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The purpose of this paper is to address the practically important problem of the load dependence of transverse vibrations for helical springs. At the beginning, the author develops the equations for transverse vibrations of the axially loaded helical springs. The method is based on the concept of an equivalent column. Second, the author reveals the effect of axial load on the fundamental frequency of transverse vibrations and derive the explicit formulas for this frequency. The fundamental natural frequency of the transverse vibrations of the spring depends on the variable length of the spring. The reduction of frequency with the load is demonstrated. Finally, when the frequency nullifies, the side buckling spring occurs.

Helical springs constitute an integral part of many mechanical systems. A coil spring is a special form of spatially curved column. The center of each cross-section is located on a helix. The helix is a curve that winds around with a constant slope of the surface of a cylinder. An exact stability analysis based on the theory of spatially curved bars is complicated and difficult for further applications. Hence, in most engineering applications a concept of an equivalent column is introduced. The spring is substituted for the simplification of the basic equations by an equivalent column. Such a column must account for compressibility of axis and shear effects. The transverse vibration is represented by a differential equation of fourth order in place and second order in time. The solution of the undamped model equation could be obtained by separation of variables. The fundamental natural frequency of the transverse vibrations depends on the current length of the spring. Natural frequency is the function of the deflection and slenderness ratio. Is the fundamental natural frequency of transverse oscillations nullifies, the lateral buckling of the spring with the natural form occurs. The mode shape corresponds to the buckling of the spring with moment-free, simply supported ends. The mode corresponds to the buckling of the spring with clamped ends. The author finds the critical spring compression.

Buckling refers to the loss of stability up to the sudden and violent failure of seed straight bars or beams under the action of pressure forces, whose line of action is the column axis. The known results for the buckling of axially overloaded coil springs were found using the static stability criterion. The author uses an alternative approach method for studying the stability of the spring. This method is based on dynamic equations. In this paper, the author derives the equations for transverse vibrations of the pressure-loaded coil springs. The fundamental natural frequency of the transverse vibrations of the column is proved to be the certain function of the axial force, as well as the variable length of the spring. Is the fundamental natural frequency of transverse oscillations turns to be to zero, is the lateral buckling of the spring occurs.

The spring is substituted for the simplification of the basic equations by an equivalent column. Such a column must account for compressibility of axis and shear effects. The more accurate model is based on the equations of motion of loaded helical Timoshenko beams. The dimensionless for beams of circular cross-section and the number of parameters governing the problem is reduced to four (helix angle, helix index, Poisson coefficient, and axial strain) is to be derived. Unfortunately, that for the spatial beam models only numerical results could be obtained.

The closed form analytical formulas for fundamental natural frequency of the transverse vibrations of the column as function of the axial force, as well as the variable length of the spring are derived. The practically important formulas for lateral buckling of the spring are obtained.

In this paper, the author derives the new equations for transverse vibrations of the pressure-loaded coil springs. The author demonstrates that the fundamental natural frequency of the transverse vibrations of the column is the function of the axial force. For study of the stability of the spring the author uses an alternative approach method. This method is based on dynamic equations. The new, original expressions for lateral buckling of the spring are also obtained.

THE present series of articles is intended to cover in a general manner the whole subject of vibration in aircraft. The study of aircraft vibration is a specialized branch of engineering science, calling for considerable practical skill and experience in the design and use of measuring equipment, as well as the ability to carry out the mathematical analyses which render test results intelligible and lead to the successful solution of vibration problems. The leading features of all mechanical vibration problems are, however, of such a nature as to be readily understood, and the aim of this series is to describe these leading features so as to facilitate the appreciation of the broad principles governing the avoidance of unpleasant or dangerous vibrations.

A theoretical approach is presented with the object of determining the natural frequencies and hub characteristics of a single‐rotor helicopter in free ground vibration for use in the analysis of self‐excited mechanical oscillations of hinged rotor blades. The method is intended to obviate the experimental determination of these parameters, a technique which necessitates the construction of the helicopter before its vibrational features can be explored.

Wind power is the one of best natural resources to meet the demands of electricity in India. In this regard, one of engineering college in Visakhapatnam has procured wind turbine generators of 200 kWp and got these installed on the rooftop of the college buildings for research and power generation. After starting the mills, huge vibrations were experienced by the staff and students in the laboratories and classrooms. So, the purpose of this paper is to carry out vibration and noise studies on wind turbine generator to identify the problem for high vibrations and suggest a novel method for vibration reduction.

Experimental vibration and natural frequency investigations are carried when wind velocity around 6.0 m/s using frequency analyzer, impact hammer, condenser microphone and accelerometer. An attempt is made to reduce the vibration and noise level of wind turbine generator by inserting a steel coil spring of 300 mm length having 20 turns in series with turnbuckle D shackle assembly, which is used to connect the wind turbine generator to the hook mounted on slab.

A high vibration velocity of 9.9 mm/s was observed on at base frame of wind turbine generator. The natural frequencies of hook and slab are observed in between 15 to 20 Hz from the natural frequency test. A high noise of 94.67 dBA is observed at a distance of 1 m from the base of wind turbine generator along the rotational axis of rotor. After modification to the baseline, WTG the vibration and noise levels are reduced to 4.8 mm/sec and 77.76 dBA, respectively.

This is the first time to study the huge vibrations generated in wind turbine generators installed on the rooftop of the college. Developed a novel methodology to reduce the vibrations by inserting a steel coil springs in turnbuckle D shackle assembly of wind turbine generators. After modification, wind turbine generator are running successfully without any high vibrations.

An analytical rheological‐dynamical visco‐elastic solution of one‐dimensional longitudinal continuous vibration of bars has been developed and used to evaluate the validity of the classical analytical elastic solutions. As it is well known, the resonance occurs only in the continuous or singledegree‐of‐freedom ideal elastic system when the excitation frequency ωP is equal to the one of the natural frequency of the bar. However, owing to the visco‐elastic nature of materials and frequency dependence of the damping factor it is useful to consider separately the situations arising when the is positive (system is stable) and when it is negative. Negative damping factor means that the complementary solution of the response would not die away (system is unstable because of the factor e). Rheologic behavior of the bar can be characterized by one parameter, i.e. dynamic time of retardation TK D=1/ω, like in a single‐degree‐of‐freedom spring mass system. RDA model has the same phase angle as a simple single‐degree‐of‐freedom spring mass system with damping in the steady state vibration and from that the damping factor is obtained. This paper provides description of the dynamic magnification factor and the transmissibility of several metallic materials using RDA similitude and could be concluded that an ideally effective antivibration mount material should satisfy at least two requirements: first, it should posses a relatively large damping factor; and second, it should possess a damping factor that either remains constant or decreases only slowly with frequency.

This study aims to perform dynamic response analysis of damaged rigid-frame bridges under multiple moving loads using analytical based transfer matrix method (TMM). The effects of crack depth, moving load velocity and damping on the dynamic response of the model are discussed. The dynamic amplifications are investigated for various damage scenarios in addition to displacement time-histories.

Timoshenko beam theory (TBT) and Rayleigh-Love bar theory (RLBT) are used for bending and axial vibrations, respectively. The cracks are modeled using rotational and extensional springs. The structure is simplified into an equivalent single degree of freedom (SDOF) system using exact mode shapes to perform forced vibration analysis according to moving load convoy.

The results are compared to experimental data from literature for different damaged beam under moving load scenarios where a good agreement is observed. The proposed approach is also verified using the results from previous studies for free vibration analysis of cracked frames as well as dynamic response of cracked beams subjected to moving load. The importance of using TBT and RLBT instead of Euler–Bernoulli beam theory (EBT) and classical bar theory (CBT) is revealed. The results show that peak dynamic response at mid-span of the beam is more sensitive to crack length when compared to moving load velocity and damping properties.

The combination of TMM and modal superposition is presented for dynamic response analysis of damaged rigid-frame bridges subjected to moving convoy loading. The effectiveness of transfer matrix formulations for the free vibration analysis of this model shows that proposed approach may be extended to free and forced vibration analysis of more complicated structures such as rigid-frame bridges supported by piles and having multiple cracks.

Modelling methods can be helpful for understanding vibrations of beam structures including cracks, as well as for early detection of crack. This study aims to provide an analytical modelling approach for a cantilever beam considering a slant vertical crack along its height. However, previous uniform crack methods cannot be used for describing this case. The results from the analytical, finite element (FE) and experimental methods are compared to verify the vibration problem.

A massless rotational spring model is adopted to describe the crack. An extended method based on the calculation method for a uniform vertical edge crack is proposed to obtain the stiffness of the slant case. The beam is divided into a series of independent thin slices along the beam height. An Euler–Bernoulli beam model is applied to formulate each slice. The crack in each slice is considered as a uniform one. The transfer matrix method in the literature is used to obtain the beam vibration frequencies and mode shapes. Influences of crack location and sizes on the natural frequencies for the cantilever beam, as well as the mode shapes, are analysed. An established FE model and test results in the listed references are used to validate the developed method.

The numerical results show that the rotational stiffness at the cracked section and the natural frequencies of the beam decrease by increasing the crack sizes; the natural frequencies for the beam are greatly influenced by the crack sizes and location; the first natural frequency decreases with the distance from the beam fixed end to the crack location; the value of the first natural frequency reaches a minimum value when the crack is at the beam fixed end; the value of the second natural frequency is a minimum value when the crack is at the beam middle; and the value of the third natural frequency is a minimum value when the crack is at the beam free end. Saltation is observed in some mode shapes at the crack location, especially for larger crack depths; but, the mode shapes of the beam are slightly influenced by the vertical crack.

This study gives a useful analytical modelling method for free vibration analysis for the cantilever beam with a vertical crack, which can overcome the disadvantages of the previous uniform crack methods.

High-rise tower structures supported by side frame structure and viscous damper in chemical industry can produce plasticity under dynamic loads, such as wind and earthquake, which will heavily influence the long-term safety operation. This paper aims to systematically study the optimization design of these structures by free vibration and dynamic shakedown analysis.

The transfer matrix method and Euler–Bernoulli beam vibration are used to study the free vibration characteristic of the simplified high-rise tower structure. Then the extended stress compensation method is used to construct the self-equilibrated stress by using the dynamic load vertexes and the lower bound dynamic shakedown analysis for the structure with viscous damper. Using the proposed method, comprehensive parametric studies and optimization are performed to examine the shakedown load of high-rise tower with various supported conditions.

The numerical results show that the supported frame stiffness, attached damper or spring parameters influence the free vibration and shakedown characters of high-rise tower very much. The dynamic shakedown load is lowered down quickly with external load frequency increasing to the fundamental natural frequency of the structure under spring supported condition, while changed little with the damping connection. The optimized location and parameter of support are obtained under dynamical excitations.

In this study, the high-rise tower structure is simplified as a cantilever beam supported by a short cantilever beam and a damper under repeated dynamic load, and linear elasticity for solid is assumed for free vibration analysis. The current analysis does not account for effects such as large deformation, stochastic external load and nonlinear vibration conditions which will inevitably be encountered and affect the load capacity.

This study provides a comprehensive method for the dynamical optimization of high-rise tower structure by combining free vibration and shakedown analysis.

An improved analytical method for calculating the natural frequencies of a switched reluctance motor (SRM) stator is proposed in this paper. The method is different from traditional analytical methods, which only consider the influence of mass of the stator poles and windings on the natural frequencies of the SRM stator. This paper aims to consider the influence of stiffness and mass of the stator poles and windings simultaneously and reasonably.

An innovated analytical method based on the electromechanical analogy method is presented. In the proposed analytical formulae for calculating the natural frequencies, the influence of the windings on natural frequencies is considered by using the springs to simulate the flexible connection between the stator core and windings, and the stator poles are treated as both additional mass and additional equivalent stiffness. Both three-dimensional (3D) finite-element analysis (FEA) and experimental modal analysis results validate the improved method.

The influence of the mass and stiffness of stator winding is considered by using the springs to simulate the flexible connection between the stator core and windings, and the stator poles are treated as both additional mass and additional equivalent stiffness. The traditional analytical method only considers the influence of mass. Therefore, the calculation results are comparatively lower than 3D FEA results and may lead to a large error. The 3D FEA and experimental modal analysis confirm that the proposed method has good precision for low-order natural frequency calculation of SRMs.

An improved analytical method for calculating the natural frequencies of an SRM stator is proposed. Unlike the traditional analytical method, the proposed method can consider the influence of stiffness and mass of the stator poles and windings. This method is valuable for designers to predict the natural frequencies accurately.

In the vibration reduction field, constrained stand-off layer damping cylindrical shell plays an important role. However, due to the lack of accurate analysis of its damping characteristics, this hinders its further research and application. Therefore, the purpose of this paper is concerned with an accurate solution for the vibration-damping characteristics of a constrained stand-off-layer damping cylindrical shell (CSDCS) under various classical boundary conditions and conducts a further analysis.

Based on the Rayleigh–Ritz method and the Hamilton principle, a dynamic model of CSDCS is established. Then the loss factor and the frequency of CSDCS are obtained. The correctness and convergence behavior of the present model are verified by comparing the calculation results with the literature. By using for various classical boundary conditions without any special modifications in the solution procedure, the characteristics of CSDCS with S-S, C-C, C-S, C-F and S-F boundaries are discussed.

The Rayleigh–Ritz method is effective in handling the problem of CSDCS with different boundaries and an accurate solution is obtained. The boundary conditions have an important influence on the vibration and damping behavior of the CSDCS.

Based on the Rayleigh–Ritz method and Hamilton principle, a dynamic model of CSDCS is established for the first time, and then the loss factor and frequency of CSDCS are obtained. In addition, the effectiveness of adding the stand-off layer between the base shell and the viscoelastic layer is confirmed by discussing the characteristics of CSDCS with S-S, C-C, C-S, C-F and S-F boundaries.