Use of Polynomial Mode Shape Function in Free Vibration Analysis of Cantilever Pelton Turbine

Abstract

In this study, we investigate the use of polynomial mode shape functions in the free
vibration analysis of a cantilever Pelton turbine and compare it with the conventional transcendental
mode shape function. The mathematical model is developed using the assumed mode method and
Lagrange’s equation, incorporating rotational inertia and centrifugal effects. Variable mode shapes and
critical frequencies are determined using both polynomial and transcendental functions, and are
compared exclusively. The results show that polynomial mode shape functions closely approximate
the first mode shape but the deviation increases in higher modes, with a critical frequency deviation of
2.23% in the first mode and 14.58% in the second mode and 20.53% in the third mode. These findings
suggest that while the polynomial mode shape functions offer computational simplicity, their accuracy
decreases in complex rotor-dynamic systems. We can use polynomial mode shape function is case of
simple dynamics and problems in which higher modes are negligible. Being the Cantilever Pelton
Turbine complex, presently designed mathematical model and its outcomes are fully vowed to explain
its vibrational consequences for first mode. The authors believe that the findings obtained through this
theoretical work are applicable not only to Cantilever Pelton turbines but also to other rotor-dynamic
systems with similar boundary conditions, such as turbine blades, flexible shafts, and cantilevered rotor
structures in hydro and wind energy applications.