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The Effect of Boundary Conditions on the Natural Vibration Characteristics of Deep-hole Bulkhead Gate Beier Luo*1, Junxing Wang2, Wen Wang3

State Key Laboratory of Water Resource and Hydropower Engineering Science, Wuhan University Wuhan, Hubei, 430072, P.R. China *1lbe415@whu.edu.cn; 2jxwang@whu.edu.cn; 3wangwen@whu.edu.cn Received 28 March 2014; Accepted 2 April 2014; Published 4 June 2014 © 2014 Science and Engineering Publishing Company Abstract

This paper studied the effects of boundary bearing conditions and fluid-solid interaction two factors on the natural vibration characteristics of a deep hole bulkhead gate. Calculated with Block Lanczos method in model analysis of ANSYS, the results show that the influence on gate’s natural vibration characteristics is very significant when the restraints of forward flow direction, transverse direction or vertical direction changes within a certain range. Its rule is in accordance with the rule of transverse vibration of rectangle thin plate. The natural frequencies are closely related to the restraints of the modal vibration direction, and have nothing to do with other directions. Therefore, the corresponding direction of vibration modes can be obtained by analysing the rule of gate’s natural frequency changing with the restraints. In considering the effect of fluid-solid interaction, water in front of the gate is simulated by added mass, the result shows that the impact created by upstream water level changing within a certain range on gate’s natural frequency can be neglected, and the restraints have a relatively small effect.

Keywords

Deep Hole Bulkhead Gate; Natural Vibration Characteristics; Restraint Condition; Fluid-Solid Interaction

Introduction

Deep-hole bulkhead gate is widely used in the field of hydraulic engineering but the issues of flow-induced vibration are often problematic to the engineers. Strong vibration is likely to occur in some situation leading to the failure of the structure. Therefore, the study of the flow-induced vibration of deep-hole bulkhead gate is necessary. The natural vibration characteristics of the bulkhead gate are the foundation

of the study of the kinetic response and stability. Further analysis can only be conducted when the impact factors and variation trend of the natural vibration characteristics of the bulkhead gate are fully evaluated and recommendations are proposed for actual engineering practice.

In actual engineering practice, Wu et al. proposed the fundamental equations and definite condition for fluid-structure interaction system for common hydraulic structure; Li simulated the water in front of the bulkhead gate using fluid elements in ANSYS and results showed that the effect of fluid-structure interaction on the natural vibration characteristics of bulkhead gate was substantial; relevant research was conducted on the vibration characteristics of fluid-structure interaction of plate structure under different constraint: Haddara et al. evaluated the kinetic characteristics of rectangular plate immersed in the water; Ergin et al. studied the kinetic characteristics of vertical cantilever plate partly immersed in the water; Kerboua et al. evaluated the vibration problem of horizontal and vertical plate immersed in or floating on the water. It is clear that studies of the vibration characteristics of plate structure mainly focused on the problem of fluid-structure interaction but the influence of variation of boundary conditions was neglected. Fixing of the support or water-sealing may cause pre-tension to the bulkhead gate or produce gaps, leading to the variation of boundary conditions for the vibration and change the characteristics of natural vibration, affecting the kinetic behaviour of the gate and the systematic nonlinear kinetic behaviour is complex.

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Recent studies on the variation of boundary constraint conditions focused on the transverse vibration characteristics of the plate: Saha et al. studied the free vibration problem of rectangular Mindlin plate with uniform elastic constraint boundary condition; Zhang et al. studied the vibration problem of rectangular thin plate with arbitrary non-uniform elastic constraint. The aforementioned studies were focused on plates with regular shape and the distribution of boundary constraint was regular. Theoretical solutions could be obtained but those methods cannot be used on bulkhead gate with complex size and constraint conditions. Therefore, this study used a sliding deep-hole bulkhead gate with continuously variable constraint stiffness as research object to study the effect of boundary conditions on the natural vibration characteristics of the gate. The variation of natural vibration of the gate with constraint stiffness was evaluated by calculation using ANSYS software. The condition in the possible scenario of discontinuous variation was analysed and the fluid-structure interaction was studied to provide recommendation for vibration reduction design of the gate in engineering practice.

Research Object and Fundamental Theory

Research Object

A working gate located at the exit of the tail-end of the sand duct of a hydraulic power plant was selected as the research object. The hole had a dimension of 3.2×4.2m (width× height) and a hydraulic head under normal operation of 74m. The working gate was a sliding bulkhead gate with top and side water-seals located at the downstream face and the gate was operated by a hydraulic pressure headstock gear. Line contact existed between the downstream sliding block and the chute slide way in the narrow door with P type side and top water-seals positioned in area. On the upstream face, two arc-shaped elastic sliding blocks were used to contact with the inner surface of the chute of the narrow door at the left and right hand side of the gate to prevent the gate from incline or shift towards the upstream face without getting to tight in the chute to avoid difficulties in operating the gate. Two rectangular blocks were positioned at each side of the gate to avoid side shifting. A 5mm gap was maintained between the sliding block and the chute slide way. The size of the gate and the form of support are shown in Fig. 1.

Due to the existence of arch-shape elastic sliding

blocks, the sliding blocks at the downstream face could escape the slide way during forward flow vibration leading to shifting of the gate towards upstream and forward flow deformation of the gate. The constraints of sliding blocks at downstream face are ineffective and the arch-shape elastic sliding blocks were compressed. In addition to the variation of the pre-compressed amount of water-seal with vibration, repeated impact of contact, separate and re-contact is occurred to the gate resulting to the variation of constraint stiffness in forward flow direction. In terms of transverse vibration of the gate, due to the fact that there is gap between the side sliding blocks and the side chute slide way, the transverse constraint is not unlimited line displacement but segmental. Whether or not its effect on the vibration characteristics of the gate can be ignored needs further investigation. For vertical vibration of the gate, the lever of the hydraulic pressure headstock gear can deliver simultaneously tensile and compressive force and constraint stiffness depends on the stiffness of the piston rod and the fixing condition of hydraulic pressure headstock gear.

FIG. 1 PLAN OF THE GATE AND DETAILED VIEW OF THE

SUPPORT FORM

Fundamental Theory

The bulkhead gate is analysed in a three-dimensional system. In finite element method, the differential equation of the linear constant coefficient matrix of gate structure considering fluid-structure interaction is expressed as:

( ) ( ) ( ) ( ) ( ) ( ) ( )a a aM M x t C C x t K K x t P t+ + + + + = (1)

where M , C , K are respectively mass, damping and stiffness matrix of the gate structure; aM , aC , aK are respectively added mass, damping and stiffness matrix of the gate vibration system under fluid-structure interaction; ( )P t is the pulsed compressive

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force of the flow which varies with t and relates to the condition of the flow. When studying the natural vibration of the bulkhead gate, damping has little influence on the frequency and type of natural vibration in a light damping system such as gate structure, and ignoring the effect of damping significantly simplifies the calculation of characteristics of natural vibration of the structure, the effect of damping was ignored in the study, i.e.

0aC C+ = . In addition, the added stiffness imposed by flow to the gate is normally small compared with the stiffness of the gate, 0aK = was assumed in this study. Therefore, Eq. (1) can be simplified as:

( ) ( ) ( ) 0aM M x t Kx t+ + = (2) As the boundary of the gate belongs to a bounded domain, separation of variables in Eq. (2) yields the typical eigenvalue problem in Eq. (3):

( )2n a n nM M Kω + Φ = Φ (3)

where, nω is the natural frequency in the n order modal, nΦ is the vector of the natural mode of vibration in the n order modal which does not vary with time but satisfy the boundary condition.

Method of Calculation

Real symmetric matrix is very significant in the application of engineering and related field, especially in the field of structural vibration engineering. Depending on different types of matrix eigenvalue problem, many methods can be used for calculation including: deflation method, sub-space iteration method, Lanczos Method, QL decomposition and nonsymmetry method. Block Lanczos method was adapted in this study.

Block Lanczos can be used to obtain the eigenvalue and eigenvector for large symmetric matrix. The method is based on blocking Lanczos theory with combination of Sturm theory by using automatic shifting technique to obtain the required eigenvalue and eigenvector. Block Lanczos is a transformation of the typical Lanczos method which uses multiple vectors simultaneously to enable Lanczos recursion to obtain the tridiagonal matrix similar to the original matrix. The eigenvalue and eigenvector of the tridiagonal matrix can be solved using QL method. The eigenvalue is the approximation of the eigenvalue of the original matrix and the eigenvector can be transform to obtain the approximate eigenvector of the original matrix. The extraction of the eigenvalue and eigenvector for n order real symmetric matrix can be

summaries as:

(1) Initialisation, i.e., randomly select n b× matrix 0R to let 0 0Q = , 1 0R AR= and 1 1 1R Q B= ;

(2) Loop 1,2, ,i m= , calculation the recursion using Eq.(4):

1 1 1T

i i i i i iR AQ Q A Q B+ − −= − − (4)

where Ti i iA Q AQ= , 1 1 1i i iR Q B+ + += , iQ is n b× matrix

with each column mutually orthogonal; iA is b b× matrix, iB is b b× upper triangular matrix, b is the size of the block, the tridiagonal matrix in block form calculated by above process is:

1 1

1 2 2

2

1

1

0

0

T

T

mTm

m m

A B

B A BT B

BB A

−

−

=

(3) By means of Given transformation, mT is simplified to a constant tridiagonal matrix MT . Select a series of spin matrix iG and calculate using Eq(5):

1 1 1 2T T T

r r m r MG G G T G G G T− = (5) where ( )M mb M n= .

(4) Repeatedly decompose MT by QL method ultimately yields ( )1, , MD diag λ λ= in diagonal form and the diagonal elements, 1, , Mλ λ and the required eigenvalues.

The Effect of Constraint Conditions

The effect of added mass is ignored in this section. Due to the deformation of the arch-shape elastic sliding blocks in the upstream face, the contact and escape process of the sliding blocks in the downstream face and the gaps in the side sliding blocks, a certain level of variation of constraint stiffness is likely to occur. As a result, the natural mode of vibration of the gate is formed with a part which is similar to rigid body displacement and the mode of vibration can be viewed as a combination of rigid body displacement mode and elastic deformation mode. The two modes are relative and it is hard to satisfy the displacement boundary condition in calculation. One simply way is to incorporate the variable degrees of freedom of the constraints in the global stiffness matrix of the structure to enable the implementation of variable boundary conditions in the software.

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ANSYS software is used in the calculation. Base on the geometry and force properties, the main structure including plates in the upstream and downstream face, main and secondary beams were modelled by SHELL63 shell element. Movable supports located in the chute were modelled by SOLID45 element. Material parameters of the gate includes: elastic modulus E=200GPa, Poisson’s ratio 0.3 and density 7800kg/m3.

The Effect of Constraint in Forward Flow Direction

The arch-shaped sliding blocks at the upstream face is elastic to some extend and remained contact but the sliding blocks at the downstream face is one-way linearly contacted with the chute and can escape the slide way. Therefore, it is possible that for the gate at a certain moment that the supports at the downstream face in the extreme conditions of fully contact or fully escape. In order to evaluate the variation trend of the frequency of natural vibration between the two extreme conditions, the forward flow constraints was set as four constant stiffness elastic constraints in the upstream face with an elastic modulus of 10GPa; the constraint at the slide way in the downstream face was set to be linear displacement variable stiffness elastic constraints; transverse and vertical constraints were set as one-way fixed constraints. The effect of the forward flow constraints on the frequency of natural vibration of the gate was studied by continuous variation of constraint stiffness method. The frequencies of the first 6th order were extracted. As the model of natural vibration in each order will exhibit different rate of variation with the variation of the stiffness of the constraint, the variation trend of respective frequency of each 6 order with the elastic modulus of the forward flow constraint ES is shown in Fig. 2; where S1 is the forward flow arch-shaped swing, S2 is the forward flow incline forward and backward vibration, H is the vertical straight movement, C is the transverse rotation, S3 is the forward flow pulse, SC is the forward flow S-shaped twist accompanied with transverse rotation.

It is clear from Fig. 2 that the variation of the stiffness of constraints mainly affected the mode of vibration in forward flow direction and there was little effect on the vertical vibration. The variation trend was similar to an S-shaped curve and increased with the increase of ES. When ES varied between 107~1010(Pa), the rate of variation of frequencies in each order was significant and the rate of variation was stable and move towards the two extreme conditions when ES lied outside the

range. As the variation range of ES was smaller than the elastic modulus of the gate E, the frequency of natural vibration of the gate in forward flow direction was relatively stable. In addition, the variation of constraint stiffness in forward flow direction did not change the mode of vibration but only the relative deformation in each mode.

FIG. 2 VARIATION TRENDS OF FREQUENCIES OF NATURAL

VIBRATION WITH FORWARD FLOW CONSTRAINTS

FIG. 3 VARIATION TRENDS OF FREQUENCIES OF NATURAL

VIBRATION WITH TRANSVERSE CONSTRAINTS

FIG. 4 VARIATION TRENDS OF FREQUENCIES OF NATURAL

VIBRATION WITH VERTICAL CONSTRAINTS

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The Effect of Transverse and Vertical Constraints

In order to evaluate the effect of transverse and vertical constraints on the frequency of natural vibration of the gate, the forward flow constraints were fixed and the transverse and vertical constraints were set as variable elastic modulus elastic constraints. Calculate the variation trend of the frequency of natural vibration of the gate f with the elastic modulus of transverse constraints EC and vertical constraints EH by extracting the first 6th orders, the results are given in Fig. 3 and Fig. 4.

Results showed that the variation of the stiffness of transverse constraints affect significantly the frequency of the transverse vibration. The variation trend appeared to be S-shaped and the mode of vibration was similar to transverse rigid body rotation. When EC varied between 108~1011(Pa), the variation of the frequency of transverse vibration was obvious but the forward flow vibration and vertical vibration were not affected. During flow induced vibration, the transverse vibration of the gate is less significant and therefore gaps in the side slide way do not essentially affect the vibration characteristics of the gate.

Moreover, the variation of the stiffness of vertical constraints affects significantly the frequency of the vertical vibration and the trend was similar to the case of transverse constraints. The variation trend appeared to be S-shaped and when EH varied between 108~1011(Pa), the variation of the frequency of vertical vibration was obvious. The mode of vertical vibration was similar to rigid body motion and the frequency was determined by the stiffness of the vertical constraints. Due to the effect of dowel bars, the stiffness of vertical constraints would not vary in a large range. Nevertheless, if the dowel bars are damaged or the connection to the hydraulic gear is loosen which results in a sudden reduction of stiffness of vertical constraint and thus the frequency of vertical vibration of the gate, the gate will vibrate strongly in vertical direction if induced by high water flow. Therefore, the assurance of the safety of the strength and stiffness of the dowel bars reduces the probability of the excess vertical vibration of the gate. In addition, drum type hoist is connected with the gate by steel rods and the stiffness of vertical constraints for this type of hoist is smaller, resulting in a reduced frequency of vertical vibration which is easier for vertical induced vibration to occur.

The Effect of Global Constraints

Fix the vertical constraints, the variation of the first 6

orders of frequencies of natural vibration of the gate with ES and EC were evaluated and are given in Fig. 5.

FIG.5 – continued (caption on next page)

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FIG. 5 VARIATION TRENDS OF FREQUENCIES OF NATURAL

VIBRATION WITH FORWARD FLOW AND TRANSVERSE CONSTRAINTS: (a) S1; (b) S2; (c) H; (d) C; (e) S3; (f) SC.

For bulkhead gates, the mode of vibration in the forward flow direction are richer than that in transverse or vertical directions, which is similar to the natural mode of vibration of rectangular plate in the transverse direction, whose frequency of natural vibration can be calculated by Eq. (6). In Eq. (6), mnλ is the frequency factor of the rectangular plate determined by the boundary conditions of the supporting plate. Different value of mnλ , corresponds different natural mode of vibration in (m,n) order which is related to m, n and the shape of vibration. Normally, m and n respectively represent the number of semi-waves in each direction of the rectangular.

2

2mn

mnDha

λω

ρ= (6)

It can be seen from Fig. 5 that (a) (b) (e) varied significantly with ES with did not varied with EC, which matches the mode of vibration in forward flow direction. The shape was similar to the transverse vibration of thin plate which is simply-supported at both ends. For instance, S1 in Fig. 5(a) is similar to the mode of vibration of rectangular plate with simply-supported ends in the (1,0) order, S2 in Fig. 5(b) is

similar to that in the (0,2) order, S3 in Fig. 5(e) is similar to that in the (0,3) order and SC in Fig. 5(f) is similar to that in the (2,0) order, as shown in Fig. 6.

(a)S1 (b)S2 (c)S3 (d)SC

FIG. 6 MODE OF VIBRATION OF FORWARD FLOW VIBRATION IN EACH ORDER FOR THE GATE

In addition, Fig. 5(d) shows the mode of transverse vibration controlled by the transverse constraints. Transverse vibration is basically similar to rigid body motion and is a secondary factor. Therefore, in the condition that constraint is not damaged, substantial transverse vibration would not occur. Fig. 5(f) shows the mode of vibration controlled simultaneously but forward flow and transverse constraints which when the stiffness of transverse constraints are large, the mode of vibration is shown in Fig. 6(d). Fig. 5(c) is the vertical vibration which is basically not related to the forward flow and transverse constraints.

It is clear that the frequency and mode of natural vibration are closely related to the condition of constraints in the vibrating direction but no related to the constraints in other directions. A general picture of the relevant mode of vibration can be obtained by analysing the variation trend of the frequency of structure with the constraints. For bulkhead gates, the characteristics of natural vibration are similar to the case of transverse vibration of rectangular thin plates and the forward flow vibration is the controlling model of vibration and the transverse vibration is secondary.

The Effect of Fluid-structure Interaction

The characteristics of natural vibration of the gate are not only related to the constraints but also the contact with water. In this section, the effect of water was added as a certain type of boundary condition into the gate system. In other words, the effect of water was equalised to added mass to simplify the model.

In order to evaluate the effect of added mass, the eigenvalue equation is given in Eq. (3) and the added mass can be calculated by classical Westergaard hydrodynamic pressure approximation.

Added mass force is essential a fluid inertia force caused by counter action of the structure from the

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variation of the flow force induced by vibration of the structure. With different mode of vibration, the added mass will change. For bulkhead gate, transverse vibration has only relative slipping at the water-solid surface and no added mass is formed in the assumption that perturbed motion has potential. Moreover, the forward flow vibration is the controlling vibration and it is significantly affected by added mass. Therefore, the variation of forward flow vibration with added mass is discussed in this section.

First, the forward flow, transverse and vertical constraints were fixed and the added mass was reflected by different amount of hydraulic head H from 5m to 74m (the designed water level) and mass element MASS21 was used to add the mass to the plate in the upstream face. The relationship of the frequency of the first three orders of natural vibration and the water level H is given in Fig. 7.

FIG. 7 VARIATION OF FREQUENCIES OF NATURAL

VIBRATION IN FORWARD FLOW VIBRATION OF THE GATE WITH UPSTREAM WATER LEVEL

It can be seen from Fig. 7 that the added mass increased with the increase of H; the frequency of natural vibration in each order of the gate reduced with the increase of added mass; when the hydraulic head at the gate was smaller than 20m, the rate of variation of the frequencies were large and the effect of added mass was obvious; when the hydraulic head exceeded 60m, the variation of the frequencies became smooth and the effect of added mass was minimum. It can be seen that the frequency of natural vibration of swallow-hole gate is very sensitive to the variation of added mass but the frequency of natural vibration of deep-hole gate is stable toward the variation of added mass (the variation of upstream water level). In other words, upstream water level does not affect the characteristics of forward flow vibration for deep-hole gate in a certain range.

In the conditions of different added mass, the variation of the three forward flow vibrations with the stiffness of forward flow constraints is given in Fig. 8.

FIG. 8 VARIATION TRENDS OF FREQUENCIES OF NATURAL

VIBRATION WITH FORWARD FLOW CONSTRAINTS FOR DIFFERENT UPSTREAM WATER LEVEL: (a) S1; (b) S2; (c) S3.

It is clear from Fig. 8 that with the increase of water level H, the variation trend of the frequency of natural vibration with the stiffness of forward flow constraints was identical when the added mass was not considered but the rate of variation reduced. In other words, the sensitivity of variation of the frequency of natural vibration with the stiffness of forward flow constraints reduced.

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Conclusions

This study used a deep-hole bulkhead gate as research object to evaluate the effect of boundary support conditions and fluid-structure interaction on the characteristics of natural vibration by means of constraint stiffness variation and added mass using Block Lanczos modal analysis in ANSYS software. Results showed that the effect of constraint stiffness is not negligible on the characteristics of natural vibration and the variation trend that the frequency of natural vibration of the gate reduces with the increase of hydraulic head was established. The conclusions can be referred by engineering practice in similar fields. The main conclusions can be drawn:

(1) The variation of forward flow, transverse and vertical constraint stiffness is substantially effective on the characteristics of natural vibration in a certain range. The frequency is stable outside this range. For the gate in this study, when ES varied between 107~1010(Pa), the variation of frequencies in each order was large and for EC and EH, the range was 108~1011. This range is smaller than the elastic modulus of the gate. Therefore in normal conditions, the frequency of natural vibration of the gate is relatively stable. At the same time, the stiffness of supports and moving components of the gate as well as doweling bars should be maintained to prevent reduction of the frequency of vibration resulting in flow-induced vibration.

(2)The frequency of natural vibration is related closely to the constraints in the direction of vibration but not related to constraints in other directions. The general direction of vibration in the relevant mode can be obtained by analysing the variation trend of frequency of natural vibration of the gate. For bulkhead gate, the characteristics of natural vibration are similar to the case of the transverse vibration of rectangular thin plate. The forward flow vibration is controlling and the transverse vibration is secondary comparably.

(3) When the upstream water level H lied within 20m, the variation of frequencies of natural vibration was obvious; when the hydraulic head increased to above 60m, the variation of frequencies of natural vibration was stable. For deep-hole gates, the added mass induced by the variation of hydraulic head within a

certain range can be ignored. With the increase of added mass, the sensitivity of variation of the frequency of natural vibration with the stiffness of forward flow constraints reduced. In other words, the effect of constraints reduces when fluid-structure interaction is considered.

(4) This study evaluated the characteristics of natural vibration of the gate. The effect of boundary conditions on the kinetic response requires further research.

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