Zhou
Mingde, Cai Guozheng and
Zhou Yiren
Nanjing Hydraulic Research Institute, Nanjing, 210029 China
Abstract: In this paper, from dimensional analysis, the model similarity law for gate vibration is derived. By the aid of a practical example, a systematic experimental technique is introduced, which includes the measurement of frequencies of air, still water and flowing water, virtual masses of submerging gates and flow disturbing forces, etc.
Keywords: experimental method, vibration, cylinder gate
Practice shows that the cylinder gate fixed in the tower-type intake of flood- relief tunnel always produces a global vibration in operation, which mainly appears three types of longitudinal extensive, transverse bending and tangential torsion of gate lifting rod. It is well known that it is difficult to achieve hydro-elastic similarity in a model owing to the limitation of model material. In view of the fact, a comparatively practicable approach is to separate the experiment into two parts, one is a hydraulic model, another is a structure dynamic model, and then to synthesize all data from two tests. This method is so effective in dealing with the similarity problems in resonance model.
For a gate vibrating in water flow, the major acting forces governing the motion include the gravitational force of flow, elastic force of material, and fluid and solid inertial forces. Therefore, it is necessary to build a so-called hydraulic-structural dynamic model, which scale must be designed to satisfy the similarity of gravity and elastic forces in order to provide the dynamic similarity between model and prototype. Factors that have influence on gate vibration may now be interrelated by following equations.
(1)
in which D is size of gate [L], g is unit weight of gate [F/L3], g is gravitational acceleration [L/T2], I is inertial force [F], E is elastic modulus of material [F/L2], S is rupture strength [F/L2], k is viscous damping factor [FT/L], and m is Coulomb friction factor [0].
The F, L and T above represent the units of force, length and time respectively. So by making use of the “p ” theorem, six variables may be obtained with the three basic units.
(2)
Obviously, there are a variety of p combinations. Solving Eq.2 with I, E and k as basic variables yields
,
,
(3)
It is to say, the values of dimensionless numbers I/D3g, E/Dg, k/D5/2gg-1/2, E/S and m must be identical in model and prototype. Scale relations between model and prototype physical quantities relevant to vibrations can be obtained on these grounds, as shown in Table 1, in which l represents the scale of length, and the subscript r refers to the model- to- prototype ratios.
Table 1 Scale relations between model and prototype
|
Physical quantity |
Symbol |
Scale conversion |
|
Gate size |
D |
Dr=l |
|
Unite weight |
g |
gr |
|
Inertial force |
I |
Ir=l3gr |
|
Time |
T |
Tr=l1/2 |
|
Acceleration |
A |
Ar=1 |
|
Modulus of elasticity |
E |
Er=lgr |
|
Rupture strength |
S |
Sr=lgr |
|
Viscous damping factor |
k |
kr=l5/2gr |
|
Coulomb friction factor |
m |
mr=1 |
|
Strain |
e |
er=1 |
|
stress |
s |
sr=1 |
|
Natural period of vibration |
t |
tr=l1/2 |
|
Natural frequency of vibration |
f |
fr=1/l1/2 |
As can be seen from Table 1, in carrying experiments in a Froude scale model, the choice of model material should satisfy following condition
(4)
Besides, if similarity of hydrodynamic pressure is taken into account, the following additional condition is also necessary
(5)
in which gw is unit weight of fluid. Because water is usually used as model fluid, the condition of model material can be written as
(6)
It is clearly that the condition greatly limits the section of model material, even in some case, there is no kind of material to be selected to satisfy strict dynamic similarity. In view of the facts, the method, which separates the experiment into two parts of a hydraulic model and a structure dynamic model, is so effective in dealing with the similarity problems in resonance model.
Measurement of natural frequency
The cylinder gate arrangement and experimental arrangement were shown in Fig.1 and Fig.2 respectively.
Plan view of cylinder gate Section A-A
Fig. 2 Experimental arrangement of cylinder gate
From the structural dynamic model, the natural frequency of vibration can be obtained. The data measured from the experiment can be converted into prototype values according to the similar criterion of gravity force- elastic force, i.e.
(7)
The common methods of measuring natural frequencies include resonance method and impact method. However, for the global vibrations of cylinder gates including lifting rods, it is convenient to use the impact method. The connecting pattern of resistance strain gages should be appropriately modified in experiment, shown in Fig.3.
(a)
Transverse bending (b) Longitudinal
extensive (c) Tangential torsion
Fig. 3 Connection pattern of resistance strain gage for various modes
Measurement of damping test
For example, a practical record of transverse vibration in experiment is shown in Fig.4. The viscous damping factor is defined as in which h is logarithmic decrement of amplitude, M is gate mass, and t is natural vibration period. Substituting h, M and t measured in tests into Eq.8, the value of k will be calculated. Then, according to the similarity relationship shown in Table 1, this value can be extrapolated to the prototype by using the following equation
(8)
(9)
Fig. 4 Measured waveform of transverse vibration
It is just as well to point out that the impact method is also suitable for measuring in-water natural frequency and damping factor.
Determination of gate virtual mass
It is well known that the natural frequency of a gate vibrating in water is somewhat lower than that in vacuum or in air due to the influence of fluid inertia. According to the theory of hydro-elastic, it may be imagined that part of water mass is “frozen” on the gate, namely virtual mass. Now take the longitudinal vibration test of a cylinder gate as an example, whose arrangement is shown in Fig.5. The gate is made of perspex tube with the outer diameter of 14cm, 0.5cm thick and 5.0cm high, in which a steel ring with the thickness of 0.1cm keeps close to the internal surface. The total weight of the gate system (including lifting rods) is 840kg. According to the predetermined test runs, we utilized a small hammer to knock the end of support bar softly in vertical direction to cause vibrations, and then measured the in-water and in-air natural frequencies of the gate system respectively. The record of the waveform of natural frequencies is shown in Fig.6 and Fig.7.
(a) Gate opening 1/4 (a) Head=60 cm opening 1/4
(b) Gate opening 3/4 (b) Head=12 cm opening 1/4
Fig.
6 Record of in-air
natural
Fig. 7 Record of in-water forced
vibration
waveforms
vibration waveforms
After taking above procedural steps, the virtual mass can be calculated by following equation
(10)
or alternatively
(11)
in which Mv is virtual mass, Ma is mass of the gate in air, C= Mv – Ma is added mass, fa is natural frequency in air, fw is natural frequency in water. Substituting the fa and fw into to Eq.10 or Eq.11, the virtual mass or added mass of the gate can be calculated.
Measurement of frequency of disturbing force
Before tests, calibrated resistance wire strain gages were stuck directly on the lifting rods. In experiment, the signals of disturbing forces in lifting rods are transferred to an oscilloscope. The waveforms recorded by the oscilloscope are given in Fig.8. Because the frequency of disturbing force is associated with model scale, two models with a big scale and a small scale are used for comparison study. The big model is as twice times as the small one in geometric size. The experiment results of disturbing forces obtained from both models are list in Table 2,the data in parentheses referring to the small model values.
From Table 2, it can be seen that there is no full consistency in the frequency of disturbing force between two models. It seems that the model simulation only meets the law of gravity similarity qualitatively.
Table 2 Master frequencies of disturbing force measured in big and small models
|
Gate opening |
H=132cm(66cm) |
H=82cm(41cm) |
H=30cm(15cm) |
||||||
|
No.1 |
No.2 |
No.3 |
No.1 |
No.2 |
No.3 |
No.1 |
No.2 |
No.3 |
|
|
1/8 |
10.1 -- |
10.0 -- |
10.0 -- |
11.5 -- |
10.8 -- |
11.7 -- |
10.9 -- |
10.5 -- |
9.5 -- |
|
1/4 |
9.0 (16.0) |
11.0 (16.0) |
10.0 (16.0) |
11.1 -- |
10.6 -- |
10.6 -- |
10.7 (18.0) |
10.8 (18.5) |
11.2 (19.0) |
|
1/2 |
12.0 (13.0) |
12.3 (13.5) |
12.0 (13.5) |
11.5 -- |
11.0 -- |
10.8 -- |
10.5 (15.0) |
11.5 (16.0) |
10.0 (14.5) |
|
3/4 |
11.6 (11.2) |
11.2 (11.0) |
10.0 (11.0) |
-- -- |
-- -- |
-- -- |
11.5 (14.5) |
11.5 (14.5) |
9.5 (14.0) |
By describing an experimental process of a practical work, this paper presents an experimental method and measuring technique for vibration problem of a cylinder gate. However, it is clear that this method is only effective in solving resonance problem.
References
[1] R. D. Blevins, Flow- induced vibration, New York, 1977.
[2] Zhou Mingde, Theory and practice of global vibration of cylinder gate, Journal of Vibration, Measurement & Diagnosis, Vol. 10, 1990 (in Chinese).
[3] Zhou Mingde, An experimental study on vibration of a cylinder gate installed in a tower-type intake conduit, Nanjing Hydraulic Research Institute, No. 9,1963 (in Chinese).
[4] Zhou Mingde, An experimental technique on vibration model of the cylinder gate, Journal of Hydraulic Engineering, No.4, 1984 (in Chinese).