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Singiresu S.Rao 저/이시복 역, "기계진동학 SI 단위", 6판
기계진동학 Chapter 3
기계진동학 (Dong-A University)
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These solutions represent a preliminary version
of the Instructors' Solutions Manual (ISM). It is
possible and even likely that at this preliminary
stage of preparing the ISM there are some
omissions and errors in the draft solutions.
These will be corrected and this manual will be
republished.
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Total stiffness with steel specimen:
keq = k1 + k2 = (1.8  106) + (135  106) = 136.8  106 N/m
Force in specimen due to magnets (static) due to elongation X = k2X. Force in specimen due
to a.c. current in magnets (dynamic) due to elongation X  k eq x  m 2 x.
Ratio of stresses 

k2 x
keq x  m x
2

k2 x
1
1

 . sp
2
2

k
x
m

2
eq
135  10
1

6
2
2
(136.8  10 )  20
6
Squaring both sides of this equation and rearranging gives:
400 4  5.47e9 2  5.42e16  0
 2  2.033e7 (positive value)
  4510 rad/s or 718 Hz
 2  0.18653761  108 (positive value)
  4508.8801 rad/s  687.3890 Hz
Equation of motion: m eq x  k eq x  F(t)
where meq = mass of valve and valve rod plus mass of spring at end = 12 + 8/3 = 14.67 kg
F(t) = Aρ0 sin ωt = (0.065)(65  103)sin 8t = (4225 sin 8t)N
Response of valve at steady state = xρ(t) = (X sin 8t)m
where X 
4225
 0.066 m or 6.6 cm
(65  10 )  (14.67)(8)2
3
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 st  static radial displacement of shaft under weight of turbine
  radial deflection of shaft under rotation
48EI
 stiffness of centrally loaded simply supported beam
l3


k
m 2  k(   st ) or m 2  k  k  st or


 st k  m 2
k
Let cri 
k
be the critical speed
m
If critical speed is
1
of operating speed op
5th
1
k
 
m 5 op
(E.1)
Here m = 250 kg and   3000 
k
2
 314.16 rad / s
60
48EI
(stiffness of centrally loaded simply supported beam)
l3
(E.2)
Substituting Equations (E.2) into (E.1) and squaring both sides gives:
48 EI  2

25
ml3
E  200 Gpa I 
 d4
64
and l  30d
(E.3)
Rearranging Equation (E.3) gives
(250)(27000)(314.6)2
 d  0.057 m or 5.7 cm
(470)(25)(109 )
Since l = 30 d,
l = 1.71 meters
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I
k

64
(D4  d4 ) 

64
(0.110)
4

 (0.095)4  3.189  106 m 4
48EI (48)(200  109 )(3.189  106 )

 1.96  106 N/m
l3
(2.5)3
k
1.96  106

 88.54 rad/s
m
250
Eccentricity, r1 = 0.045m, m0 = 0.25 kg
m  250 kg  n 
Radial Force F0 = m0 r2 at resonance
= 88.2 N

At resonance, Eq 3.15 gives, for x 0 = x 0  0 ,
1
Displacement of turbine is x(t)   stn  sin n t
2
FD
88.54
where  st 

 45  106 m
k
1.96  106
To activate the limit switch, x(t) = 0.012 m, and hence
1
(45  106 )(88.54)t sin 88.54t
2
0.002t sin 88.4t = 0.012
0.012 
or
i.e., t sin88.54t = 6
(E.1)
Eq. (E1) is solved by trial and error (assuming values of t = 10.0, 9.0, 8.0, etc.) as
t  6.024 s
Tip mass = m0 = 0.04 kg
(0.006)(0.0018)3
 2.916  1012 m 4
12
3  I (3)(200  109 )(2.916  1012 )
k 3 
 64.8 N/m
(0.3)3
l
I
m = mass of beam = (0.3)(0.006)(0.0018)(8000) = 0.0259 kg
1
1

2 
2
k
64.8
n  
 
  37.55 rad/s
 m 0  0.23 m 
 0.04  (0.23)(0.0259) 
1
1
2 

2
1  (0.02 )2
1  (2r )2
X 60


 50  


2 2
2
2 2
2 
Y 1.2
 (1   )  (2r ) 
(1   )  (0.02 ) 
i.e., r4 – 1.9996 r2 + 0.9996 = 0
r

 0.9999
n
   0.9999n  37.56 rad/s.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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lOMoARcPSD|17472938
© 2018 Pearson Education Ltd. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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