Arbitrary Dynamic Loads Research Articles

Free and Forced Vibrations of the Carrier Beam of the Vehicle Chassis

The article is dealing with the stress-strain state of the carrier beam of the two-axle vehicle chassis for a dynamic perturbing load that occurs in the process of driving along the unevenness of the road, taking into account the elastic characteristics of the spring suspension. When analyzing the ride smoothness, it is necessary to take into account low-frequency forced vibrations caused by road irregularities, as well as free low-frequency vibrations. The dynamic calculation was carried out by the force method. In this case, the expansion of a given arbitrary dynamic load is performed in terms of the main modes of vibration. The calculation for free and forced vibrations was made by an exact analytical method, taking into account the elastic pliability of the axial supports of the beam. In the course of the study, the external dynamic forces acting on the structure under consideration, reduced to three-point masses, were determined and three forms of free vibrations were obtained. The results of the study can be used to optimize the design of the truck chassis and to improve their reliability and safety in operation. By understanding the behavior of a suspension system under dynamic loads, engineers can design stronger, more reliable designs that can withstand the stress of transporting heavy loads over rough roads. In general, this study contributes to the development of more efficient and safer transport systems

ON THE ISSUE OF DYNAMIC CALCULATION OF THE ROADBED BY DIRECT INTEGRATION METHOD

The article considers the methods of dynamic calculations of the roadbed under the action of arbitrary dynamic loads. When analyzing elastic systems, it is convenient to separately consider the static and dynamic components of the applied load to separately determine the reaction from each type of load, and then combine both reaction components to evaluate the overall reaction. Following this approach, static and dynamic calculation methods can be considered fundamentally different in nature. Within the framework of this approach, the concept of dynamic can be defined as time-varying. Then a dynamic load is any load whose magnitude, direction, and location of application change over time. The article shows that similarly, the reaction of the system under dynamic load also varies over time and is dynamic.

Integral equations of the second kind in dynamic analysis of nonlinear systems with a finite number of degrees of freedom under arbitrary dynamic loading and material dependencies

Introduction. Development of computational methods for nonlinear systems possesses a significant potential, considering that linear theory sometimes fails to accurately describe the properties of dynamic systems, and the linear approximation gives only a very rough idea of real processes for a number of cases.Aim. Calculating linear systems and deriving the resulting equations for nonlinear systems involves impulse response and transfer functions of linear “generating” systems of differential equations. Such an approach in comparison with the traditional method of the so-called normal forms enables the calculation algorithm to be simplified considerably, avoiding several stages and presenting the solution by means of the normal mode method for linear systems directly with respect to the generalized coordinates.Materials and methods. The paper presents a method and an algorithm developed for the calculation of nonlinear systems with a finite number of degrees of freedom under arbitrary dynamic loading and material nonlinearity. Systems of nonlinear differential equations were reduced to nonlinear integral equations of the second kind, considered as resulting equations. The solution was developed in time steps, the value of which, among other things, determines the accuracy of the solution and the nature of the computational algorithm.Results. The paper presents main computational dependencies in a generalized form, convenient for numerical simulation. The author provides solutions for a nonlinear system with one degree of freedom and a cubic reactiondisplacement relation, as well as for a system with one and two degrees of freedom with a viscous damper. In both cases, the developed solution contains all properties of nonlinear systems, including the jump (transition) from the upper ascending branch to the lower, stable one, and the associated excitation of free oscillations.Conclusions. According to the calculations, the occurrence of nonlinear effects in oscillating systems makes positive impact on their behavior, in resonant modes in particular.

Dynamic analysis of segmental linings of shield tunnels using a state space method and its application in physical test interpretation

Based on the state space method and D'Alembert's principle, a new analytical method is established to study the dynamic response of circular segmental linings of shield tunnels. The tunnel lining segments are considered to be curved Euler–Bernoulli beams, the joints connecting the segments are modeled by a series of distributed springs, and the lining–surrounding soil interaction is represented by distributed reaction springs with viscous dampers. With the state space method, the governing equations can be concisely expressed in matrix form, and a separate variable method in conjunction with a frequency scanning strategy is employed to obtain the natural frequencies and corresponding modal shapes of the free vibration of the linings. The orthogonality of the vibration modes of the linings is demonstrated using a symplectic inner product, and the response of the linings under arbitrary dynamic loads is solved using the modal superposition method. Numerical and model experimental examples are presented to verify the efficiency and robustness of the proposed method. We prove that the proposed method can solve the dynamic response of segmental linings by considering the effect of joint and asymmetric loading conditions with high efficiency and accuracy.

Robust design of monopiles for offshore wind turbines considering uncertainties in dynamic loads and soil parameters

Complex marine environment causes significant variations in the dynamic loads and soil parameters that affect the design of a monopile. Despite these uncertainties, a multi-objective optimization technique is developed to produce the optimal designs. This optimization is based on the robust design concept, in which the design robustness, cost, and safety are considered as the optimal objectives. The responses of a monopile to arbitrary dynamic loads are initially determined using a transient finite element technique, which considers unknown soil parameters. These responses include the maximum rotation and displacement at the midline, maximum axial stress, and fatigue damage. Next, the variations (regarding standard deviations) in these responses, which are determined by the Monte–Carlo simulations, are adopted to define the design robustness. Finally, the cost and safety requirements of the monopile are evaluated, and robust design optimization is performed. The aforementioned robust design protocols are demonstrated with a 5 MW offshore wind turbine, which produces the optimal design. Therefore, the optimal designs for the monopile are investigated based on various safety requirements. Moreover, the design parameters that promote the design robustness with optimum efficiency are studied and proposed.

Generalization of Duhamel's integral to multi-degree-of-freedom systems

This article presents a generalization to multi-degree-of-freedom (MDOF) systems of the classical convolution formula for single-degree-of-freedom (SDOF) systems that is widely known as Duhamel's integral. Remarkably, the extended convolution formula for a MDOF system is ultimately cast directly in terms of the system stiffness, mass and damping matrices in a manner that fully parallels the SDOF case, and without resorting to real or complex modal superposition. Still, although the proof based on the use of the Laplace transform might suggest that the obtained formulae are applicable to any type of viscous damping matrix, a more careful evaluation demonstrates that they contain a rather subtle constraint that implies that damping is of the classical type. The free vibration elicited by non-vanishing initial conditions is also considered. As a result, we ultimately obtain the general solution for MDOF systems with arbitrary dynamic loads and initial conditions. For undamped systems, the equivalence of the classical modal analysis with the direct approach proposed herein is demonstrated analytically. Special closed-form solutions, singular solutions and numerical examples are also investigated to a limited extent so as to further illustrate the application of the proposed formulae in the context of practical problems.

Analytical solution for vibrations of a curved tunnel on viscoelastic foundation excited by arbitrary dynamic loads

The paper proposes a new analytical solution for the dynamic response of a curved tunnel resting on a viscoelastic foundation subjected to dynamic loads in arbitrary forms. For the derivation, the governing differential equations and boundary conditions of the problem are established based on the Hamilton principle and the dynamic equilibrium theory. The modal superposition method is employed to obtain the explicit formulations of tunnel response, including deflection, velocity, acceleration, bending moment and shear force. The proposed solution is verified by providing comparisons between its results and those from the Finite Element program ABAQUS. Further parametric analysis explore the influence of radius of curvature and soil-structure relative stiffness ratio on the dynamic response of the tunnel. The proposed solution can be used by practitioners for the design of curved tunnels or pipelines.

NONLINEAR PROBLEM OF FRACTURE MECHANICS OF AN INTERFACE CRACK SUBJECTED TO SHEAR WAVE

Solution for the problem for an interface crack under the action of a harmonic shear wave in normal direction is presented. The contact of the crack faces is put into consideration. The problem is solved by the boundary integral equations method, the vector components in the boundary integral equations are presented by Fourier series. The unilateral Signorini boundary conditions are involved to prevent the interpenetration of the crack faces and the emergence of tensile forces in the contact zone. Amonton-Coulomb Friction Law included allows to put into consideration relative resting of the crack opposite faces or their relative motion when one crack face is slipping or sliding across another face. The contact boundary restrictions are implemented using the iterative correction algorithm. The mathematical model adequacy is checked by comparing with classical model solution for statics problems that takes into account the crack faces contact. Numerical researches of friction influence on the displacement and contact forces distribution, size of contact zone are carried out. Influence of the faces contact and value of the friction coefficient on the distribution of stress intensity coefficients of normal rupture and transverse shear, which are the parameters of the biomaterial fracture, are presented and analyzed. It is shown that the nature of change in the distribution of the stress intensity coefficients for the conditions of tensile and shear waves is fundamentally different. It is concluded that it is possible to extend the approach proposed to the problems of three-dimensional fracture mechanics for composites with interfacial cracks at arbitrary dynamic loading.

Local stress constraints in topology optimization of structures subjected to arbitrary dynamic loads: a stress aggregation-free approach

Local stress constraints in topology optimization of structures subjected to arbitrary dynamic loads: a stress aggregation-free approach

Topology optimization of dynamic problems based on finite deformation theory

AbstractThis study proposes a topology optimization method for dynamic problems based on the finite deformation theory to derive a structure that can reduce deformations for arbitrary dynamic loads that have large deformations. To obtain a structure that can minimize deformations due to dynamic loading for the isotropic hyperelastic model, the square norm of dynamic compliance is set as the objective function. A sensitivity analysis method of the equation of motion, based on Newmark's method for unknown displacements, is presented in the current study. The analysis is carried out by developing a general design sensitivity equation that can accurately account for the response of the structure to dynamic loading and simultaneously display a high affinity for the general constitutive law by using the adjoint variable method. The accuracy of the obtained sensitivity is verified by using the finite difference method as a benchmark. Numerical examples are then used to demonstrate the validity of the proposed method. The results show that the proposed method is able to derive optimization results according to the magnitude of the applied load.

DYNAMIC PROBLEM FOR A THIN-WALLED BAR WITH A MONOSYMMETRIC PROFILE

The paper presents an analytical solution to the dynamic problem for a thin-walled elastic rod, thecross-section of which has one axis of symmetry. The solution is constructed for an arbitrary dynamic load and two types of boundary conditions: hinged support in constrained torsion and free warping of the end sections of the rod; rigid fastening with constrained torsion and absence of warping. The peculiarity of the mathematical model lies in the fact that the differential equations of motion contain a complete system ofinertial terms. Spectral expansions obtained as a result of using the method of integral transformations are represented as an effective method for solving linear non-stationary problems in mechanics. The structuralalgorithm of the method of finite multicomponent integral transformations proposed by Yu.E. Senitsky is used.

Modeling a Dynamic Response at Resonant Vibrations of an Elongated Plate with an Integral Damping Coating

Classical methods of surface damping using free and constraining damping layers are discussed. The structure of a perspective integrated version of a damping coating is presented. This integral damping coating consists of two layers of a material with pronounced viscoelastic properties, between which there is a thin reinforcing layer of a high modulus material. A generalization of the Thompson-Kelvin-Voigt model is given for the description of viscoelastic properties of the material under tension-compression in the case of a complex stress state. A finite-element method was developed to determine the dynamic response of an elongated plate with the integral damping coating. This method is based on a four-layer finite element with 14 degrees of freedom: the main material is within the Kirchhoff-Love's model, the damping layers are in a flat stress state, the reinforcing layer perceives tension and compression. This model allows us to take into account the effect of transverse compression of the damping layers of the plate, which significantly increases its damping properties at high vibration frequencies. The stiffness matrices, the damping matrices, and the mass matrices of the constituent layers aim at obtaining similar complete matrices of a finite element. A system of resolving equations was obtained on the basis of the Lagrange equations of the second kind with respect to the vector of nodal displacements of the finite element model of the plate with an arbitrary dynamic load. In the case of a harmonic load with a frequency that coincides with one of the frequencies of free vibrations of the plate, a transition to a modal equation with respect to the normal coordinate corresponding to the given frequency is possible. Numerical experiments were carried out to test the developed finite element method using the example of a hingedly supported elongated plate with an integral damping coating. The numerical experiments showed a qualitative change in the composition of stresses in the damping layers of the plate at high vibration frequencies, which significantly affects its damping properties.

Structural Monitoring of Underground Structures in Multi-Layer Media by Dynamic Methods.

The actual problem of structural monitoring and modeling of dynamic response from buried building is considered in the framework of arbitrary dynamic load. The results can be used for designing underground transport constructions, crossings, buried reservoirs and foundations. In existing methods, the system of sensors that register the response to a dynamic action does not allow for effective interpretation of the signal without understanding the dynamic features and resonance phenomena. The analytical and numerical solution of the problem of the dynamics of a buried object in a layered medium is considered. A multilayer half-space is a set of rigidly interconnected layers characterized by elastic properties. At a distance, an arbitrary dynamic load acts on the half-space, which causes oscillations in the embedded structure, and the sensor system registers the response. The problem of assessing the dynamic stress-strain state (DSSS) is solved using Fourier transforms with the principle of limiting absorption. As an example, the behavior of an embedded massive structure of an underground pedestrian crossing under the influence of a dynamic surface source on a multilayer medium is considered, as well as instrumental support of the sensor system. The solution in the form of stress, strain and displacement fields is obtained and compared with the experimental data. The frequency-dependent characteristics of the system are determined and the possibility of determining the DSSS by a shock pulse is shown.

A generalized bond-based peridynamic model for quasi-brittle materials enriched with bond tension–rotation–shear coupling effects

A generalized bond-based micropolar peridynamic model is proposed to simulate the nonlinear deformation and mixed-mode crack propagation of quasi-brittle materials under arbitrary dynamic loads. The mechanical behaviors of the material points are reformulated by incorporating the bond tension–rotation–shear coupling effects, which makes the model suitable for complex discontinuous problems in both three-dimensional spatial and two-dimensional plane conditions. The governing equations of the bond are established by employing the Timoshenko beam theory to simulate the interaction between material points as well as the bond coupling effect. Three kinds of peridynamic parameters, corresponding to the compressive, shear and bending stiffness of the bond, are introduced to keep the consistence of the strain energy obtained from the proposed peridynamic model and from the continuum mechanics under arbitrary deformation fields. Moreover, a novel energy-based failure criterion, involving the maximum stretch, shear strain and rotation angle limits of the bond, is proposed to describe the nonlinear behaviors and progressive failure for general quasi-brittle materials. The proposed model is verified by providing comparisons between its results and those from known analytical solutions and experimental observations. The influences of the bond tension–rotation–shear coupling effect as well as the applicability of the proposed model for the wave propagation, complex deformation and mix-mode fracture problems are also investigated. Results show that the proposed model with the bond coupling effect will greatly improve the simulation accuracy for dynamic problems of quasi-brittle materials under complex loading conditions. Results also indicate that the proposed model can well capture the nonlinear deformation, crack propagation, as well as progressive failure of materials with variable Poisson’s ratios under complex combined loads

НЕЛИНЕЙНАЯ ДИНАМИКА ТОПОЛОГИЧЕСКИ ОПТИМАЛЬНОЙ НАНО БАЛКИ ТИМОШЕНКО НА ОСНОВЕ МОДИФИЦИРОВАННОЙ МОМЕНТНОЙ ТЕОРИИ

Актуальность исследования. Для контроля данных технических операций бурения скважин, различного вида ремонта скважин, геологоразведочного бурения в нефтяной и газовой промышленности на буровых и ремонтных установках всех типов используются технологические датчики (датчики нагрузки, давления, температуры жидкости, выхода раствора, плотности жидкости). Современные датчики имеют малые размеры и для повышения чувствительности изготавливаются на базе наноэлектромеханических систем, которые включают в себя составляющие элементы, нанобалки и нанопластины. Эти элементы работают при высоких температурах и подвергаются механическим нагрузкам различного вида. Увеличение прочности этих элементов является несомненно актуальной задачей. Цель: построить математическую модель на базе кинематической гипотезы Тимошенко и модифицированной моментной теории чувствительного элемента в виде балки, нано электромеханического датчика под действием механических и тепловых полей;создать методологию получения оптимальной топологии нанобалки для произвольнойстатической и динамической нагрузкии различных граничных условий с целью увеличения ее жесткости;провести сравнительный анализ статики и нелинейной динамики оптимальной и неоптимальной балок. Объекты: элемент наноэлектромеханических систем в виде балки с учетом оптимальной микроструктуры. Методы: методы топологической оптимизации, вариационные методы, метод конечных разностей второго порядка, методытипа Рунге–Кутта, Фурье и вейвлет анализ, фазовый портрет и сечение Пуанкаре. Результаты. Построена методология получения оптимальной микроструктуры нанобалки на основе топологической оптимизации.На основе принципа Гамильтона–Остроградского построена математическаямодель неоднородной в двух направлениях (по толщине и длине) нанобалки Тимошенко на базе модифицированной моментной теории. Проведен сравнительный анализ статического изгиба и нелинейной динамики оптимальной и неоптимальной нанобалок.

Analytical Solution for Dynamic Response of Underground Rectangular Fluid Tank Subjected to Arbitrary Dynamic Loads

AbstractIn this paper, the dynamic response of underground rectangular fluid tank resting on an elastic foundation and subjected to arbitrary dynamic loads is developed in the form of analytical so...

Reduced model for the surface dynamics of a generally anisotropic elastic half-space

Near-surface resonance phenomena often arise in semi-infinite solids. For instance, when a moving load with a speed v close to the surface wave speed v R is applied to the surface of an elastic half-space, it will give rise to a large-amplitude disturbance inversely proportional to v − v R . The latter can be determined by a multiple-scale approach using an extra slow time variable. It has also been shown for isotropic elastic half-spaces that the reduced governing equation thus derived is capable of describing the surface wave contribution even for arbitrary dynamic loading. In this paper, we first derive the analogous evolution equation for a generally anisotropic elastic half-space, and then assess its applicability in the study of travelling waves in a half-space that is coated with a continuous array of spring-like vertical resonators or bonded to an elastic layer of different properties. Our results are validated by comparison with previously known results, and illustrative calculations are carried out for a fibre-reinforced half-space and a coated half-space that is subjected to a finite deformation.

Exact analytical solution to the 3D Navier-Lame equation for a curved beam of constant curvature subject to arbitrary dynamic loading

This paper presents an exact analytical solution to the 3D transient dynamics of a linear elastic, isotropic homogeneous, curved beam, with uniform rectangular cross-section. The solution technique uses vector identities to decouple the governing equations. The decoupled equations are then solved by method of separation of variables. The solutions to the decoupled equations can be recombined to form a new equation. Solving this new equation yields the displacement field. To demonstrate the capabilities of the proposed solution technique, a generic case study was modeled and computed. A curved beam is subjected to a longitudinal impulse loading and the transient displacement field is calculated. This solution technique is valid for any curvature so long as the curvature is the same throughout the beam. This includes the limiting case where the inner radius of the beam goes to zero and the curved beam becomes a section of a disk. The presented solution results were compared with an approximate solution from the literature and experimental data from the literature. The solution is also compared with an explicit FEM solution conducted by the Authors. The presented solution agrees with the results from the literature and from the FEM solutions conducted by the Authors. This paper demonstrates that the accuracy and robustness of the proposed solution technique meets the needs of many potential applications.

Analytical solution for a finite Euler–Bernoulli beam with single discontinuity in section under arbitrary dynamic loads

• Analytical solution is derived for finite beam on viscoelastic foundation to arbitrary loads. • The foundation has a step change at the discontinuous section of the beam. • The modal shape superposition scheme is used to obtain explicit solution. • The distinct beam element method is employed for free vibration analysis. • Degraded solutions for two special loads are expressed and validated. In this paper, the dynamic response of a finite Euler–Bernoulli beam with single discontinuity in section lying on viscoelastic foundation subjected to arbitrary dynamic loads is investigated. The viscoelastic foundation has a step change at the discontinuous section of the beam. Based on the modal superposition method, the explicit formulations of the problem are obtained for beam deflection, velocity, acceleration, bending moment and shear force. The natural frequencies and corresponding modal shape functions of the beam are obtained by imposing continuity at the contact between different components of the beam. The solution is verified by providing comparisons between its results and those from the Finite Element program ABAQUS.

Comparative study on damping models of linear system subjected to arbitrary dynamic loading

Linear and nonlinear damping models, correlated through equivalent damping coefficient, have been developed for discrete vibrating system employing energy equivalence principle at steady state response of the system. Application of, so developed, damping models to derive response of discrete vibrating system subjected to arbitrary dynamic loading is not obvious due to transient nature of the system. Thus, the equivalent damping coefficient of damping models needs to be redefined to derive realistic transient response of the system. In the present study, first cycle damping coefficient (FCDC) approach is proposed that uses the first cycle transient response of the system to derive equivalent damping coefficient for linear and nonlinear damping models through genetic algorithm propelled energy equivalence with that of linear viscous damping model. A single degree of freedom (SDOF) system with linear and nonlinear damping models subjected to arbitrary dynamic loading are solved. Efficacy of proposed FCDC approach is established through the experimental study on a lightly damped single storey building model. Velocity squared damping model yields higher peak displacement response for the system vis-a-vis system with linear viscous damping model. Proposed FCDC approach can be effectively utilised if the transient response of the system is available experimentally.