THE KIMMEL EQUATION FOR HYDRAULICALLY DAMPED AXIAL ROTOR OSCILLATIONS

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1 The 9th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery Honolulu, Hawaii, February -4, THE KIMMEL EQUATION FOR HYDRAULICALLY DAMPED AXIAL ROTOR OSCILLATIONS Philip R. LeGoy Process Engineer ESB International Ltd Dublin, Ireland Stephan Fuelling Department of Physics University of Nevada, Reno Reno, Nevada 89557, USA ABSTRACT Liquefied gas fluid machinery, pumps as well as liquid turbines utilize thrust-equalizing mechanisms to accommodate high axial forces generated by the hydraulic fluid. Common thrust balancing devices use the hydraulic pressure of the fluid to equalize the axial thrust. Measurements on thrust-balanced pumps and turbines have shown the rotor oscillates around a stable axial equilibrium position. A recently published mathematical model of these axial oscillations, the Kimmel Equation represents a non-linear second order differential equation, which was solved for the ideal case with no damping and free oscillation: the natural frequency can be expressed through an elliptic integral. The actual case for hydraulic damping includes a non-linear quadratic velocity term. In the absence of exciting forces the equation with quadratic damping can be analytically solved in form of integrals. A solution is presented and discussed for oscillations with different damping factors. The axial movement depends on the initial displacement and its positive or negative direction. Solutions of the Kimmel Equation can be used to predict the axial rotor oscillations with hydraulic damping. INTRODUCTION Hydraulic turbines and pumps generate a relatively high axial thrust, which can reach several tons on multistage turbines and pumps, resulting in heavy bearing loads with reduced efficiency due to friction losses (Kimmel, 998). A variety of mechanisms have been developed to balance or reduce these high axial forces and are published in standard literature. The design of most of these mechanisms allows small axial movements of the rotor and consists of a fixed and a variable orifice. The opening of the variable orifice depends on the momentary axial position of the rotor. There is a certain neutral position of the rotor where the sum of all axial forces is zero. These mechanisms are self-adjusting thrust balancing devices and can be compared to a spring with a non-linear spring characteristic. The mass of the rotor assembly and the hydraulic spring form an axially oscillating mechanical system with nonlinear damping and spring characteristics. As reported by Evrensel et al. 999 and Evrensel, Finley, any fluctuation in the distribution of the fluid pressure or in the electromagnetic field of the induction generator or electric motor results in an excitation of axial rotor oscillations. Kimmel, developed a mathematical model of these axial rotor oscillations in form of a non-linear second order differential equation with quadratic damping. d z/dt + D(dz/dt)( dz/dt ) + k[( z) ]=F(t) () The natural frequency of an oscillating system occurs only for the ideal case of free oscillation with no damping. Kimmel, presented the solution for the natural frequency in form of an elliptic integral (Appendix ). (t t ) = (k) ½ [K+ z ( z) ] - ½ dz () Figure shows an example for the values k=.5 and K=.5, corresponding to the positive amplitude A=.5 and the negative amplitude.. The example demonstrates the asymmetric oscillation of the system. Figure Figure shows the graph for the velocity dz[t]/dt over the displacement z[t] for the same example. The closed egg-shaped curve indicates a stable solution in form of a double periodic function: the difference of the elliptic integral of the first and second kind.

2 Figure This integral is of the form ( F[ R [u], exp[u], Ei[u]] ) ½ du (8) HYDRAULIC DAMPING The described thrust balancing mechanism is a hydraulic system with fluid friction acting against the flow direction and proportional to the square of the fluid velocity. In the case of free oscillation the Kimmel Equation (), a nonlinear second order differential equation modeling the hydraulically damped axial rotor oscillations can be solved in analytical terms. For free oscillations the equation () reduces to d z/dt + D(dz/dt)( dz/dt ) + k[( z) ]= () with the conditions: z <, the damping factor D > and the spring constant per mass k >. Substituting the velocity dz[t]/dt by y[z] as a function of the displacement z[t] and substituting the acceleration d z[t]/dt by y[z](dy[z]/dz) reduces the order of the Kimmel Equation (): (dy[z]/dz) y[z]+ Dy[z] y[z] +k z (-z) (-z) = (4 ) The reduced equation (4) is a Bernoulli Equation and can be transformed into a linear differential equation with the substitution x[z] = (y[z] ) : dx[z]/dz + SDx[z] + k z (-z) (-z) = (5) and cannot be solved with elementary functions. It represents the analytic solution of the Kimmel Equation in integral form (Appendix ). The inverse function Kim[u] of the integral is a periodic function and defined with Kim[u] = F [ ( F[ R [u], exp[u], Ei[u]] ) ½ du ] (9) Kim [u] = ( F[ R [u], exp[u], Ei[u]] ) ½ du () Figure and Figure 4 show an example of the solution for the hydraulically damped free axial rotor oscillation with the spring constant k = and the initial value t =, z = A =.5. Figure Figure is the graph for the velocity dz[t]/dt over the displacement z[t] presenting a stable periodic solution. 4 Figure 4 S is equal to one for positive velocities and equal to minus one for negative velocities. S= for y[z]> and S= for y[z]< (6) The solution of equation (5) x[z] involves exponential integral functions Ei[u] (Appendix ) and is equal to the square of the displacement velocity y[z]. The final solution of equation () is received through the integral t t = ± (x[z] ) ½ dz (7) Figure 4 shows the first period of the hydraulically damped oscillations with the time t[z] over the displacement z. It represents the graph of t[z] = Kim [z] ( with t = t[z] for.5 z >.5)

3 NON-EXISTENCE OF CRITICAL DAMPING The solutions of the hydraulically damped axial rotor oscillation with hydraulic non-linear spring characteristic depend on the spring constant k, the initial amplitudes A or B and the damping factor D. For all technical applications it is important to know if an oscillatory system has aperiodic solutions with critical damping or over damping. In the case of the Kimmel Equation with the solution in integral form, an aperiodic solution, if existing, has to meet the following condition: dz[t]/dt = for z = and for all A and < D < This condition is true if x[z] = for z = and A and < D < ( Appendix 4 ). Assuming for any given damping factor D that there exists an initial amplitude A for which the condition x[z = ] = is true, then there exists an aperiodic solution for the given values of D and A. The function x[z = ] = x[] for A= is always equal to zero for all values of D. The partial differential x[]/ A is always positive for all values of A and D within the given range x[]/ A= ((-A)A) / ( (-A) exp[ad]) () and x [] > for all < A < and < D <. Figure 5 and 6 show the function x[] for A and for the ranges: < D. (Figure 5) and. D (Figure 6) Figure The function x [ ] is always larger than for all values of < A < and < D <. Only for A= or for infinite large damping constants D the velocity dz[t]/dt approaches asymptotically zero for z =. In Figure 5 and 6 the x-axis represents the damping factor D and the y-axis the initial amplitude A. The z-axis represents the square of the velocity dz[t]/dt for z= Figure Equation () and its conclusions show that the equation (4) for free oscillation has no aperiodic solutions for finite values of the damping factor D and the initial amplitude A. CONCLUSION In the absence of exciting forces the Kimmel Equation with quadratic damping can be analytically solved in form of integrals. The solutions are periodic and stable. Critically damped or over damped solutions are non-existent and the hydraulically damped axial oscillations are theoretically oscillating for unlimited time, approaching smaller and smaller amplitudes. REFERENCES Kimmel, H.E. 998, Power Generation using Thrust Balanced Hydraulic Turbines, Proceedings of the American Power Conference, Vol.6, pg. ff, Illinois Institute of Technology, Chicago Evrensel, C.E., Kimmel, H.E. and Cullen, D.M., 999, Axial Rotor Oscillations in Cryogenic Fluid Machinery, Proceedings of the rd ASME/JSME Joint Fluids Engineering Conference, San Francisco, California Kimmel, H.E.,, Cryogenic Francis Turbines, Proceedings of the 8th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, Honolulu, Hawaii Evrensel, C.E., Finley, C.D.,, Verification of Axial Rotor Oscillations in Cryogenic Turbine Generators, Proceedings of the 9 th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, Honolulu, Hawaii,.

FEDSM99 S-291 AXIAL ROTOR OSCILLATIONS IN CRYOGENIC FLUID MACHINERY

Proceedings of the 3 rd ASME/JSME Joint Fluids Engineering Conference 1999 ASME Fluids Engineering Division Summer Meeting July 18-23 1999, San Francisco, California FEDSM99 S-291 AXIAL ROTOR OSCILLATIONS