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On theoretical and practical aspects of Duhamel’s integral

Różański Michał, Sikora Beata, Smuda Adrian, Wituła Roman

Archives of Control Sciences

2021

Vol. 31, No. 4

815--847

The paper is a new approach to the Duhamel integral. It contains an overview of formulas and applications of Duhamel’s integral as well as a number of new results on the Duhamel integral and principle. Basic definitions are recalled and formulas for Duhamel’s integral are derived via Laplace transformation and Leibniz integral rule. Applications of Duhamel’s integral for solving certain types of differential and integral equations are presented. Moreover, an interpretation of Duhamel’s formula in the theory of operator semigroups is given. Some applications of Duhamel’s formula in control systems analysis are discussed. The work is also devoted to the usage of Duhamel’s integral for differential equations with fractional order derivative.

A methodology for online rotor stress monitoring using equivalent Green’s function and steam temperature model

Banaszkiewicz M., Dominiczak K.

Transactions of the Institute of Fluid-Flow Machinery

2016

nr 133

13--37

The requirement of high operational ﬂexibility of utility power plants creates a need of using online systems for monitoring and control of damage of critical components, e.g., steam turbine rotors. Such systems make use of diﬀerent measurements and mathematical models enabling calculation of thermal stresses and their continuous control. The paper presents key elements of the proposed system and discusses their use from the point of view of thermodynamics and heat transfer. Thermodynamic relationships, well proven in design calculations, were applied to calculate online the steam temperature at critical locations using standard turbine measurements as input signals. The model predictions were compared with operational data from a real power plant during a warm start-up and show reasonably good accuracy. The eﬀect of variable heat transfer coeﬃcient and material properties on thermal stresses was investigated numerically by ﬁnite element method (FEM) on a cylinder model, and a concept of equivalent Green’s function was introduced to account for this variability in thermal stress model based on Duhamel’s integral. This approach was shown to produce accurate results for more complicated geometries by comparing thermal stresses at rotor blade groove computed using FEM and Duhamel’s integral.