Reflections and Conclusions

In this chapter, reflection on the discovery of gate instability mechanisms is provided. Stemming from Ishii's encounter as an undergraduate with the Wachi gate failure and his subsequent development of the theory for eccentricity instability, a framework for future analyses of other gate instabilities was established. The study of two degrees-of-freedom instabilities of long-span gates created a paradigm for mode coupling in hydraulic gate vibrations. The Folsom failure occurred with eyewitness testimony claiming to have heard and felt vibration. The path to understanding the mechanism that could produce vibration of the Folsom gate was the realization that the skinplate can be easily excited to undergo streamwise vibration. To counteract such vibration, dynamic design criteria for Tainter gates are needed. A draft formulation of dynamic guidelines for Tainter gate design is developed. We hope for feedback from those who use the guidelines to provide for the continuing improvement of the guidelines.

Theory of Self-Excited Coupled-Mode Vibration of Tainter Gates

In general, any mechanism that produces an unbalanced moment may also serve to initiate a rigid body rotation of a Tainter gate about the trunnion pin. From the modal analysis testing on an intact Folsom Dam Tainter gate, and an understanding of the concepts of flow-rate variation pressure and push-and-draw pressure presented in Chapters 4 and 5, respectively, a conceptual model of the vibration mechanism can be formulated. The whole gate rotation induces a flow-rate variation pressure and a coupled inertia torque on the skinplate, as presented in Chapter 4. Both the flow-rate-variation pressure and the inertia torque excite the skinplate to rotate in a bending mode shape about a horizontal nodal line. In the present chapter we will develop the theory behind such an instability mechanism, called the self-excited coupled-mode instability, culminating in the graphical representation of the Folsom Dam gate instability in terms of a dynamic stability criterion diagram under the conditions at which failure occurred.

Analogies Between Coupled-Mode Gate Vibration and Coupled-Mode Flutter

In this chapter the similarities between the Tacoma Narrows Bridge failure in 1940 and the Folsom Dam gate failure in 1995 are examined. In both cases, static design guidelines were followed in the design of the structure under the assumption that large, massive structures would not be susceptible to dynamic excitation. Fundamentals of two-dimensional coupled mode flutter are presented. The frequency mode coalescence that occurs in two-dimensional flutter is noted. It is seen to have some resemblance to the mode-coupling in the coupled-mode instability of Tainter gate. The need for development of dynamic design guidelines for Tainter gates is argued to be parallel to the need for dynamic design guidelines for suspension bridges in the wake of the Tacoma Narrows failure.

Flow-Induced Vibration of Long-Span Gates

In this chapter the theoretical equations for fluctuating pressures due to vertical and streamwise gate motions developed in Chapters 4 and 5 are used to derive equations of motion for long-span gates with underflow, overflow and simultaneous over- and underflow. Theoretical development of analysis methods is supported by laboratory and full-scale measurements. Specifically, this chapter considers long-span gate instabilities including one degree-of-freedom vibration of gates with underflow and free discharge, one degree-of-freedom vibration of a gate with submerged discharge and vortex shedding excitation, a two degree-of-freedom vibration of long-span gates with only underflow, and two degrees-of-freedom vibration of long-span gates with simultaneous over and underflow. A method is developed to predict pressure loading on the crest of the gate with overflow.

Fluid Dynamics

A review of basic fluid dynamics is presented in this chapter. Fluid static loading of hydraulic gates is examined. The focus in the present context will be on one-dimensional, incompressible flow of Newtonian fluids (air and water). Viscous effects will be included as loss coefficients in pressure drop calculations through ducts and channels. Discharge coefficients of hydraulics gates are presented to account for viscous effects in the flow past these gates. More advanced concepts related to the instabilities of boundary layers and free shear layers, and transition to turbulence will be introduced briefly and references provided for further investigation by the interested reader. Readers are encouraged to review additional fluid dynamic concepts using the text with which they are most comfortable.

Vibrations

Vibration concepts are reviewed. Single degree-of-freedom vibration (SDOF) are analyzed. Subsequently, the analysis is extended to two degrees-of-freedom (2DOF) systems and coupling in a 2DOF system. The analysis of parametric coupling is introduced. Two sections on energy flow and the modeling of damping follow. Normal modes and mode shapes for systems with multiple degrees-of-freedom (MDOF) will then be considered. By generalizing MDOF systems to continuous systems, we can analyze bending modes in plates. Experimental modal analysis is introduced to prepare the reader for later application of this technique to full-scale operational gates. The second section of this chapter reviews fundamental concepts of fluid-structure systems with resonance. The chapter concludes with a short discussion of stability concepts.

Pressure on a Submerged Plate Undergoing Streamwise Rotation

In this chapter theoretical expressions are developed for the pressure loading on a vertical flat plate bounding an upstream reservoir with the plate undergoing rotational streamwise oscillation in close proximity to a bounding surface beneath the plate. The flow field in the reservoir can be decomposed into two parts. The first component of the flow field is due to only the rotational vibratory motion of the rigid weir plate (about some point on the weir plate), while the gate remains entirely closed with no discharge. The second component of the flow field results from the up-and-down vibration of the weir plate. Data from model scale testing in the laboratory and field tests on full-scale Tainter gates show excellent to good agreement with the theoretical predictions, validating the use of the theory.

Streamwise Gate Vibration

Streamwise vibrations of gates due to the bending flexibility of the skinplate of Tainter gates or the weir plate of long-span gates result in pushing-and-drawing of the water in the reservoir. During each cycle of vibration, the gate's motion must accelerate and then decelerate the water mass in contact with the vibrating gate surface, resulting in a substantial added mass effect. From simple single degree-of-freedom mass-spring-damper vibration theory, one understands that the effect of added mass is to lower the frequency of gate vibration. In addition to the push-and-draw effect, streamwise motion can also result in discharge fluctuation for inclined gates, providing a source of gate excitation. Rayleigh's wave theory analysis from the previous chapter is applied to provide an analysis framework for determining the magnitude of wave radiation damping and to calculate the added mass.

Wachi Dam and Folsom Dam

The failures at the Wachi Dam on the Yura River in Japan on July 2, 1967 and at the Folsom Dam on the American River near Sacramento, on July 17, 1995 are analyzed in light of the development of the self-excited coupled-mode instability mechanism. In both failures vibration of the gate was either suspected or noted by eyewitnesses. An exploratory study of how the predictions from the theory of the self-excited coupled-mode instability mechanism fit with the limited known facts is undertaken. The theoretical predictions from self-excited coupled-mode instability theory are found to explain well these two failures and are consistent with the reported circumstances surrounding the failures.

Overflow Gates

This chapter focusses on instabilities associated with overflowing gates -- nappe oscillations and flap gate vibrations. A detailed mapping of potential initiating and sustaining energy sources is developed to help understand the mechanism of nappe oscillations. The precise mechanism of nappe oscillations remains an unsolved problem in fluid mechanics, even though we understand the process well enough to provide effective countermeasures to prevent nappe oscillations. We do not yet have the ability to definitively predict the onset criteria for nappe oscillations nor can we predict which mode of nappe oscillations will predominate and what produces changing modes with changing flow conditions. A method of determining added mass and in-water vibration frequency of flap gates is presented at the conclusion of the chapter.