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Unformatted text preview: 628 CHAPTER 22 VlBRATlONS PROBLEMS 22—1. When a ZDlb weight is suspended from a spring, the
spring is stretched a distance of 4 in. Determine the natural
frequency and the period of vibration for a 10—lb weight
attached to the same spring. 22—2. A spring has a stiffness of 600 N/m. If a 4—kg block is
attached to the spring. pushed 50 mm above its equilibrium
position. and released from rest. determine the equation
which describes the block’s motion. Assume that positive
displacement is measured downward. 22—3. When a 3kg block is suspended from a spring. the
spring is stretched a distance of 60 mm. Determine the
natural frequency and the period of vibration for a 0.2kg
block attached to the same spring. *22—4. An 8kg block is suspended from a spring having a
stiffness k = 80 N/m. If the block is given an upward
velocity of 0.4 m/s when it is 90 mm above its equilibrium
position. determine the equation which describes the
motion and the maximum upward displacement of the
block measured from the equilibrium position. Assume that
positive displacement is measured downward. 22—5. A 2lb weight is suspended from a spring having a
stiffness k = 2 lb/in. If the weight is pushed l in. upward
from its equilibrium position and then released from rest,
determine the equation which describes the motion. What is
the amplitude and the natural frequency of the vibration? 2%. A 6—lb weight is suspended from a spring having a
stiffness k = 3lb/in. If the weight is given an upward
velocity of 20 ft/s when it is 2 in. above its equilibrium
position. determine the equation which describes the
motion and the maximum upward displacement of the
weight. measured from the equilibrium position. Assume
positive displacement is downward. I22—7. A spring is stretched 175 mm by an 8kg block. If the
block is displaced 100 mm downward from its equilibrium
position and given a downward velocity of 1.50 m/s,
determine the differential equation which describes the
motion. Assume that positive displacement is measured
downward. Use the Runge—Kutta method to determine the
position of the block. measured from its unstretched position. when I = 0.22 3. (See Appendix B.) Use a time increment of
A! = 0.02 3. *22—8. If the block in Prob. 227 is given an upward
velocity of 4 m/s when it is displaced downward a distance
of 60 mm from its equilibrium position, determine the
equation which describes the motion. What is the amplitude
of the motion? Assume that positive displacement is
measured downward. 22—9. Determine the frequency of Vibration for the block.
The springs are originally compressed A. k k
mm mm nun &._.._M
Prob. 22—9 22—10. A pendulum has a 0.4mlong cord and is given
a tangential velocity of 0.2 m/s toward the vertical from a
position 6 = 0.3 rad. Determine the equation which describes
the angular motion. 22—11. A platform. having an unknown mass, is supported
by four springs, each having the same stiffness it. When
nothing is on the platform, the period of vertical vibration is
measured as 2.35 s; whereas if a 3kg block is supported on
the platform, the period of vertical vibration is 5.23 s.
Determine the mass of a block placed on the (empty)
platform which causes the platform to vibrate vertically
with a period of 5.62 5. What is the stiffness k of each of
the springs? Prob. 22—1 1 *22—12. If the lower end of the 30kg slender rod is
displaced a small amount and released from rest, determine
the natural frequency of vibration. Each spring has a
stiffness of k = 500 N/m and is unstretched when the rod is
hanging vertically. k Prob. 22—12 22—13. The body of arbitrary shape has a mass m, mass
center at G, and a radius of gyration about G of kc. If it is
displaced a slight amount 0 from its equilibrium position
and released, determine the natural period of vibration. Prob. 22—13 PROBLEMS 62 9 22—14. Determine to the nearest degree the maximum
angular displacement of the bob if it is initially displaced
0 = 0.2 rad from the vertical and given a tangential velocity
of 0.4 m/s away from the vertical. Prob. 22—14 22—15. The semicircular disk weighs 20 lb. Determine the
natural period of vibration if it is displaced a small amount
and released. Prob. 22—15 *22—16. The square plate has a mass m and is suspended
at its corner by the pin 0. Determine the natural period of
vibration if it is displaced a small amount and released. Prob. 22—16 630 CHAPTER 22 VIBRATIONS 22—17. The disk has a weight of 10 lb and rolls without
slipping on the horizontal surface as it oscillates about its
equilibrium position. If the disk is displaced, by rolling it
counterclockwise 0.4 rad, determine the equation which
describes its oscillatory motion when it is released. k = 1001b/ft Prob. 2217 22—18. The pointer on a metronome supports a 0.41b
slider A, which is positioned at a fixed distance from the
pivot O of the pointer. When the pointer is displaced, a
torsional spring at O exerts a restoring torque on the
pointer having a magnitude M = (1.26) lb'ft, where 6
represents the angle of displacement from the vertical,
measured in radians. Determine the natural period of
vibration when the pointer is displaced a small amount 8
and released. Neglect the mass of the pointer. \k =1.21bft/rad Prob. 22—18 22—19. The block has a mass m and is supported by a rigid
bar of negligible mass. If the spring has a stiffness k,
determine the natural period of vibration for the block. Prob. 22—19 *22—20. The disk, having a weight of 15 lb, is pinned at its
center 0 and supports the block A that has a weight of 3 lb.
If the belt which passes over the disk is not allowed to slip at
its contacting surface, determine the natural period of
vibration of the system. Prob. 2220 PROBLEMS 6 31 22—21. While standing in an elevator, the man holds a
pendulum which consists of an 18in. cord and a 0.5lb bob. If
the elevator is descending with an acceleration a = 4 ft/sz,
determine the natural period of Vibration for small amplitudes
of swing. V/ Prob. 22—23 *22—24. The bar has a length l and mass m. It is supported at
its ends by rollers of negligible mass. If it is given a small displacement and released, determine the natural frequency
of vibration. Prob. 2221 22—22. The 50lb spool is attached to two springs. If the
spool is displaced a small amount and released, determine
the natural period of vibration.The radius of gyration of the
spool is k0 = 1.5 ft. The spool rolls without slipping. Prob. 22—24 22—25. The 25lb weight is fixed to the end of the rod
assembly. If both springs are unstretched when the assembly is
in the position shown, determine the natural period of
vibration for the weight when it is displaced slightly and
released. Neglect the size of the block and the mass of the rods. {‘76 in.~>l«i6 in!» k = 21b/in. k = 2 lb/in. Prob. 22—22 22—23. Determine the natural frequency for small
oscillations of the 10lb sphere when the rod is displaced a
slight distance and released. Neglect the size of the sphere and the mass of the rod. The spring has an unstretched
length of 1 ft. Prob. 22—25 636 CHAPTER 22 VIBRATIONS PROBLEMS, 2226. Solve Prob. 22—13 using energy methods.
22—27. Solve Prob. 22—15 using energy methods.
*22—28. Solve Prob. 22—16 using energy methods.
22—29. Solve Prob. 22—20 using energy methods. 22—30. The uniform rod of mass m is supported by a pin at
A and a spring at B. If the end B is given a small downward
displacement and released, determine the natural period of
vibration. Prob. 22—30 2231. Determine the differential equation of motion of
the 3kg block when it is displaced slightly and releasedThe
surface is smooth and the springs are originally unstretched. Prob. 2231 >1‘22—32. Determine the natural period of vibration of the
10lb semicircular disk. Prob. 22—32 22—33. The 7kg disk is pinconnected at its midpoint.
Determine the natural period of Vibration of the disk if the
springs have sufficient tension in them to prevent the cord
from slipping on the disk as it oscillates. Hint: Assume that
the initial stretch in each spring is 60. This term will cancel
out after taking the time derivative of the energy equation. k = 600 N/m . k = 600 N/m Prob. 22—33 22—34. The 5lb sphere is attached to a rod of negligible mass
and rests in the horizontal position. Determine the natural
frequency of vibration. Neglect the size of the sphere. In 1ft 4 Prob. 22—34 PROBLEMS 6 3 7 22—35. The bar has a mass of 8 kg and is suspended from two 22—38. Determine the natural frequency of Vibration of
springs such that when it is in equilibrium, the springs make an the 20lb disk. Assume the disk does not slip on the
angle of 45° with the horizontal as shown. Determine the inclined surface. natural period of Vibration if the bar is pulled down a short distance and released. Each spring has a stiffness of
k = 40 N/m. 101b/in..
1 1"
i“ Prob. 22—38
Pmb' 22—35 22—39. If the disk has a mass of 8 kg. determine the natural
.9246. Determine the natural period of Vibration Of the frequency of vibrationThe springs are originally unstretched. 3kg sphere. Neglect the mass of the rod and the size of
the sphere. g300 mm ~47 300 mm Prob. 22—36 Prob. 22—39 22—37. The slender rod has a weight of 4lb/ft. If it is *22—40. Determine the differential equation of motion of supported in the horizontal plane by a balland—socketjoint the 3kg spool. Assume that it does not slip at the surface
at A and a cable at B, determine the natural frequency of of contact as it oscillates. The radius of gyration of the spool
vibration when the end B is given a small horizontal about its center of mass is kG = 125 mm. displacement and then released. k:400N/m
~———I/I///////////.——~ 1.5 ft Prob. 22—37 ﬂ Prob. 22—40 PROBLEMS 2241. Use a blockand—spring model like that shown in
Fig. 22—1401 but suspended from a vertical position and
subjected to a periodic support displacement of
5 = 50 cos wot, determine the equation of motion for the
system, and obtain its general solution. Define the
displacement y measured from the static equilibrium
position of the block when t = 0. 22—42. The 20lb block is attached to a spring having a
stiffness of 201b/ft. A force F = (6 cos 2t) lb, where I is in
seconds, is applied to the block. Determine the maximum
speed of the block after frictional forces cause the free
vibrations to dampen out. ZF=6c0521 0:01.10 Prob. 22—42 22—43. A 4—lb weight is attached to a spring having a
stiffness k = 101b/ft. The weight is drawn downward a
distance of 4 in. and released from rest. If the support
moves with a vertical displacement 6 = (0.5 sin 4:) in,
where t is in seconds, determine the equation which
describes the position of the weight as a function of time. PROBLEMS 649 *22~44. If the block is subjected to the impressed force
F = F0 cos wot, show that the differential equation of
motion is j} + (k/m)y = (FD/m) cos (not, where y is
measured from the equilibrium position of the block. What
is the general solution of this equation? F = F0 cos war Prob. 22—44 22—45. The light elastic rod supports a 4kg sphere. When
an lSN vertical force is applied to the sphere, the rod
deflects 14 mm. If the wall oscillates with harmonic
frequency of 2 Hz and has an amplitude of 15 mm,
determine the amplitude of vibration for the sphere. Prob. 22—45 650 CHAPTER 22 VIBRATIONS 22—46. Use a blockandspring model like that shown in
Fig. 22l4a, but suspended from a vertical position and
subjected to a periodic support displacement 5 = 60 sin wot,
determine the equation of motion for the system, and obtain
its general solution. Define the displacement y measured
from the static equilibrium position of the block when I = 0. 22—47. A Skg block is suspended from a spring having a
stiffness of 300 N/ m. If the block is acted upon by a vertical
force F = (7 sin St) N, where [is in seconds, determine the
equation which describes the motion of the block when it is
pulled down 100 mm from the equilibrium position and
released from rest at t = 0. Assume that positive displacement
is downward. .. k = 300N/m F=7sin8t Prob. 22—47 *22—48. The 4kg circular disk is attached to three springs,
each spring having a stiffness k = 180 N/m. If the disk is
immersed in a ﬂuid and given a downward velocity of 0.3 m/s
at the equilibrium position, determine the equation which
describes the motion. Assume that positive displacement is
measured downward, and that fluid resistance acting on the disk furnishes a damping force having a magnitude
F = (60M) N, wherevis in m/s. Prob. 22—48 22—49. The instrument is centered uniformly on a platform P,
which in turn is supported by four springs, each spring having a
stiffness k = 130 N/m. If the ﬂoor is subjected to a vibration
(00 2 7 Hz, having a vertical displacement amplitude
50 = 0.17 ft, determine the vertical displacement amplitude of
the platform and instrument. The instrument and the platform
have a total weight of 18 lb. Prob. 22—49 22—50. The 450kg trailer is pulled with a constant speed over
the surface of a bumpy road, which may be approximated by a
cosine curve having an amplitude of 50 mm and wave length of
4 m. If the two springs s which support the trailer each have a
stiffness of 800 N/ m, determine the speed 11 which will cause
the greatest vibration (resonance) of the trailer. Neglect the
weight of the wheels. L m~i~2m~—l Prob. 22—50 22—51. Determine the amplitude of vibration of the trailer
in Prob. 22—50 if the speed 1) = 15 km/h. F2m—+~—2m—~i Prob. 22—51 *22—52. The electric motor turns an eccentric ﬂywheel
which is equivalent to an unbalanced 0.25—lb weight located
10 in. from the axis of rotation. If the static deﬂection of the
beam is 1 in. because of the weight of the motor, determine
the angular velocity of the ﬂywheel at which resonance will
occur‘ The motor weights 150 lb. Neglect the mass of
the beam. PROBLEMS 6 5 1 22—53. What will be the amplitude of steadystate vibration of the motor in Prob. 22—52 if the angular velocity
of the ﬂywheel is 20 rad/s? 22—54. Determine the angular velocity of the flywheel in Prob. 22—52 which will produce an amplitude of vibration of
0.25 in. Probs. 22—53/54 22—55. The engine is mounted on a foundation block
which is springsupported. Describe the steadystate
vibration of the system if the block and engine have a total
weight of 1500 lb and the engine, when running, creates an
impressed force F = (SOsin21)1b, where t is in seconds.
Assume that the system vibrates only in the vertical
direction, with the positive displacement measured
downward, and that the total stiffness of the springs can be
represented as k = 2000 lb/ft. Prob. 22—52 Prob. 22—55 652 CHAPTER 22 VIBRATIONS *22—56. Determine the rotational speed (0 of the engine in
Prob. 22—55 which will cause resonance. Prob. 2256 22—57. The block, having a weight of 12 lb, is immersed in
a liquid such that the damping force acting on the block has
a magnitude of F = (0.7lvl) lb, where v is in ft/s. If the
block is pulled down 0.62. ft and released from rest,
determine the position of the block as a function of time.
The spring has a stiffness of k = 53lb/ft. Assume that
positive displacement is downward. Prob. 22—57 2258. A 7lb block is suspended from a spring having a
stiffness of k = 75 lb/ ft. The support to which the spring is
attached is given simple harmonic motion which may be
expressed as 6 = (0.15 sin 2:) ft. where tis in seconds. If the
damping factor is c/cc = 0.8, determine the phase angle d)
of forced vibration. 22—59. Determine the magnification factor of the block,
spring. and dashpot combination in Prob. 2258. *22—60. The bar has a weight of 6 1b. If the stiffness of the
spring is k = 8 ib/ft and the dashpot has a damping
coefficient (2 = 60 lb  s/ft. determine the differential equation
which describes the motion in terms of the angle 6 of the bar’s
rotation. Also, what should be the damping coefficient of the
dashpot if the bar is to be critically damped? l~ ~——2 ft + 3ft i
Prob. 22—60 2261. A block having a mass of 7 kg is suspended from a
spring that has a stiffness k = 600 N/m. If the block is given
an upward velocity of 0.6 m/s from its equilibrium position
at t = 0, determine its position as a function of time.
Assume that positive displacement of the block is
downward and that motion takes place in a medium which
furnishes a damping force F = (50M) N, where v is in m/s. 22—62. The damping factor, c/c(., may be determined
experimentally by measuring the successive amplitudes
of vibrating motion of a system. If two of these
maximum displacements can be approximated by x,
and x2, as shown in Fig. 22—17, show that the ratio In xl/xz = 2w(c/c(.)/ V1 ~ (c/c(,)2.The quantity 1n xl/xg is caiied the logarithmic decrement. 22—63. Determine the differential equation of motion for the
damped vibratory system shown.What type of motion occurs? (=200Ns/m *22—64. The 20kg block is subjected to the action of the
harmonic force F = (90 cos 6t) N, where I is in seconds,
Write the equation which describes the steadystate motion. PROBLEMS 653 22—66. The 10kg blockspringdamper system is continually damped. If the block is displaced to x = 50 mm
and released from rest, determine the time required for it to
return to the position x = 2 mm. Prob. 22—64 22—65. Draw the electrical circuit that is equivalent to the
mechanical system shown. What is the differential equation
which describes the charge q in the circuit? Prob. 22—65 Prob. 22—66 ...
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This note was uploaded on 04/16/2008 for the course CE 325 taught by Professor Docwong during the Spring '08 term at USC.
 Spring '08
 DocWong

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