Parametric Evaluation and Dynamic Analysis of Turbine Blades–Damper Assembly Using Bond Graph Technique

Abstract

Purpose

Gas turbine blades are subjected to fluctuating gas forces and centrifugal forces, which result into large resonant stresses and further lead to the failure of blades. Nowadays, dry friction damping is frequently applied in gas turbines, especially at hot locations, to reduce resonant stresses. The novelty of the paper lies in simulation of an assembly consists of two gas turbine blades and an under-platform damper through bond graph modelling technique. In this research work, dynamic behaviour of turbine blade and under-platform friction damper are obtained by evaluating their parameters.

Method

A lumped parameter model is developed for turbine blades and damper assembly. Further, the values of equivalent damping coefficient (C_eq) for an under-platform friction damper are evaluated through Lazan’s law. For simulation, a discrete bond graph model of the turbine blades–damper assembly is created and simulated through Runge–Kutta method.

Results and Conclusions

The dynamic parameters viz. displacement, velocity and force on turbine blades, damper’s displacement, velocity and frictional force of under-platform damper for all varying values of equivalent damping coefficients (C_eq) are obtained. These evaluated dynamic parameters of turbine blades and under-platform friction damper are further analysed for optimum designing of under platform friction damper to reduce resonant stresses and hence increase the fatigue life of turbine blades.

Appendix A

The best way to study the dynamics of a system residing in multi-energy domain is to start with a schematic diagram, which includes its important components and portrays how they are connected together. Bond graph is an explicit graphical tool for capturing the common energy structure of systems. Through Bond Graph approach, a physical system can be represented by symbols and lines, identifying the power flow paths. The lumped parameter elements of resistance, capacitance and inertance are interconnected in an energy conserving way by bonds and junctions resulting in a network structure. From the pictorial representation of the bond graph, the derivation of system equations is so systematic that it can be algorithmized. In bond graphs, one needs to recognise basic symbols, i.e. three basic one port passive elements inertance (I), capacitance (C), and resistance (R); two basic active elements source of effort (SE), and source of flow (SF); and two basic junctions i.e. constant effort junction (0), and constant flow junction (1). The basic variables are effort (e), flow (f), time integral of effort (P) and the time integral of flow (Q).

The constitutive equations obtained from bond graph model of turbine blades with under-platform friction damper are presented as

$$d\left( {{\text{P22}}} \right) \, = \, - {\text{R2}}0 \times {\text{P22}}/{\text{M22 }} - {\text{ K21}} \times {\text{Q21 }} + {\text{ K27}} \times {\text{Q27 }} - {\text{ K3}}0 \times {\text{Q3}}0 \, - {\text{ R31}} \times \left( {{\text{P7}}/{\text{M7 }} + {\text{ P22}}/{\text{M22}}} \right),$$

(11)

$$d\left( {{\text{P17}}} \right) \, = \, - {\text{K27}} \times {\text{Q27 }} + {\text{ SE18 }} - {\text{ R26}} \times {\text{P17}}/{\text{M17,}}$$

(12)

$$d\left( {{\text{P7}}} \right) \, = {\text{ K24}} \times {\text{Q24 }} - {\text{ R8}} \times {\text{P7}}/{\text{M7 }} - {\text{ K9}} \times {\text{Q9 }} - {\text{ K3}}0 \times {\text{Q3}}0 \, - {\text{ R31}} \times \left( {{\text{P7}}/{\text{M7}} + {\text{P22}}/{\text{M22}}} \right),$$

(13)

$$d\left( {{\text{P1}}} \right) \, = {\text{ SE2 }} - {\text{ K24}} \times {\text{Q24 }} - {\text{ R23}} \times {\text{P1}}/{\text{M1,}}$$

(14)

$$d\left( {{\text{Q3}}0} \right) \, = {\text{ P7}}/{\text{M7 }} + {\text{ P22}}/{\text{M22,}}$$

(15)

$$d\left( {{\text{Q27}}} \right) \, = {\text{ P17}}/{\text{M17 }} - {\text{ P22}}/{\text{M22,}}$$

(16)

$$d\left( {{\text{Q24}}} \right) \, = {\text{ P1}}/{\text{M1}} - {\text{P7}}/{\text{M7,}}$$

(17)

$$d\left( {{\text{Q21}}} \right) \, = {\text{ P22}}/{\text{M22,}}$$

(18)

$$d\left( {{\text{Q9}}} \right) \, = {\text{ P7}}/{\text{M7}}{.}$$

(19)

In Eqs. 11–19, ‘d’ represents the time derivative of the state variable within the first parenthesis.