p sin 2 cos W g r 2 p Find p p 76.3 high precession Problem 21-71 The top consists of a thin disk that has weight W and radius r. The rod has a negligible mass and length L. If the top is spinning with an angular velocity s, determine the steady-state precessional angular velocity p. 777 Engineering Mechanics - Dynamics Chapter 21 Given: W r 8 lb 0.3 ft g 90 deg 32.2 ft 2 L 0.5 ft s s rad 300 s Solution: Mx I ' sin 2 cos Iz ' sin ' cos ' Guess p 1 rad s r 4 2 Given 2 p sin p cos s W L sin W g W 2 L g rad s p sin 2 cos W g r 2 p Find p p 1.19 *Problem 21-72 The top has weight W and can be considered as a solid cone. If it is observed to precess about the vertical axis at a constant rate of y, determine its spin s. Given: W y 3 lb 5 rad s 30 deg L r g 6 in 1.5 in 32.2 ft s 2 778 Engineering Mechanics - Dynamics Chapter 21 Solution: I 3 W 80 g 4r 2 L 2 W g 3L 4 2 Iz 3 W 2 r 10 g I ' sin 2 Mx cos 2 Iz ' sin ' cos ' W 3L sin 4 I y sin cos Iz y sin y cos ' ' 1 3WL 4 4 I y cos Iz y 2 4 Iz y cos 2 ' 652 rad s Problem 21-73 The toy gyroscope consists of a rotor R which is attached to the frame of negligible mass. If it is observed that the frame is precessing about the pivot point O at rate p determine the angular velocity R of the rotor. The stem OA moves in the horizontal plane. The rotor has mass M and a radius of gyration kOA about OA. Given: p 2 rad s M kOA a g Solution: 200 gm 20 mm 30 mm 9.81 m s 2 Mx Iz y z 779 Engineering Mechanics - Dynamics Chapter 21 Mga M kOA ga 2 p R R kOA R 2 p 368 rad s Problem 21-74 The car is traveling at velocity vc around the horizontal curve having radius . If each wheel has mass M, radius of gyration kG about its spinning axis, and radius r, determine the difference between the normal forces of the rear wheels, caused by the gyroscopic effect. The distance between the wheels is d. Given: vc 100 km hr kG r d 300 mm 80 m M 16 kg 400 mm 1.3 m Solution: I 2M kG vc s 2 I 2.88 kg m 2 r vc s 69.44 rad s p p 0.35 rad s M I s p I s p F I s p Fd d F 53.4 N 780 Engineering Mechanics - Dynamics Chapter 21 Problem 21-75 The projectile shown is subjected to torque-free motion. The transverse and axial moments of inertia are I and Iz respectively. If represents the angle between the precessional axis Z and the axis of symmetry z, and is the angle between the angular velocity and the z axis, show that and are related by the equation tan = (I/Iz)tan . Solution: From Eq. 21-34 HG sin y I Iz I and HG cos z Iz Hence y z tan However, y y z sin Iz I I tan Iz and z cos tan tan tan Q.E.D *Problem 21-76 While the rocket is in free flight, it has a spin s and precesses about an axis measured angle from the axis of spin. If the ratio of the axial to transverse moments of inertia of the rocket is r, computed about axes which pass through the mass center G, determine the angle which the resultant angular velocity makes with the spin axis. Construct the body and space cones used to describe the motion. Is the precession regular or retrograde? Given: s 3 rad s 10 deg 781 Engineering Mechanics - Dynamics Chapter 21 r 1 15 Solution: Determine the angle tan r from the result of prob.21-75 tan atan r tan Thus, 0.673 deg 9.33 deg Regular Precession Since Iz I Problem 21-77 The projectile has a mass M and axial and transverse radii of gyration kz and kt, respectively. If it is spinning at s when it leaves the barrel of a gun, determine its angular momentum. Precession occurs about the Z axis. Given: M kz kt 0.9 kg 20 mm 10 deg 25 mm s 6 rad s Solution: I Iz M kt 2 2 I Iz 5.625 3.600 10 10 4 kg m 2 2 M kz s 4 kg m 782 Engineering Mechanics - Dynamics Chapter 21 I Iz I Iz HG cos 2 HG I Iz cos I Iz HG 6.09 10 3 kg m s Problem 21-78 The satellite has mass M, and about axes passing through the mass center G the axial and transverse radii of gyration are kz and kt, respectively. If it is spinning at s when it is launched, determine its angular momentum. Precession occurs about the Z axis. Units Used: Mg Given: M kz kt 1.8 Mg 0.8 m 1.2 m s 10 kg 3 6 rad s 5 deg Solution: I Iz ' M kt 2 2 I Iz 2592 kg m 2 2 M kz s 1152 kg m I ' Iz I Iz 'I HG cos Iz m s 2 HG cos I Iz HG 12.5 Mg Problem 21-79 The disk of mass M is thrown with a spin precession about the Z axis. z. The angle is measured as shown. Determine the 783 Engineering Mechanics - Dynamics Chapter 21 Given: M 4 kg 160 deg r z 125 mm 6 rad s Solution: 1 2 Mr 4 1 2 Mr 2 I Iz Applying Eq.21 - 36 I ' z Iz I Iz HG cos 2 HG I Iz z cos I Iz HG 0.1995 kg m s ' HG I ' 12.8 rad s Note that this is a case of retrograde precession since Iz > I *Problem 21-80 The radius of gyration about an axis passing through the axis of symmetry of the space capsule of mass M is kz, and about any transverse axis passing through the center of mass G, is kt. If the capsule has a known steady-state precession of two revolutions per hour about the Z axis, determine the rate of spin about the z axis. Units Used: Mg Given: M kz 1.6 Mg 1.2 m 784 10 kg 3 Engineering Mechanics - Dynamics Chapter 21 kt 1.8 m 20 deg Solution: I Iz M kt 2 2 M kz Using the Eqn. I tan Iz Iz I tan atan tan 9.19 deg 785 ...
View Full Document

This homework help was uploaded on 04/13/2008 for the course ME 2580 taught by Professor Vandenbrink during the Spring '08 term at Western Michigan.

Ask a homework question - tutors are online