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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.

Owing to the cable flexibility, it is practically a lot easier to measure the high‐vibration frequencies of the cable than the fundamental vibration frequency. The objective of this study is to present a method to determine the cable tension based on frequency differences so that the effects of cable sag and bending stiffness can be included.

The paper includes theoretical derivation, laboratory study to verify the method and practical application in a real bridge.

It is suggested to measure the high‐vibration frequencies, and to use the vibration frequency difference to determine the fundamental vibration frequency of the cable and then to estimate the cable tension. The reliability of the method is verified by laboratory tests and the method is then applied to determine cable tensions in a real bridge.

This paper provides theoretical derivations to demonstrate that under certain conditions, the frequency difference of a cable with sag and bending is almost equal to the natural frequency of the same cable when it is taut. This unique characteristic of cable vibration is used to determine the cable tension similar to the fundamental frequency‐based taut‐string formula that is commonly used in practice.

Extends the evolutionary structural optimization method to the solution for the natural frequency optimization of a two‐dimensional structure with additional non‐structural lumped masses. Owing to the significant difference between a static optimization problem and a structural natural frequency optimization problem, five basic criteria for the evolutionary natural frequency optimization have been established. The inclusion of these criteria into the evolutionary structural optimization method makes it possible to solve structural natural frequency optimization problems for two‐dimensional structures with additional non‐structural lumped masses. Gives two examples to demonstrate the feasibility of the extended evolutionary structural optimization method when it is used to solve structural natural frequency optimization problems.

In this paper, an approximate analytical approach is developed for the determination of natural longitudinal frequencies of a cantilever-cracked beam based on the Lagrange inversion theorem.

The crack is modeled by an equivalent axial spring with stiffness according to Castigliano's theorem. Thus, an implicit frequency equation corresponding to cantilever-cracked bar is obtained. The resulting equation is solved using the Lagrange inversion theorem.

Effect of different crack depths and crack positions on natural frequencies of the cracked beam is analyzed. It is shown that an increase in the crack depth ratio produces a decrease in the fundamental longitudinal natural frequency of a cracked bar. Furthermore, approximate analytical results are compared with those obtained numerically as well as from experimental tests.

A new approximate analytical expression of a fundamental longitudinal frequency, as a function of crack depth and crack location, is obtained.

This paper aims to improve the life of the printed circuit boards (PCB) used in computers based on modal analysis by increasing the natural frequency of the PCB assembly.

In this work, through experiments and numerical simulations, an attempt has been made to increase the fundamental natural frequency of the PCB assembly as high as practically achievable so as to minimize the impacts of dynamic loads acting on it. An optimization tool in the finite element software (ANSYS) was used to search the specified design space for the optimal support location of the six fastening screws.

It is observed that by changing the support locations based on the optimization results the fundamental natural frequency can be raised up to 51.1% and the same is validated experimentally.

Manufacturers of PCBs used in computers fix the support locations based on symmetric feature of the board not on the dynamic behavior of the assembly. This work might lead manufacturers to redesign the location of other surface mount components.

This work provides guidelines for PCB manufacturers to finalize their support locating points which will improve the dynamic characteristics of the PCB assembly during its functioning.

This study provides a novel method to improve the life of PCB based on support locations optimization which includes majority of the surface mount components that contributes to the total mass the PCB assembly.

Considering the randomness of physical parameters of structural material, dynamic characteristic topology optimization mathematical model based on reliability of planar continuum structures is built in this paper. In which topology information variables of the structure are taken as design variables, minimizing the mean value of total structural weight as objective function and satisfying the reliability requirement of structural dynamic characteristic as constraints. In the process of optimization, the ESO method based on probability is adopted as solution strategy. At the same time, distribution function method is utilized to convert the reliability constraints into conventional constraints formally. A square thin plate with four sides fixed is used as an example to demonstrate the rationality and validity of the presented model.

In reinforced concrete (RC) structures, an evidence of damage is the presence of cracking. In order to evaluate the effect of damage on cracking pattern and natural frequency in RC slabs, two of such structures with different dimensions and reinforcement ratios were tested, in which cracks were induced through application of static load, followed by modal tests using impact excitation. The paper aims to discuss this issue.

The gradient of the fundamental natural frequency along the decay, the crack opening rate and also a global damage index based on changes of the fundamental natural frequency were evaluated.

The behaviour of the aforementioned gradient was distinct for both slabs, increasing monotonically with the cracking level for the slab with lowest reinforcement ratio, and increasing until 33 per cent of the collapse load and then decreasing afterwards for the slab with the highest ratio. Changes of the gradient were consistent with changes of the crack opening rate. Both results of gradient changes and cracking pattern brought evidence that the balance between open (old) and breathing (new) cracks differed between the slabs, and may be responsible for such differences.

Damage assessment in RC structures using vibration tests is mostly concentrated on beams. In this work, an advance is made by investigating slabs. The lack of a unique pattern of changes of the gradient implies that its absolute value is not generally suitable for the association with the damage level. However, the impact tests can be effectively used to detect early damage on slabs using this proposed parameter.

The purpose of this paper is to develop a topology optimization algorithm considering natural frequencies.

To incorporate natural frequency as design criteria of shell-infill structures, two types of design models are formulated: (1) type I model: frequency objective with mass constraint; (2) type II model: mass objective with frequency constraint. The interpolation functions are constructed by the two-step density filtering approach to describe the fundamental topology of shell-infill structure. Sensitivities of natural frequencies and mass with respect to the original element densities are derived, which will be used for both type I model and type II model. The method of moving asymptotes is used to solve both models in combination with derived sensitivities.

Mode switching is one of the challenges faced in eigenfrequency optimization problems, which can be overcome by the modal-assurance-criterion-based mode-tracking strategy. Furthermore, a shifting-frequency-constraint strategy is recommended for type II model to deal with the unsatisfactory topology obtained under direct frequency constraint. Numerical examples are systematically investigated to demonstrate the effectiveness of the proposed method.

In this paper, a topology optimization method considering natural frequencies is proposed by the author, which is useful for the design of shell-infill structures to avoid the occurrence of resonance in dynamic conditions.

Condition assessment on reinforced concrete (RC) structures is one of the critical issues as a result of structure degradation due to aging in many developed countries. The purpose of this paper is to examine the sensitivity and reliability of the conventional dynamic response approaches, which are currently applied in the RC structures. The key indicators include: natural frequency and damping ratio. To deal with the non-linear characteristics of RC, the concept of random decrement is applied to analyze time domain data and a non-linear damping curve could be constructed to reflect the condition of RC structure.

A full-scale RC structure was tested under ambient vibration and the impact from a rubber hammer. Time history data were collected to analyze dynamics parameters such as natural frequency and damping ratio.

The research demonstrated that the measured natural frequency is not a good indicator for integrity assessment. Similarly, it was revealed that the traditional theory of viscous damping performed poorly for the RC with non-linear characteristics. To address this problem, a non-linear curve is constructed using random decrement and it can be used to retrieve the condition of the RC structure in a scientific manner.

The time domain analysis using random decrement can be used to construct a non-linear damping curve. The results from this study revealed that the damage of structure can be reflected from the changes in the damping curves. The non-linear damping curve is a powerful tool for assessing the health condition of RC structures in terms of sensitivity and reliability.

Based on the asymptotic solution for predicted natural frequencies of a two‐dimensional elastodynamic problem from the finite element analysis, presents the concept of the asymptotic error, which is an approximate error but tends to the exact error when the characteristic length of elements approaches zero, and a practical error estimator. The present practical error estimator contains two criteria: one is the error estimator criterion, the other the finite element mesh design criterion. Using this practical error estimator, not only can the accuracy of a finite element solution for natural frequencies of a two‐dimensional elastodynamic problem be directly evaluated without any further finite element calculation, but also a new target finite element mesh for the desired accuracy of solution can be immediately designed from the relevant information of an original finite element solution. Generally, for the purpose of designing a new target finite element mesh, this original finite element solution is obtainable from a very coarse mesh of a few elements and usually does not satisfy the accuracy requirement. Since the new target finite element mesh could result in a finite element solution with a desire accuracy, the finite element solution so obtained can be used for a structural design in engineering practice. The related numerical results from vibration problems of three representative plates of different shapes under plane stress conditions have demonstrated the correctness and applicability of the present practical error estimator.