Mechanism and quantitative criterion of free vibration characteristics of hydraulic systems using the water hammer reflection coefficient

Hydraulic vibration is a periodic hydraulic transient in piping systems, which can result in local damage and operating accidents involving hydraulic, mechanical, and electrical systems. However, traditional hydraulic vibration theory is limited in practical application because of its complex mathematical form and low computational efficiency. This study presents a free vibration analysis method of hydraulic systems using the water hammer reflection coefficient. A physically meaningful criterion (i.e., the product of the water hammer reflection coefficients at the upstream and downstream boundaries of the pipe, denoted as rurd), is proposed for quantifying free vibration characteristics, and it is verified via traditional hydraulic vibration theory. The modulus of rurd determines the damping factor of free vibration, and the argument of rurd determines the natural frequency. Because of the elasticity of water column, there are multiple vibration modes that correspond to multiple water hammer reflection coefficients at the pipe boundary. For a single pipe, the stability condition requires the modulus of rurd for all vibration modes to be less than 1. For a multi-pipe system, the stability condition requires every single pipe is stable. Furthermore, the applicability of this new method is demonstrated using a reservoir-pipe-valve-turbine system. The influence mechanism of some dominant factors on the free vibration characteristics of the hydraulic system is revealed. The results indicates that the free vibration stability can be improved by reducing the wave velocity or increasing the valve local loss (which changes the water hammer reflection characteristics of the hydraulic system), or by increasing the pipe friction loss to dissipate the energy of the water hammer waves as they propagate. Overall, this study presents a new perspective for understanding hydraulic vibration and offers potential significance in engineering applications.