Frequency Of Forced Vibration Research Articles

Гармонический анализ крутящего момента двигателя ЯМЗ-5340

The purpose of the study is to find a technique and conduct on its basis a harmonic analysis of the torque of an internal combustion engine for the subsequent determination of resonant vibrations of the crankshaft. The method of calculating the harmonics of the engine torque is given on the example of a YaMZ-5340 diesel engine with a power of 100 kW at a crankshaft rotation speed of 2300 min-1. Based on the results of thermal and dynamic calculation of the YaMZ-5340 engine, a graph of changes in its torque is constructed and harmonics of the first and second orders are calculated. The frequencies of natural and forced vibrations of the crankshaft and its resonant critical rotation frequency are determined. The number of the torque harmonic has been determined, which, according to the frequency of forced vibrations, coincides with the frequency of natural vibrations and causes resonance. The results of the study can be applied in the design and fine-tuning of engines in order to calculate and reduce torsional vibrations.

Nonlinear Forced Vibration of Functionally Graded Graphene-Reinforced Composite (FG-GRC) Laminated Cylindrical Shells under Different Boundary Conditions with Thermal Repercussions

The key aim of this research is to develop an analytical model for the forced vibration of graphene-reinforced composite (GRC) cylindrical shell with viscous damping and thermal environment effects under several boundary conditions. The prescribed mechanical and thermal characteristics of the GRC layer are evaluated utilizing a micromechanical extended Halpin–Tsai technique. Further, the governing equations are determined to be grounded on the shear deformation theory (SDT) with the von Kármán-form in terms of the geometric nonlinearity. The basic nonlinear partial differential governing equation is transformed into a nonlinear ordinary differential equation with the Galerkin-based method. The resulting ordinary governing differential equations are systematically solved based on the multiple scales scheme in order to obtain the nonlinear forced vibration frequency response of the laminated GRC cylindrical shell in the presence of damping impact and subjected to multiple boundary conditions. The validation of the obtained expressions is achieved by comparing them with the available literature data where the comparison revealed a clear consistency between them. Moreover, a parametric investigation is provided to illustrate the impacts of the distribution of graphene layers, elastic foundation coefficients, damping ratios, temperature effects and dimensionless radial excitation amplitude on the forced nonlinear amplitude-frequency ratio responses for a functionally-graded graphene-reinforced composite (FG-GRC)-layered cylindrical shells. The analytical outcomes indicate that the different distribution types of graphene, elastic foundation coefficients, damping ratios, temperature, and radial excitation parameters have a noteworthy impact on the frequency–amplitude behaviors of an FG-GRC-layered cylindrical shell. In addition, the new insights gained from this research might contribute to a deeper understanding of the nonlinear forced vibration responses in subsequent analysis and design techniques for giving appropriate benchmark findings.

Forced vibrations of a finite length metabeam with periodically arranged internal hinges and external supports

The paper investigates a problem of forced vibrations of two finite length metabeams: one is with periodically arranged internal hinges and the second one with periodically arranged internal hinges and external supports. Based on the Euler–Bernoulli beam theory and the transfer matrix method, general solutions of the periodic beams are obtained. Applying the Bloch-Floquet theory, the explicit expressions are derived defining the metabeams band gap structures. The corresponding band gap dispersion curves are plotted and analysed. It is shown that when the frequency of forced vibrations coincides with the band gap frequencies a strong localization of flexible waves occurs at the interfaces of the periodic beams. The localization of flexible waves increases significantly with the number of hinges and supports.

Analytical modeling and numerical analysis of thermoelastic damping in ultrathin elastic films due to surface effects

Analytical techniques used for estimating thermoelastic damping by incorporating both mechanical and thermal interactions between surfaces and the rest of the bulk are intricate and challenging due to the limited understanding of the damping mechanisms in extra-thin films subjected to forced vibrations. This paper proposes a modified model to analytically calculate the thermoelastic damping of ultrathin elastic films due to surface effects and analyzes the thermoelastic damping variation with different factors through numerical experiments on two materials. The model considers surface stresses derived from the elastic surface theory using Kirchhoff's kinetic hypothesis and determines thermoelastic damping by considering thermal dissipation and elastic potential energy. The results show that surface effects significantly influence the thermoelastic damping of the film, and the specific behavior of a thin film’s thermoelastic damping with respect to film thickness is impacted by various factors, including material property, the variation range of film thickness, and the forced vibration frequency. This study provides insights into the thermoelastic damping behavior of thin films and has important implications for the development of nanoscale oscillators in MEMS or NEMS systems.

Experimental and analytical investigation on forced and free vibration of sandwich structures with reinforced composite faces in an acidic environment

The main objective of this study is to investigate the impact of nanoparticles as reinforcement material on the vibrational behavior of sandwich structures in an acidic medium. The glass fiber reinforced polymer (GFRP) faces were fabricated with and without the addition of 3 wt% nanoclay and nanosilica to determine the mechanical behaviors of the GFRP faces in the presence of an acidic medium. The obtained results showed adding 3 wt% of nanoclay caused better durability and less mass variation of composite specimens in sulfuric acid. The “Coefficient of acidic immersion expansion” (βacid) is determined by measuring the length and mass variation of GFRP specimens in the immersion, and applied to low order piecewise shear deformation theory (LOPSDT) for the first time; Also the frequency results of LOPSDT have been shown good agreement in validation with the ANSYS numerical solution. It is shown that acidic environment reduces the frequency of the first mode of sandwich plates with reinforced face by 3 wt% nanosilica, and nanoclay has increased by 6.81 % and 4.66 %, respectively. This study indicates after one month of immersion, the natural frequency of the sandwich with pure, and 3 wt% nanoclay reduces about 1 %, and the natural frequency of the sandwich with the faces reinforced with 3 wt% nanosilica reduces by more than 3 %; Moreover, the frequency of forced vibrations, caused by acidic immersion expansion, was improved significantly by 10.04 % and 6.54 % in the first mode by incorporating 3 wt% of nanoclay, and nanosilica into the faces of the sandwich in one month of immersion compared to the sandwich with pure faces.

Effectiveness of Friction Force Reduction in Sliding Motion Depending on the Frequency of Longitudinal Tangential Vibrations, Sliding Velocity and Normal Pressure

Abstract The article presents the results of experimental research and simulation analyses of the influence of slip velocity, normal pressures and vibration frequency on the effectiveness of friction force reduction carried out in sliding motion in the presence of forced tangential vibrations. In experimental studies, changes in the driving force were measured during the slip of the upper body over the vibrating lower body. The direction of these vibrations was parallel both to the contact plane and to the direction of movement of the shifted body. The simulation tests were carried out in the Matlab/Simulink environment through the use of numerical procedures that were specially created for this purpose. Dynamic friction models considering the tangential compliance of contact and the phenomenon of pre-sliding displacement were used for calculations. The paper presents the designated values of the so-called coefficient of average friction force reduction in sliding motion for the following friction pairs: steel C45–steel C45, steel C45–cast iron GGG40 and steel C45–polytetrafluoroethy-lene PTFE (Teflon). The results of numerical analyses were in good agreement with those of experimental tests. A significant dependence of the level of average friction force reduction on the frequency of forced vibrations, sliding velocity as well as the kind of sliding pair material, and normal pressures was shown.

On the primary resonance of laminated moderately-thick plates containing of heterogeneous nanocomposite layers considering nonlinearity

In this study, the primary resonance of laminated plates consisting of heterogeneous nanocomposite layers is investigated comparatively within first order shear deformation theory (FSDT) and classical lamination plate theory (CLPT). One of the features of the work is the generalization of the FSDT for homogeneous anisotropic laminated plates to laminated functionally graded anisotropic nanocomposite plates. The mechanical properties of the matrix reinforced with carbon nanotube (CNT), and fundamental relations of laminated moderately thick plates consisting of heterogeneous anisotropic nanocomposite layers are modeled theoretically within FSDT using von Kármán type nonlinear theory. It then derives equations of motion in the form of nonlinear partial differential equations (NL-PDEs) within FSDT. NL-PDEs equations are transformed into nonlinear ordinary differential equations (NL-ODEs) by Galerkin method and are solved by multi-scale method. The nonlinear forced vibration frequency as the function of amplitude at primary resonance within both theories are obtained for the first time. In addition, backbone curve and nonlinear frequency/linear frequency ratio are found. The reliability and accuracy of the proposed formulation is verified by comparing with the results in the literature. Finally, the effects of external excitation, non-linearity, and variation of CNT patterns on the forced vibration frequencies are examined.

Numerical study of the parametric optimization of the forced oscillation frequencies of the shell of a minimal surface on a trapezoidal contour under thermal and power loading

This research paper discusses various methods and approaches to the optimal design of structures. Methods for solving the optimization problem can be divided into two large groups. The first group includes methods that are based on the use of the necessary conditions for the extremes of the objective function. The second group consists of mathematical programming methods: linear, convex, dynamic programming, and random search. In mathematical terms, optimal design problems are optimization problems - the search for an extremum of the objective function and the values of the parameters at which the extremum is achieved. The choice of the optimality criterion is one of the main problems of optimal design. The most widely developed problems are those that have the optimization criterion of weight or volume of the structure while satisfying the conditions of strength, rigidity and stability.
 Optimal design problemsare also divided into three large groups. The first group is parametric optimization problems, which involve the optimization of one or more parameters, called design variables, to minimize or maximize the objective function. The second group is topological optimization, in which unnecessary material is discarded, where the Mises stress is zero, thereby minimizing the objective function. The third group is optimization of the shape of the object under study, when the shape corresponds to internal forces, the shell with the smallest area is modeled on a given cone (shells of minimal surfaces), as well as methods of applied geometry, where the surface shape is modeled for a certain load.
 To perform the parametric optimization of the forced vibrations of the shell of the minimum surface on a trapezoidal contour, the objective function is the weight of the spatial structure. The variables in the parametric optimization problem are the thickness of the finite elements from 1 to 100 mm. The structure constraint is imposed on the first forced oscillation frequency of 0.250 Hz. This type of problem is used to prevent resonance from process equipment that can affect the natural frequencies of the structure under external load.Subject of this study is an interesting applied problem for construction mechanics, as it is the first time to display the application of two types of optimization on one research object.
 The results of a numerical study of the parametric optimization of the minimum surface shell on a trapezoidal cage under thermal power loading. The parametric optimization helped to reduce the weight of the shell by 13.4%, which is 1810 kg of sheet steel. The first forced oscillation frequency meets the constraint of the optimization calculation. We constructed 10 forced vibration frequency shapes of the shell before and after optimization, and also presented the distribution of the shell thickness after the optimization calculation.

Optimal design of the forced oscillation frequencies of the minimum surface shell on a circular contour consisting of two inclined ellipses under thermal and power loading

The problem of choosing the optimal design solution is a multi-purpose complex task, the solution of which requires taking into account a wide range of requirements for different stages of implementation. There are several levels of solving optimal design problems. The first level deals with general design models, including the main structural elements, taking into account the physical and geometric features of their behavior. This level can be called the "global" level of optimization. Due to the fact that the automated calculation of structures is impossible without the use of finite element models, and at the next "detailed" level, the parameters that make up this model (coordinates of nodes, physical and geometric properties of individual elements, etc.)
 The choice of a finite element model for optimal design is directly related to the complexity (the possibility of computer implementation) of the optimal design problem. To reduce the computational procedures associated with solving the problem of numerical analysis, an approach based on the approximation of system parameters (displacements, forces, frequencies, waveforms, etc.) can be used.
 The problem of forced oscillations for shells of minimal surfaces is solved by the finite element method using the Lagrange's variational principle. The potential and kinetic deformation energies of the shell of a minimal surface are defined in matrix form as the sum of the potential and kinetic energies of each finite element.
 The optimization algorithm for single-criterion parametric optimization is performed as follows: the objective function is the weight of the shell of the minimum surface on a circular contour consisting of two inclined ellipses, the design variables are the thickness of the shell, and the constraints are represented by the first forced vibration frequency. The results of changing the objective function are a decrease in the weight of the shell by 3050 kg of C240 steel, which is a percentage equivalent of 13.4% without loss of strength and stability of the shell of the minimum surface on a circular contour consisting of two inclined ellipses, which proves the effectiveness of the authors' methodology.

Calculation of optimal parameters of the foundation for woodworking machines with large dynamic loads

The groundwork for machines with dynamic loads should meet the calculation requirements regarding strength and suitability for regular operation and the need for occupational safety standards regarding acceptable vibration levels. In addition, fluctuations in the groundwork should not cause a harmful impact on technological processes, equipment, and devices located on the foundation or outside it.A mathematical model of foundation vibrations and a methodology for calculating the foundation parameters are presented, taking into account dynamic loads from unbalanced masses and dynamic characteristics of the ground base.The mathematical model is a solution to the differential equation of the foundation vibrations.The article refutes a number of conventional concepts regarding the body of the foundation, its base, the geometric center and the damper between the foundation and the ground.The basis for determining the vibration damping modulus is the logarithmic decrement of vibration damping.It is constant and depends on the elastic and damping properties of the soil, as well as the installation mass. The more significant the logarithmic decrement, the faster the extinction occurs. It was found that an increase in the vibration resistance coefficient allows reducing the mass of the installation with a simultaneous increase in the area of its base, so the mass acts as a limitation.It was established that to ensure vibration resistance and prevent the resonance of the foundation, the ratio of the frequency of its natural vibrations to the frequency of forced vibrations should be at least 1.5.The calculation of the optimal mass, the base area and the depth of foundation laying is carried out according to the criterion of the minimum cost of its construction.An application software has been developed for the calculation. The projection methodology for the foundation for the machine with large dynamic loads is for a two-story sawmill.

ОЦЕНКА РЕЗОНАНСНЫХ РЕЖИМОВ ПРИВОДА ГЛАВНОГО ДВИЖЕНИЯ ТОКАРНОГО СТАНКА С БЕССТУПЕНЧАТЫМ РЕГУЛИРОВАНИЕМ

The study objective is to define and evaluate the resonant drive modes of the lathe primary motion continuously rated. The problem to which the paper is devoted is to draw up a design scheme of a dynamic torsional drive system for the lathe primary motion with the spindle speed continuously rated and to make a calculation that allows finding out the lowest natural torsional vibration frequencies of the drive. Research methods: analysis of a multi-mass analytical model of the lathe drive primary motion, followed by its reduction to models with a smaller number of masses (inertial elements). The novelty of the work is in the development of software that allows simplifying linear dynamic systems and finding out natural frequencies. Study results: the dynamic quality of the drive is analyzed, which gives the opportunity to find non-resonant zones of the drive by the rotational and toothed frequencies of forced vibrations with the help of the frequency ranges of shafts. Conclusions: the developed software makes it possible at the design stage to simplify the dynamic system, find the lowest natural frequencies of torsional vibrations and take measures to exclude resonant phenomena.

Representation of In-Service Performance for Cable-Stayed Railway-Highway Combined Bridges Based on Train-Induced Response's Sensing Data and Knowledge.

Real-time representation of the current performance of structures is an important task for perceiving potential danger in in-service bridges. Methods driven by the multisource sensing data of structural health monitoring systems are an effective way to achieve this goal. Due to the explicit zero-point of signals, the live load-induced response has an inherent advantage for quantitatively representing the performance of bridges. Taking a long-span cable-stayed railway–highway combined bridge as the case study, this paper presents a representation method of in-service performance. First, the non-stationary sections of train-induced response are automatically extracted by wavelet transform and window with threshold. Then, the data of the feature parameter of each non-stationary section are automatically divided into four cases of train load according to the calculational theory of bridge vibration under train effect and clustering analysis. Finally, the performance indexes for structural deformation and dynamics are determined separately, based on hierarchical clustering and statistical modeling. Fusing the real variability of massive data from monitoring and the knowledge of mechanics of theoretical calculations, accurate and robust indexes of bridge deflection distribution and forced vibration frequency are obtained in real time. The whole process verifies the feasibility of the representation of bridge in-service performance from massive multisource sensing data. The presented method, framework, and analysis results can be used as a reference for the design, operation, and maintenance works of long-span railway bridges.

Investigation of a full-size damper for an electrically driven centrifugal pump for oil production

The article describes a way to combat fatigue effects in the details of connecting modules of an electric driven centrifugal pump unit for oil production. A constructive solution for implementing the method in relation to complex downhole conditions in the form of a multifunctional damper using a differential piston to transfer it from the transport position when lowering into the well into the working one is shown. For a full-size damper, experimental studies of its vibration- isolating characteristics have been carried out when used in the form of substrates for supporting arms of elastomers of various densities and compositions. The preferred characteristics of elastomers and their ranking for various frequencies of forced vibrations are determined. Keywords: module, connection parts; electrically driven centrifugal pump unit; electrocentrifugal pumping unit; differential piston; damper; sbstrate; vibration velocity.

On the primary resonance of non-homogeneous orthotropic structures with viscous damping within shear deformation theory

This study is one of the first attempts on the nonlinear forced vibration behaviors of nonhomogeneous orthotropic (NHO) structural members with linear viscous damping at primary resonance within the shear deformation theory (SDT). First, mechanical properties of double curved systems consisting of NHO materials are mathematically modeled and nonlinear basic relations are established. Using these relations, nonlinear basic partial differential equations are derived and reduced to ordinary differential equations with second and third order nonlinearities by Galerkin procedure. Multiple-scales method is used to obtain the nonlinear forced vibration frequency–amplitude dependence of double curved NHO structural members with damping. After testing the correctness of the proposed methodology, the influences of non-homogeneity, damping, transverse shear deformations and anisotropy on nonlinear forced vibration frequencies for various structural members at the primary resonance are investigated and interpreted in detail.

Ceramic-Based Piezoelectric Material for Energy Harvesting Using Hybrid Excitation.

This paper analyzes the energy efficiency of a Micro Fiber Composite (MFC) piezoelectric system. It is based on a smart Lead Zirconate Titanate material that consists of a monolithic PZT (piezoelectric ceramic) wafer, which is a ceramic-based piezoelectric material. An experimental test rig consisting of a wind tunnel and a developed measurement system was used to conduct the experiment. The developed test rig allowed changing the air velocity around the tested bluff body and the frequency of forced vibrations as well as recording the output voltage signal and linear acceleration of the tested object. The mechanical vibrations and the air flow were used to find the optimal performance of the piezoelectric energy harvesting system. The performance of the proposed piezoelectric wind energy harvester was tested for the same design, but of different masses. The geometry of the hybrid bluff body is a combination of cuboid and cylindrical shapes. The results of testing five bluff bodies for a range of wind tunnel air flow velocities from 4 to 15 m/s with additional vibration excitation frequencies from 0 to 10 Hz are presented. The conducted tests revealed the areas of the highest voltage output under specific excitation conditions that enable supplying low-power sensors with harvested energy.

Influences of material gradient and nonlinearity on the forced vibration of orthotropic shell structures

In this study, the influences of material gradient and nonlinearity on the forced vibration of orthotropic shell structures under external excitations are investigated for first time. The mathematical model of inhomogeneous orthotropic double-curved shallow shells is built using the Hamilton principle and von Karman-type nonlinearity. The basic equations are reduced to nonlinear ordinary differential equations using the Galerkin procedure. Using the multiscale method, the frequency-amplitude relations of double-curved shallow shells and the nonlinear frequency response of forced vibrations are obtained for first time. The reliability of the obtained expressions is checked by comparison with the literature data. In numerical analysis, the influence of inhomogeneity, orthotropy, nonlinearity, and the external excitation parameter on the frequency of forced vibrations is investigated in detail by performing unique numerical calculations taking into account various profiles of inhomogeneous orthotropic shallow spherical and hyperbolic paraboloidal (or hypar) shells.

An approach to the solution of nonlinear forced vibration problem of structural systems reinforced with advanced materials in the presence of viscous damping

In this study, the nonlinear forced vibration of composite structural systems such as plates, panels and shells reinforced with advanced materials in the presence of linear viscous damping is investigated. Hamilton principle and von Kármán-type nonlinear theory are used to obtain the theoretical model of double-curved shells reinforced by carbon nanotubes (CNTs). The nonlinear partial differential equations are reduced to ordinary differential equations using Galerkin method. By using the multiscale method, the frequency-amplitude relation and nonlinear forced vibration frequency of structural systems are obtained for the first time. Since double-curved shells can be transformed into other structural systems such as spherical and hyperbolic-paraboloid shells, rectangular plate and cylindrical panel in special cases, the expressions for nonlinear frequencies can also be used for them. In additional, the backbone curve and the nonlinear frequency/linear frequency ratio are determined as a function of the amplitude in primary resonance for the first time. The results are verified by comparing the reliability and accuracy of the proposed formulation with those in the literature. Finally, a systematic study is aimed at controlling the influence of nonlinearity and types of distribution of CNTs on the frequencies and their quantitative and qualitative variation in the presence of external excitation and viscous damping.

Comparative Analysis of Vibrations of Ring-Shaped Ultrasonic Concentrators

The paper provides a comparative analysis of the influence of ring concentrator shape in ultrasonic systems on their amplitude-frequency characteristics. Devices are known in which elastic elements are used either as resonators or as working tools of ultrasonic technological systems. However, the use of elastic elements as concentrators of ultrasonic vibrations is insufficiently studied and requires comprehensive research and development of recommendations for their practical application. For this purpose, a theoretical analysis has been carried out in the paper while using the ANSYS computer program, which made it possible to perform modal and harmonic analysis of ring models with various shapes. The round ring has a nominal outer diameter of 50 mm and a variable cross-section. Three ring models have been analyzed: one round model and two models of oval shape. To conduct a comparative analysis and identify frequencies at which resonance occurs, the vibration characteristics of the rings have been considered in the frequency range from 1 to 26 kHz. Results of the analysis show that, depending on the frequency of forced vibrations, bending vibrations are formed in the rings, which act in different coordinate planes. In this case, a change in the shape of the rings is accompanied by a change in the amplitude of the bending vibrations. The most intense ring vibrations along the vertical axis have been achieved in circular rings. It has been found that with an increase in the frequency of forced oscillations, an increase in the number of oscillation periods is observed. If in the region of low vibration frequencies only a one-period vibration mode is formed in the ring, then in the region of ultrasonic vibrations the number of vibration periods increases to two and three. All the considered ring models have several natural vibration frequencies with a certain periodicity in different coordinate planes depending on the shape of the rings. The intensity of the vibrations is different in different directions and depends on the shape and frequency of the forced vibrations. Examples of vibration mode variations for various ring shapes are demonstrated in the paper.

EXPERIMENTAL AND COMPUTATIONAL STUDY OF LIQUID OSCILLATION DAMPING IN A FREE SURFACE VESSEL

Studying the issues related to solving the problems of the movement of fluid in partially filled structures of various shapes has always been relevant.This is due, in particular, to the need to ensure the longitudinal and lateral stability of the movement of objects in which the liquid is transported. Contemporarymethods and research tools that allow describing the movement of liquid in a container are overly complex and require in-depth knowledgefrom a scientist. Therefore, development of a mathematical algorithm that would be simple and at the same time ensure sufficient accuracy in determiningthe oscillatory motion in a container is advisable. Each development of a new mathematical algorithm requires experimental research to verifyits adequacy. The purpose of this work is to confirm the feasibility of using the created mathematical apparatus for determining the main parameters offree vibrations depending on the level of liquid in a rectangular prismatic container. The experimental research methodology provides for checking theadequacy of the mathematical algorithms in determining the frequencies of free oscillations of liquid in a container by comparing the theoretical andexperimental values of the oscillation periods, as well as determining the damping decrement of oscillations for liquids of different viscosity. As a result,the adequacy of the formulas for determining the frequencies of free oscillations of liquid in a container is proved; the maximum error does notexceed 4.35%. The decrements of vibration damping are determined experimentally, as well as using theoretical models of the motion of viscous liquidin the so-called partial surface layers, for three liquids of different viscosity, namely, for water, for 20% sugar and water solution, and for vegetable oil.On the basis of experimental studies, the amplitudes and frequencies of forced vibrations of various types of liquids are determined by constructing anamplitude-frequency characteristic. It is shown by calculation how the damping decrement of the liquid affects the value of the horizontal displacementof the surface layer of the liquid in the tank during its transportation by a wheeled tractor.

ПОПЕРЕЧНЫЕ КОЛЕБАНИЯ БАЛКИ СО СВОБОДНЫМИ КРАЯМИ НА УПРУГОМ ОСНОВАНИИ ПРИ ДЕЙСТВИИ ДИНАМИЧЕСКОЙ НАГРУЗКИ

It is known that in the case of dynamic influences, the important dynamic characteristics of the structures of a structure are the frequencies and forms of natural vibrations, with which its response to dynamic effects is associated. These structures include structures lying on the ground, which can be considered when calculating them as beams on an elastic foundation. Using the method of initial parameters for a beam with free edges located on an elastic base, the problem of determining the influence of the attached mass m1 on the circular frequency of natural transverse vibrations (without taking into account the resistance forces) is solved, the first three frequencies and forms of natural transverse vibrations of the beam are determined. The problem of determining the efforts in the beam, taking into account its own weight and dynamic load (perturbing force) F (t), with the frequency of forced vibrations corresponding to the resonance and interresonance zones applied at an arbitrary point d, was solved. The conditions are formulated that must be taken into account when analyzing the dynamic behavior of a structure under the action of variable loads in various design situations.