Mechanical Engineering Chapter 21 Problem The Mass Each Slender Bar The Frequency Small Vibrations The System

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page-pf1
Problem 21.34 The mass of each slender bar is 1 kg.
If the frequency of small vibrations of the system is
0.935 Hz, what is the mass of the object A?
350
mm
280
mm
A
0.280 m, m=1 kg, and Mthe mass of A. The moments of inertia
about the xed point is the same for the two vertical bars. The kinetic
energy is
T=1
2I˙
θ2+1
2I˙
θ2+1
2mv2+1
2Mv2
A,
+MgLA(1cos θ),
V=(MgLA+2mgL)(1cos θ).
=const.
A
θ
is I=mL2
3, from which
g(2+η)
Solve:
page-pf2
Problem 21.35* The 4-kg slender bar is 2 m in length.
It is held in equilibrium in the position θ0=35by a tor-
sional spring with constant k. The spring is unstretched
when the bar is vertical. Determine the period and fre-
quency of small vibrations of the bar relative to the
equilibrium position shown.
θ
k
0
Solution: The total energy is
Solving for k,
k=mgL
sin θ0
sin 35
(4)(2)2(3)(9.81)
(2)(2)cos 35
so τ=2π
748
page-pf3
Problem 21.36 The mass m=2 slug, the spring con-
stant is k=72 lb/ft, and the damping constant is c=
8 lb-s/ft. The spring is unstretched when x=0. The
mass is displaced to the position x=1 ft and released
from rest.
(a) If the damping is subcritical, what is the frequency
of the resulting damped vibrations?
(b) What is the value of xat t=1s?
k
c
x
m
page-pf4
Problem 21.37 The mass m=2 slug, the spring con-
stant is k=72 lb/ft, and the damping constant is c=
32 lb-s/ft. The spring is unstretched with x=0. The
mass is displaced to the position x=1 ft and released
from rest.
(a) If the damping is subcritical, what is the frequency
of the resulting damped vibrations?
(b) What is the value of xat t=1s?
k
c
x
m
750
page-pf5
x=0.
(a) What value of the damping constant ccauses the
system to be critically damped?
(b) Suppose that chas the value determined in part (a).
At t=0,x=1 ft and dx/dt =4 ft/s. What is the
value of xat t=1s?
k
c
x
m
Problem 21.39 The mass m=2 kg, the spring con-
stant is k=8 N/m, and the damping coefcient is c=
12 N-s/m. The spring is unstretched when x=0. At
t=0, the mass is released from rest with x=0. Deter-
mine the value of xat t=2s.
k
c
x
page-pf6
Problem 21.40 The mass m=0.15 slugs, the spring
constant is k=0.5 lb/ft, and the damping coefcient is
c=0.8 lb-s/ft. The spring is unstretched when x=0.
We identify
752
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Problem 21.41 A 2570-lb test car moving with velo-
city v0=5 mi/h collides with a rigid barrier at t=0. As
a result of the behavior of its energy-absorbing bumper,
the response of the car to the collision can be simulated
by the damped spring-mass oscillator shown with k=
8000 lb/ft and c=3000 lb-s/ft. Assume that the mass
is moving to the left with velocity v0=5 mi/h and the
spring is unstretched at t=0. Determine the car’s posi-
tion (a) at t=0.04 s and (b) at t=0.08 s.
k
c
x
x
y
y
Car colliding with a rigid barrier
Simulation model
v0
v0
Solution: The equation of motion is
page-pf8
Problem 21.42 A 2570-lb test car moving with velo-
city v0=5 mi/h collides with a rigid barrier at t=0. As
a result of the behavior of its energy-absorbing bumper,
the response of the car to the collision can be simulated
by the damped spring-mass oscillator shown with k=
8000 lb/ft and c=3000 lb-s/ft. Assume that the mass
is moving to the left with velocity v0=5 mi/h and the
spring is unstretched at t=0. Determine the car’s decel-
eration (a) immediately after it contacts the barrier; (b) at
t=0.04 s and (c) at t=0.08 s.
x
y
y
Car colliding with a rigid barrier
v0
v0
Problem 21.43 The motion of the car’s suspension
shown in Problem 21.42 can be modeled by the damped
springmass oscillator in Fig. 21.9 with m=36 kg, k=
22 kN/m, and c=2.2 kN-s/m. Assume that no external
forces act on the tire and wheel. At t=0, the spring is
unstretched and the tire and wheel are given a velocity
dx/dt =10 m/s. Determine the position xas a function
of time.
Shock absorber
Coil spring
x
kc
m
x
Solution: Calculating ωand d, we obtain
The time derivative is
754
page-pf9
Problem 21.44 The 4-kg slender bar is 2 m in length.
Aerodynamic drag on the bar and friction at the sup-
port exert a resisting moment about the pin support of
magnitude 1.4(dθ/dt ) N-m, where /dt is the angular
velocity in rad/s.
(a) What are the period and frequency of small vibra-
tions of the bar?
(b) How long does it take for the amplitude of vibration
to decrease to one-half of its initial value?
θ
Solution:
1.4
dt mg L
2sin θ=1
3mL2α.
The (linearized) equation of motion is
d2θ
dt2+4.2
mL2
dt +3g
2Lθ=0.
This is of the form of Eq. (21.16) with
d=4.2
2mL2=4.2
2(4)(2)2=0.131 rad/s,
ω=3g
2L=3(9.81)
2(2)=2.71 rad/s.
From Eq. (21.18), ωd=ω2d2=2.71 rad/s, and from
Eqs. (21.20),
τd=2π
ωd=2.32 s,
fd=1
τd=0.431 Hz.
(b) Setting edt =e0.131t=0.5, we obtain t=5.28 s.
mg
θ
O
Problem 21.45 The bar described in Problem 21.44 is
given a displacement θ=2and released from rest at
t=0. What is the value of θ(in degrees) at t=2s?
page-pfa
Problem 21.46 The radius of the pulley is R=
100 mm and its moment of inertia is I=0.1 kg-m2. The
mass m=5 kg, and the spring constant is k=135 N/m.
The cable does not slip relative to the pulley. The
coordinate xmeasures the displacement of the mass
relative to the position in which the spring is unstretched.
Determine xas a function of time if c=60 N-s/m and
the system is released from rest with x=0. x
c
k
R
m
756
page-pfb
Problem 21.47 For the system described in Prob-
lem 21.46, determine xas a function of time if c=
120 N-s/m and the system is released from rest with
x=0.
Solution: From the solution to Problem 21.46 the canonical form
of the equation of motion is
The system is supercritically damped, since d2
2. The homogenous
k=0.3633. The solution is
Apply the initial conditions, at t=0, x=0, dx
dt =0, from which
x(t) =mg
Problem 21.48 For the system described in Prob-
lem 21.46, choose the value of cso that the system is
critically damped, and determine xas a function of time
if the system is released from rest with x=0.
R
k=0.3633 m. The solution:
Apply the initial conditions at t=0, x=0, dx
page-pfc
Problem 21.49 The spring constant is k=800 N/m,
and the spring is unstretched when x=0. The mass of
each object is 30 kg. The inclined surface is smooth.
The radius of the pulley is 120 mm and it moment of
inertia is I=0.03 kg-m2. Determine the frequency and
period of vibration of the system relative to its equilib-
rium position if (a) c=0, (b) c=250 N-s/m.
k
x
c
20
Solution: Let T1be the tension in the rope on the left of the
pulley, and T2be the tension in the rope on the right of the pulley.
The equations of motion are
Problem 21.50 The spring constant is k=800 N/m,
and the spring is unstretched when x=0. The damping
0.03 kg-m2.Att=0,x=0 and dx/dt =1 m/s. What
is the value of xat t=2s?
Solution: From Problem 21.49 we know that the damping is sub-
critical and the key parameters are
The solution is
758
page-pfd
Problem 21.51 The homogeneous disk weighs 100 lb
and its radius is R=1 ft. It rolls on the plane surface.
The spring constant is k=100 lb/ft and the damping
constant is c=3 lb-s/ft. Determine the frequency of
small vibrations of the disk relative to its equilibrium
position.
c
k
R
Solution: Choose a coordinate system with the origin at the center
page-pfe
Problem 21.52 In Problem 21.51, the spring is
unstretched at t=0 and the disk has a clockwise angular
velocity of 2 rad/s. What is the angular velocity of the
disk when t=3s?
Solution: From the solution to Problem 21.51, the canonical form
of the equation of motion is
Problem 21.53 The moment of inertia of the stepped
disk is I. Let θbe the angular displacement of the disk
relative to its position when the spring is unstretched.
Show that the equation governing θis identical in form
to Eq. (21.16), where d=R2c
2Iand ω2=4R2k
I.
θ
2R
ck
R
Solution: The sum of the moments about the center of the stepped
θ
760
page-pff
Problem 21.54 In Problem 21.53, the radius R=
250 mm, k=150 N/m, and the moment of inertia of
the disk is I=2 kg-m2.
(a) At what value of cwill the system be critically
damped?
page-pf10
Problem 21.55 The moments of inertia of gears Aand
Bare IA=0.025 kg-m2and IB=0.100 kg-m2. Gear
Ais connected to a torsional spring with constant k=
10 N-m/rad. The bearing supporting gear Bincorporates
a damping element that exerts a resisting moment on
gear Bof magnitude 2(dθB/dt) N-m, where B/dt is
the angular velocity of gear Bin rad/s. What is the fre-
quency of small angular vibrations of the gears?
140 mm
200 mm
AB
d2θB
M=6.622 rad/s
762
page-pf11
Problem 21.56 At t=0, the torsional spring in
Problem 21.55 is unstretched and gear Bhas a coun-
terclockwise angular velocity of 2 rad/s. Determine the
Solution: From the solution to Problem 21.55, the canonical form
of the equation of motion for gear Bis
M=6.622 rad/s,
Problem 21.57 For the case of critically damped
motion, conrm that the expression x=Cedt +
Solution: Eq. (21.16) is
page-pf12
Problem 21.58 The mass m=2 slug and the spring
constant is k=72 lb/ft. The spring is unstretched when
x=0. The mass is initially stationary with the spring
unstretched, and at t=0 the force F(t) =10 sin 4tlb is
applied to the mass. What is the position of the mass at
t=2s?
k
F(t)
x
m
Problem 21.59 The mass m=2 slug and the spring
F(t) =10 sin 4t+10 cos 4tlb is applied to the mass.
Solution: The equation of motion is
mx=F(t)
m⇒¨x+72
2x=10
2sin 4t+10
2cos 4t
764
page-pf13
Problem 21.60 The damped springmass oscillator is
initially stationary with the spring unstretched. At t=0,
a constant force F(t) =6 N is applied to the mass.
(a) What is the steady-state (particular) solution?
12 N/m
x
F(t)
dt2+2dx
dt +4x=F(t)
3.(1)
(a) If F(t) =6 N, we seek a particular solution of the form xp=
A0, a constant. Substituting it into Equation (1), we get 4xp=
F(t)
3=2 and obtain the particular solution: xp=0.5m.
(b) Comparing equation (1) with Equation (21.26), we see that d=
1 rad/s and ω=2 rad/s. The system is subcritically damped and
the homogeneous solution is given by Equation (21.19) with
ωd=ω2d2=1.73 rad/s. The general solution is
x=xh+xp=e1(A sin 1.73t+Bcos 1.73t) +0.5m.
Problem 21.61 The damped springmass oscillator
shown in Problem 21.60 is initially stationary with the
spring unstretched. At t=0, a force F(t) =6 cos 1.6tN
is applied to the mass.
(a) What is the steady-state (particular) solution?
0.234 cos 1.6t(m).
(b) The system is subcritically damped so the homogeneous solution
0.234, so the solution is
page-pf14
Problem 21.62 The disk with moment of inertia I=
3 kg-m2rotates about a xed shaft and is attached to
a torsional spring with constant k=20 N-m/rad. At
t=0, the angle θ=0, the angular velocity is /dt =
4 rad/s, and the disk is subjected to a couple M(t) =
θ
766

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