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Problems and Solutions Section 1.5 (1.82 through 1.93)
1.82
A bar of negligible mass fixed with a tip mass forms part of a machine used to
punch holes in a sheet of metal as it passes past the fixture as illustrated in Figure
P1.82. The impact to the mass and bar fixture causes the bar to vibrate and the
speed of the process demands that frequency of vibration not interfere with the
process. The static design yields a mass of 50 kg and that the bar be made of steel
of length 0.25 m with a cross sectional area of 0.01 m2. Compute the systemโs
natural frequency.
Figure P1.82 A bar model of a punch fixture.
Solution: From equation (1.63)
ฯn =
( 2.0 ร 10 )( 0.01) (N/m )m = 1.26 ร 10 rad/s
EA
=
lm
11
2
50 ( 0.25)
kg โ
m
2
4
This is about 2000 Hz, which is likely too high to be a problem but could cause
some undesirable noise.
1.83
Consider the punch fixture of Figure P1.82. If the system is giving an initial
velocity of 10 m/s, what is the maximum displacement of the mass at the tip if the
mass is 1000 kg and the bar is made of steel of length 0.25 m with a cross
sectional area of 0.01 m2?
Solution: First compute the frequency:
ฯn =
EA
=
lm
( 2.0 ร 10 )( 0.01) (N/m )m = 2.828 ร 10 rad/s
11
2
1000 ( 0.25)
2
3
kg โ
m
From equation (1.9) the maximum amplitude is
Amax =
ฯ 2n x02 + v02
ฯn
=
v0
10 m/s
=
= 0.0035 m ,
ฯ n 2828 1/s
or about 0.35 mm, not much.
ยฉ 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained
from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying,
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1.84
Consider the punch fixture of Figure P1.82. If the punch strikes the mass off
center it is possible that the steel bar may vibrate in torsion. The mass is 1000 kg
and the bar 0.25 m-long, with a square cross section of 0.1 m on a side. The mass
polar moment of inertia of the tip mass is 10 kg/m2. The polar moment of inertia
for a square bar is b4/6, where b is the length of the side of the square. Compute
both the torsion and longitudinal frequencies. Which is larger?
Solution: First compute the longitudinal frequency of the bar:
ฯn =
EA
=
lm
( 2.0 ร 10 )( 0.01) (N/m )m = 2.828 ร 10 rad/s
11
2
1000 ( 0.25)
2
3
kg โ
m
Next compute the torsional frequency of the bar (square cross section):
ฯn =
GJ p
8 ร 108 (0.14 / 6)
=
= 73.03 rad/s
0.25 ร 10
lJ
In this case the torsional frequency is lower and should be considered in any
design.
1.85
A helicopter landing gear consists of a metal framework rather than the coil
spring based suspension system used in a fixed-wing aircraft. The vibration of the
frame in the vertical direction can be modeled by a spring made of a slender bar
as illustrated in Figure 1.23, where the helicopter is modeled as ground. Here l =
0.4 m, E = 20 ร 1010 N/m2, and m = 100 kg. Calculate the cross-sectional area that
should be used if the natural frequency is to be fn = 500 Hz.
Solution: From equation (1.63)
ฯn =
k
=
m
EA
lm
(1)
and
โ 2ฯ rad โ
= 3142 rad/s
ฯ n = 500 Hzโ
โ 1 cycle โ
Solving (1) for A yields:
ฯ n2lm ( 3142 )2 (.4 ) (100 )
A=
=
= 0.001974
E
20 ร 1010
A โ 0.0020 m 2 = 20cm 2
ยฉ 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained
from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying,
recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
1.86
The frequency of oscillation of a person on a diving board can be modeled as the
transverse vibration of a beam as indicated in Figure 1.26. Let m be the mass of
the diver (m = 100 kg) and l = 1.5 m. If the diver wishes to oscillate at 3 Hz, what
value of EI should the diving board material have?
Solution: From equation (1.67),
ฯ 2n =
3EI
3
ml
and
โ 2ฯ rad โ
= 6ฯ rad/s
ฯ n = 3Hz โ
โ 1 cycle โ
Solving for EI
ฯ n2 ml 3 ( 6ฯ ) (100 ) (1.5)
=
= 3.997 ร 104 Nm 2
EI =
3
3
2
1.87
3
Consider the spring system of Figure 1.32. Let k1 = k5 = k2 =100 N/m, k3 = 50
N/m, and k4 = 1 N/m. What is the equivalent stiffness?
Solution: Given: k1 = k2 = k5 = 100 N/m,k3 = 50 N/m, k4 = 1 N/m
From Example 1.5.4
k eq = k1 + k2 + k 5 +
k3 k4
k3 + k 4
โ keq = 300.98 N/m
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from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying,
recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
1.88
Springs are available in stiffness values of 10, 100, and 1000 N/m. Design a
spring system using these values only, so that a 100-kg mass is connected to
ground with frequency of about 1.5 rad/s.
Solution: Using the definition of natural frequency:
ฯn =
keq
m
With m = 100 kg and ฯn = 1.5 rad/s the equivalent stiffness must be:
k eq = mฯ n = (100)(1.5) = 225 N/m
There are many configurations of the springs given and no clear way to determine
one configuration over another. Here is one possible solution. Choose two 100
N/m springs in parallel to get 200 N/m, then use four 100 N/m springs in series to
get an equivalent spring of 25 N/m to put in parallel with the other 3 springs since
1
1
k eq = 1
=
= 25
1 1 1 4 100
+ + +
k1 k2 k 3 k4
Thus using six 100 N/m springs in the following arrangement will produce an
equivalent stiffness of 225 N/m
2
2
1
5
6
2
3
4
ยฉ 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained
from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying,
recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
1.89
Calculate the natural frequency of the system in Figure 1.32(a) if k1 = k2 = 0.
Choose m and nonzero values of k3, k4, and k5 so that the natural frequency is 100
Hz.
Solution: Given: k1 = k2 = 0 and ฯ n = 2ฯ (100) = 628.3 rad/s
From Figure 1.29, the natural frequency is
ฯn =
k5 k3 + k5 k4 + k3 k4
m(k3 + k4 )
and
โ
kk โ
keq = โ k5 + 3 4 โ
โ
k3 + k4 โ
Equating the given value of frequency to the analytical value yields:
k k +k k +k k
2
ฯ 2n = (628.3) = 5 3 5 4 3 4
m( k3 + k 4 )
Any values of k3, k4, k5, and m that satisfy the above equation will do. Again, the
answer is not unique. One solution is
k 3 = 1 N/m, k 4 = 1 N/m, k 5 = 50,000 N/m, and m = 0.127 kg
1.90* Example 1.4.4 examines the effect of the mass of a spring on the natural
frequency of a simple spring-mass system. Use the relationship derived there and
plot the natural frequency (normalized by the natural frequency, ฯn, for a massless
spring) versus the percent that the spring mass is of the oscillating mass.
Determine from the plot (or by algebra) the percentage where the natural
frequency changes by 1% and therefore the situation when the mass of the spring
should not be neglected.
Solution: The solution here depends on the value of the stiffness and mass ratio
and hence the frequency. Almost any logical discussion is acceptable as long as
the solution indicates that for smaller values of ms, the approximation produces a
reasonable frequency. Here is one possible answer. For
ยฉ 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained
from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying,
recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
From this plot, for these values of m and k, a 10 % spring mass causes less then a
1 % error in the frequency.
ยฉ 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained
from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying,
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1.91
Calculate the natural frequency and damping ratio for the system in Figure P1.91
given the values m = 10 kg, c = 100 kg/s, k1 = 4000 N/m, k2 = 200 N/m and k3 =
1000 N/m. Assume that no friction acts on the rollers. Is the system overdamped,
critically damped or underdamped?
c
k1
m
k2
k3
Figure P1.91
Solution: Following the procedure of Example 1.5.4, the equivalent spring
constant is:
keq = k1 +
k 2 k3
(200)(1000)
= 4000 +
= 4167 N/m
k 2 + k3
1200
Then using the standard formulas for frequency and damping ratio:
keq
4167
=
= 20.412 rad/s
m
10
c
100
ฮถ=
=
= 0.245
2mฯ n 2(10)(20.412)
ฯn =
Thus the system is underdamped.
1.92
Calculate the natural frequency and damping ratio for the system in Figure P1.92.
Assume that no friction acts on the rollers. Is the system overdamped, critically
damped or underdamped?.
Figure P1.92
Solution: Again using the procedure of Example 1.5.4, the equivalent spring
constant is:
kk
2 ร3
k eq = k1 + k2 + k3 + 4 5 = (10 + 1 + 4 +
)kN/m = 16.2 kN/m
k4 + k5
2 +3
Then using the standard formulas for frequency and damping ratio:
ยฉ 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained
from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying,
recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
keq
16.2 ร 10 3
=
= 40.25 rad/s
m
10
c
1
ฮถ=
=
= 0.001242 โ 0.001
2mฯ n 2(10)(40.25)
Thus the system is underdamped, in fact very lightly damped.
ฯn =
1.93
A manufacturer makes a cantilevered leaf spring from steel (E = 2 x 1011 N/m2)
and sizes the spring so that the device has a specific frequency. Later, to save
weight, the spring is made of aluminum (E = 7.1 x 1010 N/m2). Assuming that the
mass of the spring is much smaller than that of the device the spring is attached
to, determine if the frequency increases or decreases and by how much.
Solution: Use equation (1.67) to write the expression for the frequency twice:
ฯ al =
3Eal
and ฯ steel =
m๏ฌ3
3Esteel
rad/s
m๏ฌ3
Dividing yields:
ฯ al
=
ฯ steel
3Eal
m๏ฌ 3 =
3Esteel
m๏ฌ 3
7.1 ร 1010
= 0.596
2 ร 1011
Thus the frequency is decreased by about 40% by using aluminum.
ยฉ 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained
from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying,
recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

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