95s

EXPERIMENTAL AND NUMERICAL STUDIES ON STEEL PENSTOCK VIBRATIONS

CAUSED BY PRESSURE PULSATIONS

Kosuke YAMAMOTO1 and Hideharu NAKAMURA2

1Member of JSCE, Dr. Eng., Central Research Institute of Electric Power Industry (1646 Abiko, Abiko-Shi, Chiba, 270-1194, Japan)

2Fellow of JSCE, Dr. Eng., Professor, Graduate School of Engineering, Hiroshima University (1-4-1 Kagamiyama, Higashi-Hiroshima-Shi, Hiroshima, 739-8527, Japan)

In our previous paper, we described vibrations with circumferential mode number n = 2 when a frequency of a pressure pulsation in a turbine coincides with the natural frequency of a steel penstock. We proposed the use of natural frequency calculation formulae based on numerical analyses that target n = 2, 3 vibrations. Because a limited number of papers are available describing experimental or analytical vibration studies, we performed 1/4 scale model experiments on pressure pulsations with the parameters of plate thickness and the presence of stiffeners. Based on a comparative study of experiments and numerical analyses, we proposed modification of the vibration prevention specifications stipulated in the current technical standard.

Key Words : steel penstock, pressure pulsation, vibration test, numerical analysis, design standard

1. INTRODUCTION Hydraulic power stations sometimes experience momentary pressure pulsation in the hydraulic turbine or draft tube. When frequency of this pulsation approaches the natural frequency of the steel penstock, the pipe shell causes a resonance which can result in significant vibrations. Measures should therefore be taken at the design stage to prevent the primary natural frequency of the steel penstocks from approaching the rotational speed of hydraulic turbines. However, instances of vibrations with circumferential mode number n = 2 (Fig.1) are sometimes observed in newly installed or recently renewed steel penstocks. Since natural frequency calculation formulae specified in the current Technical Standards for Gates and Penstocks1) are not considered well suited for handling low-order cross-sectional deformation vibration modes, we proposed new natural frequency calculation formulae2) derived from numerical analysis based on three-dimensional This paper is translated into English from the Japanese paper, which originally appeared on J. Struct. Mech. Earthquake Eng., JSCE, No.759/I-67, pp121-130, 2004.4.

finite element analysis. The Steel Penstocks Manual3) published by ASCE recommends a three-dimensional finite element analysis including actual penstock profiles and supporting conditions as the most appropriate method for determining the natural frequency of the steel penstock. While this agrees basically with our method of investigation, the problem remains that results of finite element analyses are rarely proven in verification tests. Studies in this field made in Japan in the 1950s and early 1960s include pioneering studies by

Circumferential mode: 2=n 3=n

Axial mode: 1=k 2=k 3=k

Fig.1 Vibration modes of steel penstocks

Structural Eng./Earthquake Eng., JSCE, Vol.22, No. 2, 95s-106s, 2005 October(Translated from a paper originally appeared on Journal of Structural Mechanics

and Earthquake Engineering, JSCE, No. 759/I-67, pp.121-130, 2004 April.)

96s

Fig.2 Shape parameters of existing steel penstocks and models used in experiments

Kito4)-10). A series of these studies focused on derivation of theoretical formulae relating to the natural frequencies of steel penstocks, factoring in the added mass of internal fluid and internal pressure. A comparison was made of theoretical values and actual measurements taken at several suspect resonance points on the steel penstocks. Kito’s formulae for natural frequency calculation were adopted in the Technical Standards for Gates and Penstocks revised in 1973, and have been widely used in routine calculations. However, as we pointed out in the previous paper2), reduction in the accuracy of circumferential mode number n = 2, which is considered the most important element in steel penstock analysis, is caused inevitably by the simplification of equations in the process of obtaining theoretical solutions. On the other hand, Shiraki et al. conducted theoretical and experimental investigations of vibrations in steel penstock reinforced by stiffeners or ring girders11), 12). For these investigations, they produced two sets of 3-span models with a 50 cm diameter, 3.3 mm plate thickness, and 3 m support intervals. They introduced water into the steel pipe, applied forced mechanical vibrations externally to the bottom face of the middle of the center span and exerted internal pressure in order to obtain the natural frequency mode and vibration frequencies. However, this study posed the problem that the pressure change caused by internal fluid was removed from the vibration source and that this example was slightly outside the parameters (Fig.2) where vibrations actually posed a problem in penstock profiles. We conducted vibration experiments, based on the prior studies2) of actual conditions of existing steel penstocks (total 99 power stations), using a penstock model targeting the region of shape

Fig.3 Conceptual illustration of vibration experiments using model parameters where vibrations caused actual problems. Vibration characteristics were determined, and experimental results were compared with results of finite element analysis. In this experiment, in order to produce vibration conditions as close as possible to the actual conditions, changes in internal fluid pressure were used as an excitation condition, and clarification was made of vibration characteristics relevant to changes in penstock plate thickness and the presence of intermediate stiffeners. Considerations were given to the following points: (i) To confirm the generation of vibrations with

circumferential mode number n = 2 and n = 3 in the penstock model and to verify that their natural frequencies can be evaluated by numerical analysis with fairly good accuracy.

(ii) To confirm the effects of circumferential stiffeners on the restriction of cross-sectional deformation vibrations.

This paper is divided into five sections. Section 2 outlines models and excitation apparatus used in the experiments and describes measured items. Section 3 discusses the results of the knocking test, sweeping test, and mode measurement test. Section 4 compares the results of the vibration experiments and the finite element analyses. Section 5 describes our proposal for revision of vibration prevention regulations which has been derived from reference 2) and is based on results of the investigation presented in this paper. We recommend this revision be included in the Technical Standards for Gates and Penstocks.

2. EXPERIMENTS OF VIBRATIONS OF STEEL PENSTOCK USING MODELS (1) Method of experiments We produced an approximately 1/4 scale model of a penstock with both ends supported (one fixed, the other supported by a roller) as shown in Fig.3. A pulsation generation cylinder exerted periodic volumetric (pressure) changes on condition that

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without stiffener with stiffeners vibration occurence present study Shiraki et al.

・

Relief valve

Penstock model(pressurized)

Vibrationproof rubber

Pulsation generation cylinderCrank pulley

Steel saddle (fixed) Steel saddle(roller)

Flange

Valve

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2400

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500

4000250 250

Air vent valve

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Table 1 Material properties of steel penstock models

Photo 1 Full view of penstock model constant internal pressure was previously applied to the penstock model. Two methods to obtain constant internal pressure exertion, an elevated water tank and a pressurizing pump, were checked to determine which was more effective. The elevated water tank was selected for experimental measurements because of its ability to maintain constant internal pressure and to permit volumetric changes during cross-sectional vibration more easily. (2) Outline of penstock model Two carbon steel penstock models were produced; one with a plate thickness 2.22 mm (t2.2 model) and one with 1.57 mm (t1.6 model). Both were 4 m in length and 50 cm in inside diameter. In order to check reinforcement effects by circumferential stiffeners, an intermediate stiffener (height 25 mm, plate thickness 2.13 mm) was attached to the t2.2 model at a point 1 m inward from both ends of the model flange (referred to as the t2.2-s model). Fig.4 shows the shapes and dimensions of the penstock models, Table 1 shows the particulars of the steel materials used (results of the material test), and Photo 1 shows a full view of each model. (3) Outline of experimental apparatus As shown in Fig.3, a steel saddle support for

Fig.4 Shapes and dimensions of penstock model

Photo 2 Pulsation generation apparatus fixing purpose was provided at both ends of the penstock model. One end was secured to the reaction floor using a PC steel bar while fastening bolts for the other end were left loose to allow axial displacement anticipating support by the rollers. The pulsation generation cylinder (Photo 2) was provided to the support at the fixed end, and a piping was provided to the support at the roller supported end to enable connection of the elevated water tank. The penstock model was slightly tilted (one degree inclination) to prevent air bubble accumulation at the top of the piping. An air vent valve was provided at the upper portion of the piston (cylinder) of the pulsation generation apparatus and at the upper portion of both end supports. The pulsation generation apparatus is controllable in the range of 0 - 50 Hz, and vibration frequency can be raised and lowered at a constant rate. The excitation stroke of the piston was determined to be double amplitude of 0.342 mm after confirming the stable control region of the pulsation generation apparatus by preliminary tests. Since the piston diameter was 300 mm, volumetric change due to excitation of the piston was 24.2 cm3. The elevated water tank (inside diameter 500 mm, height 1,000 mm) was installed 2.4 m above the floor considering that past vibration instances were observed in the low pressure region.

Saddle support

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Test Material Young's Poisson's Yield Tensilemodel type modulus ratio stress stress

(GPa) (MPa) (MPa)JIS

G3131 201 0.279 245 392SPHC

JISt1.6 G3101 205 0.267 347 431

SS400

t2.2t2.2-s

98s

Fig.5 Section of measurement and sensor arrangement (4) Method of experiments and measuring items The following experimental measurements (one type of static test, three types of dynamic tests) were taken for each model. Fig.5 shows the arrangement of measurement sensors. a) Static test (To determine initial imperfections of the penstock model) (i) Measurement of circumferential strain

distribution at pressurizing b) Dynamic test (To determine vibration characteristics of the penstock model) (ii) Knocking test (To knock penstock model

flange using a plastics hammer) (iii) Sweep test (Excitation frequency was

increased monotonously in the range of 0 - 44 Hz)

(iv) Mode measurement test (Vibration mode was measured at sinusoidal wave excitation while fixed to the vibration frequency that resulted in large responses)

3. RESULTS OF EXPERIMENTS (1) Measurement of circumferential strain distribution at pressurizing Initial imperfections of the penstock model can

be determined approximately through investigation of circumferential strain distribution at pressurizing. A pressurizing test was then conducted up to 250 kPa using a pressurizing pump. (Vibration experiments use internal pressure 26.5 kPa generated by the elevated water tank.) Measurement was taken at span center of the penstock model (Section C’ in Fig.5). Fig.6 shows measurement results. Since the plate thickness of the t1.6 model was approximately 2/3 that of the t2.2 model (with approximately 1.5 times the circumferential strain due to internal pressure), the scale of the strain distribution diagram was set to 1.5 times that of the t2.2 model. As noticed from Fig.6 (a), with the t2.2 model (without stiffener), strain distribution presented an ellipse with its line of apsides in the direction of the welded joint, and it was supposed that a flattening due to the initial shape of the penstock, its own weight and water weight was caused in the direction of 90 - 270 degrees, which was orthogonal to the line of apsides. With an internal pressure of 26.5 kPa used in vibration experiments, the outside diameter in the penstock vertical direction (0 - 180 degree direction) was compared to that of the horizontal direction (90 - 270 degree direction). The results showed that the horizontal case was 3 mm

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Welded jointWelded joint

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Displacement gauge- STH:circumferential- STA:axial

+

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(a) t2.2 model (b) t2.2-s model (c) t1.6 model Fig.6 Distribution of circumferential strain at pressurizing (Section C’)

(a) t2.2 model (b) t2.2-s model (c) t1.6 model

Fig.7 Example of results of frequency analysis in knocking test (DISP3) longer and that a remarkable flattening was left in the lower pressure region. A similar flattening in the 90 - 270 degree direction was also noticed with the t1.6 model (Fig.6 (c)). The pipe shell of the t2.2 model was the same with or without a stiffener. With the t2.2-s model, the installation of a stiffener resulted in a slight reduction in strain and strain distribution was somewhat rounded (Fig.6 (b)). (2) Knocking test The knocking test was carried out by striking a plastic hammer against the upper portion of the flange joint on the side containing the pulsation generation apparatus. The target strength of knocking was a 200 - 300 gal response acceleration of the penstock. It was anticipated that the knocking test would produce only a minor oval vibration accompanying cross-sectional change (volumetric change) and easy extraction of the beam oscillation mode since it is free from cross-sectional change. An example of test results is shown in Fig.7 which shows the frequency analysis of one channel of a displacement gauge (DISP3). (See Fig.5 for symbols for measurement points.) Internal pressure was 26.5 kPa for all cases. Responses of the pressure gauge and strain gauge were generally small. Pulses due to knocking and damping of free vibrations were recorded as responses of the accelerometer and displacement

gauge. Remarkable dominant vibration frequencies were observed in the results of frequency analysis of both the accelerometer and displacement gauges, and it was judged that these represent vibration frequencies in the beam mode. The beam mode frequency was assumed to be 25.9 Hz for the t2.2 model, 26.4 Hz for the t2.2-s model and 23.8 Hz for the t1.6 model. (3) Sweep test and mode measurement test The pulsation generation apparatus was used in the sweep test while excitation frequency was increased from 0 to 44 Hz. Control of the pulsation generation apparatus involved volumetric changes due to movement of the piston cylinder as mentioned in 2. (3) and therefore, control of the internal pressure was not constant. After response of the penstock was noticed and cross-sectional deformation vibration was amplified, the internal pressure inevitably increased. We therefore considered that efforts to maintain constant pressure control are meaningless. Meanwhile, initial internal pressure was 26.5 kPa, and resulting circumferential stress was 2.98 MPa for the t2.2 model and 4.22 MPa for the t1.6 model. The major measurement results of response waveforms of one channel of the accelerometer (ACC6) with the highest sensitivity and frequency analysis are shown in Fig.8 and Fig.9, respectively.

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(a) t2.2 model (b) t2.2-s model (c) t1.6 model

Fig.8 Example of response acceleration waveform in sweep test (ACC6) (a) t2.2 model (b) t2.2-s model (c) t1.6 model

Fig.9 Results of frequency analysis of response acceleration waveform in sweep test (ACC6) The horizontal axis in Fig.8, represents time. Since vibration frequency was changed at a constant rate by the pulsation generating apparatus, it may be considered that the horizontal axis increased monotonously (linear change) from 0 to 44 Hz. Although the response waveform of the pressure gauge exhibited shapes similar to those shown in Fig.8, the amount of response decreased extremely at the points of beam oscillation mode that were free from cross-sectional changes. In the sweep test, behaviors of response acceleration, displacement, pressure, and strain were analyzed in detail, the natural frequency of each model was obtained, and excitation was carried out on the steady-state sinusoidal wave attached to each natural frequency in order to identify the shape of vibration mode (vibration mode order). The order of the identified vibration mode is shown in the response spectrum shown in Fig.9, where n denotes the order of the circumferential vibration mode and k denotes that of the axial vibration mode. The case where n = 1 and k = 1 then denotes the beam oscillation mode. Further, n = 0 indicates that uniform expansion and the contraction mode in the radial direction and results from attributes of the pulsation generation apparatus. Distribution of the circumferential strain in each model (Sec. C’) is shown in Fig.10 through Fig.12 as a typical vibration mode diagram. One example of the axial direction vibration mode, k = 1,

is shown in Fig.10. These diagrams show the distribution at the point when the maximum circumferential strain was recorded and after half of the cycle had elapsed. Such conditions repeated alternately during excitation of the steady-state sinusoidal wave. Distribution of circumferential strain of the t1.6 model involved a deflection as noticed in Fig.12. This was considered attributable to that fact the shape of vibration mode was affected by the non-uniform curvature of the model surface due to difficulties involved in processing such a thin plate during the penstock pipe manufacture. Results obtained from the sweep test and mode measurement test are shown in Table 2 and summarized as follows: (i) Distinct vibration modes n = 2 and k = 1, and

natural frequency were obtained in all tests. (ii) Vibration mode n = 3 and k = 1 was confirmed

only when no stiffener was provided. (iii) Vibration modes n = 1 and k = 1 (beam

oscillation mode) had the natural frequencies slightly less than 25 Hz for all the models. Although approximately 1 Hz difference was recognized compared to results of the knocking test, it was judged that results of the sweep test could accurately represent vibration behavior of the whole penstock model since the knocking test was based on local excitation and input was limited accordingly. Therefore,

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(a) n = 2 (21.9Hz) (b) k = 1 (21.9Hz) (c) n = 3 (34.8Hz)

Fig.10 Distribution of circumferential strain during excitation of the steady-state sinusoidal wave (t2.2 model) Table 2 Summary of the sweep test and mode measurement test n = 2 (25.9Hz) n = 2 (18.4Hz) Fig.11 Distribution of circumferential strain Fig.12 Distribution of circumferential strain during excitation of the steady-state during excitation of the steady-state sinusoidal wave (t2.2-s model) sinusoidal wave (t1.6 model) the value obtained from the sweep test was

referred for all subsequent analyses. (iv) By adding two circumferential stiffeners to the

t2.2 model, natural frequency of n = 2 and k = 1 mode increased from 21.9 Hz to 25.9 Hz while vibration mode n = 3 and k = 1 disappeared.

A propagation rate of approximately 800 m/s was measured for the pressure wave in the penstock in the t2.2 model. From this, it was judged that with the current test system (total pipe length 4 + 1 m), the resonance of pressure pulsation was above 40 Hz, and there was little chance for mutual interference between the pressure wave and the peak response of the penstock model. 4. COMPARISON WITH RESULTS OF FINITE ELEMENT ANALYSIS (1) Outline of analysis The finite element method (FEM) using thin shell elements with axial symmetry condition was adopted for the numerical analysis. Effects of the

internal fluid were handled by including added mass in the mass matrix of the shell. Details can be found in our previous report2) regarding theoretical equations for the cylindrical shell and fluids, formulation of the element matrix, shell-fluid contact conditions, motion equations for the shell (including fluid coupling factors), and eigen value analysis. Actual analysis procedures were as follows. In the first step, internal pressure was generated to check stresses generated in the penstock pipe shell and these values were used as the initial stress. In the second step, the natural frequency was analyzed to obtain the vibration frequency and vibration mode. No time history analysis was attempted. (2) Analysis model The analysis object was set as shown in Fig.13 based on the model used in the vibration experiments. Dimensions used are as follows: a) Penstock Length: L = 4,000 mm (body) + 500 mm×2 (support pipe) Inside diameter: Di = 500 mm

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Test Natural Mode No.model frequency n : circum.

(Hz) k : axial21.9 n =2, k =1

t2.2 24.8 n =1, k =131.1 n =034.8 n =3,k =124.7 n =1, k =1

t2.2-s 25.9 n =2, k =131.7 n =018.4 n =2, k =1

t1.6 24.9 n =1, k =130.1 (n =3,k =1)

102s

Flange joint Fig.13 Outline of structure to be analyzed (a) Vibration mode n = 2, k = 1 (27.2Hz) (b) Vibration mode n = 3, k = 1 (42.8Hz）

Fig.15 Vibration mode diagram of t2.2-s model obtained by finite element analysis Plate thickness: (Two types) Body t = 2.22 mm (t2.2 model) Body t = 1.57 mm (t1.6 model) Support pipe t = 9.00 mm

(t2.2 and t1.6 models) b) Circumferential stiffener (t2.2-s model only) Plate thickness: ts = 2.13 mm Height: b = 25.00 mm Place of installation: Each at 1,500 mm ( = Ls)

from end cover c) Flange joint Plate thickness: tf = 15.00 mm Height: H = 75.00 mm Properties of the materials and internal pressure were based on those used in the previous experiments. The boundary conditions were set based on the experimental conditions as follows: (i) Fixed end: Axial displacement u = 0 Out-of-plane displacement w = 0 Circumferential displacement v = 0 Rotationθ= xu ∂∂ free (ii) Roller end: w = 0, v = 0, u and θ are free Two separate cases were investigated for the exterior of the support pipe, one with a rigid cover (with internal pressure applied to the cover factored

in) and without (free fluid surface). The investigation showed that with internal pressure of this order, effects of the end cover were negligible. Therefore the discussion stated below applies the values obtained for the case where no end cover was provided. Fig.14 shows the finite element model. (3) Identification of elastic support conditions Since ideal fixing conditions could not be realized in the experiments, we used an elastic support configuration that allows perpendicular movement. In order to identify these support conditions, leaf spring elements (ring-shape) which allow expansion and contraction in the radial direction only were incorporated to the nodes (total six) at the fixed end and the roller end where boundary conditions were established. Rigidity of the leaf spring was decreased gradually to obtain a state where the beam mode vibration frequency agrees with experimental values. The spring constant 329 kN/cm showed a beam mode frequency for the t1.6 model that agrees with the experiment (24.9 Hz). The beam mode vibration frequencies for the t2.2 and t2.2-s models were obtained analytically using this spring constant. The resultant value of 25.3 Hz for both models was evidently close to the experimental values. This spring constant was therefore considered to have

Body

Stiffener

Support pipe

Fig.14 Meshed model with stiffener for finite element analysis (Although axial symmetry elements are used, parting lines are provided in the circumfe- rential direction at regular intervals.)

Support pipe

t

Di

500

LRoller endFixed end

Internal pressure p

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b

Ls 1500

ts

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500Support pipe

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Htf

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(Unit : mm)

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Table 3 Comparison of natural frequencies obtained by analyses and experiments high reliability and was applied throughout the analyses. It was further confirmed that the natural frequency of cross-sectional deformation vibration is not affected even if this spring constant is changed. (4) Results of analysis and discussions Table 3 shows comparison between vibration frequencies obtained by vibration eigen value analyses and results of experiments. Fig.15 shows an example (t2.2-s model) of the vibration mode obtained. Comparison of vibration frequencies obtained by analyses and experiments reveals that the difference between the two at the most important mode of n = 2 and k = 1 was 10% at the maximum (analysis value is higher). The cases with thicker plate or the stiffener had smaller influence on the difference, and the difference with t2.2-s model was within 5%. For n = 3 and k = 1, the analysis and experiment agreed fairly well with the t2.2 model while there was a significant difference with the t1.6 model. With the t1.6 model, the result of the analysis for vibration mode n = 3 and k = 1 was 24.7 Hz which was in close agreement with the vibration frequency for n = 1 and k = 1 (beam vibration mode) in the experiment. It was then considered likely that the beam vibration with the largest response in the experiment could not be distinguished from the vibration mode n = 3 and k = 1. (See area in the vicinity of 25 Hz on the x-axis in Fig.9 (c).) Cross-Sectional deformation for the vibration mode n = 3 alternated in the direction of 90 to 270 degrees in a manner similar to that in Fig.10 (c), which shows results for the t2.2 model. Therefore, with the arrangement of displacement gauges DISP1 - DISP5 (displacement measurement at penstock bottom face), the axial mode number could not be accurately determined. For this reason, we cannot ignore the possibility that 30.1 Hz applies specifically to n = 3 and k = 2. Data related to this possibility are shown in the brackets in Tables 2 and 3.

5. REFLECTION TO DESIGNING Vibrations of the steel penstock greatly depend on the characteristics of the hydraulic turbine and the condition of entire waterway, and the shape of the penstock is not the sole factor to be investigated. It is therefore difficult to forecast the generation of vibrations as early as the design stage. Technical Standards for Gates and Penstocks1) published by the Japan Hydraulic Gate and Penstock Association in 1960, along with the revisions of 1968, 1973, 1981 and 1993, is officially recognized as the accepted standard in Japan’s Electric Utility Law. This standard is regarded as the basic technical guideline for working-level designers and manufacturing staff in the electric industry. In the current standard the issue of vibration is dealt with in the section related to steel penstocks. Article 21, Considerations for vibrations, deals with prevention measures to be taken at the design stage, and Article 98, Reduction in vibration, covers measures to be taken when vibration occurs. In order for results of the investigation shown in this paper to be reflected in actual design works, a proposal for a review of Article 21 should be considered. It should be emphasized that of all the resonance phenomena related to steel penstocks, the greatest attention should be given to the circumferential mode number n = 2 and axial mode number k = 1. This proposal is derived from a survey of past vibration troubles, and from experimental and analytical investigations. Awareness of the significance of this vibration mode should be emphasized, appropriate equations for vibration calculation should be presented, and vibration prevention characteristics of circum- ferential stiffeners should be clarified. No vibrations of circumferential mode number greater than n = 3 or of axial mode number greater than k = 2 have been reported for actual penstocks to date. However, analyses and model experiments have confirmed generation of vibrations of this order. Since there is a significant difference of vibration prevention effects by circumferential stiffeners between n = 2 and n = 3, it would be appropriate to handle these cases separately. Reference 2) shows basic concepts based on a survey of actual conditions and of parameter analyses for cases involving fixed spacing and equivalent plate thickness for circumferential stiffeners. No remarkable effects of vibration prevention by circumferential stiffeners are available for circumferential mode number n = 2. It is therefore reasonable to consider that the space between ring girders with high rigidity is equal to the fixed

Mode No.n : circum. FEM Exp. FEM Exp. FEM Exp.

k : axial (Hz) (Hz) (Hz) (Hz) (Hz) (Hz)n =1, k =1 25.3 24.8 25.3 24.7 24.9 24.9n =2, k =1 23.8 21.9 27.2 25.9 20.5 18.4n =2, k =2 56.5 60.2 49.7n =3, k =1 33.6 34.8 42.8 24.7 (30.1)n =3, k =2 45.4 73.6 36.5 (30.1)n =4, k =1 63.8 67.6 43.6n =4, k =2 67.3 88.0 47.3

t2.2 t2.2-s t1.6

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Table 4 Evaluation procedures for natural frequencies of steel penstock

Mode No. n：circum.

k：axial

Calculation formulae for natural frequency Arrangement of fixed intervals for cases

with circumferential stiffener

n = 1

k = 1

- Calculation is not required, if the vibration frequency as a beam where the space between ring girders is considered as a fixed interval is sufficiently higher than the vibration frequency for n = 2.

- If calculation is required, the entire penstock should be considered as a continuous beam, and vibration analysis should be made to obtain the vibration frequency at the point in question.

- Circumferential stiffeners should be neglected, and the space between the ring girders should be considered as a fixed interval.

n = 2

k = 1

(Calculation equation shown in the reference 2))

25

3

42 83701420773ER

P.Rt.

LtR.Ef ++=

γ

- Considering the effects of circumferential stiffeners on rigidity reinforcement, the wall thickness of the penstock shell is replaced by an equivalent wall thickness according to the procedures shown in reference 2), and the space between ring girders is considered as a fixed interval. However, when the equivalent wall thickness is more than twice the original, a separate investigation by finite element analysis, etc. is necessary.

n = 3

k = 1

(Calculation equation shown in the reference 2))

25

3

43 823781091ER

P.Rt.

LtR.Ef ++=

γ

n≧4

k = 1

(Calculation equation shown in the current Technical Standards for Gates and Penstocks)

( )( ) ( )

2

22

2

2242

2 22 2

1 1

2 11

1

11

1

E nf

R nn

n

n RPn

Etn

π γε

αβ

να

=+

++

−+ + −

−+

- Circumferential stiffeners with rigidity applicable to ordinary installation may be considered basically rigid. Therefore, the space between the ring girder and the circumferential stiffener or between the circumferential stiffeners is considered as a fixed interval.

NOTES: 1) For k = 2, the fixed interval may be 1/2. In reality, however, no calculation is necessary since the vibration frequency increases. 2) For symbols relating to f2, f3 and f, the following figures should be used: E : Elastic modulus of steel [206kN/mm2→2.06×108kg/mm･sec2], γ : Density of steel [7.85×10-6kg/mm3], ν : Poisson's ratio of steel, t : Wall thickness of penstock [mm], L : Span length of penstock [mm], R : Radius to center of wall thickness [mm], P : Internal pressure [MPa],

E

γ7105120 ×= . [mm/sec], /k R Lα π= , 2 2/12t Rβ = , ε= (Density of water/Density of steel)

1Rt n

⋅ ⋅

spacing. The effects of circumferential stiffeners should be considered as the equivalent wall thickness which is calculated by the ratio of the moment of inertia for a pipe shell including a stiffener that is within the effective width to the moment of inertia of the pipe shell alone2). No vibration instance has been reported for n = 3 and it

has been confirmed by the results of parameter analysis that sufficient vibration prevention effects can be developed by the rigidity of the circumferential stiffeners. From the above discussion, we intend to propose a revision of Article 21 of the current Technical Standards for Gates and Penstocks regarding the

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vibration frequency calculation procedures to be applied at the design stage. These are summarized in Table 4. In this table, n = 1 denotes a beam oscillation which is not investigated here since the vibration frequency is higher than that of the cross-sectional deformation vibration n = 2. If any resonance is anticipated at the vibration frequency of a simple beam, it is appropriate to re-evaluate using a continuous beam that accurately reflects actual conditions. Further, for n ≧ 4, it is appropriate to apply the equations defined in the current standard. The definition of the resonance region should take into consideration the accuracy of the vibration calculation equations and damping coefficients. At present, however, sufficient numerical data are not available to allow quantitative discussions and it is difficult to define this region with adequate technical grounds. Guidance should be taken from the description stated in ASCE’s Steel Penstock Manual3) that reads “Natural frequency of the penstock should be separated from the pulsation component of the hydraulic turbine or draft tube by more than ±20%”. 6. CONCLUSION This paper introduced a series of vibration experiments and finite element analyses to determine natural frequencies and vibration modes for cases where n = 1 - 3 and k = 1. We confirmed the vibration characteristics of steel penstocks, discussed these accomplishments together with the results shown in the reference 2), and referred to design standards. Conclusions reached are as follows: (1) Natural frequencies for mode n = 2 and k = 1,

which are considered the most important, showed a maximum 10% difference between analysis and experiment regardless of the presence of stiffeners (analysis value was higher). The difference was smaller with greater thickness and with a stiffener. The t2.2-s model (with stiffener) showed a little less than 5% difference. It was judged that natural frequencies obtained by experiments and by analyses exhibited good agreement in view of the facts that with experiments, the response spectrum did not show a clear peak and the natural frequency could not be identified accurately, and the boundary conditions of analysis model which were applied uniformly in the circumferential direction was more rigid than experimental model. We then concluded that there was justifiable evidence to

support the finite element analysis that was used substantially in the previous report for derivation of equations for natural frequency calculations.

(2) Natural frequency for mode n = 3 and k = 1 showed good agreement between analysis and experiment with the t2.2 model, and significant difference with the t1.6 model. It was judged that vibration mode n = 3 and k = 1 could not be distinguished clearly in the t1.6 experiments, because the vibration frequency in the beam mode came close to the analytical natural frequency for n = 3 and k = 1.

(3) Beam vibration mode of n = 1 and k = 1 is easily affected by the supporting rigidity of the penstock while cross-sectional deformation vibrations are hardly affected. Therefore, results of cross-sectional deformation vibration obtained by experiments can be utilized without modification.

(4) As mentioned in (1) above, both experiments and analyses showed that circumferential stiffeners, to a certain extent, effect an increase in the natural frequency for n = 2 and k = 1, though their effect is not remarkable. It was revealed further that for n = 3 and k = 1, stiffeners are effective in preventing the occurrence of the vibration mode itself.

(5) In order to reflect the results of this investigation, revision of the vibration prevention guidelines specified in the current Technical Standards for Gates and Penstocks is recommended as a possible option. We then proposed natural frequency calculation equations for every circumferential mode number, equivalent wall thickness that factors in the rigidity of the circumferential stiffeners, and arrangement of fixed intervals.

ACKNOWLEGEMENT: This study was conducted as one of the surveillance studies conducted by the Research committee on the prevention of vibration in steel penstocks (fiscal 2000 - 2002) sponsored by the Japan Hydraulic Gate and Penstock Association. The authors wish to express thanks to the Japan Hydraulic Gate and Penstock Association, and the personnel in the electric power companies and steel penstock manufacturers for their co-operation in promoting these experiments and analyses. REFERENCES 1) Hydraulic Gate and Penstock Association: Technical

Standards for Gates and Penstocks, First Edition, 1960, Second Edition, 1968, Third Edition, 1973, Fourth Edition,

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1981, Fifth Edition, 1993. (in Japanese) 2) Nakamura, H. and Yamamoto, K.: A vibration finding survey

of existing steel penstocks and natural frequency calculation formulae, Structural Eng. / Earthquake Eng., JSCE, Vol.20, No.2, pp.163s-178s, 2003.10.

3) ASCE: Steel Penstocks, ASCE Manuals and Reports on Engineering Practice, No.79, 1993.

4) Kito, F.: On deformation of a cylindrical shell which is suddenly subjected to a uniform external pressure, Transactions of JSME, Vol.18, No.66, pp.174-182, 1952. (in Japanese)

5) Kito, F.: On the vibration of a penstock, Transactions of JSME, Vol.18, No.69, pp.96-99, 1952. (in Japanese)

6) Kito, F.: On the vibration of a penstock - II, Transactions of JSME, Vol.19, No.79, pp.30-35, 1953. (in Japanese)

7) Kito, F.: Report on investigation of vibration of penstocks in hydro-electric power stations, Journal of JSME, Vol.55, No.403, pp.513-517, 1952.8. (in Japanese)

8) Kito, F.: On vibration of a thin cylindrical shell, which is immersed in water and subjected to a combined stress, Transactions of JSME, Vol.22, No.115, pp.205-211, 1956.3. (in Japanese)

9) Kito, F.: The vibration of penstocks, Water Power, pp.379-385, 1959.10.

10) Kito, F.: Some remarks on the penstock vibrations, Hydraulic Gate and Penstock, No.20, pp.13-20, 1961.9. (in Japanese)

11) Shiraki, K., Honma, T. and Nagata, K.: Penstock vibration characteristics, Mitsubishi Juko Giho, Vol.11, No.2, pp.105-114, 1974.3. (in Japanese)

12）Shiraki, K., Honma, T., Takayama, H. and Kito, F.: Dynamic characteristics of penstocks, Hydraulic Gate and Penstock, No.126, pp.22-36, 1981.1. (in Japanese)

（Received November 25, 2004）