• Advances in aircraft and spacecraft science
• /
• 제11권1호
• /
• pp.55-81
• /
• 2024

This paper presents the exact solutions and closed forms for of nonlinear stability and vibration behaviors of straight and curved beams with nonlinear viscoelastic boundary conditions, for the first time. The mathematical formulations of the beam are expressed based on Euler-Bernoulli beam theory with the von Karman nonlinearity to include the mid-plane stretching. The classical boundary conditions are replaced by nonlinear viscoelastic boundary conditions on both sides, that are presented by three elements (i.e., linear spring, nonlinear spring, and nonlinear damper). The nonlinear integro-differential equation of buckling problem subjected to nonlinear nonhomogeneous boundary conditions is derived and exactly solved to compute nonlinear static response and critical buckling load. The vibration problem is converted to nonlinear eigenvalue problem and solved analytically to calculate the natural frequencies and to predict the corresponding mode shapes. Parametric studies are carried out to depict the effects of nonlinear boundary conditions and amplitude of initial curvature on nonlinear static response and vibration behaviors of curved beam. Numerical results show that the nonlinear boundary conditions have significant effects on the critical buckling load, nonlinear buckling response and natural frequencies of the curved beam. The proposed model can be exploited in analysis of macrosystem (airfoil, flappers and wings) and microsystem (MEMS, nanosensor and nanoactuators).

• Structural Engineering and Mechanics
• /
• 제68권5호
• /
• pp.563-573
• /
• 2018

Temperature-induced responses, such as strains and displacements, are related to the boundary conditions. Therefore, it is required to determine the boundary conditions to establish a reliable bridge model for temperature-induced responses analysis. Particularly, bridge bearings usually present nonlinear behavior with an increase in load, and the nonlinear boundary conditions cause significant effect on temperature-induced responses. In this paper, the bridge nonlinear boundary conditions were simulated as bilinear translational or rotational springs, and the boundary parameters of the bilinear springs were identified based on the measured temperature-induced responses. First of all, the temperature-induced responses of a simply support beam with nonlinear translational and rotational springs subjected to various temperature loads were analyzed. The simulated temperature-induced strains and displacements were assumed as measured data. To identify the nonlinear translational and rotational boundary parameters of the bridge, the objective function based on the temperature-induced responses is then created, and the nonlinear boundary parameters were further identified by using the nonlinear least squares optimization algorithm. Then, a beam structure with nonlinear translational and rotational springs was simulated as a numerical example, and the nonlinear boundary parameters were identified based on the proposed method. The numerical results show that the proposed method can effectively identify the parameters of the nonlinear boundary conditions. Finally, the boundary parameters of a real arch bridge were identified based on the measured strain data and the proposed method. Since the bearings of the real bridge do not perform nonlinear behavior, only the linear boundary parameters of the bridge model were identified. Based on the bridge model and the identified boundary conditions, the temperature-induced strains were recalculated to compare with the measured strain data. The recalculated temperature-induced strains are in a good agreement with the real measured data.

• 대한수학회지
• /
• 제39권2호
• /
• pp.319-330
• /
• 2002

The method of quasilinearization generates a monotone iteration scheme whose iterates converge quadratically to a unique solution of the problem at hand. In this paper, we apply the method to two families of three-point boundary value problems for second order ordinary differential equations: Linear boundary conditions and nonlinear boundary conditions are addressed independently. For linear boundary conditions, an appropriate Green\`s function is constructed. Fer nonlinear boundary conditions, we show that these nonlinearities can be addressed similarly to the nonlinearities in the differential equation.

• Structural Engineering and Mechanics
• /
• 제86권3호
• /
• pp.361-371
• /
• 2023

Boundary condition is an important factor affecting the vibration characteristics of structures, under different boundary conditions, structures will exhibit different vibration behaviors. On the basis of the previous work, this paper extends to the nonlinear resonance behavior of axially moving graphene platelets reinforced metal foams (GPLRMF) plates with geometric imperfection under different boundary conditions. Based on nonlinear Kirchhoff plate theory, the motion equations are derived. Considering three boundary conditions, including four edges simply supported (SSSS), four edges clamped (CCCC), clamped-clamped-simply-simply (CCSS), the nonlinear ordinary differential equation system is obtained by Galerkin method, and then the equation system is solved to obtain the nonlinear ordinary differential control equation which only including transverse displacement. Subsequently, the resonance response of GPLRMF plates is obtained by perturbation method. Finally, the effects of different boundary conditions, material properties (including the GPLs patterns, foams distribution, porosity coefficient and GPLs weight fraction), geometric imperfection, and axial velocity on the resonance of GPLRMF plates are investigated.

• Steel and Composite Structures
• /
• 제50권2호
• /
• pp.149-158
• /
• 2024

Considering that different boundary conditions can have an important impact on structural vibration characteristics. In this paper, the nonlinear forced vibration behavior of functionally graded material (FGM) doubly curved shells with initial geometric imperfections under different boundary conditions is studied. Considering initial geometric imperfections and von Karman geometric nonlinearity, the nonlinear governing equations of FGM doubly curved shells are derived using Reissner's first order shear deformation (FOSD) theory. Three different boundary conditions of four edges simply supported (SSSS), four edges clamped (CCCC), clamped-clamped-simply-simply (CCSS) were studied, and a system of nonlinear ordinary differential equations was obtained with the help of Galerkin principle. The nonlinear forced vibration response of the FGM doubly curved shell is obtained by using the modified Lindstedt Poincare (MLP) method. The accuracy of this method was verified by comparing it with published literature. Finally, the effects of curvature ratio, power law index, void coefficient, prestress, and initial geometric imperfections on the resonance of FGM doubly curved shells under different boundary conditions are fully discussed. The relevant research results can provide certain guidance for the design and application of doubly curved shell.

• 대한수학회논문집
• /
• 제27권1호
• /
• pp.97-106
• /
• 2012

In this paper, the main purpose is to study existence of the global attractors for the weakly damped wave equation with nonlinear boundary conditions. To this end, we first show that the existence o a bounded absorbing set by the perturbed energy method. Secondly, we utilize the decomposition of the solution operator to verify the asymptotic compactness.

• 한국소음진동공학회:학술대회논문집
• /
• 한국소음진동공학회 1998년도 춘계학술대회논문집; 용평리조트 타워콘도, 21-22 May 1998
• /
• pp.392-398
• /
• 1998

In order to check the validity of nonlinear normal modes of continuous, systems by means of the energy-based formulation, we consider a beam with a nonlinear boundary condition. The initial and boundary e c6nsl of a linear partial differential equation and a nonlinear boundary condition is reduced to a linear boundary value problem consisting of an 8th order ordinary differential equations and linear boundary conditions. After obtaining the asymptotic solution corresponding to each normal mode, we compare this with numerical results by the finite element method.

• Computers and Concrete
• /
• 제30권6호
• /
• pp.433-443
• /
• 2022

In the current paper, the nonlinear resonance response of functionally graded graphene platelet reinforced (FG-GPLRC) beams by considering different boundary conditions is investigated using the Euler-Bernoulli beam theory. Four different graphene platelets (GPLs) distributions including UD and FG-O, FG-X, and FG-A are considered and the effective material parameters are calculated by Halpin-Tsai model. The nonlinear vibration equations are derived by Euler-Lagrange principle. Then the perturbation method is used to discretize the motion equations, and the loadings and displacement are all expanded, so as to obtain the first to third order perturbation equations, and then the asymptotic solution of the equations can be obtained. Then the nonlinear amplitude-frequency response is obtained with the help of the modified Lindstedt-Poincare method (Chen and Cheung 1996). Finally, the influences of the distribution types of GPLs, total GPLs layers, GPLs weight fraction, elastic foundations and boundary conditions on the resonance problems are comprehensively studied. Results show that the distribution types of GPLs, total GPLs layers, GPLs weight fraction, elastic foundations and boundary conditions have a significant effect on the nonlinear resonance response of FG-GPLRC beams.

• Structural Engineering and Mechanics
• /
• 제81권1호
• /
• pp.59-68
• /
• 2022

Prestressed prefabricated hollow core concrete slabs with spans of 5 m and 10 m are commonly used since last century and still in service due to the advantage of construction convenience and durability. However, the end slabs are regularly subjected to cracks at the top and fail with brittleness due to the asymmetric boundary conditions. To better maintain such widely used type of hollow core slabs, the effect of asymmetric constraint in the end slabs are systematically studied through detailed nonlinear finite element analyses and experimental data. Experimental tests of slabs with four prestressed tendons and seven prestressed tendons with different boundary conditions were conducted. Results observe three failure modes of the slabs: the bending failure mode, shear and torsion failure mode, and transverse failure mode. Detailed nonlinear finite element models are developed to well match the failure modes and to reveal potential damage scenarios with asymmetric boundary conditions. Recommendations regarding ultimate capacity of the slabs with asymmetric boundary conditions are made to ensure a safe and rational design of prestressed concrete hollow slabs for short span bridges.

• Structural Engineering and Mechanics
• /
• 제86권6호
• /
• pp.751-764
• /
• 2023

In practical engineering such as aerial refueling pipes, the boundary of the fluid-conveying pipe is difficult to be completely immovable. Pipes under movable and immovable boundaries are controlled by different dominant nonlinear factors, where the boundary mobility will affect the nonlinear dynamic characteristics, which should be focused on for adopting different strategies for vibration suppression and control. The nonlinear dynamic instability characteristics of functionally graded fluid-conveying pipes lying on a viscoelastic foundation under movable and immovable boundary conditions are systematically studied for the first time. Nonlinear factors involving nonlinear inertia and nonlinear curvature for pipes with a movable boundary as well as tensile hardening and nonlinear curvature for pipes with an immovable boundary are comprehensively considered during the derivation of the governing equations of the principal parametric resonance. The stability boundary and amplitude-frequency bifurcation diagrams are obtained by employing the two-step perturbation- incremental harmonic balance method (TSP-IHBM). Results show that the movability of the boundary of the pipe has a great influence on the vibration amplitude, bifurcation topology, and the physical meanings of the stability boundary due to different dominant nonlinear factors. This research has guidance significance for nonlinear dynamic design of fluid-conveying pipe with avoiding in the instability regions.