Structural mobility is useful for the estimation of structural power flows in coupled ... Although the methods of measuring structural mobilities are easily found for ...
The measurement of structural mobilities of a circular cylindrical shell Ruisen Ming, Jie Pan, and Michael P. Norton Department of Mechanical and Materials Engineering, The University of Western Australia, Nedlands 6907, Australia
共Received 14 April 1999; revised 17 November 1999; accepted 21 November 1999兲 Structural mobility is useful for the estimation of structural power flows in coupled systems. Although the methods of measuring structural mobilities are easily found for one-dimensional beam structures, few are available for cylindrical shells. In this paper, a new method is proposed for the measurement of the structural mobilities of a circular cylindrical shell. A point force excitation is used instead of circumferential modal forces which are difficult to implement in practice. This method utilizes the least squares technique to obtain the transfer function components of different circumferential modes from the measured data. Experiments were carried out on a circular cylindrical shell with different end conditions excited by a point force to verify the feasibility of this proposed method. © 2000 Acoustical Society of America. 关S0001-4966共00兲00603-2兴 PACS numbers: 43.40.Ey, 43.20.Ye 关CBB兴
INTRODUCTION
Circular cylindrical shells are important elements of many types of industrial and defense structures. For the control of noise and vibration in coupled cylindrical shell systems, it is necessary to characterize the structural wave field and the vibrational energy transmission across structural joints. This characterization can be made by using structural mobility functions.1 The theoretical calculation of structural mobilities of a circular cylindrical shell has been studied by several authors.1–4 However, the experimental measurement of the structural mobilities has received scant attention, perhaps due to the difficulties in practically implementing a desirable circumferential modal force or in experimentally decomposing the wave components of different circumferential modes. To date few methods are available for measuring the structural mobilities of a circular cylindrical shell. The structural vibration of a circular cylindrical shell exhibits a two-dimensional modal pattern, presented by the superposition of axial and circumferential modes. The vibrational components of different circumferential modes are orthogonal to each other and are required to be decomposed in the analysis. Jong and Verheij have proposed a method5 to experimentally decompose the acceleration components of n⫽0,1,2 circumferential modes at frequencies below the n ⫽3 cutoff frequency in the wave field where the higher order circumferential modal responses become insignificant. This method utilizes the symmetrical nature of a circular cylindrical shell and a group of phase matched accelerometers on the shell surface to obtain the circumferential modal amplitudes and phases. This method cannot be applied to the near wave field or the frequencies above the n⫽3 cutoff frequency because higher order (n⬎2) circumferential modal responses are significant and the simple addition and subtraction of the signals measured on eight symmetrical positions cannot eliminate the effect of the higher order (n⬎2) circumferential modal responses. Recently the authors6 have successfully applied the method of least squares to decompose the vibra1374
J. Acoust. Soc. Am. 107 (3), March 2000
tion components of different circumferential modes. Theoretically this method has no frequency limit and can be applied to the case where the vibration is the superposition of many circumferential modes. It has been demonstrated that in the far wave field at frequencies below the n⫽3 cutoff frequency, these two methods give comparable results.6 In this paper, a new method is proposed and its theoretical basis is outlined for the measurement of structural mobilities in a circular cylindrical shell. A point force excitation is employed instead of a desirable circumferential modal force excitation which is difficult to implement in practice. Therefore, the response of the shell is the superposition of different circumferential modes. This proposed method uses an array of phase matched accelerometers to simultaneously measure dynamical responses at several positions around the cross section of interest in a circular cylindrical shell. Then the method of least squares is used during the data processing to decompose the transfer function components of different circumferential modal accelerations to the input force. A series of experiments was carried out under laboratory conditions to verify the feasibility of this proposed method.
I. STRUCTURAL MOBILITIES A. Prediction
Consider a circular cylindrical shell of thickness h, radius a and length L. Let the shell be referenced to a cylindrical coordinate system (r, ,x) where x is taken in the axial direction of the shell, measures the angle in the circumferential direction, and the r axis is directed outward along the radial direction, as shown in Fig. 1. If only a radial force per unit area, F, acts on the shell surface and if u, v and w represent the displacement components of the shell middle surface in the axial, tangential and radial directions, respectively, the equation of motion 共Reissner–Naghdi–Berry theory兲 for an element of shell can be written as7,8
0001-4966/2000/107(3)/1374/9/$17.00
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⬁
v⫽
兺
n⫽1
⬁
v n⫽
8
兺兺
V ns sin共 n ⫹ n0 兲
n⫽1 s⫽1
⫻exp共 ⫺ik ns x⫹i t 兲 , ⬁
w⫽
⬁
兺
w n⫽
n⫽0
m
兺兺
n⫽0 s⫽1
W ns cos共 n ⫹ n0 兲
⫻exp共 ⫺ik ns x⫹i t 兲 ,
FIG. 1. Coordinates for a circular cylindrical shell.
冋
册
2 w 1⫺ 2 1 2 1⫹ 2 v ⫽0; ⫹ ⫺ u⫹ ⫹ 2 2 2 2 2 x 2a 2a x a x cp t 共1a兲
冋
册
1⫺ 2 1 2 1 2 1⫹ 2 u ⫹ 共 1⫹  2 兲 ⫹ ⫺ v 2a x 2 x 2 a 2 2 c 2p t 2 ⫹
冋
冉
1 1 3 3 2 ⫺  ⫹ a2 a2 3 x2
冋
冉
冊册 冊册
u 1 1 3 3 ⫹ 2 ⫺2 2 3 ⫹ 2 a x a a x ⫹
冋
册
共1b兲
w⫽0; v
1 a2 2 F 2 4 , 2 1⫹  ⵜ ⫹ 2 2 w⫽⫺ a t B cp
共1c兲
where is the Poisson’s ratio, c p is the phase speed of an extensional wave propagating in a thin plate,  2 ⫽h 2 /12a 2 , ⌬ 2 ⫽a 2 ( 2 / x 2 )⫹ 2 / 2 is a Laplacian type operator, B ⫽Eh/(1⫺ 2 ) is the extensional rigidity. At cross section x, the effective transverse force S x , the bending moment M x , the extensional stress N x and the effective shear stress T x can be expressed as8 S x ⫽D
冉
冊
2⫺ 2 v 3w 4⫺3 3 w ⫺ ⫺ ; a2 x a2 x 2 x3
冉 冉 冊 冊 冉 冉 冊冊 冉
共2a兲
v 2w 2w ⫺ 2 ⫺ 2 ; M x ⫽D 2 a x
共2b兲
u v ⫹ ⫹w N x ⫽B x a
共2c兲
T x⫽
冊
B 共 1⫺ 兲 1 u v w ⫹ 共 1⫹2  2 兲 ⫺4  2 , 2) a x x 2
共2d兲
where D⫽Eh /(12(1⫺ )) is the bending rigidity of the shell. The displacement components of the wave propagating in a finite circular cylindrical shell can be expressed in the following forms:9 3
⬁
u⫽
兺 u n ⫽ n⫽0 兺 s⫽1 兺 U ns cos共 n ⫹ n0 兲 n⫽0
J. Acoust. Soc. Am., Vol. 107, No. 3, March 2000
共3a兲
Mx
Nx
T x 兴 T 兩 x⫽0
⬁
⫽
兺
n⫽0
J n 关 G n ,⫺G n 兴 A 0n X n ; Mx
Nx
共4a兲
T x 兴 T 兩 x⫽L
⬁
⫽
兺
n⫽0
J n 关 G n ,⫺G n 兴 A Ln X n ,
共4b兲
where G n (1,s)⫽⫺ (iD/a 3 ) ((k ns a) 3 ⫹(4⫺3 )n 2 k ns a⫹(2 ⫺ )nk ns a ␣ v ns ); G n 共 2,s 兲 ⫽
D 共共 k ns a 兲 2 ⫹ n 2 ⫹n ␣ v ns 兲 ; a2
G n 共 3,s 兲 ⫽
B 共 k a ␣ ⫹ n ␣ v ns ⫹ 兲 ; a ns uns
G n 共 4,s 兲 ⫽
1⫺ B 共 n ␣ uns ⫹ 共 1⫹2  2 兲 k ns a ␣ v ns 2 a
m
⫻exp共 ⫺ik ns x⫹i t⫹i /2兲 , 1375
Q 兩 x⫽0 ⫽ 关 S x
2
⬁
共3c兲
where n0 is a constant number representing the polarization angle of nth circumferential mode; is the radian frequency; n is the circumferential modal number; k ns is the axial wave number and subscript s corresponds to the axial wave number solutions. In general, n0 ⫽n 10 does not hold. For a single force excitation, however, this relation will hold. For the breathing mode 共n⫽0兲, the tangential component of displacement is zero 共actually the pure torsional wave exists but is uncoupled with other wave components in circular cylindrical shells兲 and the characteristic equation of the system is sixth order. Therefore, three pairs 共m⫽6兲 of axial wave number solutions exist for a finite circular cylindrical shell. Each pair of solutions has the same magnitude but different phases representing the axial waves simultaneously propagated along cylindrical shells in both positive and negative directions. For n⬎0, however, the characteristic equation of the system is eighth order and four pairs 共m⫽8兲 of axial wave number solutions are possible. A pure imaginary or complex wave number corresponds to an evanescent wave which is important at positions near the source but becomes insignificant at positions more than one wavelength away from the source. Only the wave with a pure real wave number can propagate along cylindrical shells. For each n and s pair, the coefficient ratios, ␣ uns ⫽U ns /W ns and ␣ v ns ⫽V ns /W ns , can be obtained by solving Eqs. 共1a兲 and 共1b兲. Substituting Eq. 共3兲 into 共2兲 gives the stress and moment resultant vectors at the two ends, x⫽0 and x⫽L, of the cylindrical shell as
Q 兩 x⫽l ⫽ 关 S x
;
共3b兲
⫹4n  2 k ns a 兲
共 s⫽1,2,3,4兲 ;
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J n ⫽diag兵 cos(n ⫹ n0), cos(n⫹ n0), cos(n⫹ n0), sin(n ⫹n0)其, A 0n ⫽diag兵 E m ,B m 其 , A Ln ⫽diag兵 B m ,E m 其 and B m ⫽diag兵 exp(⫺ikn1L),exp(⫺ikn2L), . . . ,exp(⫺iknmL)其 are diagonal matrices; E m is an m⫻m unit matrix (m⫽3 for n⫽0 and m⫽4 for n⬎0兲; X 0 ⫽ 关 W 01 ,W 02 , . . . ,W 06兴 T and X n ⫽ 关 W n1 ,W n2 , . . . ,W n8 兴 T (n⬎0). X n can be determined by solving the matrix equations resulting from the end boundary conditions. For the calculation of structural mobilities of a finite cylindrical shell, a free end condition is usually assumed at the end of x⫽0 and a desirable external force vector is assumed to act on it. For example, for the calculation of the radial force mobilities, only one external radial force is assumed to act on the end of x⫽0 and this external force vector may consist of many circumferential modal forces, ⬁ that is, F⫽ 兺 n⫽0 关 F n ,0,0,0兴 T . The force equilibrium at the end of x⫽0 gives a matrix equation, Q 兩 x⫽0 ⫽F. The boundary condition at the other end (x⫽L) will depend on practical considerations. For a free end condition, Q 兩 x⫽l ⫽0. For a simply supported end condition, 关 M x ,N x , w/ x ,u 兴 T 兩 x⫽l ⫽0, where M x and N x can be obtained from Eq. 共4b兲. For a clamped end condition, 关 u, v ,w, w/ x 兴 兩 x⫽l ⫽0. After X n is determined, all the components of displacement at any position of the shell can be calculated and then the structural mobilities can be evaluated from the ratios of the corresponding velocity components at the desirable position to the acting external circumferential force.
B. Measurement
It is difficult in practice to generate a force which only excites a single circumferential mode. If a point force is applied to a circular cylindrical shell, the response of the shell will be the superposition of different circumferential modal components 共n⫽0,1,2, . . . 兲. This is because a point force is the sum of different circumferential modal forces 共theoretically the circumferential modal number ranges from 0 to infinite兲. For example, if a point force f 0 acts at the position (x⫽0, ⫽ 0 ), the radial force per unit area can be expressed as
Y nv F ⫽
vF v˙ n ⌳ n v¨ n ⌳ n H n ⫽ ⫽ , Fn i f 0 i
where H nv F is the transfer function of the tangential acceleration component of the nth circumferential mode to the input point force. The above equations show that the measurement of the structural mobility for the nth circumferential mode is the measurement of the transfer function of the nth circumferential modal acceleration component to the input point force. In order to determine the structural mobility, it is necessary to extract the corresponding transfer function component of the desirable circumferential mode from the measured transfer function signal. Since the accelerometer is attached on the shell outer surface during the measurement, its output signal is not equal to the acceleration component of the shell middle surface unless the accelerometer main axis is lying along the radial direction. The relationships between the measured transfer function signals and the transfer function of the acceleration components on the shell middle surface to the input point force are given by7 uF Hm ⫽H uF ⫺d
vF Hm ⫽
冉 冊
兺
n⫽0
F n cos共 n 共 ⫺ 0 兲兲 ,
共5兲
where F n ⫽(1/⌳ n ) f 0 ␦ (x⫺0); ⌳ n is equal to 2 for n⫽0 and for n⬎0. By definition, the structural mobility of the nth circumferential mode is the ratio of the nth circumferential modal velocity to the nth circumferential modal force. For example, the input radial force mobility of the nth circumferential mode can be expressed as Y wF n ⫽
˙ n ⌳ nw ¨ n ⌳ n H wF w n ⫽ ⫽ , Fn i f 0 i
H wF n
共6兲
is the transfer function of the nth circumferential where modal radial acceleration component to the input point force. Similarly, the cross mobility Y nv F , the ratio of the tangential velocity component to the radial force component of the nth circumferential mode, can be expressed as 1376
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H wF ; x
d d H wF 1⫹ H v F ⫺ ; a a
共8兲 wF Hm ⫽H wF ,
where d is the distance between the main axis of the rotauF vF , Hm tional accelerometer and the shell middle surface; H m wF and H m are the measured transfer function signals when the main axis of the accelerometer is lying along the axial, tangential and radial directions, respectively. To calculate H uF and H v F , it is necessary to know H wF and its first order derivative first. The first order derivative of H wF can be estimated from the data measured at two adjacent crosssections close to the cross-section of interest. Substituting Eq. 共3兲 into 共8兲 gives ⬁
uF Hm ⫽
兺
n⫽0
⬁
F⫽ f 0 ␦ 共 x⫺0 兲 ␦ 共 a ⫺a 0 兲 ⫽
共7兲
冉
H uF n ⫺d
冊
H wF n cos关 n ⫹ n0 兴 x
⬁
⫽
兺
n⫽0 ⬁
vF ⫽ Hm
兺 n⫽1
uF cos关 n ⫹ n0 兴 ; H mn
冋冉 冊 1⫹
共9a兲
册
d v F nd wF H ⫹ sin关 n ⫹ n0 兴 H a n a n
⬁
⫽
兺
n⫽0
vF H mn sin关 n ⫹ n0 兴 ;
共9b兲
H wF n cos关 n ⫹ n0 兴 .
共9c兲
⬁
wF ⫽ Hm
兺
n⫽0
uF Note that the modal transfer function components, H mn , wF vF H mn and H n , are complex numbers and they are characterized by an amplitude and phase 共or real and imaginary parts兲. When the structural wave field is dominated by several circumferential modes, the measured transfer functions will be approximately equal to the sum of those dominating transfer
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FIG. 2. Predicted magnitudes of the radial force mobilities 关共a兲 Y wF; 共b兲 Y uF; 共c兲 Y vF] of a finite steel circular cylindrical shell under the free–free end conditions for different circumferential modes: n⫽0 共- - - - -兲; n⫽1,4,6 共------兲; n⫽2,5,8 共 — —兲 and n⫽3,7 共—–—-兲.
FIG. 3. Predicted magnitudes of the radial force mobilities 关共a兲 Y wF; 共b兲 Y uF; 共c兲 Y vF] of a semi-infinite steel circular cylindrical shell for different circumferential modes: n⫽0 共- - - - -兲; n⫽1,4,6 共------兲; n⫽2,5 共— — —兲 and n⫽3 共—–—-兲.
function components 共that is, the summations in the above equations are taken by only considering these dominating circumferential modes兲. For a circular cylindrical shell at frequencies below the mth cutoff frequency, the responses of the higher order (n⬎m) circumferential modes at the source position do not display any resonance peak and they decrease with increasing circumferential number n. Therefore, the measured transfer functions in a frequency range can be approximately decomposed into a finite number of dominating circumferential modal components. In order to determinate all three orthogonal modal transfer function components at frequencies where m circumferential modes dominate, it is required to perform the measurements on at least N⫽2m 关or N⫽2m⫺1 if the breathing 共n⫽0兲 mode is included兴 positions 共for a single force excitation, n0 ⫽n 10 holds, N will reduces to m⫹1) in the three orthogonal directions at the cross-section of interest. The required minimum measurement position number depends on the unknown coefficient number of the circumferential modes of interest. It is also noted that the separation distance between adjacent accelerometers should be less than a/m to satisfy the Nyquist spatial sampling criterion.10
Although theoretically a vector containing N unknown quantities can be determined from N experimental data, the solution is usually unreliable because of measurement errors and ill conditioned coefficient matrices due to the selection of inappropriate measurement locations. A reliable solution needs more than N measurement positions.11 The method of uF vF , H mn and H wF least squares11 can be used to extract H mn n uF vF from the measured real and imaginary parts of H m , H m and wF at more than N positions around a cross-section. For Hm example, at frequencies where m circumferential modes dominate, these dominating radial transfer function components can be obtained from the minimization of its error function e w
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J. Acoust. Soc. Am., Vol. 107, No. 3, March 2000
N
e w⫽
兺
i⫽1
冋
wF H m,i ⫺
兺m
册
2
H wF n cos共 n ⫹ n0 兲 ,
共10兲
where N(⬎(2m⫺1)) is the total number of the measurewF is the ment positions on the cross section of interest and H m,i measured radial transfer function component at position i. To obtain a minimum value of the error function, both H wF n and n0 need to be adjusted. The minimization of e w is divided for a into two steps. The first step is to find optimal H wF n Ming et al.: Mobilities of shells
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FIG. 4. Measured 共-------兲 and predicted 共— — — –兲 real parts of the radial force mobilities of a finite steel circular cylindrical shell under the free–free uF end conditions for n⫽1: 共a兲 Re兵Y wF 1 其; 共b兲 Re兵Y 1 其; 共c兲 negative values of Re兵Y v1F其.
FIG. 5. Measured 共--------兲 and predicted 共— — — –兲 imaginary parts of the radial force mobilities of a finite steel circular cylindrical shell under the uF vF free–free end conditions for n⫽1: 共a兲 Im兵Y wF 1 其; 共b兲 Im兵Y 1 其; 共c兲 Im兵Y 1 其.
given n0 . The second step is to determine the optimal n0 by finding the minimum value of the error e w ( n0 ). The final result of the two minimizations gives rise to the modal amplitude of the transfer function and the true value of the polarization angle n0 .
should be negligible. For the case of free-damped end conditions, the shell was supported at a position 0.5 m away from its free end by a piano wire with a very soft foam pad and its another end was buried 共0.5 m兲 in a dry sand filled box which intended to provide an effectively absorptive termination for all circumferential modes and wave types. For both cases, the 共free兲 end of interest was driven in the radial direction by a mechanical shaker fed with a pseudo-random noise signal. To measure the driving force, an impedance head was mounted at the driving location. The shaker and the impedance head were connected by a steel rod of 30 mm in length and 1 mm in diameter to avoid possible axial and tangential force excitations. Since the excitation was stable, the responses at different positions around the cross section of interest were measured by one 共B&K 4375兲 accelerometer at different times. Therefore, the measured data should contain no phase matching error. For the measurement of axial and tangential acceleration components, an aluminum cube of 10 mm side dimensions was used to construct a rotational accelerometer. The masses of the accelerometer and the cube are 2.4 g and 2.7 g, respectively. The predicted normalized mass loading errors of measured accelerations are negligible
II. RESULTS AND DISCUSSIONS A. Experimental setup
A 2.2-m-long steel circular cylindrical shell of a⫽32.5 mm and h⫽1.7 mm was used in the experiment. Two types of end conditions were chosen: 共1兲 free–free end conditions; 共2兲 free-damped end conditions. For the case of free–free end conditions, the cylindrical shell was supported 共nearly point contact兲 at positions 0.5 m away from both ends by two piano wire fixed on a frame via two very soft foam pads. Due to very low damping in the shell structure, light (1.6 kg/m2 ) damping strips 共Idikell兲 were attached to the shell outer surface 共covering about 60% of the total surface area兲 to increase the energy dissipation in the shell. The mass loading of the damping strips is small 共surface mass density ratio is less than 8%兲 and the effect on the structural wave field 1378
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FIG. 6. Measured 共--------兲 and predicted 共— — — –兲 real parts of the radial force mobilities of a finite steel circular cylindrical shell under the free-free uF end conditions for n⫽2: 共a兲 Re兵Y wF 1 其; 共b兲 Re兵Y 1 其; 共c兲 negative values of vF Re兵Y 1 其.
in the frequency range of interest. The transverse sensitivity of the accelerometer was measured using a B&K 4294 Calibration Exciter and an aluminum cube. Two points were marked on the accelerometer in the two orthogonal transverse axes, one of them has a minimum transverse sensitivity reading. The transverse sensitivity components in the two orthogonal transverse axes were 0.91% and 2.67%, respectively, at 159 Hz. During the measurements, the two orthogonal transverse axes 共or the marked points兲 were directed along the axial, tangential or radial directions, respectively, depending on which acceleration component is of primary interest. The errors due to the presence of accelerometer transverse sensitivity were eliminated using the following equation during data processing:
冋册冋 u
1
v ⫽ ␣ x w ␣ rx
␣ x
␣ xr
1
␣ r
␣ r
1
册冋册 ⫺1
um
vm , wm
共11兲
where ␣ x is the tangential component of the transverse sensitivity of the accelerometer whose main axis is directed along the axial direction; (u m , v m ,w m ) are the measured acceleration signals. 1379
J. Acoust. Soc. Am., Vol. 107, No. 3, March 2000
FIG. 7. Measured 共-------兲 and predicted 共— — — –兲 imaginary parts of the radial force mobilities of a finite steel circular cylindrical shell under the uF vF free–free end conditions for n⫽2: 共a兲 Im兵Y wF 1 其; 共b兲 Im兵Y 1 其; 共c兲 Im兵Y 1 其.
The frequency range of analysis was set up to 3 kHz 共from the Love–Timoshenko theory,8 the predicted cutoff frequencies of the n⫽2,3 circumferential modes are 1079 Hz and 3053 Hz, respectively兲. In order to know the dominating circumferential mode number at frequencies of interest and then to determine the measurement position number at the cross-section of interest, the magnitudes of low order circumferential modal mobility components are predicted, as shown in Fig. 2 for a finite steel cylindrical shell with free– free end conditions and in Fig. 3 for a semi-infinite steel cylindrical shell. It can be seen that the mobility components of n⫽1,2,3 circumferential modes dominate at frequencies of interest. The measured transfer functions in Eq. 共9兲 should be approximately equal to the sum of the n⫽1,2,3 circumferential modal components. Due to the point force excitation, n0 ⫽n 10 holds. Therefore, at least four unknown quantities are required to be determined for the estimation of circumferential modal mobility components at frequencies of interest. To assess the effect of the summation number in Eq. 共9兲 on the measurement accuracy, different sets of circumferential modes 共n⫽1,2,3; n⫽1,2,3,4, n⫽1,2,3,4,5 and Ming et al.: Mobilities of shells
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FIG. 8. Measured 共-------兲 and predicted 共— — — –兲 real parts of the radial force mobilities of a finite steel circular cylindrical shell under the freeuF damped end conditions for n⫽1: 共a兲 Re兵Y wF 1 其; 共b兲 Re兵Y 1 其; 共c兲 negative values of Re兵Y v1F其.
n⫽0,1,2,3,4,5兲 are considered in the date processing for estimating the n⫽1,2 circumferential modal mobility components from the same measurement data. It is shown that all the results agree well, especially at frequencies close to the resonance frequencies. The results shown in the following figures are obtained by considering 4 共n⫽1,2,3,4兲 circumferential modes only from data measured on 12 positions. Twelve measurement positions were uniformly distributed 共the radial angle between the neighboring positions was 30°兲 around the free end cross section. For the calculation of the axial transfer function component, the first order derivative was estimated from the data measured around the of H wF n cross-section of the free end and that of 15 mm away from the free end. Every measured transfer function signal 共both real and imaginary parts兲 was recorded in a dual-channel real-time frequency analyzer. B. Measurement of structural mobilities
For the cylindrical shell under test, the mobility component of the breathing 共n⫽0兲 mode is quite small compared with those of n⫽1,2 circumferential modes at most frequencies, as shown in Figs. 2 and 3. The measured n⫽0 mobility components were much higher than the predicted ones at 1380
J. Acoust. Soc. Am., Vol. 107, No. 3, March 2000
FIG. 9. Measured 共-------兲 and predicted 共— — — –兲 imaginary parts of the radial force mobilities of a finite steel circular cylindrical shell under the uF vF free-damped end conditions for n⫽1: 共a兲 Im兵Y wF 1 其; 共b兲 Im兵Y 1 其; 共c兲 Im兵Y 1 其.
most frequencies and they were the residues. Therefore, the measured mobility curves of the breathing 共n⫽0兲 mode will not be shown in the following. Only the mobility components of n⫽1,2 circumferential modes were considered in the analysis. Figures 4–7 show the comparisons of the measured and predicted real and imaginary parts of the input and cross radial force mobilities of the finite circular cylindrical shell under the free–free end conditions for n⫽1 and 2, respectively. In Figs. 4共c兲 and 6共c兲 the negative values of the real parts of the cross mobility Y vF n are shown. It can be seen that both the predicted and measured mobility curves show peaks at the resonant frequencies. The predicted resonant frequencies are slightly lower at all frequencies for n⫽1 and at frequencies above 1.82 kHz for n⫽2 but little higher than the measured ones at frequencies below 1.82 kHz for n⫽2. The measured n⫽2,3 cutoff frequencies are 1028 Hz and 2884 Hz which are slightly lower than the predicted ones. These could result from the errors in the assumption of the shell material properties in the predictions. At frequencies below the n⫽2 cutoff frequencies, the measured n⫽2 circumferenMing et al.: Mobilities of shells
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FIG. 10. Measured 共-------兲 and predicted 共— — — –兲 real parts of the radial force mobilities of a finite steel circular cylindrical shell under the uF free-damped end conditions for n⫽2: 共a兲 Re兵Y wF 1 其; 共b兲 Re兵Y 1 其; 共c兲 negative vF values of Re兵Y 1 其.
tial modal mobility components do not agree well with the predicted ones. This is because the residual effects of resonances of the n⫽1 circumferential mode. At frequencies above 2884 Hz, the measured curves shown in those figures were not correct because the dominating circumferential modes are different. From Figs. 4 and 6, it can be seen that a large error could be present in the measured mobilities at nonresonant frequency, especially for cross mobilities. At some nonresonant frequencies especially for Y vn F , the measured real parts of the mobilities are in opposite signs with or much higher than the predicted values 共for clearness, the negative values uF vF of Re兵Y wF n 其 and Re兵Y n 其 and the positive values of Re兵Y n 其 are not shown in the figures兲. This is because the measurement accuracy of a translational acceleration is usually higher than that of a rotational acceleration, and because the dissipation loss factor of the shell was so small that the wave field was very reactive at nonresonant frequencies. In a very reactive wave field, the measured signals could contain large errors. This indicates that in a reactive wave field, the axial and tangential mobilities at the nonresonant frequencies may not be accurately measured by using this proposed method. Figures 8–11 show the comparisons of the measured 1381
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FIG. 11. Measured 共-------兲 and predicted 共— — — –兲 imaginary parts of the radial force mobilities of a finite steel circular cylindrical shell under the uF vF free-damped end conditions for n⫽2: 共a兲 Im兵Y wF 1 其; 共b兲 Im兵Y 1 其; 共c兲 Im兵Y 1 其.
and predicted real and imaginary parts of the input and cross radial force mobilities of the finite circular cylindrical shell under the free-damped end conditions for n⫽1 and 2, respectively. Figures 8共c兲 and 10共c兲 show the negative real parts of the cross mobility Y nv F . It can be seen that the measured mobility curves under the free-damped end conditions is smoother than those under the free–free end conditions shown in Figs. 4–7, even at the troughs, except for the frequencies very close to the cutoff frequencies where a big jump or fluctuation is observed. The reason is that the damped end gave a very effectively absorptive termination. Again, at frequencies above the n⫽3 cutoff frequency, the mobility curves are not correct because of the same reason for Figs. 4–7. The measured mobility curves fluctuate while the predicted ones do not. The frequency averages of the measured real parts of the mobilities are slightly smaller than their predicted ones while those of the measured imaginary parts of the mobilities are a little larger than their predicted ones. This is because the predictions were made based on the semi-infinite circular cylindrical shell model but this assumed model does not accurately represent the actual cylindrical shell system. The damped end absorbed most but not Ming et al.: Mobilities of shells
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all of the incident energy. From these figures it may be concluded that for cylindrical shells with free-damped end conditions this proposed method gives accurate results.
ACKNOWLEDGMENT
Support for this work from Australian Research Council is gratefully acknowledged.
III. CONCLUSIONS
A new method is proposed in this paper to measure the structural mobilities of a circular cylindrical shell. This method utilizes a point force excitation instead of circumferential modal forces which are difficult to implement in practice. The method of least squares is employed to obtain the transfer functions of different circumferential modal acceleration components to the input point force. The outlined theory has been experimentally verified on a steel circular cylindrical shell of different end conditions. The measured results show that this proposed method is successful in measuring the structural mobilities of a circular cylindrical shell. The measurement accuracy of this method, however, depends on the acceleration components of interest and the properties of structural wave field. A radial force mobility is usually more accurately measured than the axial or tangential ones because the measurement accuracy of a translational acceleration is higher than that of a rotational one. The absorption in the nonexcitation end will reduce measurement error especially at nonresonant frequencies. Although only a radial force excitation was demonstrated in the experiment, this new method can be applied in principle to the cases of other force excitations such as axial force excitation or acoustical source excitation.
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1
R. S. Ming, J. Pan, and M. P. Norton, ‘‘The mobility functions and their application in calculating power flow in coupled cylindrical shells,’’ J. Acoust. Soc. Am. 105, 1702–1713 共1999兲. 2 P. A. Franken, ‘‘Input impedances of simple cylindrical structure,’’ J. Acoust. Soc. Am. 32, 473–477 共1960兲. 3 M. Heckl, ‘‘Vibrations of point-driven cylindrical shells,’’ J. Acoust. Soc. Am. 34, 1553–1557 共1962兲. 4 C. R. Fuller, ‘‘The input mobility of an infinite circular cylindrical elastic shell filled with fluid,’’ J. Sound Vib. 87, 409–427 共1983兲. 5 C. A. F. de Jong and J. W. Verheij, ‘‘Measurement of energy flow along pipes,’’ Second International Congress on Recent Developments in Air and Structure-borne Sound and Vibration, March 1992, Auburn University, pp. 577–585 共1992兲. 6 J. Pan, R. Ming, and M. P. Norton, ‘‘The measurement of structure-borne sound power flows on an elastic cylindrical shell,’’ Technical Report, Centre for Vibration and Noise Control, Department of Mechanical and Materials Engineering, The University of Western Australia, Australia 共1999兲 共submitted to J. Sound Vib.兲. 7 M. C. Junger and D. Feit, Sound, Structures and Their Interaction 共The MIT Press, Cambridge, MA, 1972兲. 8 A. W. Leissa, Vibration of Shells 共NASA SP-288, Washington DC, 1973兲. 9 C. R. Fuller, ‘‘The effects of wall discontinuities on the propagation of flexural waves in cylindrical shells,’’ J. Sound Vib. 75, 207–228 共1981兲. 10 C. R. Fuller, S. J. Elliott, and P. A. Nelson, Active Control of Vibration 共Academic, New York, 1996兲. 11 D. G. Rees, Foundations of Statistics 共Chapman and Hall, New York, 1987兲.
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