Vibrations and sound radiation of a cylindrical shell under a circumferentially moving force

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VIBRATIONS AND SOUND RADIATION OF A CYLINDRICAL SHELL UNDER A
CIRCUMFERENTIALLY MOVING FORCE
Raym on d Panneton, Alain Berry, Frédér ic Laville
GAUS, Mechanical Engineering
University of Sherbrooke, Sherbrooke (Quebec) J1K 2R1
1. IN TR OD UC TION
Vibrations and sound radiation by finite cylindrical shells have
been extensively studied in the past few years. Usually, the
authors have studied the vibrations and the sound radiation by
cylinders in the case of a non-moving harmonic driving force
[1]. Most papers dealing with a moving force on a cylindrical
shell (axially [2] or circumferientially [3]) were only concerned
about the mechanical vibrations. This is a presentation of the
work under progress to develop a model including both the
vibrations and the sound radiation of a simply supported
cylindrical shell excited by a circumferentially moving radial
po int force. The motivation behind this work is the
modélisation of the "pressure screens" used in the pulp and
paper industry. The theoretical formulation presented in section
2 is based on a v ariational approach. The case of a
homogeneous cylindrical shell in air is treated as a first step
towards more complex structures. Numerical results in terms of
quadratic velocity and radiated sound power are presented, and
principal phenomena related to the moving force rotational
speed are discussed in section 3.
2. THEO R E T ICAL F OR M UL A TIO N
The studied system consists of a baffled thin cylindrical shell
with the simply supported boundary conditions (Fig. 1). In the
case of a finite cylinder, and with a variational approach, one
can find the governing equations of motion for the studied
system using the Hamilton's function, which has the form:
(1)
where Tshell and E s he ll are respectively the shell kinetic and
deformation energy, is the energy related to the exterior
acoustic pressure field, and E forc e is the energy of the rotational
driving force. Using the thin shell theory and under Donnell's
assumptions, the three first terms are expressed as in reference
[1]. The energy term related to the radial force is
Efoi = {P(M ) )* {U(M) } dV (2)
where V is the volume of the cylinder, F (M) is the radial force at
a point M on the shell, and U(M ) is the displacement of point
M . A radial point force located at xa and travelling around the
circumference is expressed as:
P(M,t) = P(x,> >,t) = -P-S(x
aL xa)8( (p - Q-t) (3)
where 8 is the Dirac distribution, and Q is the rotational speed
of the force. Applying the Poisson's summation formula on (3)
one can separate the space and time variables:
$
sf
:•}
V „W ^ h
V L * *
o
Xo Baffle
Fig. 1: Schem atic of the cylindrical shell excited by a
circumferentially moving radial point force
P(M,t) = £ P(M )P(t)
N=-~
, P(M) = 1— o(x - x0)-e (4)
2n aL
P(t) = e m <
Integrating P(M) in (2) and developing on the modes of a
simply supported in vac uo circular cylindrical shell, one can
minim ize this energy function with respect to the modal
amplitudes and obtain the expression of the generalized force
vector:
(5)
where
8n,= 4 2
0 n*N
0n=N=0,a=0
2 n=N=0, a = l
-j n=N *0, a=0
1 n=N ï 0, a=l
Applying the same development for the three first energy terms
of equation (1) gives finally, for an e/NOt rotational excitation,
the following modal equation of motion:
oo
J
M-nmj ( COnmj ( 1 ~j?l) ~ (N£2) ~j(N£2) ^ ^ hvnq &nqk ~ ^Nnmj(6)
q= l k= l
where N O represents the N * harmonic of the rotational speed, n
the circumferential order, m and q the longitudinal orders, j and
k the type of mode (torsional, radial, axial), (Onmj the eigen-
angular frequencies, a a nmj the modal amplitudes, 7] the structural
damping, and Znm<? the modal radiation impedances.
For a better understanding of equation (6), let's neglect Z nmq.
Then, one can observe that maxima for modal amplitudes will
occur when
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(7)n < 0.2 71 (8)
where £2C is named the critical speed. In fact, there are as many
critical speeds as eigen-angular frequencies.
2. NUMER ICAL RESULTS
The results for a 0.003 m thick steel shell with a length of 1.2 m
and a radius of 0.48 m are presented in Figs. 2, 3 and 4 for two
different rotational speeds (25 Hz and 75 Hz), for the first
longitudinal order (/n=l), and for the type of mode (torsional,
radial, axial) having the lowest eigen-angular frequency.
Fig. 2 represents critical speeds versus circumferential orders.
As one can see, the first critical speed occurs at 28 Hz, for the
fifth circumferential order and the first longitudinal order (i.e.
mode (5,1)).
Because the 25 Hz rotational speed is very close to the first
critical speed of 28 Hz associated with the mode (5,1), the
quadratic velocity amplitude presents a significant single peak
for the 25 Hz fifth harmonic (i.e. 125 Hz or the fifth '+' in
Fig. 3). Since only frequencies around mode (5,1) (75-250 Hz)
are very excited, a low sound power will be radiated (see Fig. 4).
If the rotational speed is increased up to 75 Hz, one can predict,
by looking at Fig. 2, that a first peak will occur at its third
harmonic (mode (3,1)) and a second at its sixteenth harmonic
(mode (16,1)). The result predicted is verified in Fig. 3. The
bandwidth excited is now very large and the final result will be
an important increase of the radiated sound power (see Fig. 4).
For the 75 Hz rotational speed, the previous results include only
the first longitudinal order. If the m first longitudinal orders are
included, the quadratic velocity and the radiated sound power will
be radically different because more than two critical speeds will
occur.
For the 25 Hz rotational speed, including m longitudinal modes
will not change appreciably the curves because no other critical
frequency will occur.
Finally, as one can see, on Fig. 3 and 4, or by the mean of
equation (6), to obtain the quadratic velocity and the radiated
sound power at 2000 Hz, for a radial force rotating at 25 Hz, the
circumferential order has to be equal to 80 (80^ harmonic). In
that case, we need to ensure that thin shell theory is still
applicable by using the following criteria:
circumferential order (n)
Fig. 2: Critical speed versus circumferential orders
3. CONCLUSION
The model developed in the case of a simply supported
cylindrical shell has allowed us to draw some preliminary
conclusions useful in design such as the low level of sound
radiation when the force rotational speed is lower than the
critical frequency associated with the first mode. The use of the
variationnal approach will allow the integration of more
complex parameters such as stiffeners, visco-elastic layers,
internal pressure and heavy fluid.
ACKNOWLEDGEM ENTS
This work was supported by N.S.E.R.C. (Natural Sciences and
Engineering Research Council of Canada), I.R.S.S.T. (Quebec
Occupational Health and Safety Institute), and Andritz Sprout
Bauer inc.
REFEREN CES
[1] B. Laulagnet, J.L. Guyader, "Sound radiation by finite
cylindrical ring stiffened shells", J.S.V., 1 3 8 (2),
pp. 173-191, 1990
[2] P.Mann-Nachbar, "On the role of bending in the dynamic
response of thin shells to moving discontinuous loads",
J. Aerospace Sc., pp.648-657, june 1962
[3] S.C. Huang, W. Soedel, "On the forced vibration of simply
supported rotating cylindrical shells", J.A.S.A., 84(1),
pp. 275-285, July 1988
S ' 100
3
£ 80
0
■§ 60
>
Ü
‘S 40
TJ .
aJ
Ô 20
o
0 500 1000 1500 2000
Frequency (Hz)
Fig. 3: Quadratic velocity (each V corresponds to an harmonic
of the 25 or 75 Hz rotational speed)
S ' 80
T3
| 60
&
x) 40
1 20
O
0 500 1000 1500 2000
Frequency (Hz)
Fig. 4: Radiated sound power (each V corresponds to an
harmonic of the 25 or 75 Hz rotational speed)
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