The dynamic imbalance of a certain crankshaft is approximated by the physical model

Place your textbook on your desk, with fixed axes oriented as shown. Rotate the book about the x-axis through a 90 angle and then from this new position rotate it 90 about the y-axis. Sketch the final position of the book. Repeat the process but reverse the order of rotation. From your results, state your conclusion concerning the vector addition of finite rotations. Reconcile your observations with Fig. 7/4. Problem 7/1

Repeat the experiment of Prob. 7/1 but use a small angle of rotation, say, . Note the near-equal final positions for the two different rotation sequences. What does this observation lead you to conclude for the combination of infinitesimal rotations and for the time derivatives of angular quantities? Reconcile your observations with Fig. 7/5.

The solid cylinder is rotating about the fixed axis OA with a constant speed rev/min in the direction shown. If the x- and y-components of the velocity of point P are 12 ft/sec and ft/sec, determine its z-component of velocity and the radial distance R from P to the rotation axis. Also find the magnitude of the acceleration of P.

A timing mechanism consists of the rotating distributor arm AB and the fixed contact C. If the arm rotates about the fixed axis OA with a constant angular velocity 30(3i 2j 6k) rad/s, and if the coordinates of the contact C expressed in millimeters are (20, 30, 80), determine the magnitude of the acceleration of the tip B of the distributor arm as it passes point C. Problem 7/4 30 mm z A C

The rotor and shaft are mounted in a clevis which can rotate about the z-axis with an angular velocity . With  0 and  constant, the rotor has an angular velocity 0 4j  3k rad/s. Find the velocity vA of point A on the rim if its position vector at this instant is r 0.5i 1.2j 1.1k m. What is the rim speed vB of any point B? Problem 7/5

The disk rotates with a spin velocity of 15 rad/s about its horizontal z-axis first in the direction (a) and second in the direction (b). The assembly rotates with the velocity N 10 rad/s about the vertical axis. Construct the space and body cones for each case. Problem 7/6

The rotor B spins about its inclined axis OA at the speed rev/min, where . Simultaneously, the assembly rotates about the vertical z-axis at the rate . If the total angular velocity of the rotor has a magnitude of 40 rad/s, determine . Problem 7/7

A slender rod bent into the shape shown rotates about the fixed line CD at a constant angular rate . Determine the velocity and acceleration of point A. Problem 7/8

The rod is hinged about the axis O-O of the clevis, which is attached to the end of the vertical shaft. The shaft rotates with a constant angular 0 as shown. If  is decreasing at the constant rate p, write expressions for the angular velocity and angular acceleration  of the rod. Problem 7/9

The panel assembly and attached x-y-z axes rotate with a constant angular velocity  0.6 rad/sec about the vertical z-axis. Simultaneously, the panels rotate about the y-axis as shown with a constant rate 0 2 rad/sec. Determine the angular acceleration  of panel A and find the acceleration of point P for the instant when  90. Problem 7/10 z y x O A P B 18 16 20 0

The motor of Sample Problem 7/2 is shown again here. If the motor pivots about the x-axis at the constant rate 3 rad/sec with no rotation about the Z-axis (N 0), determine the angular acceleration  of the rotor and disk as the position  30 is passed. The constant speed of the motor is 120 rev/min. Also find the velocity and acceleration of point A, which is on the top of the disk for this position. Problem 7/11

If the motor of Sample Problem 7/2, repeated in Prob. 7/11, reaches a speed of 3000 rev/min in 2 seconds from rest with constant acceleration, determine the total angular acceleration of the rotor and disk second after it is turned on if the turntable is rotating at a constant rate N 30 rev/min. The angle  30 is constant.

The spool A rotates about its axis with an angular velocity of 20 rad/s, first in the sense of a and second in the sense of b. Simultaneously, the assembly rotates about the vertical axis with an angular velocity 1 10 rad/s. Determine the magnitude of the total angular velocity of the spool and construct the body and space cones for the spool for each case. Problem 7/13

In manipulating the dumbbell, the jaws of the robotic device have an angular velocity p 2 rad/s about the axis OG with  fixed at 60 . The entire assembly rotates about the vertical Z-axis at the constant rate  0.8 rad/s. Determine the angular velocity and angular acceleration  of the dumbbell. Express the results in terms of the given orientation of axes x-y-z, where the y-axis is parallel to the Y-axis. Problem 7/14

Determine the angular acceleration  of the dumbbell of Prob. 7/14 for the conditions stated, except that  is increasing at the rate of 3 rad/s2 for the instant under consideration.

The robot shown has five degrees of rotational freedom. The x-y-z axes are attached to the base ring, which rotates about the z-axis at the rate 1. The arm O1O2 rotates about the x-axis at the rate 2 The control arm O2A rotates about axis O1-O2 at the rate 3 and about a perpendicular axis through O2 which is momentarily parallel to the x-axis at the rate 4 Finally, the jaws rotate about axis O2-A at the rate 5. The magnitudes of all angular rates are constant. For the configuration shown, determine the magnitude of the total angular velocity of the jaws for  60 and  45 if 1 2 rad/s, 1.5 rad/s, and 3 4 5 0. Also express the angular acceleration  of arm O1O2 as a vector.

For the robot of Prob. 7/16, determine the angular velocity and angular acceleration  of the jaws A if  60 and  30, both constant, and if 1 2 rad/s, 2 3 4 0, and 5 0.8 rad/s, all constant.

The wheel rolls without slipping in a circular arc of radius R and makes one complete turn about the vertical y-axis with constant speed in time . Determine the vector expression for the angular acceleration  of the wheel and construct the space and body cones. Problem 7/18 z A r x O y R

Determine expressions for the velocity v and acceleration a of point A on the wheel of Prob. 7/18 for the position shown, where A crosses the horizontal line through the center of the wheel.

The circular disk of 120-mm radius rotates about the z-axis at the constant rate z 20 rad/s, and the entire assembly rotates about the fixed x-axis at the constant rate x 10 rad/s. Calculate the magnitudes of the velocity v and acceleration a of point B for the instant when  30. Problem 7/20

The crane has a boom of length 24 m and is revolving about the vertical axis at the constant rate of 2 rev/min in the direction shown. Simultaneously, the boom is being lowered at the constant rate 0.10 rad/s. Calculate the magnitudes of the velocity and acceleration of the end P of the boom for the instant when it passes the position  30. Problem 7/21

The design of the rotating arm OA of a control mechanism requires that it rotate about the vertical Z-axis at the constant rate rad/s. Simultaneously, OA oscillates according to sin , where radians and t is in seconds measured from the time when . Determine the angular velocity and the angular acceleration  of OA for the instant (a) when s and (b) when s. The x-y reference axes rotate in the X-Y plane with the angular velocity . Problem 7/22

For the rotating and oscillating control arm OA of Prob. 7/22, determine the velocity v and acceleration a of the ball tip A for the condition when . Distance and sin as defined in Prob. 7/22 with rad/s and rad.

If the angular velocity rad/s of the rotor in Prob. 7/5 is constant in magnitude, determine the angular acceleration of the rotor for (a) and rad/s (both constant) and (b) and rad/s (both constant). Find the magnitude of the acceleration of point A in each case, where A has the position vector r 0.5i 1.2j 1.1k m at the instant represented.

The vertical shaft and attached clevis rotate about the z-axis at the constant rate  4 rad/s. Simultaneously, the shaft B revolves about its axis OA at the constant rate 0 3 rad/s, and the angle  is decreasing at the constant rate of /4 rad/s. Determine the angular velocity and the magnitude of the angular acceleration  of shaft B when  30. The x-y-z axes are attached to the clevis and rotate with it. Problem 7/25

The right-circular cone A rolls on the fixed rightcircular cone B at a constant rate and makes one complete trip around B every 4 seconds. Compute the magnitude of the angular acceleration  of cone A during its motion. Problem 7/26

The pendulum oscillates about the x-axis according to  sin 3t radians, where t is the time in seconds. Simultaneously, the shaft OA revolves about the vertical z-axis at the constant rate z 2 rad/sec. Determine the velocity v and acceleration a of the center B of the pendulum as well as its angular acceleration  for the instant when t 0. Problem 7/27

The solid right-circular cone of base radius r and height h rolls on a flat surface without slipping. The center B of the circular base moves in a circular path around the z-axis with a constant speed v. Determine the angular velocity and the angular acceleration of the solid cone. Problem 7/28 x z y A O r h

The solid cylinder has a body cone with a semivertex angle of 20. Momentarily the angular velocity has a magnitude of 30 rad/s and lies in the y-z plane. Determine the rate p at which the cylinder is spinning about its z-axis and write the vector expression for the velocity of B with respect to A. Problem 7/29

The helicopter is nosing over at the constant rate q rad/s. If the rotor blades revolve at the constant speed p rad/s, write the expression for the angular acceleration  of the rotor. Take the y-axis to be attached to the fuselage and pointing forward perpendicular to the rotor axis. Problem 7/30

The collar at O and attached shaft OC rotate about the fixed x0-axis at the constant rate  4 rad/s. Simultaneously, the circular disk rotates about OC at the constant rate p 10 rad/s. Determine the magnitude of the total angular velocity of the disk and find its angular acceleration .

If the angular rate p of the disk in Prob. 7/31 is increasing at the rate of 6 rad/s per second and if  remains constant at 4 rad/s, determine the angular acceleration  of the disk at the instant when p reaches 10 rad/s.

For the conditions of Prob. 7/31, determine the velocity vA and acceleration aA of point A on the disk as it passes the position shown. Reference axes x-y-z are attached to the collar at O and its shaft OC.

An unmanned radar-radio controlled aircraft with tilt-rotor propulsion is being designed for reconnaissance purposes. Vertical rise begins with  0 and is followed by horizontal flight as  approaches 90. If the rotors turn at a constant speed N of 360 rev/min, determine the angular acceleration  of rotor A for  30 if is constant at 0.2 rad/s. Problem 7/34

End A of the rigid link is confined to move in the x-direction while end B is confined to move along the z-axis. Determine the component n normal to AB of the angular velocity of the link as it passes the position shown with vA 3 ft/sec. Problem 7/35

The small motor M is pivoted about the x-axis through O and gives its shaft OA a constant speed p rad/s in the direction shown relative to its housing. The entire unit is then set into rotation about the vertical Z-axis at the constant angular velocity  rad/s. Simultaneously, the motor pivots about the x-axis at the constant rate for an interval of motion. Determine the angular acceleration  of the shaft OA in terms of . Express your result in terms of the unit vectors for the rotating x-y-z axes.

The flight simulator is mounted on six hydraulic actuators connected in pairs to their attachment points on the underside of the simulator. By programming the actions of the actuators, a variety of flight conditions can be simulated with translational and rotational displacements through a limited range of motion. Axes x-y-z are attached to the simulator with origin B at the center of the volume. For the instant represented, B has a velocity and an acceleration in the horizontal y-direction of 3.2 ft/sec and 4 ft/sec2 , respectively. Simultaneously, the angular velocities and their time rates of change are x 1.4 rad/sec, 2 rad/sec2 , y 1.2 rad/sec, 3 rad/sec2 , z 0. For this instant determine the magnitudes of the velocity and acceleration of point A. Problem 7/37

The robot of Prob. 7/16 is shown again here, where the coordinate system x-y-z with origin at O2 rotates about the X-axis at the rate Nonrotating axes X-Y-Z oriented as shown have their origin at O1. If 2 3 rad/s constant, 3 1.5 rad/s constant, 1 5 0, 1.2 m, and 0.6 m, determine the velocity of the center A of the jaws for the instant when  60. The angle  lies in the y-z plane and is constant at 45. Problem 7/38

For the instant represented collar B is moving along the fixed shaft in the X-direction with a constant velocity vB 4 m/s. Also at this instant X 0.3 m and Y 0.2 m. Calculate the velocity of collar A, which moves along the fixed shaft parallel to the Y-axis. Solve, first, by differentiating the relation X2 Y2 Z2 L2 with respect to time and, second, by using the first of Eqs. 7/4 with translating axes attached to B. Each clevis is free to rotate about the axis of the rod.

The spacecraft is revolving about its z-axis, which has a fixed space orientation, at the constant rate p rad/s. Simultaneously, its solar panels are unfolding at the rate which is programmed to vary with  as shown in the graph. Determine the angular acceleration  of panel A an instant (a) before and an instant (b) after it reaches the position  18. Problem 7/40 x y z A p 0 0 2 18 ( ) (/s) 90

The disk has a constant angular velocity p about its z-axis, and the yoke A has a constant angular velocity 2 about its shaft as shown. Simultaneously, the entire assembly revolves about the fixed X-axis with a constant angular velocity 1. Determine the expression for the angular acceleration of the disk as the yoke brings it into the vertical plane in the position shown. Solve by picturing the vector changes in the angular-velocity components. Problem 7/41

The collar and clevis A are given a constant upward velocity of 8 in./sec for an interval of motion and cause the ball end of the bar to slide in the radial slot in the rotating disk. Determine the angular acceleration of the bar when the bar passes the position for which z 3 in. The disk turns at the constant rate of 2 rad/sec. Problem 7/42

The circular disk of 100-mm radius rotates about its z-axis at the constant speed p 240 rev/min, and arm OCB rotates about the Y-axis at the constant speed N 30 rev/min. Determine the velocity v and acceleration a of point A on the disk as it passes the position shown. Use reference axes x-y-z attached to the arm OCB. Problem 7/43

Solve Prob. 7/43 by attaching the reference axes x-y-z to the rotating disk.

For the conditions described in Prob. 7/36, determine the velocity v and acceleration a of the center A of the ball tool in terms of .

The circular disk is spinning about its own axis ( y-axis) at the constant rate p 10 rad/s. Simultaneously, the frame is rotating about the Z-axis at the constant rate  4 rad/s. Calculate the angular acceleration  of the disk and the acceleration of point A at the top of the disk. Axes x-y-z are attached to the frame, which has the momentary orientation shown with respect to the fixed axes X-Y-Z. Problem 7/46

The center O of the spacecraft is moving through space with a constant velocity. During the period of motion prior to stabilization, the spacecraft has a constant rotational rate  rad/sec about its z-axis. The x-y-z axes are attached to the body of the craft, and the solar panels rotate about the y-axis at the constant rate rad/sec with respect to the spacecraft. If is the absolute angular velocity of the solar panels, determine Also find the acceleration of point A when  30. Problem 7/47

The thin circular disk of mass m and radius r is rotating about its z-axis with a constant angular velocity p, and the yoke in which it is mounted rotates about the X-axis through OB with a constant angular velocity 1. Simultaneously, the entire assembly rotates about the fixed Y-axis through O with a constant angular velocity 2. Determine the velocity v and acceleration a of point A on the rim of the disk as it passes the position shown where the x-y plane of the disk coincides with the X-Y plane. The x-y-z axes are attached to the yoke.

For the conditions specified with Sample Problem 7/2, except that  is increasing at the steady rate of 3 rad/sec, determine the angular velocity and the angular acceleration  of the rotor when the position  30 is passed. (Suggestion: Apply Eq. 7/7 to the vector to find . Note that in Sample Problem 7/2 is no longer the complete angular velocity of the axes.)

The wheel of radius r is free to rotate about the bent axle CO which turns about the vertical axis at the constant rate p rad/s. If the wheel rolls without slipping on the horizontal circle of radius R, determine the expressions for the angular velocity and angular acceleration  of the wheel. The x-axis is always horizontal. Problem 7/50

The gyro rotor shown is spinning at the constant rate of 100 rev/min relative to the x-y-z axes in the direction indicated. If the angle  between the gimbal ring and the horizontal X-Y plane is made to increase at the constant rate of 4 rad/s and if the unit is forced to precess about the vertical at the constant rate N 20 rev/min, calculate the magnitude of the angular acceleration  of the rotor when  30. Solve by using Eq. 7/7 applied to the angular velocity of the rotor. Problem 7/51

For a short interval of motion, collar A moves along its fixed shaft with a velocity vA 2 m/s in the Y-direction. Collar B, in turn, slides along its fixed vertical shaft. Link AB is 700 mm in length and can turn within the clevis at A to allow for the angular change between the clevises. For the instant when A passes the position where y 200 mm, determine the velocity of collar B using nonrotating axes attached to B and find the component n, normal to AB, of the angular velocity of the link. Also solve for vB by differentiating the appropriate relation x2 y2 z2 l 2 .

The three small spheres, each of mass m, are rigidly mounted to the horizontal shaft which rotates with the angular velocity as shown. Neglect the radius of each sphere compared with the other dimensions and write expressions for the magnitudes of their linear momentum G and their angular momentum about the origin O of the coordinates. Problem 7/53

The spheres of Prob. 7/53 are replaced by three rods, each of mass m and length l, mounted at their centers to the shaft, which rotates with the angular velocity as shown. The axes of the rods are, respectively, in the x-, y-, and z-directions, and their diameters are negligible compared with the other dimensions. Determine the angular momentum of the three rods with respect to the coordinate origin O. Problem 7/54

The aircraft landing gear viewed from the front is being retracted immediately after takeoff, and the wheel is spinning at the rate corresponding to the takeoff speed of 200 km/h. The 45-kg wheel has a radius of gyration about its z-axis of 370 mm. Neglect the thickness of the wheel and calculate the angular momentum of the wheel about G and about A for the position where  is increasing at the rate of 30 per second. Problem 7/55

The bent rod has a mass per unit length and rotates about the z-axis with an angular velocity . Determine the angular momentum HO of the rod about the fixed origin O of the axes, which are attached to the rod. Also find the kinetic energy T of the rod. Problem 7/56

Use the results of Prob. 7/56 and determine the angular momentum HG of the bent rod of that problem about its mass center G using the given reference axes.

The slender rod of mass m and length l rotates about the y-axis as the element of a right-circular cone. If the angular velocity about the y-axis is , determine the expression for the angular momentum of the rod with respect to the x-y-z axes for the particular position shown. Problem 7/58

The solid half-circular cylinder of mass m revolves about the z-axis with an angular velocity as shown. Determine its angular momentum H with respect to the x-y-z axes. Problem 7/59

The solid circular cylinder of mass m, radius r, and length b revolves about its geometric axis at an angular rate p rad/s. Simultaneously, the bracket and attached shaft revolve about the x-axis at the rate rad/s. Write the expression for the angular momentun of the cylinder about O with reference axes as shown.

The elements of a reaction-wheel attitude-control system for a spacecraft are shown in the figure. Point G is the center of mass for the system of the spacecraft and wheels, and x, y, z are principal axes for the system. Each wheel has a mass m and a moment of inertia I about its own axis and spins with a relative angular velocity p in the direction indicated. The center of each wheel, which may be treated as a thin disk, is a distance b from G. If the spacecraft has angular velocity components x, y, and z, determine the angular momentum HG of the three wheels as a unit. Problem 7/61

The gyro rotor is spinning at the constant rate p 100 rev/min relative to the x-y-z axes in the direction indicated. If the angle  between the gimbal ring and horizontal X-Y plane is made to increase at the rate of 4 rad/sec and if the unit is forced to precess about the vertical at the constant rate N 20 rev/min, calculate the angular momentum HO of the rotor when  30. The axial and transverse moments of inertia are Izz 5(103 ) lb-ft-sec2 and Ixx Iyy 2.5(103 ) lb-ft-sec2 . Problem 7/62

The slender steel rod AB weighs 6.20 lb and is secured to the rotating shaft by the rod OG and its fittings at O and G. The angle  remains constant at 30, and the entire rigid assembly rotates about the z-axis at the steady rate N 600 rev/min. Calculate the angular momentum HO of AB and its kinetic energy T. Problem 7/63

The rectangular plate, with a mass of 3 kg and a uniform small thickness, is welded at the 45 angle to the vertical shaft, which rotates with the angular velocity of 20 rad/s. Determine the angular momentum H of the plate about O and find the kinetic energy of the plate. Problem 7/64

The circular disk of mass m and radius r is mounted on the vertical shaft with an angle between its plane and the plane of rotation of the shaft. Determine an expression for the angular momentum H of the disk about O. Find the angle  which the angular momentum H makes with the shaft if 10. Problem 7/65

The right-circular cone of height h and base radius r spins about its axis of symmetry with an angular rate p. Simultaneously, the entire cone revolves about the x-axis with angular rate . Determine the angular momentum HO of the cone about the origin O of the x-y-z axes and the kinetic energy T for the position shown. The mass of the cone is m. Problem 7/66

Each of the slender rods of length l and mass m is welded to the circular disk which rotates about the vertical z-axis with an angular velocity . Each rod makes an angle  with the vertical and lies in a plane parallel to the y-z plane. Determine an expression for the angular momentum HO of the two rods about the origin O of the axes. Problem 7/66

The spacecraft shown has a mass m with mass center G. Its radius of gyration about its z-axis of rotational symmetry is k and that about either the x- or y-axis is k. In space, the spacecraft spins within its x-y-z reference frame at the rate p Simultaneously, a point C on the z-axis moves in a circle about the z0-axis with a frequency (rotations per unit time). The z0-axis has a constant direction in space. Determine the angular momentum HG of the spacecraft relative to the axes designated. Note that the x-axis always lies in the z-z0 plane and that the y-axis is therefore normal to z0. Problem 7/68

The uniform circular disk of Prob. 7/48 with the three components of angular velocity is shown again here. Determine the kinetic energy T and the angular momentum HO with respect to O of the disk for the instant represented, when the x-y plane coincides with the X-Y plane. The mass of the disk is m. Problem 7/69 Y y A x X z Z p

The 4-in.-radius wheel weighs 6 lb and turns about its y-axis with an angular velocity p 40 rad/sec in the direction shown. Simultaneously, the fork rotates about its x-axis shaft with an angular velocity 10 rad/sec as indicated. Calculate the angular momentum of the wheel about its center O. Also compute the kinetic energy of the wheel. Problem 7/70

The assembly, consisting of the solid sphere of mass m and the uniform rod of length 2c and equal mass m, revolves about the vertical z-axis with an angular velocity . The rod of length 2c has a diameter which is small compared with its length and is perpendicular to the horizontal rod to which it is welded with the inclination  shown. Determine the combined angular momentum HO of the sphere and inclined rod.

In a test of the solar panels for a spacecraft, the model shown is rotated about the vertical axis at the angular rate . If the mass per unit area of panel is , write the expression for the angular momentum HO of the assembly about the axes shown in terms of . Also determine the maximum, minimum, and intermediate values of the moment of inertia about the axes through O. Problem 7/72

Each of the two rods of mass m is welded to the face of the disk, which rotates about the vertical axis with a constant angular velocity . Determine the bending moment M acting on each rod at its base. Problem 7/73

The slender shaft carries two offset particles, each of mass m, and rotates about the z-axis with the constant angular rate as indicated. Determine the x- and y-components of the bearing reactions at A and B due to the dynamic imbalance of the shaft for the position shown. Problem 7/74

The uniform slender bar of length l and mass m is welded to the shaft, which rotates in bearings A and B with a constant angular velocity . Determine the expression for the force supported by the bearing at B as a function of . Consider only the force due to the dynamic imbalance and assume that the bearings can support radial forces only. Problem 7/75

If a torque M Mk is applied to the shaft in Prob. 7/75, determine the x- and y-components of the force supported by the bearing B as the bar and shaft start from rest in the position shown. Neglect the mass of the shaft and consider dynamic forces only.

The paint stirrer shown in the figure is made from a rod of length 7b and mass per unit length. Before immersion in the paint, the stirrer is rotating freely at a constant high angular velocity about its z-axis. Determine the bending moment M in the rod at the base O of the chuck. Problem 7/77 b

The 6-kg circular disk and attached shaft rotate at a constant speed 10 000 rev/min. If the center of mass of the disk is 0.05 mm off center, determine the magnitudes of the horizontal forces A and B supported by the bearings because of the rotational imbalance. Problem 7/78

Determine the bending moment M at the tangency point A in the semicircular rod of radius r and mass m as it rotates about the tangent axis with a constant and large angular velocity . Neglect the moment mgr produced by the weight of the rod. Problem 7/79

If the semicircular rod of Prob. 7/79 starts from rest under the action of a torque MO applied through the collar about its z-axis of rotation, determine the initial bending moment M in the rod at A.

The large satellite-tracking antenna has a moment of inertia I about its z-axis of symmetry and a moment of inertia IO about each of the x- and y-axes. Determine the angular acceleration of the antenna about the vertical Z-axis caused by a torque M applied about Z by the drive mechanism for a given orientation . Problem 7/81

The plate has a mass of 3 kg and is welded to the fixed vertical shaft, which rotates at the constant speed of 20 rad/s. Compute the moment M applied to the shaft by the plate due to dynamic imbalance. Problem 7/82 x z y O 45 200 mm 200 mm 100 mm 100 mm = 20 rad/

Each of the two semicircular disks has a mass of 1.20 kg and is welded to the shaft supported in bearings A and B as shown. Calculate the forces applied to the shaft by the bearings for a constant angular speed N 1200 rev/min. Neglect the forces of static equilibrium. Problem 7/83

Solve Prob. 7/83 for the case where the assembly starts from rest with an initial angular acceleration 900 rad/s2 as a result of a starting torque (couple) M applied to the shaft in the same sense as N. Neglect the moment of inertia of the shaft about its z-axis and calculate M

The uniform slender bar of mass per unit length is freely pivoted about the y-axis at the clevis, which rotates about the fixed vertical z-axis with a constant angular velocity . Determine the steady-state angle assumed by the bar. Length b is greater than length c. Problem 7/85

The circular disk of mass m and radius r is mounted on the vertical shaft with a small angle between its plane and the plane of rotation of the shaft. Determine the expression for the bending moment M acting on the shaft due to the wobble of the disk at a shaft speed of rad/s. Problem 7/86

The thin circular disk of mass m and radius R is hinged about its horizontal tangent axis to the end of a shaft rotating about its vertical axis with an angular velocity . Determine the steady-state angle  assumed by the plane of the disk with the vertical axis. Observe any limitation on to ensure that   0. Problem 7/87

Determine the normal forces under the two disks of Sample Problem 7/7 for the position where the plane of the curved bar is vertical. Take the curved bar to be at the top of disk A and at the bottom of disk B.

The uniform square plate of mass m is welded at O to the end of the shaft, which rotates about the vertical z-axis with a constant angular velocity . Determine the moment applied to the plate by the weld due only to the rotation. Problem 7/89

For the plate of mass m in Prob. 7/89, determine the y- and z-components of the moment applied to the plate by the weld at O necessary to give the plate an angular acceleration starting from rest. Neglect the moment due to the weight.

The uniform slender rod of length l is welded to the bracket at A on the underside of the disk B. The disk rotates about a vertical axis with a constant angular velocity . Determine the value of which will result in a zero moment supported by the weld at A for the position  60 with b l/4. Problem 7/91

The half-cylindrical shell of radius r, length 2b, and mass m revolves about the vertical z-axis with a constant angular velocity as indicated. Determine the magnitude M of the bending moment in the shaft at A due to both the weight and the rotational motion of the shell. Problem 7/92

The homogeneous thin triangular plate of mass m is welded to the horizontal shaft, which rotates freely in the bearings at A and B. If the plate is released from rest in the horizontal position shown, determine the magnitude of the bearing reaction at A for the instant just after release. Problem 7/93

If the homogeneous triangular plate of Prob. 7/93 is released from rest in the position shown, determine the magnitude of the bearing reaction at A after the plate has rotated 90.

A dynamics instructor demonstrates gyroscopic principles to his students. He suspends a rapidly spinning wheel with a string attached to one end of its horizontal axle. Describe the precession motion of the wheel. Problem 7/95

The student has volunteered to assist in a classroom demonstration involving a momentum wheel which is rapidly spinning with angular speed p as shown. The instructor has asked her to hold the axle of the wheel in the horizontal position shown and then attempt to tilt the axis upward in a vertical plane. What motion tendency of the wheel assembly will the student sense?

A car makes a turn to the right on a level road. Determine whether the normal reaction under the right rear wheel is increased or decreased as a result of the gyroscopic effect of the precessing wheels.

The 50-kg wheel is a solid circular disk which rolls on the horizontal plane in a circle of 600-mm radius. The wheel shaft is pivoted about the axis O-O and is driven by the vertical shaft at the constant rate rev/min about the Z-axis. Determine the normal force R between the wheel and the horizontal surface. Neglect the weight of the horizontal shaft.

The special-purpose fan is mounted as shown. The motor armature, shaft, and blades have a combined mass of 2.2 kg with radius of gyration of 60 mm. The axial position b of the 0.8-kg block A can be adjusted. With the fan turned off, the unit is balanced about the x-axis when b 180 mm. The motor and fan operate at 1725 rev/min in the direction shown. Determine the value of b which will produce a steady precession of 0.2 rad/s about the positive y-axis. Problem 7/99 x z y b A O

An airplane has just cleared the runway with a takeoff speed v. Each of its freely spinning wheels has a mass m, with a radius of gyration k about its axle. As seen from the front of the airplane, the wheel precesses at the angular rate  as the landing strut is folded into the wing about its pivot O. As a result of the gyroscopic action, the supporting member A exerts a torsional moment M on B to prevent the tubular member from rotating in the sleeve at B. Determine M and identify whether it is in the sense of M1 or M2. Problem 7/100

An experimental antipollution bus is powered by the kinetic energy stored in a large flywheel which spins at a high speed p in the direction indicated. As the bus encounters a short upward ramp, the front wheels rise, thus causing the flywheel to precess. What changes occur to the forces between the tires and the road during this sudden change? Problem 7/101

The 210-kg rotor of a turbojet aircraft engine has a radius of gyration of 220 mm and rotates counterclockwise at 18 000 rev/min as viewed from the front. If the aircraft is traveling at 1200 km/h and starts to execute an inside vertical loop of 3800-m radius, compute the gyroscopic moment M transmitted to the airframe. What correction to the controls does the pilot have to make in order to remain in the vertical plane?

A small air compressor for an aircraft cabin consists of the 3.50-kg turbine A which drives the 2.40-kg blower B at a speed of 20 000 rev/min. The shaft of the assembly is mounted transversely to the direction of flight and is viewed from the rear of the aircraft in the figure. The radii of gyration of A and B are 79.0 and 71.0 mm, respectively. Calculate the radial forces exerted on the shaft by the bearings at C and D if the aircraft executes a clockwise roll (rotation about the longitudinal flight axis) of 2 rad/s viewed from the rear of the aircraft. Neglect the small moments caused by the weights of the rotors. Draw a free-body diagram of the shaft as viewed from above and indicate the shape of its deflected centerline. Problem 7/103

The two solid cones with the same base and equal altitudes are spinning in space about their common axis at the rate p. For what ratio h/r will precession of their spin axis be impossible? Problem 7/104 p h h r

The blades and hub of the helicopter rotor weigh 140 lb and have a radius of gyration of 10 ft about the z-axis of rotation. With the rotor turning at 500 rev/min during a short interval following vertical liftoff, the helicopter tilts forward at the rate 10 deg/sec in order to acquire forward velocity. Determine the gyroscopic moment M transmitted to the body of the helicopter by its rotor and indicate whether the helicopter tends to deflect clockwise or counterclockwise, as viewed by a passenger facing forward. Problem 7/105

The 4-oz top with radius of gyration about its spin axis of 0.62 in. is spinning at the rate p 3600 rev/min in the sense shown, with its spin axis making an angle  20 with the vertical. The distance from its tip O to its mass center G is 2.5 in. Determine the precession of the top and explain why  gradually decreases as long as the spin rate remains large. An enlarged view of the contact of the tip is shown.

The figure shows a gyro mounted with a vertical axis and used to stabilize a hospital ship against rolling. The motor A turns the pinion which precesses the gyro by rotating the large precession gear B and attached rotor assembly about a horizontal transverse axis in the ship. The rotor turns inside the housing at a clockwise speed of 960 rev/min as viewed from the top and has a mass of 80 Mg with radius of gyration of 1.45 m. Calculate the moment exerted on the hull structure by the gyro if the motor turns the precession gear B at the rate of 0.320 rad/s. In which of the two directions, (a) or (b), should the motor turn in order to counteract a roll of the ship to port? Problem 7/107 A B Vertical (a) (b) Forward Port (left) Starboard (right)

Each of the identical wheels has a mass of 4 kg and a radius of gyration kz 120 mm and is mounted on a horizontal shaft AB secured to the vertical shaft at O. In case (a), the horizontal shaft is fixed to a collar at O which is free to rotate about the vertical y-axis. In case (b), the shaft is secured by a yoke hinged about the x-axis to the collar. If the wheel has a large angular velocity p 3600 rev/min about its z-axis in the position shown, determine any precession which occurs and the bending moment MA in the shaft at A for each case. Neglect the small mass of the shaft and fitting at O. Problem 7/108

If the wheel in case (a) of Prob. 7/108 is forced to precess about the vertical by a mechanical drive at the steady rate 2j rad/s, determine the bending moment in the horizontal shaft at A. In the absence of friction, what torque MO is applied to the collar at O to sustain this motion?

The figure shows the side view of the wheel carriage (truck) of a railway passenger car where the vertical load is transmitted to the frame in which the journal wheel bearings are located. The lower view shows only one pair of wheels and their axle which rotates with the wheels. Each of the 33-in.-diameter wheels weighs 560 lb, and the axle weighs 300 lb with a diameter of 5 in. Both wheels and axle are made of steel with a specific weight of 489 lb/ft3 . If the train is traveling at 80 mi/hr while rounding an 8 curve to the right (radius of curvature 717 ft), calculate the change R in the vertical force supported by each wheel due only to the gyroscopic action. As a close approximation, treat each wheel as a uniform circular disk and the axle as a uniform solid cylinder. Also assume that both rails are in the same horizontal plane.

The primary structure of a proposed space station consists of five spherical shells connected by tubular spokes. The moment of inertia of the structure about its geometric axis A-A is twice as much as that about any axis through O normal to A-A. The station is designed to rotate about its geometric axis at the constant rate of 3 rev/min. If the spin axis A-A precesses about the Z-axis of fixed orientation and makes a very small angle with it, calculate the rate at which the station wobbles. The mass center O has negligible acceleration. Problem 7/111 A A Z O

The uniform 640-mm rod has a mass of 3 kg and is welded centrally to the uniform 160-mm-radius circular disk which has a mass of 8 kg. The unit is given a spin velocity p 60 rad/s in the direction shown. The axis of the rod is seen to wobble through a total angle of 30. Calculate the angular velocity of precession and determine whether it is or Problem 7/112

The electric motor has a total weight of 20 lb and is supported by the mounting brackets A and B attached to the rotating disk. The armature of the motor has a weight of 5 lb and a radius of gyration of 1.5 in. and turns counterclockwise at a speed of 1725 rev/min as viewed from A to B. The turntable revolves about its vertical axis at the constant rate of 48 rev/min in the direction shown. Determine the vertical components of the forces supported by the mounting brackets at A and B. Problem 7/113

The spacecraft shown is symmetrical about its z-axis and has a radius of gyration of 720 mm about this axis. The radii of gyration about the x- and y-axes through the mass center are both equal to 540 mm. When moving in space, the z-axis is observed to generate a cone with a total vertex angle of 4 as it precesses about the axis of total angular momentum. If the spacecraft has a spin velocity about its z-axis of 1.5 rad/s, compute the period  of each full precession. Is the spin vector in the positive or negative z-direction? Problem 7/114

The 8-lb rotor with radius of gyration of 3 in. rotates on ball bearings at a speed of 3000 rev/min about its shaft OG. The shaft is free to pivot about the X-axis, as well as to rotate about the Z-axis. Calculate the vector for precession about the Z-axis. Neglect the mass of shaft OG and compute the gyroscopic couple M exerted by the shaft on the rotor at G. Problem 7/115 9 70 i K Z O

The housing of the electric motor is freely pivoted about the horizontal x-axis, which passes through the mass center G of the rotor. If the motor is turning at the constant rate p, determine the angular acceleration which will result from the application of the moment M about the vertical shaft if 0. The mass of the frame and housing is considered negligible compared with the mass m of the rotor. The radius of gyration of the rotor about the z-axis is kz and that about the x-axis is kx. Problem 7/116

The thin ring is projected into the air with a spin velocity of 300 rev/min. If its geometric axis is observed to have a very slight precessional wobble, determine the frequency of the wobble. Problem 7/117

A boy throws a thin circular disk (like a Frisbee) with a spin rate of 300 rev/min. The plane of the disk is seen to wobble through a total angle of 10. Calculate the period  of the wobble and indicate whether the precession is direct or retrograde. Problem 7/118

The figure shows a football in three common inflight configurations. Case (a) is a perfectly thrown spiral pass with a spin rate of 120 rev/min. Case (b) is a wobbly spiral pass again with a spin rate of 120 rev/min about its own axis, but with the axis wobbling through a total angle of 20. Case (c) is an end-over-end place kick with a rotational rate of 120 rev/min. For each case, specify the values of p, , , and as defined in this article. The moment of inertia about the long axis of the ball is 0.3 of that about the transverse axis of symmetry. Problem 7/119 2

The rectangular bar is spinning in space about its longitudinal axis at the rate p 200 rev/min. If its axis wobbles through a total angle of 20 as shown, calculate the period  of the wobble. Problem 7/120

The 5-kg disk and hub A have a radius of gyration of 85 mm about the -axis and spin at the rate rev/min. Simultaneously, the assembly rotates about the vertical z-axis at the rate rev/min. Calculate the gyroscopic moment M exerted on the shaft at C by the disk and the bending moment in the shaft at O. Neglect the mass of the shaft but otherwise account for all forces acting on it. Problem 7/121

The uniform slender bar of mass m and length l is centrally mounted on the shaft A-A, about which it rotates with a constant speed . Simultaneously, the yoke is forced to rotate about the x-axis with a constant speed . As a function of , determine the magnitude of the torque M required to maintain the constant speed . (Hint: Apply Eq. 7/19 to obtain the x-component of M.) Problem 7/122

The solid circular disk of mass m and small thickness is spinning freely on its shaft at the rate p. If the assembly is released in the vertical position at  0 with 0, determine the horizontal components of the forces A and B exerted by the respective bearings on the horizontal shaft as the position  /2 is passed. Neglect the mass of the two shafts compared with m and neglect all friction. Solve by using the appropriate moment equations. Problem 7/123 b A B

The earth-scanning satellite is in a circular orbit of p period . The angular velocity of the satellite about its y- or pitch-axis is 2/, and the angular rates about the x- and z-axes are zero. Thus, the x-axis of the satellite always points to the center of the earth. The satellite has a reaction-wheel attitude-control system consisting of the three wheels shown, each of which may be variably torqued by its individual motor. The angular rate z of the z-wheel relative to the satellite is 0 at time t 0, and the x- and y-wheels are at rest relative to the satellite at t 0. Determine the axial torques Mx, My, and Mz which must be exerted by the motors on the shafts of their respective wheels in order that the angular velocity of the satellite will remain constant. The moment of inertia of each reaction wheel about its axis is I. The x and z reaction-wheel speeds are harmonic functions of the time with a period equal to that of the orbit. Plot the variations of the torques and the relative wheel speeds x, y, and z as functions of the time during one orbit period. (Hint: The torque to accelerate the x-wheel equals the reaction of the gyroscopic moment on the z-wheel, and vice versa.) Problem 7/124

The two solid homogeneous right-circular cones, each of mass m, are fastened together at their vertices to form a rigid unit and are spinning about their axis of radial symmetry at the rate p 200 rev/min. (a) Determine the ratio h/r for which the rotation axis will not precess. (b) Sketch the space and body cones for the case where h/r is less than the critical ratio. (c) Sketch the space and body cones when h r and the precessional velocity is 18 rad/s.

The solid cylindrical rotor weighs 64.4 lb and is mounted in bearings A and B of the frame which rotates about the vertical Z-axis. If the rotor spins at the constant rate p 50 rad/sec relative to the frame and if the frame itself rotates at the constant rate  30 rad/sec, compute the bending moment M in the shaft at C which the lower portion of the shaft exerts on the upper portion. Also compute the kinetic energy T of the rotor. Neglect the mass of the frame. Problem 7/126 Z C A B z X Y y 4 r = 3 4 60

The cylindrical shell is rotating in space about its geometric axis. If the axis has a slight wobble, for what ratios of l/r will the motion be direct or retrograde precession? Problem 7/127

The solid cube of mass m and side a revolves about an axis M-M through a diagonal with an angular velocity . Write the expression for the angular momentum H of the cube with respect to the axes indicated. Problem 7/128

An experimental car is equipped with a gyro stabilizer to counteract completely the tendency of the car to tip when rounding a curve (no change in normal force between tires and road). The rotor of the gyro has a mass m0 and a radius of gyration k, and is mounted in fixed bearings on a shaft which is parallel to the rear axle of the car. The center of mass of the car is a distance h above the road, and the car is rounding an unbanked level turn at a speed v. At what speed p should the rotor turn and in what direction to counteract completely the tendency of the car to overturn for either a right or a left turn? The combined mass of car and rotor is m.

The wheels of the jet plane are spinning at their angular rate corresponding to a takeoff speed of 150 km/h. The retracting mechanism operates with  increasing at the rate of 30 per second. Calculate the angular acceleration  of the wheels for these conditions.

The motor turns the disk at the constant speed rad/sec. The motor is also swiveling about the horizontal axis B-O (y-axis) at the constant speed rad/sec. Simultaneously, the entire assembly is rotating about the vertical axis C-C at the constant rate rad/sec. For the instant when , determine the angular acceleration of the disk and the acceleration a of point A at the bottom of the disk. Axes x-y-z are attached to the motor housing, and plane O- is horizontal. Problem 7/131

The collars at the ends of the telescoping link AB slide along the fixed shafts shown. During an interval of motion, vA 5 in./sec and vB 2 in./sec. Determine the vector expression for the angular velocity n of the centerline of the link for the position where yA 4 in. and yB 2 in. Problem 7/132

The solid cone of mass m, base radius r, and altitude h is spinning at a high rate p about its own axis and is released with its vertex O supported by a horizontal surface. Friction is sufficient to prevent the vertex from slipping in the x-y plane. Determine the direction of the precession  and the period  of one complete rotation about the vertical z-axis.

The rectangular steel plate of mass 12 kg is welded to the shaft with its plane tilted from the plane (x-y) normal to the shaft axis. The shaft and plate are rotating about the fixed z-axis at the rate rev/min. Determine the angular momentum of the plate about the given axes and find its kinetic energy T. Problem 7/134

The circular disk of radius r is mounted on its shaft which is pivoted at O so that it may rotate about the vertical z0-axis. If the disk rolls at constant speed without slipping and makes one complete turn around the circle of radius R in time , determine the expression for the absolute angular velocity of the disk. Use axes x-y-z which rotate around the z0-axis. (Hint: The absolute angular velocity of the disk equals the angular velocity of the axes plus (vectorially) the angular velocity relative to the axes as seen by holding x-y-z fixed and rotating the circular disk of radius R at the rate of 2/.) Problem 7/135

Determine the angular acceleration  for the rolling circular disk of Prob. 7/135. Use the results cited in the answer for that problem.

Determine the velocity v of point A on the disk of Prob. 7/135 for the position shown

Determine the acceleration a of point A on the disk of Prob. 7/135 for the position shown

A top consists of a ring of mass m 0.52 kg and mean radius r 60 mm mounted on its central pointed shaft with spokes of negligible mass. The top is given a spin velocity of 10 000 rev/min and released on the horizontal surface with the point O remaining in a fixed position. The axis of the top is seen to make an angle of 15 with the vertical as it precesses. Determine the number N of precession cycles per minute. Also identify the direction of the precession and sketch the body and space cones. Problem 7/139

The uniform circular disk of 4-in. radius and small thickness weighs 8 lb and is spinning about its y- axis at the rate N 300 rev/min with its plane of rotation tilted at a constant angle  20 from the vertical x-z plane. Simultaneously, the assembly rotates about the fixed z-axis at the rate p 60 rev/min. Calculate the angular momentum HO of the disk alone about the origin O of the x-y-z coordinates. Also calculate the kinetic energy T of the disk.

Rework Prob. 7/140 if , instead of being constant at 20, is increasing at the steady rate of 120 rev/min. Find the angular momentum HO of the disk for the instant when  20. Also compute the kinetic energy T of the disk. Is T dependent on

The dynamic imbalance of a certain crankshaft is approximated by the physical model shown, where the shaft carries three small 1.5-lb spheres attached by rods of negligible mass. If the shaft rotates at the constant speed of 1200 rev/min, calculate the forces RA and RB acting on the bearings. Neglect the gravitational forces. Problem 7/142 8 8 8 8 x y z A B 1 1 2 2 3 3 120 E

Each of the two right-angle bent rods weighs 2.80 lb and is parallel to the horizontal x-y plane. The rods are welded to the vertical shaft, which rotates about the z-axis with a constant angular speed N 1200 rev/min. Calculate the bending moment M in the shaft at its base O. Problem 7/143

Each of the quarter-circular plates has a mass of 2 kg and is secured to the vertical shaft mounted in the fixed bearing at O. Calculate the magnitude M of the bending moment in the shaft at O for a constant rotational speed N 300 rev/min. Treat the plates as exact quarter-circular shapes.

Calculate the bending moment M in the shaft at O for the rotating assembly of Prob. 7/144 as it starts from rest with an initial angular acceleration of 200 rad/s2 .

The half-cylindrical shell of mass m, radius r, and length b revolves about one edge along the z-axis with a constant rate as shown. Determine the angular momentum H of the shell with respect to the x-y-z axes. Problem 7/146