TRANSVERSE VIBRATION ANALYSIS OF A PRESTRESSED THIN CIRCULAR PLATE IN CONTACT WITH AN ACOUSTIC CAVITY
Daniel G Gorman[1] ,Chee K Lee and Ian A Craighead
Department of Mechanical Engineering, James Weir Building,
University of Strathclyde, Glasgow G1 1XJ
Jaromír Horáček
Institute of Thermomechanics, Academy of Sciences of the Czech Republic, Dolejškova 5,182 00 Prague 8, Czech Republic
ABStRACT
This paper describes the free transverse vibration analysis of a thin circular plate, subjected to in plane stretching, whilst in interaction with a cylindrical acoustic cavity. An analysis is performed which combines the equations describing the plate and the acoustic cavity to form a matrix equation which, when solved, produces the natural frequencies (latent roots) of the coupled system and associated latent vectors which describe the mode shape coefficients of the plate. After assessing the numerical convergence of the method, results are compared with those from a commercial finite element code (ANSYS). The results analysis is then extended to investigate the effect of stressing upon the free vibration of the coupled system.
Keywords Vibrations, vibro-acoustic interaction, structural/acoustic.
1. INTRODUCTION
Owing to their wide application in mechanical systems ranging from musical instruments to structural elements in industrial and space applications, the transverse vibration of circular plates and membranes has been the subject of many investigations from the end of nineteenth century. Of particular interest has been the effect upon the natural frequencies and associated mode shapes of these structural elements due to the inclusion of in-plane stressing as a result of thermal gradients and more general forms of hydrostatic loading. An excellent and extensive overview of much of this work is presented in reference [1]. In all of these studies it has been demonstrated that the inclusion of in-plane stressing can have a significant effect upon the natural frequencies of light thin plates where the restraining forces and moments due to the in-plane stressing becomes comparable, if not in excess of, the retraining forces and moments due to the inherent flexural rigidity of the plate. Much of the same body of work has shown that although the associated mode shapes are altered by the addition of the in-plane stressing, as compared to the in-plane stress free plate, the change is not so pronounced as the changes in the natural frequencies. However, these significant changes in natural frequencies, and less significant changes in mode shapes will no doubt result in significant changes in vibratory response to general dynamic loading of the plate as compared to the plate in a non pre-stressed state. Furthermore, often plates are in contact with enclosed acoustic cavities, obvious examples being musical percussion instruments and pressure vessel bursting discs. Frequency-modal characteristics for incompressible fluid in a rigid cylindrical container covered by a flexible circular membrane have been studied [2] and for interaction of liquid in a rigid cylindrical tank with a circular flexible bottom plate [3].
Coupled plate–liquid natural vibrations for a simply supported rectangular plate carrying liquid with reservoir conditions at its edges were studied in [4] and interaction of a rectangular flexible panel with an acoustic cavity was experimentally investigated in [5]. Recently, strong coupling between a clamped elastic rectangular plate and a quadrilateral (parallelepipedic) water–filled rigid cavity was experimentally studied in [6]. Vibroacoustic couplings and frequency modal characteristics of a rigid rectangular fluid-filled cavity with a flexible plate on one of its faces were theoretically studied [7,8]. Acoustic–structural couplings for an elastic plate in interaction with a cylindrical fluid–filled cavity was investigated [9,10], and similarly for a circular prestressed membrane [11,12].
In this paper we consider the free undamped vibration of a thin circular plate subjected to in-plane pre-stressing and in contact with a cylindrical acoustic cavity . Accordingly, an analytical/numerical treatise, based upon a combination of the Euler-Bernoulli and Helmholtz equations and the Ritz-Galerkin technique, of the system is performed and focuses upon the free vibration of the structure and how this is affected by gas coupling and stressing which can be due to pressure acting on, and/or temperature of, the structure. The analysis is confined to the modal parameters of natural frequencies and associated mode shapes.
2. Theoretical ANALYSIS
The equation of motion, describing the free small lateral vibration, = (r,,t), of a circular disc subjected to constant in-plane load intensity, N, and in interaction with the acoustic cavity, as shown in Figure 1, is
Figure 1 – Schematic diagram
, (1)
where
, , = r/a and ;
E is Young’s modulus, m is Poisson number and rd is the plate density; a and h are the radius and thickness of the plate, respectively; L is the length of the cylindrical cavity and p is the pressure inside.
Now writing , (2)
where
and , (3)
where is the natural mode shape of the disc in vacuo and ms is a constant for that mode, generally referred to as the mode shape coefficient for the mode consisting of m nodal diameters and s nodal circles. In this particular case, for a stressed disc clamped at the periphery, the mode shapes, , are according to [1]:
, (4)
where and are roots (values of s = 1, 2, 3 etc.) computed from the equation:
(5)
and , (6)
Im and Jm are the Bessel functions.
For particular values of m and s, the natural frequency of free undamped vibration, is then:
. (7)
In the case where the plate is not pre-stressed, i.e., N = 0, then and equations (4), (5) and (7) are altered accordingly. Now for a particular mode of vibration for the disc in vacuo:
. (8)
Therefore combination of equations (1), (3) and (8) gives:
. .. (9)
We shall now establish the form of the acoustic pressure, p, acting on the disc by reference to the acoustic cavity. Consider the acoustic cavity shown in Figure 1, whose velocity potential, f = f(x , r ,, t) is described by
, (10)
where c is speed of sound. Now writing
,
where (11)
and substituting equation (11) into (10) gives (for a set value of m)
, (12)
where and k is a constant. For the right hand side of equation (12) equal to –k2 we have
,
where , since must be finite when. At for each value of m
. , (13)
Therefore for a set value of m, the condition (13) has roots amq (q = 1, 2, 3 etc.), which satisfy the equation .
Similarly ,
where .
Therefore equation (11), for a set value of m, becomes:
. (14)
At , the axial component of the velocity of the gas and the lateral velocity of the plate must be equal, i.e.,
for .
Therefore from equations (2), (3) and (14) for a set value of m we have
. (15)
Multiplying both sides of equation (15) by and integrating between according to [13] gives
, (16)
where (17)
the value of which can be obtained through standard numerical integration.
Now the pressure, p, at the surface of the plate is given by:
,
where is the fluid density.
Therefore combining equations (14) and (16) we have:
. (18)
Substituting equation (18) into equation (9) gives:
.
Multiplying both sides by and integrating between we have:
, (19)
where .
Now, since
we can introduce a quantity instead of by the relation:
. (20)
Hence equation (19) can be re-written as
= 0 , (21)
where .
Equation (21) can be represented in matrix form as
, (22)
where
aqs(x) = . (23)
Hence values of x can be obtained (iterated upon) which renders the determinant of matrix (22) equal to zero. Consequently for each of these values (roots) of x we can then obtain the corresponding values of mode shape coefficients m1, m2, ……… mn., normalised to m1.
3. RESULTS AND DISCUSSION
In this study, since in all cases we are dealing with some degree of structural/fluid vibration interaction, it would be erroneous to describe any mode of vibration as either purely a structural mode or an acoustic (fluid) mode. Rather we will refer to the modes as either structural/acoustic (st/ac) to denote modes which are predominantly structural with acoustic interference and likewise acoustic/structural (ac/st) to denote modes which are predominantly acoustic but with structural interference.
Also we shall define the parameter , as reported in reference [1], as
for a circular plate clamped around its boundaries. The significance of is that it is a ratio of the lateral restraining force of the plate due to the in-plane stressing to that due to the flexural rigidity and for values of the plate will be in a buckled state. Therefore we shall consider the case where is greater than -1.
3.1 Convergence
As explained earlier, the roots x (from which the natural frequencies of the coupled system can be obtained) and the mode shape coefficients ms are obtained by iterating values of x which renders the determinant of the matrix equation (22) equal to zero. The determinant of this matrix equation is obtained by performing the LU decomposition [14], whereupon the value of the determinant is the product of the diagonal terms. Subsequently these root values of x which render the determinat zero are substituted back into equation (22) to obtain the corresponding values of the mode shape coefficients, ms, (normalised to m1, ) which describe which structural modes are present and dominate. Of immediate interest therefore is the convergence of the solution with respect to size of the square dimensions of the [A] matrix selected, i.e., the solutions obtained from the first n rows and columns of the matrix. For this convergence analysis, the following parameters were used:
radius of cylinder (a) = 38 mm
plate thickness (h) = 0.38 mm
length of cylinder (L) = 255 mm
density of air (rf )= 1.2 kg/m3
density of plate (rd) = 7800 kg/m3
Poisson ratio (m) = 0.3
Young’s modulus (E) = 2.1x 1011 Pa
speed of sound in air (c ) = 343 m/s.
These parameters were selected upon the basis that in the absence of any inplane stressing, the natural frequency of the first axisymetric mode (m = 0, s =1) of vibration of the plate in vacuo corresponds closely to the natural frequency of the first axisymetric mode (m =0, q = 1) of vibration of the acoustic cavity if the disc was assumed rigid and contained in equation (14) is π. In other words we will have strong structural/acoustic coupling between these two modes around this common frequency. Accordingly, prior to listing the table which demonstates the convergence of the technique, Table 1 lists (a) the natural frequencies of axisymetric modes of the disc if it were in vacuo, and, (b) natural frequencies of the axisymetric modes of the acoustic cavity if the disc was to be rigid.
m s / ω [Hz] / ξ2 [eq. (20)]0 1
0 2
0 3 / 671.83
2615.5
5859.8 / 10.216
39.771
89.104
(a)
m q / ω [Hz] / [eq. (14)]0 1
0 1
0 1
0 2
0 2
0 2
0 3
0 3
0 3 / 672.55
1345.1
2017.6
2891.4
3117.2
3461.0
3864.1
4035.8
4306.9 / π
2π
3π
π
2π
3π
π
2π
3π
(b)
Table 1 Calculated natural frequencies: a) for the circular plate in vacuo, b) for the acoustic cavity if the disc was rigid.
Table 2 lists natural frequencies and corresponding modal coefficients, qms , for values of n = 2, 4, 6 and 8 with m = 0 (axisymetric mode) and in-plane load intensity N = 0.
Table 2 demonstrates the convergence of the technique (with respect to n x n) for computing the natural frequencies and corresponding mode shape coefficients, m1, m2, … mn., normalised to m1 for the axisymetric modes (m = 0) of the coupled system .
n = 2 / n = 4 / n = 6 / n = 8 / Comments636.33
(1, 6.44x10-4) / 636.99
(1, 2.58x10-4, 2.17x10-6) / 637
(1, 2.49x10-4,
1.79x10-6,--
-- 4.7x10-11) / 637
(1, 2.49x10-4,-----1.4x10-14) / 1st st/ac
(s=1, q=1),
strong coupling with 1st ac/st
708.19
(1,-7.619x10-4) / 707.59
(1, -3.074x10-4,
-2.713x10-6) / 707.55
(1, -2.97x10-4,
--- -3.53x10-11) / 707.55
(1, -2.94x10-4,
--- -4x10-14) / 1st ac/st
(q=1, s=1),
strong coupling with 1st st/ac
1347.1
(1,-2.526x10-2) / 1347.3
(1, -1.032x10-2,
-7.078x10-5) / 1347.3
(1, 1x10-2, ---
-- -1.09x10-9) / 1347.3
(1, 9.9x10-3,
--- -1x10-12) / 2nd ac/st
(q=1, s=1),
weak coupling
2017.5
(1,-0.122) / 2018.4
(1, -4.99x10-2,
-2.038x10-4) / 2018.4
(1, -4.84x10-2,
--- -3.02x10-9) / 2018.4
(1, -4.8x10-2,
--- -2x10-12) / 3rd ac/st
(q=1, s=1)
2601.1
(1,-13.312) / 2607.6
(1, -12.57,
-5.189x10-4) / 2607.8
(1, -12.58,---
--- -6.8x10-9) / 2607.8
(1, -12.58,
--- -4.4x10-12) / 2nd st/ac
(s=2, q=1)
Table 2 Convergence of the natural frequencies [Hz] and the mode shapes coefficients
3.2 Comparison with results obtained from ANSYS[2]
In the construction of the finite element model, the same physical parameters of the disc were selected as that for the convergence test in 3.1 above. However in this case the length of acoustic cavity, L, was set as 350mm.
The three-dimensional model uses 6000 elements (type FLUID30) for the fluid in the cylinder and 300 elements (type SHELL63) for the plate. The cylinder walls were assumed rigid and the plate was fully fixed at the edges. The plate and fluid elements that are in contact are coupled for fluid-structure interaction. The plate is first pre-stressed by heating (or cooling) followed by a modal analysis of the combined system. The Lanczos unsymmetric eigensolver method is used for the mode extraction during the solution process. It is worth noting that, for prestressing to work correctly in ANSYS, it is necessary to select all the elements of the model (not just the plate itself) during the prestressing phase of the solution.
For the particular plate/acoustic cavity configuration the natural frequencies were computed by iteration of the determinant of the first three rows and columns of matrix equation (22) for a value of m = 0 (axisymmetric modes only). Corresponding values of natural frequencies were obtained from the ANSYS analysis described above. Tables 3a and 3b shows these values of natural frequencies. Also included in Tables 3a and 3b are the corresponding values of natural frequency associated with the plate in vacuo, i.e., in the absence of any acoustic coupling effects, and, the acoustic cavity alone if the plate was treated as a rigid boundary. From Table 3a ( =0) one can see that the modes of natural frequencies for the coupled system resemble those for the plate in vacuo and the acoustic cavity alone, in other words the system is fairly uncoupled. However for the case where = - 0.12752 (Table 3b) the fundamental natural frequency of the plate in vacuo approaches a value close to the acoustic natural frequency giving rise to strong interaction in the values for the coupled system.