This paper explores the concept of moving singularities in the boundary element analysis. The singularities are placed on an auxiliary boundary which is located outside the domain of the problem and are allowed to move as part of the solution process. This results in a highly adaptive but non‐linear method. Examples involving the two‐ and three‐dimensional Laplace's equations are solved. Excellent agreement with exact solutions is obtained using a minimal number of singularities. Also, the trajectories of the singularity motion are plotted. The behaviour seen here is that, as the solution approaches convergence, the singularities exhibit a general trend of moving away from the domain of the problem.

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In this paper, the free vibration characteristics of an elastically connected double-beam system (ECDBS) connected by an elastic layer with uniform elastic stiffness under arbitrary boundary conditions have been investigated. Timoshenko beam theory, in which the effects of shear deformation and rotational inertia are considered, is utilized to model the free vibration of the ECDBS, and displacement components can be obtained from the Haar wavelet series and their integral. Hamilton’s principle is applied to construct coupled governing equations, and the virtual spring boundary technique is applied to generalize boundary conditions at four ends of the ECDBS. The convergence and accuracy of the proposed method can be confirmed by comparing the results with those of previous literature. A lot of new features have been identified, such as the frequency characteristics of the ECDBS with different cross sections under arbitrary boundary conditions, which can be provided as reference data for future research.

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In this paper, a methodology to analyze the nonlinear vibration of a Jeffcott rotor supported on a squeeze-film damper (SFD) with centering springs, which is widely used in high-speed rotating machines, is proposed using the incremental harmonic balance (IHB) method. In this paper, the IHB method is modified to analyze the dynamic behavior of rotor systems supported on fluid-film bearings, such as SFDs. The processing of the mass matrix, stiffness matrix, and linear force matrix proceeds in exactly the same way as in classical IHB. The nonlinear force generated in the oil film of the SFD is calculated using the alternating frequency/time method and the transformation matrix and incorporated into the computational processing of the classical IHB method. This calculation method is first proposed in this paper. Solutions computed using the proposed method are compared with solutions computed using numerical integration. The results are very close. The stability of the calculated solutions is determined using the Floquet theory. Based on this, frequency–response curves according to the change in various parameters are constructed. The proposed method can be effectively used to analyze the nonlinear vibration characteristics of rotor systems supported on fluid-film bearings, such as SFDs.