Nonlinear Dynamical Analysis of Hydraulic Turbine Governing Systems with Nonelastic Water Hammer Effect

Table 1

Changes of system properties with the bifurcation parameter 𝐾_𝑑.

Bifurcation parameter 𝐾𝑑	𝐾𝑑 < 𝐾𝑑1^∗	𝐾𝑑 = 𝐾𝑑1^∗ = 0.313	𝐾𝑑1^∗ < 𝐾𝑑 < 𝐾𝑑2^∗	𝐾𝑑 = 𝐾𝑑2^∗ = 2.036	𝐾𝑑 > 𝐾𝑑2^∗
Jacobi matrix eigenvalues	Two complex conjugate eigenvalues, negative real part one and positive real part one	−2.46; −9.77; −4.5𝑒−5±5.98𝑖	Two complex conjugate eigenvalues with negative real parts	−2.76; −6.80; −2.7𝑒−6±6.77𝑖	Two complex conjugate eigenvalues, negative real part one and positive real part one
Property	Stable limit cycle	center of bifurcation; in a supercritical state	Stable focus, stable domain	Center of bifurcation; in a supercritical state	Stable limit cycle