Mathematical Modeling of the Amplitude of Transverse Vibrations of Homogeneous Rods Under Longitudinal Impact

Bityurin, A. A.

doi:10.3103/s0025654421020047

Cited by 5 publications

(3 citation statements)

References 3 publications

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“…In this case, equations ( 3), ( 6), and conditions ( 7)-( 10) are reduced to the forms ( 11), (12), and ( 13) -( 15)…”

Section: Methodsmentioning

confidence: 99%

“…E3S Web of Conferences 401, 02003 (2023) CONMECHYDRO -2023 https://doi.org/10.1051/e3sconf/202340102003 9) is taken into account in (12), and the defining relation ( 2) is taken into account in (13). Thus, the impact problem under consideration was reduced to solving a nonlinear problem with an unknown moving boundary -equations ( 11) and ( 12) with conditions ( 13) - (15).…”

Section: Methodsmentioning

confidence: 99%

“…Thus, the problem of finding an approximate solution to problem ( 11)-( 15) was reduced to solving the Cauchy problem for a system of three nonlinear equations of the first order (12), (20), and ( 21) under the following initial conditions…”

Section: Methodsmentioning

confidence: 99%

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Impact of rigid body and viscous-plastic rod of finite length

Begmatov

Mamatova

2023

E3S Web of Conf.

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The impact of a rigid body and a rod of finite length made of a viscoplastic incompressible material moving towards each other at constant velocity is studied in the article. One-dimensional motion is considered, i.e., velocity, stress, and other parameters are considered to be averaged over the rod section. According to the model of a viscous-plastic material, the rod is divided into two parts: the region of plastic strain and the region that moves as a solid body after impact. The boundary between these regions is unknown and needs to be determined. As a result, we arrive at a problem with a free boundary, a well-known example of which is the Stefan problem. However, the formulation of the problem under consideration differs significantly from the Stefan problem. In addition, the function to be determined that describes the unknown boundary, in contrast to the Stefan problem, is not explicitly present in the boundary conditions. There are various numerical methods for solving problems with an unknown moving boundary. However, as is known, implementing these methods is associated with significant difficulties. In this article, the method of integral relations is used - a modification of the Karman-Pohlhausen method known in the boundary layer theory. The problem is reduced to the Cauchy problem for a system of nonlinear ordinary differential equations proposed to be solved by successive approximations and the Runge-Kutta method. Numerical calculations were performed. The influence of changes in the mass and magnitude of the velocities of the rod and rigid body on the size of the plastic strain region and the change in stresses on the contact surface is revealed.

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“…In this case, equations ( 3), ( 6), and conditions ( 7)-( 10) are reduced to the forms ( 11), (12), and ( 13) -( 15)…”

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

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Impact of rigid body and viscous-plastic rod of finite length

Begmatov

Mamatova

2023

E3S Web of Conf.

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Modeling of the oscillation amplitude of a statically indeterminate flat frame subjected to the impact load with account taken of the contact deformation

Bityurin

2022

Vestnik MGSU

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Introduction. The article addresses transverse vibrations of a post in a rigidly fixed statically indeterminate flat frame subjected to the falling load, having pre-set mass and pre-impact velocity. The post of a vertical frame has initial curvature affecting the maximum amplitude of transverse vibrations triggered by the impact. Materials and methods. In the course of modeling, the frame beam impact takes account of deformations in the contact area to ensure the accuracy of calculations, because otherwise the impact force value will be overestimated. It is assumed that the falling load has the shape of a cylinder featuring a certain length, when the transverse load impact on a frame beam is simulated. The linearization of the relationship between the force and the deformation of cylindrical surfaces is used. The frame post, connected to the beam and the base, is a homogeneous rod rigidly fixed at both ends. Following the load impact on the frame beam, the top end of the post absorbs the initial bending moment and the shear force, whose values depend on the mass the and pre-impact velocity of the load, the beam attachment method, and the ratio between axial moments of inertia in beam and post sections in the plane of the frame. Using the method of initial parameters, the maximum deflection of a perfectly straight and curved post is calculated for different pre-impact velocity values of the load. The maximum deflection was modeled for curved and perfectly straight posts at different pre-impact velocities of a load having the same mass. Results. The dependence between the value of maximum deflection of a vertical frame post and the pre-impact velocity of the falling load was obtained.

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Classical solution to mixed problems from the theory of longitudinal impact on an elastic semi-infinite rod in the case of separation of the impacting body after the collision

Korzyuk,

Rudzko

2024

Vescì Akademìì navuk Belarusì. Seryâ fizika-matematyčnyh navuk

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In this work, we consider two coupled initial-boundary value problems, which, based on the Saint-Venant theory, model the longitudinal impact phenomena in a semi-infinite rod. The mathematical formulation of the problem is two mixed problems for the one-dimensional wave equation with conjugation conditions. The Cauchy conditions are specified on the spatial half-line. The initial condition for the partial derivative with respect to the time variable has a discontinuity of the first kind at one point. The boundary condition, which includes the unknown function and its first- and second-order partial derivatives, is specified on the time half-line. The solution is constructed by the method of characteristics in an explicit analytical form. The uniqueness of the solution is proved, and the conditions under which a piecewise-smooth solution exists are established. The classical solution to a mixed problem with matching conditions is considered.

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Mathematical Modeling of the Amplitude of Transverse Vibrations of Homogeneous Rods Under Longitudinal Impact

Cited by 5 publications

References 3 publications

Impact of rigid body and viscous-plastic rod of finite length

Impact of rigid body and viscous-plastic rod of finite length

Modeling of the oscillation amplitude of a statically indeterminate flat frame subjected to the impact load with account taken of the contact deformation

Classical solution to mixed problems from the theory of longitudinal impact on an elastic semi-infinite rod in the case of separation of the impacting body after the collision

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