Numerical Approach to Simulate the Effect of Corrosion Damage on the Natural Frequency of Reinforced Concrete Structures

Corrosion of reinforcing steel in concrete elements causes minor to major damage in different aspects.It may lead to spalling of concrete cover,reduction of section’s capacity and can alter the dynamic properties.For the dynamic properties,natural frequency is to be a reliable indicator of structural integrity that can be utilized in non-destructive corrosion assessment.Although the correlation between natural frequency and corrosion damage has been reflected in different experimental programs,few attempts have been made to investigate this relationship in forward modeling and/or structural health monitoring techniques.This can be attributed to the limited available data,the complex nature of corrosion,and the involvement of multidisciplinaryfields.Therefore,this study presents a numerical attempt to simulate the effect of corrosion damage on the natural frequency of the structure.The approach relies on simulating the time history response of the structure using a modified Bouc-Wen model that incorporates the nonlinear effects of corrosion.Then,modal analysis is utilized to assess the change in dynamic properties in the frequency domain.Tofinish up,regression algorithms are employed tofind optimal relationship between involved parameters,including corrosion damage as input,and natural frequency as output.The efficiency of the suggested framework is illustrated in thirteen buildings with cantilevered column lateral force-resisting system and different levels of corrosion.

A NEW METHOD FOR FINDING THE NATURAL FREQUENCY SET OF A LINEAR TIME-INVARIANT NETWORK

提出了一种求线性定常ｎ阶网络的固有频率集的新方法，给出了通用方程的推导及证明．该方法首次将ｎ阶网络的固有频率与ｎ端口网络参数相联系，方程形式简洁对偶，物理意义明确，求解简捷规范且不会产生丢根现象，具有通用性和系统性．

Theoretical analysis of natural frequency of externally prestressed concrete beam based on rigidity correction

A dynamic test on externally prestressed simply supported concrete beams separately with three typical types of tendon distributions was conducted. The results show that the natural frequencies of the beams increase with the increase in the prestressing force at the tensioning stage, and the natural frequencies decrease after the cracks occur in the beams. Following the calculation formula of natural frequency of externally prestressed beam, which was reported in a literature, the natural frequencies of the experimental beams are calculated, and big errors are found between the test results and the calculated ones of natural frequency values. As a result, this paper has tried to adopt two methods to correct the rigidity parameter of the concrete beam in the formula for natural frequency calculation, and to use the corrected formula to calculate the frequencies of the experimental beams. The calculation results indicate a good consistency with the experimental ones, which verifies the feasibility of the corrected formula.

Optimization of Bearing Locations for Maximizing First Mode Natural Frequency of Motorized Spindle-Bearing Systems Using a Genetic Algorithm

This paper has developed a genetic algorithm (GA) optimization approach to search for the optimal locations to install bearings on the motorized spindle shaft to maximize its first-mode natural frequency (FMNF). First, a finite element method (FEM) dynamic model of the spindle-bearing system is formulated, and by solving the eigenvalue problem derived from the equations of motion, the natural frequencies of the spindle system can be acquired. Next, the mathematical model is built, which includes the objective function to maximize FMNF and the constraints to limit the locations of the bearings with respect to the geometrical boundaries of the segments they located and the spacings between adjacent bearings. Then, the Sequential Decoding Process (SDP) GA is designed to accommodate the dependent characteristics of the constraints in the mathematical model. To verify the proposed SDP-GA optimization approach, a four-bearing installation optimazation problem of an illustrative spindle system is investigated. The results show that the SDP-GA provides well convergence for the optimization searching process. By applying design of experiments and analysis of variance, the optimal values of GA parameters are determined under a certain number restriction in executing the eigenvalue calculation subroutine. A linear regression equation is derived also to estimate necessary calculation efforts with respect to the specific quality of the optimization solution. From the results of this illustrative example, we can conclude that the proposed SDP-GA optimization approach is effective and efficient.

Assessment of natural frequency of installed offshore wind turbines using nonlinear finite element model considering soil-monopile interaction

A nonlinear finite element model is developed to examine the lateral behaviors of monopiles, which support offshore wind turbines(OWTs) chosen from five different offshore wind farms in Europe. The simulation is using this model to accurately estimate the natural frequency of these slender structures, as a function of the interaction of the foundations with the subsoil. After a brief introduction to the wind power energy as a reliable alternative in comparison to fossil fuel, the paper focuses on concept of natural frequency as a primary indicator in designing the foundations of OWTs. Then the range of natural frequencies is provided for a safe design purpose. Next, an analytical expression of an OWT natural frequency is presented as a function of soil-monopile interaction through monopile head springs characterized by lateral stiffness KL, rotational stiffness KRand cross-coupling stiffness KLRof which the differences are discussed. The nonlinear pseudo three-dimensional finite element vertical slices model has been used to analyze the lateral behaviors of monopiles supporting the OWTs of different wind farm sites considered. Through the monopiles head movements(displacements and rotations), the values of KL, KRand KLRwere obtained and substituted in the analytical expression of natural frequency for comparison. The comparison results between computed and measured natural frequencies showed an excellent agreement for most cases. This confirms the convenience of the finite element model used for the accurate estimation of the monopile head stiffness.

ZERO MODE NATURAL FREQUENCY AND NONLINEAR VIBRATION OF COUPLED LATERAL AND TORSION OF A LARGE TURBINE GENERATOR

Zero mode natural frequency (ZMNF) is found during experiments. The ZMNF andvibrations resulted by it are studied. First, calculating method of the ZMNF excited byelectromagnetic in vibrational system of coupled mechanics and electrics are given from the view ofmagnetic energy. Laws that the ZMNF varies with active power and exciting current are obtained andare verified by experiments. Then, coupled lateral and torsional vibration of rotor shaft system isstudied by considering rest eccentricity, rotating eccentricity and swing eccentricity. UsingLargrange-Maxwell equation when three phases are asymmetric derives differential equation of thecoupled vibration. With energy method of nonlinear vibration, amplitude-frequency characteristics ofresonance are studied when rotating speed of rotor equals to ZMNF. The results show that ZMNF willoccur in turbine generators by the action of electromagnetic. Because ZMNF varies withelectromagnetic parameters, resonance can occur when exciting frequency of the rotor speed is fixedwhereas exciting current change. And also find that a generator is in the state of large amplitudein rated exciting current.

NATURAL FREQUENCY OF COMPONENT UNDER INFLUENCE OF WELDING RESIDUAL STRESS

According to the distribution feature of welding residual stress on plate with welding seam along axis,a mathematical model of the welding residual stress is established.A formula of natural frequency of four boundary simply-supported quadrate thin plate with welding residual stress is developed.Some conclusions have been obtained.They are:(1)if there is welding residual stress,all natural frequencies of component are in increase,(2)change of high rank natural frequency under the influence of residual stress is larger than that of low rank natural frequency,and the higher the rank number is the larger the absolute amount of change of natural frequency is.

NATURAL FREQUENCY FOR RECTANGULAR ORTHOTROPIC CORRUGATED-CORE SANDWICH PLATES WITH ALL EDGES SIMPLY-SUPPORTED

A simple approach to reduce the governing equations for orthotropic corrugated core sandwich plates to a single equation containing only one displacement function is presented, and the exact solution of the natural frequencies for rectangular corrugated-core sandwich plates with all edges simply-supported is obtained. Furthermore, two special cases of practical interests are discussed in details.

Gravity Effect on the First Natural Frequency of Offshore Wind Turbine Structures

The fatigue limit state is critical for most offshore wind turbine.To minimize the development of fatigue damage,dynamic amplification of the response must be avoided.Thus,it is important to ensure that the first natural frequency of the offshore wind turbine does not coincide with the excitation frequencies related to wind turbine and wave loadings.For evaluating the self-gravity influence on the first natural frequency of wind turbine support structures,the offshore wind turbine system vibration is modeled using an Euler-Bernoulli beam with axial force and horizontal force.Real data from five wind turbines available in the market are considered.The sizes of wind turbines vary from 2.3MW to 6MW,and subsequently,the heights of tubular steel towers vary from 83 mto 100m.Results indicate that the average influence of gravity on the first natural frequency is nearly 1.8%.The first natural frequency is considered ranging from 1P(rototor frequency)to 3P(blade passing frequency).The design procedure requires an accurate evaluation of the first natural frequency.From this perspective,the first natural frequency is reduced since gravity needs to be considered for the design of offshore wind turbine support structures,especially when the first natural frequency of the offshore wind turbine is close to the lower limit,rotor frequency1 P.

Assessment of structural damage using natural frequency changes

The present paper develops a new method for damage localization and severity estimation based on the employment of modal strain energy. This method is able to determine the damage locations and estimate their severities, requiring only the information about the changes of a few lower natural frequencies. First, a damage quantification method is formulated and iterative approach is adopted for determining the damage extent. Then a damage localization algorithm is proposed, in which a damage indicator is formulated where unity value corresponds to the true damage scenario. Finally, numerical studies and model tests are conducted to demonstrate the effectiveness of the developed algorithm.

Natural Frequency of Planetary Transmission with Thin-Walled Ring Gear on Elastic Supports

A lumped parameter-rigid elastic coupled dynamic model of two-stage planetary gears for a hybrid car is established through the inter-stage coupled method,in which the supports of the ring gear of planet set Ⅱ are represented as an elastic foundation with radial and tangential uniform distributed stiffness,and the ring gear of planet set Ⅱ is modeled as an elastic continuum body. The natural frequencies based on the eigenvalue problem of dynamic model of planetary transmission are solved and the associated vibration modes are discussed. The rules are revealed which are the influences of the ring gear elastic supports stiffness and rim thickness on natural frequencies of planetary transmission. The theoretical analysis indicates that the vibration modes of planetary transmission with thin-walled ring gear on elastic supports are classified into seven types: Ⅰ/Ⅱ stage coupled rotational mode,Ⅰ stage translational mode,Ⅰ stage planet mode,Ⅱ stage translational mode,Ⅱ stage degenerate planet mode,Ⅱ stage distinct planet mode and purely ring gear mode. For each vibration mode, its properties are summarized. The numerical solutions show that the elastic supports stiffness and rim thickness of the ring gear of planet set Ⅱ have different influences on natural frequencies.

Natural Frequency and Main Vibration Mode Analysis of the Braking System of Metallurgical Crane

The safety brake system of metallurgy bridge crane is generally composed of two separated block brake, brake disc, and torsion shaft. The analysis of natural frequency and main vibration mode on this two-degree torsion vibration system is the basement to study the vibration model and vibration performance. In this work, we investigated natural frequency of the braking system of metallurgical crane with analytic method. This provides a systematic guidance towards a successful brake system design

Study on conveyor natural frequency and time intervalfor switching off resistances

A conveyor belt driven by wound rotor motors produces dynamic tension, velocity and accelerationduring starting. The terrible situation (such as resonance) in dynamic analysis and design is that system naturalfrequencies are equal to those for switching off electric resistances. This paper analyzes and determines systemnatural frequencies based on a modeling method of receptances with the analysis of sub-systems model and of theprinciple of their addition and conveyor loop closure. It also puts forward to calculate the time interval for switching off electric resistances. The starting of one conveyor is simulated by lumped-mass-spring-model software tofurther illustrate the influence of time interval for switching off electric resistances on conveyor dynamic behavior. Two methods are also compared. The receptance model is proved to be an excellent alternative.

Design for a specified natural frequency of elastically constrained axially graded bars

This study derives some closed-form analytic as well as numeric solutions for the axially graded elasticallyconstrained bar with polynomial varying modulus of elasticity, along the axial coordinate. The solutions allow for vibration tailoring in such a manner that the bar possesses the pre-selected natural frequency.

NONLINEAR NATURAL FREQUENCY OF SHALLOW CONICAL SHELLS WITH VARIABLE THICKNESS

The nonlinear dynamical variation equation and compatible equation of the shallow conical shell with variable thickness are obtained by the theory of nonlinear dynamical variation equation and compatible equation of the circular thin plate with variable thickness. Assuming the thin film tension is composed of two items. The compatible equation is transformed into two independent equations. Selecting the maximum amplitude in the center of the shallow conical shells with variable thickness as the perturbation parameter, the variation equation and the differential equation are transformed into linear expression by theory of perturbation variation method. The nonlinear natural frequency of shallow conical shells with circular bottom and variable thickness under the fixed boundary conditions is solved. In the first approximate equation, the linear natural frequency of shallow conical shells with variable thickness is obtained. In the third approximate equation, the nonlinear natural frequency of it is obtained. The figures of the characteristic curves of the natural frequency varying with stationary loads, large amplitude, and variable thickness coefficient are plotted. A valuable reference is given for dynamic engineering.

Dynamic test methods for natural frequency of footbridge

Using environmental random vibration as the excitation,traditional accelerometer method,non-contact video method and non-contact laser method were employed to determine the natural frequency of Kunyu River footbridge.All the results of these three methods are close to 2.70 Hz,which are concordant with each other and hence credible.

Analysis of natural frequency for imaging interface in liquid lens

Transmission beam can be modulated at the liquid-liquid interface inside an electrowetting liquid lens.The fluctuation characteristics of the interface has a decisive effect on the beam modulation.A closed cylinder in capillary constant scale is analyzed and the natural frequencies of a flat interface are obtained using capillary wave hydrodynamics.Results in modes 0 and 1 are in good agreement with previous experiments in the literature.The influences of the radius,the height ratio and the height-to-diameter ratio of a liquid lens on the interface eigenfrequencies are analyzed.

Experimental and theoretical analysis on the natural frequency of external prestressed concrete beams

Analyses and computations of the natural frequencies of external prestressed concrete structures are the bases for studying the dynamic properties of such structures.We carried out dynamic tests on three types of external simply supported beams,pulling the beams gradually.Then vertical loads were applied to the beams while the frequencies and related coefficients were measured at every step.We calculated natural frequencies and the results indicate that natural frequencies increase as the prestressing force rises in the pre-crack stage,and decrease significantly during the post-crack stage.Substantial incoincidences exist between the calculated and experimental results for the frequency and its tendency to change.Based on the experimental results,we modified the stiffness and other parameters in the equations.The results calculated using the modified equations agree with experimental results well,so the modified equations can be used practically.

Tunable Bending Stiffness, Buckling Force, and Natural Frequency of Nanowires and Nanoplates

This paper aims to obtain the simple closed-form results for the combined effects of surface elasticity, initial stress/ strain, and material Poisson ratio on the bending stiffness, natural frequency and buckling force of nanowires and nano-plates. The results demonstrate that all these properties of nanowires or nanoplates can be designed either very sensitive or not sensitive at all to the amplitude of an applied electric potential;show how much of those properties can be controlled to vary;and thus provide a reliable guide to the measurement of the Young’s modulus of nanowires/nanoplates and to the design of nano-devices, such as nano-sensors or the cantilever of an AFM.

Natural frequency of tapered high pier considering pile-soil interaction

In seismic design of tapered high pier, the analysis of natural vibration frequency is of great importance. According to the engineering features of tapered high pier in mountainous area, a vibration calculation model was set up considering the tapered pier characteristics and pile-soil interaction. Based on Southwell frequency composition theory, it consists of elastic deformation of bridge pier and the rigid deformation of group piles, which are respectively solved by the finite-element method and energy method, and then the natural frequency is derived. The comparison between the measured and calculated results shows that the calculation errors with and without considering pile-soil interaction are 4.9% and 14.7%, respectively. Additionally, the main parameters (pier height, section variation coefficient and lateral foundation horizontal proportional coefficient) affecting natural frequency were investigated. The result shows that natural frequency ascends with the increase of the lateral foundation horizontal proportional coefficient; and it is quite necessary to consider the pile-soil interaction in natural frequency calculation of tapered high pier.