Journal of
Vibration
and Acoustics
Discussion
Discussion: ‘‘Optimum Design of
Vibration Absorbers for Structurally
Damped Timoshenko Beams’’
„Esmailzadeh, E., and Jalili, N., 1998,
ASME J. Vib. Acoust., 120,
No. 4, pp. 833 – 841…
Mohammad R. Rastegar
MSc Student
Mehrdaad Ghorashi
1
Associate Professor
Mem. ASME
e-mail: Ghorashi@sharif.edu
Mechanical Engineering Department, Sharif University of
Technology, P. O. Box 11365-9567,Tehran, Iran
1 Introduction
In a recent paper by E. Esmailzadeh and N. Jalili the design of
optimal vibration absorbers for Timoshenko beams was discussed.
While the paper is interesting, the solution seems to be question-
able, because it does not satisfy the governing differential equa-
tion of motion. There has been no reference made to the corre-
sponding experimental results, or results obtained through the
application of other analytical methods. Therefore, the application
of the results presented in the paper seems to be misleading.
2 Analysis
In the paper, the mode summation procedure has been applied,
and the following forms for the transverse deflection y ( x , t ) and
the orientation of the beam cross-section ( x , t ) have been
adopted,
y x , t =
i =1
n
Y
i
x • q
bi
t (1)
x , t =
i =1
n
i
x • q
bi
t (2)
It is observed that the same modal amplitudes for deflection and
section rotation have been assumed. It will be shown that such an
assumption results in a contradiction. For clarity of presentation,
let’s consider a simple case, where only one vibration absorber
system with a single mass, m, stiffness, k, and damping, c is at-
tached to the beam at some location, x =l . Also the only applied
force on the beam is assumed to be g ( t ), which is applied by the
absorber system. This force can be generated, for example, as a
result of an initial condition imposed on the beam. Considering
the free-body-diagram of an element of the Timoshenko beam, the
equations of motion would be,
A
2
y
t
2
-kAG
2
y
x
2
-
x
=g t • x -l (3)
EI
2
x
2
+kAG
y
x
-
- I
2
t
2
=0 (4)
Now, one can substitute Eqs. 1 and 2 in 3 and 4 to check if
assuming similar modal amplitudes for lateral deflection and sec-
tion rotation is justifiable. Substitution results in,
A
i =1
n
q ¨
bi
t Y
i
x -kAG
i =1
n
q
bi
t Y
i
x -
i
' x
=g t • x -l (5)
i =1
n
q
bi
t • EI
i
x +kAG Y
i
' x -
i
x
- I
i =1
n
q ¨
bi
•
i
x =0 (6)
On the other hand, the free vibration analysis gives,
- A
i
2
• Y
i
x -kAG• Y
i
x -
i
' x =0 (7)
EI
i
x +kAG• Y
i
' x -
i
x =-
i
2
I •
i
x (8)
Substitution of Eqs. 7 and 8 in 5 and 6 results in,
A
i =1
n
Y
i
x • q ¨
bi
t +
i
2
• q
bi
t =g t • x -l (9)
I
i =1
n
i
x • q ¨
bi
t +
i
2
• q
bi
t =0 (10)
Equations 9 and 10 are to be valid for all t and 0 x L . Thus
one concludes from 10 that,
q ¨
bi
t +
i
2
• q
bi
t =0 (11)
Substitution of Eq. 11 in 9 results in,
0 =g t • x -l (12)
Equation 12 presents a clear contradiction because it would be
valid only if g ( t ) equals to zero, that is, no absorber system is
used at all.
3 Conclusion
The method used in the paper for generating a solution for the
optimal design of vibration absorbers for Timoshenko beams, is
questionable. The assumption of same modal amplitudes for de-
flection and section rotation results in a severe contradiction.
Since the results obtained by the application of this method have
1
Corresponding author.
548 Õ Vol. 123, OCTOBER 2001 Copyright © 2001 by ASME Transactions of the ASME
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