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Purpose: The simplified modeling of many physical processes results in a second-order ordinary differential equation (ODE) system. Often the damping of these resonating systems cannot be defined in the same simplified way as the other parameters due to the complexity of the physical effects. The purpose of this paper is to develop a mathematically stable approach for damping resonances in nonlinear ODE systems. Design/methodology/approach: Modifying the original ODE using the eigenvalues and eigenvectors of a linearized state leads to satisfying results. Findings: An iterative approach is presented, how to modify the original ODE, to achieve a well-damped solution. Practical implications: The method can be applied for every physical resonating system, where the model complexity prevents the determination of the damping. Originality/value: The iterative algorithm to modify the original ODE is novel. It can be used on different fields of the physics, where a second-order ODE is describing the problem, which has only measured or empirical damping. [ABSTRACT FROM AUTHOR]

Purpose A transformer model is used as a benchmark for testing various methods to solve 3D nonlinear periodic eddy current problems. This paper aims to set up a nonlinear magnetic circuit problem to assess the solving procedure of the nonlinear equation system for determining the influence of various special techniques on the convergence of nonlinear iterations and hence the computational time.Design/methodology/approach Using the T,ϕ-ϕ formulation and the harmonic balance fixed-point approach, two techniques are investigated: the so-called “separate method” and the “combined method” for solving the equation system. When using the finite element method (FEM), the elapsed time for solving a problem is dominated by the conjugate gradient (CG) iteration process. The motivation for treating the equations of the voltage excitations separately from the rest of the equation system is to achieve a better-conditioned matrix system to determine the field quantities and hence a faster convergence of the CG process.Findings In fact, both methods are suitable for nonlinear computation, and for comparing the final results, the methods are equally good. Applying the combined method, the number of iterations to be executed to achieve a meaningful result is considerably less than using the separated method.Originality/value To facilitate a quick analysis, a simplified magnetic circuit model of the 3D problem was generated to assess how the different ways of solutions will affect the full 3D solving process. This investigation of a simple magnetic circuit problem to evaluate the benefits of computational methods provides the basis for considering this formulation in a 3D-FEM code for further investigation. [ABSTRACT FROM AUTHOR]