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FREE VIBRATION OF A CANTILEVER TUBE PARTIALLY FILLED WITH LIQUID K.-T. C J.-Z. Z Mechanical and Marine Engineering Department, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong (Received 3 November 1992, and in final form 7 October 1993) A simple and straightforward method is presented for calculating the natural frequency changes of a partially liquid filled cantilever tube of constant diameter due to changes in liquid length. The calculated results are presented along with experimental results; the two sets of results are in good agreement. The change in vibration behaviour of the tube due to liquid addition is also described. 1. INTRODUCTION

The governing equation of motion of flexural vibration of non-uniform beam has been studied since the last century. The mathematical problem consists of a fourth order differential equation with two independent variables, the axial length, x, and the time, t. The coefficients are functions of x. They are the stiffness EI(x) and the mass per unit length m(x). According to Gladwell and Bishop [1], if the coefficients are well behaved functions, a general solution exists, consisting of four independent functions of x. The solution defines the vibration mode shapes of the non-uniform beam. The solution in the time domain is harmonic. The periods give the frequencies of vibration of various modes. The forms of the four independent spatial functions are generally known. For certain geometric shapes of non-uniform beams, the functions are readily defined as by Cranch and Adler [2]. In some cases, Bessel functions are used to describe the mode shapes. A non-uniform beam is often considered as a beam with variable cross-section. Both the stiffness and mass vary along the axis. However, a beam may be uniform in stiffness but variable in mass. A few examples are cable trays with cables, bridges with trains and tubes partially filled with liquid. All these examples are similar in structure and have abrupt changes of mass at certain sections along their length. For engineering purposes, it is often enough to know the natural frequencies. Rayleigh’s stationary principle and the Rayleigh–Ritz method have been widely used for calculating natural frequencies; the mode shapes are not required to be known exactly in the calculations. In the present paper, a method is introduced for the calculation of the natural frequency changes with liquid length of a partially filled cantilever tube with constant diameter. In the calculation, the cantilever tube is separated into two sections, one with liquid and the other empty. Two equations are derived and used to describe the vibrations of the tube’s uniform parts, with equations designating the continuity and equilibrium conditions at the dividing position. The boundary conditions of the undivided cantilever are retained. Thus, the system of equations should have a closed form solution. The method is basically the same as that of Bishop and Johnson [3], especially regarding the continuity and equilibrium conditions at the liquid level position. However, the concept of receptance is not required in the present method, allowing an inexperienced student to use the method in a more straightforward manner. 185 0022–460X/95/170185 + 06 $08.00/0

7 1995 Academic Press Limited

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This short paper describes results obtained by using the method for calculation of natural frequencies of a cantilever tube partially filled with liquid mercury. Experimental results are also presented to verify the computed results. The change in vibration behaviour of the tube due to liquid addition is described. 2. NATURAL FREQUENCIES VERSUS LIQUID LENGTH

2.1. In Figure 1 is shown the cantilever tube, which has a uniform cross-section and a constant stiffness EI. The tube is partially filled with liquid. Its mass distribution is divided into two constant parts. One is mt , the tube mass per unit length and the other is ml + mw , including the liquid mass. With the co-ordinate system and notations shown in the figure, the equation of motion can be written EI 1 4v/1x 4 + m 1 2v/1t 2 = 0,

(1)

with m given by m=

6

mt + mw , mt ,

7

−lw Q x Q 0 , 0 Q x Q L − lw

(2)

where v is the tube displacement in the transverse direction. Let the solutions of equation (1) be v1 (x, t) and v2 (x, t), respectively in the ranges −lw Q x Q 0 and 0 Q x Q L − lw . The boundary conditions become v1 (−lw ) = v'1 (−lw ) = v02 (L − lw ) = v1 2 (L − lw ) = 0.

Figure 1. The experimental set-up.

(3)

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At the liquid level, the continuity and equilibrium equations are v1 (0) = v2 (0),

v'1 (0) = v'2 (0),

v01 (0) = v02 (0),

v1 1 (0) = v1 2 (0),

(4)

v1 (x, t) = (A1 sin k1 x + B1 cos k1 x + C1 sinh k1 x + D1 cosh k1 x) eivt.

(5)

for all time t. The general solutions are

ivt

v2 (x, t) = (A2 sin k2 x + B2 cos k2 x + C2 sinh k2 x + D2 cosh k2 x) e ,

(6)

where k14 = (mt + mw )v 2/EI,

k24 = mt v 2/EI.

(7)

By defining g = {mt /(mt + mw )}1/4, g1 = 12 (1 + g 2) and g2 = 12 (1 − g 2), and after some algebraic substitutions, the characteristic equation is obtained in determinant form:

G−gg1 sin u1 − gg2 sinh u1 g1 cos u1 + g2 cosh u1 G gg1 cos u1 + gg2 cosh u1 g1 sin u1 − g2 sinh u1 G −sin u2 −cos u2 G −cos u2 sin u2 G −gg2 sin u1 − gg1 sinh u1 gg2 cos u1 + gg1 cosh u1 × sinh u2 cosh u2 Here u1 = k1 lw = {(ml + mw )/EI}1/4zvlw ,

g2 cos u1 + g1 cosh u1H g2 sin u1 − g1 sinh u1 G G = 0. cosh u2 G sinh u2 H u2 = k2 (L − lw ) = (mt /EI)1/4zv(L − lw ).

(8)

(9)

If the tube is empty, lw = 0, and equation (8) can be simplified to give 1 + cosh u2 cos u2 = 0,

(10)

where, u2 becomes (mt /EI)1/4zvL, which can be denoted by u0 . If the tube is full of liquid, lw = L. Equation (8) can be expanded into 1 + cosh u1 cos u1 = 0.

(11)

where u1 = {(mt + mw )/EI}1/4 zvL. Equation (11) represents the added mass effect, while with equation (10) one takes the tube to be a uniform beam. By normalization, equations (9) become, u1 /u0 = (lw /L)(v/v0)1/2 (1/g),

u2 /u0 = (1 − lw /L)(v/v0 )1/2.

(12)

By substituting equations (12) into equation (8), and taking g as a constant parameter, the relationship between the frequency ratio v/v0 and length ratio lw /L can be obtained numerically by using a desk-top computer. The computed results will be presented later. 2.2. 2.2.1. Experimental set-up and measurement method In Figure 1, the experimental tube is clamped at the bottom, and free at the top. It is made of copper with density 8·93 × 103 kg/m3. Its length is 830 mm from top to bottom. The tube has a wall thickness of 1·40 mm and its outer diameter is 12·75 mm. Flexible

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tubing was used to connect the copper tube to a side glass tube for level indication. Connection of the tubing did not appear to affect the modal characteristics of the cantilever tube, as proven by some preliminary tests. Mercury was used in the experiments because of its high mass loading. A light-weight accelerometer was used to measure vibration. A fast fourier transform (FFT) analyzer was employed to obtain measurement of the frequency response function (FRF). A 80386 computer played the roles of (1) controlling the analyzer to measure different functions, (2) data acquisition, (3) curve fitting and (4) modal analysis. The computer was also used to calculate natural frequencies by using Lotus 123, based on equation (8). A random series of impacts by an instrumented hammer was used to excite vibration. With the analyzer set to 50 samples, on averaged mode, Hanning smoothing and 75% overlap processing, a single FRF could be measured. All measurements were verified by observing that the coherence was nearly one. 2.2.2. Results and discussion In Figure 2 is shown the shift of modal frequencies of the tube for different lengths of the mercury column. The experimental data are plotted as shown in the legend and can be compared with the curves computed in accordance with equation (8). This comparison shows that the experimental data agree quite well with the computed curves. Several qualitative features can be seen from the graphs in the figure. A noticeable feature is the existence of ‘‘plateau regions’’. These regions indicate that modal frequencies of the tube become insensitive to change of the liquid level. It can be observed that the number of ‘‘plateau regions’’ is equal to the mode number. It can also

Figure 2. The frequency shift of a copper cantilever tube partially filled with mercury. w—w, Mode 1; r––r, mode 2; q––q, mode 3; W––W, mode 4; – · · –, mode 11. Curves for theory; symbols are experimental results.

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be seen that the width of a ‘‘plateau region’’ depends on the mode number. For higher modes, the distance between regions appears to be non-periodic, in a manner like that of the nodal distribution of cantilever vibration. It is reasonable to relate the existence of the ‘‘plateau regions’’ to the presence of vibration nodes. In fact, near the nodal positions, vibrational energy due to the mercury should be much lower than that near the antinodes, and thus there is less mass effect there. Measurements of mode shapes (results not given in this paper) have shown that the nodes of the empty tube are close to the corresponding ‘‘plateau regions’’ and shifted appreciably lower from the top by the addition of mercury. As is evident in Figure 2, frequency reduction at a mercury level is different for different modes. The curves of higher modes in Figure 2 appear to be entangled together, but their curly feature remains distinct. The feature is illustrated in Figure 3, showing the experimental curves for the 11th and 12th modes. The data seem to collapse within a well defined region around the entangled curves. It is not difficult to anticipate that the curly curve will become effectively asymptotic as the mode number is increased further. (However, this could not possibly happen in reality because, for higher modes, the elementary equation of motion is not applicable.) As shown by the gradient lines marked in Figure 3, the ‘‘asymptotic’’ curves are slightly concave upward. A curve of this kind will be likely to be of use to monitor liquid levels by measuring frequency change. This is particularly useful when conventional liquid level indicators are not feasible in certain applications. One can easily see, in Figure 2, that all curves converge asymptotically to a point, implying that all modes have a similar percentage of frequency reduction for fully filled tubes. This is consistent with results given by Collinson and Warneford [4].

Figure 3. As Figure 2 but w–w mode 11 and W–W mode 12.

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It is interesting from the present results to note that changing the liquid length is, in effect, a process of detuning transverse tube vibration. Because axial vibration can carry liquid only by wall friction, it should not be disturbed very much by the liquid. In fact, some interesting results have already been obtained by the authors in the study of axial vibration of the tube. The results will be published elsewhere later. 3. CONCLUSIONS

Experiments have been carried out to verify the method of calculation. The calculated results are in agreement with the experimental ones. The results show that the change in natural frequencies is relatively small when the liquid level is in the vicinity of the nodal positions of the corresponding vibration modes. It was found that there is asymptotic behaviour for high frequency modes. For a partially filled tube, the percentage of frequency reduction was found to be generally different for different modes. The percentage was the same, however, if the tube was fully filled. REFERENCES 1. G. M. L. G and R. E. D. B 1959 Journal of Mechanical Engineering Science 1, 78–91. The receptances of uniform and non-uniform rotating shafts. 2. E. T. C and A. A. A 1956 Journal of Applied Mechanics 23, 103–108. Bending vibrations of variable section beams. 3. R. E. D. B and D. C. J 1960 The Mechanics of Vibration. Cambridge: Cambridge University Press. 4. A. E. C and I. P. W 1978 B.N.E.S. Vibration in Nuclear Plant Session 3, 307–325. Vibration tests of single heat exchanger tubes in air and static water.