In this paper, we study the following Kirchhoff type equation with a steep potential well vanishing at infinity:−(a+b∫R2|∇u|2dx)Δu+(h(x)+μV(x))u=K(x)f(u)inR2, where a,b>0 are constants, μ>0 is a ...parameter, V decays to zero at infinity as |x|−γ with γ∈(0,2 and intV−1(0) possesses multiple disjoint bounded components. We prove the existence of multi-bump solutions for μ>0 large enough and the concentration behavior of multi-bump solutions as μ→+∞.
where I_\alpha in the spirit of Berestycki and Lions. This solution is a groundstate and has additional local regularity properties; if moreover F (0,\infty ) is of constant sign and radially ...symmetric.>
•The multi-disk rub-impact analysis of a two-spool aero-engine dual-rotor model is performed using the approximate time variational method.•The significance of multi-disk rub-impact studies compared ...to the single-disk rub-impact problems are discussed based on the response and stability analysis.•Limit point and Neimark-Sacker bifurcations are observed in the response indicating sudden jump and origin of quasi-periodic motion respectively.•The influences of rubbing parameters on the dynamic characteristics of the dual-rotor model are critically evaluated.•The onset of quasi-periodic motion is happening early as the coefficient of friction and contact stiffness are increased.
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The main aim of this paper is to propose a numerical procedure for capturing the nonlinear dynamic characteristics of a two-spool aero-engine rotor system undergoing multi-disk rub-impact. In aero-engines, the possibility for the multi-disk rub-impact is high during the fan blade-out (FBO) event and subsequent windmilling action. It intensifies the nonlinear effects on the rotor vibrations and leads to certain undesired circumstances in the engine. The dual-rotor model consists of multi-stage compressors and single-stage turbines that undergo rubbing whenever their deflection exceeds the clearance. The dynamic model of the dual-rotor system is constructed using the tapered Timoshenko beam elements, rigid disks and rolling contact bearings. A proper model reduction technique based on component mode synthesis coupled with the Craig-Bampton substructuring is utilized to reduce the size of the finite element model. A semi-analytic technique called the approximate time variational method is employed to investigate the steady-state response of the system under multi-disk rub impact. Based on the proposed method, the response characteristics of the model are obtained and are verified with the numerical integration results. Compared to the single-disk rub-impact, the nonlinearities are intensified and significant variations are observed in the response characteristics and stability of the system. Period-5, quasi-periodic, and dry friction backward whirl motions are observed in the response for different values of the system parameters. During quasi-periodic motion, some unknown fractional components such as 0.716ω1, 0.766ω1, 0.916ω1 and 0.964ω1 are appeared in the response. Moreover, the dry friction backward whirl happened with a very large amplitude and it contains a superharmonic frequency component in the response.
In this paper the Orlicz–Minkowski problem, a generalization of the classical Minkowski problem, is studied. Using the variational method, we obtain a new existence result of solutions to this ...problem for general measures.
•Variational method to model sensitivity considering more error sources as possible.•A modified Delta robot is less sensitive to the geometric variations than its original counterpart.•Confirmed ...calibration method and parameter identification model by simulated calibration.•Parameter identification by minimizing square residuals between predicted and measure positions.•Robot absolute accuracy improved from 1.4mm to 0.1-0.13mm after calibration.
In this paper, the sensitivity analysis and kinematic calibration of a modified parallel Delta robot are presented. The influence of the variations related to the manipulator components and geometric parameters onto the motion accuracy, including the parallelogram, is studied by means of the variational method. The sensitivity analysis shows that the robot under study is slightly less sensitive to the geometric variations compared to its original counterpart. The developed error model was further used in the kinematic calibration of the robot as the parameter identification model, allowing more parameters and variations to be identified compared to the differential kinematic geometry based error models, which was confirmed through simulated calibrations. The calibration process was carried out along with a laser tracker to measure the end-effector positions, to identify the parameters by making use of the least square minimization method to minimize the square residuals between the predicted and measured positions. After calibration, the positioning error of the robot end-effector is significantly enhanced, with the absolute accuracy improved from 1.4mm to 0.1–0.13mm. The presented calibration process can be applicable to other parallel robots.
In this paper, we study multiple solutions for a class of perturbed semilinear Schrödinger equations −△u+V(x)u=f(x,u)+λg(x,u), u∈H1(RN), where f(x,u) is odd in u, and λg(x,u) is regarded as a ...perturbation term with parameter λ, which is not necessarily odd in u. Under some mild conditions on f and g, we prove that they admit arbitrarily many solutions provided that the perturbation parameter |λ| is small enough.
In this paper, we consider the following Kirchhoff type problem{−(a+b∫R3|∇u|2dx)Δu+λV(x)u=|u|p−2uinR3,u∈H1(R3), where a>0 is a constant, b and λ are positive parameters, and 2<p<6. Suppose that the ...nonnegative continuous potential V represents a potential well with the bottom V−1(0), the equation has been extensively studied in the case 4≤p<6. In contrast, no existence result of solutions is available for the case 2<p<4 due to the presence of the term (∫R3|∇u|2dx)Δu. By combining the truncation technique and the parameter-dependent compactness lemma, we prove the existence of positive solutions for b small and λ large in the case 2<p<4. Moreover, we also explore the decay rate of the positive solutions as |x|→∞ as well as their asymptotic behavior as b→0 and λ→∞.
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