Multiple Natural Frequencies Research Articles

Analysis of Vibrational Characteristics of the Bank of China Tower in Hong Kong with ANSYS

Abstract. With the rapid urbanization and technological advancements in construction, modern high-rise buildings have become a symbol of urban development. However, these structures face unprecedented challenges due to their increasing height and the complex dynamic loads they must withstand. This paper presents a comprehensive analysis of the Bank of China Tower in Hong Kong, a prominent architectural landmark, using ANSYS software to perform modal and transient structural analyses. The primary structure of the tower consists of five steel-reinforced concrete columns, designed to resist dynamic responses under extreme conditions such as high wind speeds and seismic activities. The modal analysis identified multiple natural frequencies and vibration modes, providing insights into the dynamic behavior of the structure. The transient structural analysis, conducted under seismic wave loading, demonstrated the tower's admirable performance under specified seismic conditions, highlighting its sufficient safety margins. This study emphasizes the importance of advanced materials and innovative structural designs in ensuring the stability and safety of high-rise buildings. The findings contribute significantly to the field of structural engineering, offering valuable insights for the design and analysis of future high-rise structures.

On the identification of the elastic modulus and axial force of beam members with unknown boundary conditions using modal information

The axial force of beam members with unknown boundary conditions can be estimated by using vibration measurements, and this method necessitates the real elastic modulus of the members to produce accurate estimations. However, in engineering practice, the value of the elastic modulus of such members differs from the designed value owing to material degradation and structural damage. In addition, the elastic modulus of the beam structure may have a significant impact on the identification accuracy of axial force. Therefore, this study develops a methodology for simultaneously estimating axial force and elastic modulus within axial force members of structural systems, based on measurements of multiple natural frequencies and mode shapes. First, an improved vibration formula for the Timoshenko beam under axial load is obtained using the energy approach. Then, based on the proposed dynamic equation, the axial force and elastic modulus of the beam are discovered to have a linear relationship when the modal information of the structure is known. Further, using this linear relationship, a method to identify the axial force and elastic modulus of the beam member under unknown boundary constraints is proposed. Finally, the applicability and accuracy of the proposed method are validated by many numerical and experimental tests, and excellent estimates of the axial forces and elastic modulus are achieved.

Transfer Matrix Method for the Analysis of Multiple Natural Frequencies

Multiple natural frequencies may be encountered when analyzing the essential natural vibration of a symmetric mechanical system or sub-structure system or a system with special parameters. The transfer matrix method (TMM) is a useful tool for analyzing the natural vibration characteristics of mechanical or structural systems. It derives a nonlinear eigen-problem (NEP) in general, even a transcendental eigen-problem. This investigation addresses the NEP in TMM and proposes a novel method, called the determinant-differentiation-based method, for calculating multiple natural frequencies and determining their multiplicities. Firstly, the characteristic determinant is differentiated with respect to frequency, transforming the even multiple natural frequencies into the odd multiple zeros of the differentiation of the characteristic determinant. The odd multiple zeros of the first derivative of the characteristic determinant and the odd multiple natural frequencies can be obtained using the bisection method. Among the odd multiple zeros, the even multiple natural frequencies are picked out by the proposed judgment criteria. Then, the natural frequency multiplicities are determined by the higher-order derivatives of the characteristic determinant. Finally, several numerical simulations including the multiple natural frequencies show that the proposed method can effectively calculate the multiple natural frequencies and determine their multiplicities.

Discrete variable topology optimization for maximizing single/multiple natural frequencies and frequency gaps considering the topological constraint

AbstractFinding optimized structural topology design for maximizing natural frequencies and frequency gaps of continuum structures is crucial for engineering applications. However, two significant numerical issues must be addressed: non‐smoothness caused by multiple frequencies and Artificial Localized Rigid Motion (ALRM) modes due to the violation of the topological constraint related to isolated islands and point‐connections. The above two issues are solved by employing the discrete variable topology optimization method based on Sequential Approximate Integer Programming (SAIP). First, the directional differentiability and multiple frequencies preservation constraints are formulated as the linear integer constraints. And then, the integer programming with these linear integer constraints is established and solved by the discrete variable linear or quadratic programming solver with multi‐constraint Canonical Relaxation Algorithm (CRA). We also prove that the popular Average Modal Frequencies (AMF) strategy, like the Kreisselmeier‐Steinhauser (KS) function, cannot rigorously tackle this non‐smoothness caused by multiple frequencies. Furthermore, to eliminate the ALRM modes and concentrate on the real structural global modes, the burning method is employed to impose the topological constraint of the first Betti number that represents the number of isolated islands and point‐connections. Numerical examples, including 2D and 3D, two‐fold and three‐fold multiple frequencies, natural frequencies and frequency gaps, are presented to show the effectiveness of the proposed method.

A design strategy for multi-span pipe conveying fluid away from resonance by graphene platelets reinforcement

The operation of mechanical systems generates vibrations with a certain range of frequencies. Pipes conveying fluid have multiple natural frequencies. It is difficult to avoid one or more natural frequencies of pipes in the operating frequency range of the system, which can lead to resonance and fatigue failure. In this paper, a design strategy to modulate the multi-span pipe's natural frequency by reinforcing pipes with graphene platelets (GPLs) nanofiller is proposed to solve the pipe resonance in engineering. The multi-span functionally graded (FG) pipe is constructed based on an integral pipe model with GPLs uniformly or non-uniformly distributed in the metal matrix. The effective properties of the nanocomposite are determined by using the Halpin-Tsai micromechanics model and rule of mixture. Subsequently, the safety range and resonance range of the multi-span FG pipe are obtained by comparing the first tenth-order natural frequency and preserved frequency band. A comprehensive investigation on the effect of patterns, weight fraction, and geometric parameters of GPL is conducted to identify the most effective way to modulate the pipe's natural frequency. The results indicate that GPL weight fraction is able to modulate the multi-span pipe's natural frequency and safety range over a wide range. In addition, the natural frequency and safety range of the FG pipe reinforced with GPLs increase as the stiffness and number of retaining clip increase. Based on the results, a pipe design strategy with fewer retaining clips is proposed for pipes away from resonance by reinforcing pipes with GPL nanofiller. In the case of an aircraft hydraulic pipe, the natural frequency of the pipe reinforced with GPLs is far away from the preserved frequency band and without resonance in operation. In summary, the design strategy of pipe reinforced with GPLs shed important insights on reducing the vibration level of multimodal structures.

Liquid sloshing in a baffled rectangular tank under irregular excitations

Sloshing responses in vertical and horizontal baffled tanks under irregular external excitations are investigated by employing a viscous fluid flow model with the Navier–Stokes solver. After the extensive validation of the present numerical model, the numerical experiment reveals many new understandings of the problem. A three-phase variation is suggested to describe the sloshing response under irregular excitations, that is, the first-order natural period controlling stage, the second-order natural period controlling stage, and the transitional stage between them. The sloshing periods are always around the corresponding natural periods with small standard deviations at the first- and second-order natural period controlling stages; while the decreased sloshing periods with large standard deviations can be observed at the transitional stage between them. The sloshing energy concentrated at a single natural frequency or multiple natural frequencies is the major reason for the phenomena. Compared to the results under regular excitations, smaller and larger sloshing amplitudes can be observed in the results of irregular excitations around the resonant and non-resonant frequency ranges, respectively. The normalized sloshing amplitudes calculated from the samplings of the large free surface amplitudes are always smaller than the samplings of the small free surface amplitudes.

Nonlinear harmonics of gap fluid resonance with floating body motions

Resonant response of water waves in a narrow gap, the so-called gap resonance, is a hydrodynamic phenomenon with practical applications. When coupled to body motions, the physics becomes rather complicated, involving body motions, fluid resonance and damping that determines the response amplitude. Gap resonances between two identical elongated bodies, with one held fixed and the other floating, are investigated experimentally for unidirectional waves with broadside incidence. Transient wave groups with different maximum surface elevations are generated, to examine the nonlinear physics. Sub- and super-harmonics, which can be substantially larger than the linear component, are observed in the responses. When going from the diffraction problem (both bodies fixed) to the coupled problem (allowing one body to move), the gap responses are significantly amplified, i.e. by a factor of 2. The first resonant mode that dominates the diffraction problem disappears in the coupled problem. A new resonant mode, which is shown to arise from body motions, is excited inside the gap through nonlinear processes. This mode features a double-humped ‘camel-back’ shape, leading to remarkably large responses. The floating body is observed to oscillate in sway with significant amplitudes at two distinct natural frequencies that are far apart. This is of great interest for the design of mooring lines connecting the two bodies, as it appears to cause resonances at the wave frequencies in addition to the low frequencies. A semi-analytical model is developed to investigate the multiple natural frequencies in sway, yielding further insight into gap resonances.

Stability of Multidegree of Freedom Potential Systems to General Infinitesimal Positional Perturbation Forces

Abstract The stability of general linear multidegree of freedom stable potential systems that are perturbed by general arbitrary positional forces, which may be neither conservative nor purely circulatory/conservative, is considered. It has been recently recognized that such perturbed potential systems with multiple frequencies of vibration are susceptible to instability, and this paper is centrally concerned with the situation when potential systems have such multiple natural frequencies. An approach based on perturbation theory that includes nonlinear terms in the expansions of the perturbed eigenvalues is developed. Explicit conditions under which the system either remains stable or becomes unstable due to flutter are provided. These results show that the stability/instability picture that emerges is far subtler and more complex than what might be intuitively inferred. The manner in which prior results related to narrower classes of perturbation matrices, like circulatory matrices, get included in the more general results obtained here is pointed out. Several numerical examples illustrate the applicability of the analytical results. An engineering application is provided demonstrating the power of the analytical results.

Vibration and Noise Characteristics of Air-Core Reactor Used in HVDC Converter Stations

Air-core reactors used in filters are significant equipment in HVDC converter stations. They are among the most serious noise sources under the action of multi-frequency magnetic forces. The vibration and noise characteristics of air-core reactors are investigated in this paper, which is the basis for noise control of HVDC projects. Firstly, based on the characteristics of magnetic forces, the generation process of vibration is analyzed theoretically. Then, Experimental Modal Analysis (EMA) is conducted and the results indicate that multiple natural frequencies exist and their mode shapes contain distinct peaks and troughs. Vibration shapes are also measured under different excitation frequencies with dense measuring points over the winding surface. The measured vibration shapes are consistent with the corresponding mode shapes. Furthermore, the noise distribution characteristics are analyzed by Boundary Element Method (BEM). The simulation results show that the noise radiation has obvious directional effect, which are mainly dominated by the corresponding vibrations and mode shapes. Based on the results of this paper, some suggestions for noise reduction are given in the last.

Multidirectional Linear Site Response Analysis by Applying the SBSR Technique to KiK-net Data

Numerous works have highlighted the importance of the site response analysis of vertical ground motion in the earthquake engineering field. However, most studies have focused on nonlinear site response analysis. Hence, the knowledge of the linear site response analysis of vertical ground motion is inadequate. In this study, we perform multidirectional linear site response analysis by applying the surface-to-borehole spectral ratio (SBSR) technique to the Kiban–Kyoshin strong-motion observation network (KiK-net) seismic data. We find the hyperbolic relationships between multiple natural frequencies and the corresponding fundamental frequency. We propose empirical formulas, which can reduce the error in the multiple natural frequencies estimated by a single-layer model. Moreover, a brief investigation is conducted to provide an extra evidence for a previous correction method to the horizontal-to-vertical spectral ratio (HVSR) technique. The results of this study provide the empirical tools and can be utilized as a reference for the multidirectional linear site response analysis based on seismic observations.

Stretching Method-Based Damage Detection Using Neural Networks.

We present in this paper a framework for damage detection and localization using neural networks. The data we use to train the network are pixel images consisting of measurements of the relative variations of m natural frequencies of the structure under monitoring over a period of d-days. To measure the relative variations of the natural frequencies, we use the stretching method, which allows us to obtain reliable measurements amidst fluctuations induced by environmental factors such as temperature variations. We show that even by monitoring a single natural frequency over a few days, accurate damage detection can be achieved. The accuracy for damage detection significantly improves when a small number of natural frequencies is monitored instead of a single one. More importantly, monitoring multiple natural frequencies allows for damage localization provided that the network can be trained for both healthy and damaged scenarios. This is feasible under the assumption that damage occurs at a finite number of damage-prone locations. Several results obtained with numerically simulated data illustrate the effectiveness of the proposed approach.

On the stability and instability of linear potential systems subjected to infinitesimal circulatory forces

The stability of linear multi-degree-of-freedom stable potential systems with multiple natural frequencies under the action of infinitesimal circulatory forces is considered. Contrary to the received view that such systems are inherently unstable, a careful study shows that such systems have a much more complex behaviour than previously recognized and could exhibit an alternation of stability and instability that depends on the structure of the potential system and its interaction with the circulatory forces. The conditions under which stability or instability ensues and the nature of this alternation in stability are explicitly obtained. In low-dimensional stable potential systems, when the coefficients of the circulatory forces are proportional to an arbitrarily small scalar parameter, all the circulatory forces that cause flutter instability are described.

Smart nano-engineered cementitious composite sensors for vibration-based health monitoring of large structures

Cement composites, generally with high resistivity, when incorporated with conductive nanoparticles, are found to be capable of reflecting strain state due to the change in electrical resistance of the matrix. These unique properties of cement-based nanocomposites (CNC) allow the development of smart cement sensors for continuous health monitoring of large concrete structures. Due to the limitation of strain-based sensing (for local behaviour), vibration-based sensing is gaining considerable importance. The performance of cement-based nanocomposites, unlike under the static load or slow varying load, for dynamic response sensing is very scanty. In the present study, an attempt has been made to develop robust smart cement sensors for capturing the multiple natural frequencies of a structure. First, a systemic way of fabrication of sensors (smart cement sensors) using cement matrix and nano-inclusions of carbon nanotubes is presented. Three different types of multi-walled nanotubes (MWCNTs) like pristine MWCNTs (P-MWCNTs), carboxyl acid (−COOH) MWCNTs and hydroxyl (−OH) MWCNTs with four different weight percentage (0.1-0.75 wt.%) of cement matrix are considered for the study. Strain sensibility of the developed sensors is explored under cyclic compression load. Further, the efficacy of the developed sensor for dynamic sensing is investigated using a large reinforced concrete bridge girder where instrumented impulse hammer is used to dynamically excite the bridge girder. It is found that the developed CNC is capable of capturing the natural frequencies (even up to the third mode) which are found to be sufficient for vibration-based health monitoring of structures. The performance of the developed sensors is compared with off-the-shelf accelerometers and is found to be in good agreement. The study provides a demonstrable technology for using the CNC as a vibration sensor for continuous health monitoring and assessment of structures, just by using ambient vibration information.

In-situ identification of shaking frequency for adaptive vibratory fruit harvesting

The improvement of the fruit removal efficiency is important in vibratory fruit harvesting. Most relevant research has focused on the detection of dynamic parameters of the tree system by conducting experiments or simulations. The present study proposes a method of identifying multiple natural frequencies of a fruit tree in-situ during vibratory harvesting. An intelligent shaker system consisting of an eccentric shaker with adjustable speed and an acceleration signal acquisition analyzer is developed. A frequency sweeping experiment using the intelligent shaker system is carried out to measure natural frequencies by searching for extreme values of the acceleration signals. Additionally, a testing experiment for measuring acceleration responses at different shaking frequencies is performed to obtain the optimal shaking frequencies. The experiments show that the natural frequencies could be obtained effectively by the frequency sweeping experiment in-situ, and the identified natural frequencies of different branches are found to be different. All results indicate that the identified natural frequencies using the proposed method can be regarded as the optimal shaking frequencies for adaptive vibratory fruit harvesting, and that it is necessary to detect the natural frequency of each large branch of a tree during vibratory fruit harvesting.

Passive vibration suppression of plate using multiple optimal dynamic vibration absorbers

In the present paper, the optimization problem of the dynamic vibration absorbers (DVAs) for suppressing vibrations in thin plates within the wide frequency band is investigated. It is considered that the plate has simply supported edges and is subjected to a concentrated harmonic force. The vibration suppression is accomplished by the implementation of multiple mass–spring absorbers in order to minimize the plate deflection at the natural frequencies of the plate without absorbers. The governing equations of the plate equipped with DVAs for both isotropic and FG plates are derived and solved numerically and analytically. The formulation of the problem is capable of optimizing the $$L_{2}$$ norm of the plate deflection at the wide frequency band with respect to mass, stiffness and position of each absorber attachment point. In this study, the possibility of simultaneous absorption of one or multiple natural frequencies of the plate without any absorbers is also studied. Some numerical results are also presented.

Static and Dynamic Properties of Branched Self-Similar Structures: The Trichotomous Lattice

The static and dynamic properties of self-similar branched structures (lattices) in which each subsequent cell replicates the structure of the previous one on some scale are investigated. The trichotomous lattices in which each element in each subsequent line divides into three self-similar ones are analyzed. The analytical conditions of static and dynamic self-similarity are found. The conditions of the same static deformation of each element are obtained. It is proved that the lattice is a band-pass filter, and its pass band depends on the similarity factor. It is shown that in the lattice there are multiple natural frequencies equal to the partial frequency of the generating element.

Surface energy effect on nonlinear free axial vibration and internal resonances of nanoscale rods

This study aims to investigate effects of the surface energy parameters on the nonlinear axial vibration and internal resonances of nanoscale rods. The surface energy parameters considered are the surface density and the surface Lamé constants. The nonlinear governing equation of motion is derived using the Hamilton's principle which includes the strain energy and the kinetic energy of the nanorod surface and bulk. The strain and kinetic energies of the nanorod bulk are obtained using the classical theory of elasticity and those of the nanorod surface are obtained using the surface elasticity theory. Then, the nonlinear governing equation of motion is solved using the method of multiple scales and natural frequencies are obtained for fixed-fixed and fixed-free boundary conditions. Effects of the surface energy parameters on the frequencies are examined for various cases. It is observed that the type of effect of the surface energy parameters depends on their sign. It is also seen that the type of effect of the surface energy parameters on nonlinear frequencies may become different at higher frequency numbers. The observations of this study can be a useful reference for more accurate design and fabrication of nano-electro-mechanical systems in which rod-shaped nano-structures, carbon nanotubes, play a significant role.

Modification of boundary condition for the optimization of natural frequencies of plate structures with fluid loading

A finite element method, boundary element method, and genetic algorithm combined method is developed for the optimization of natural frequencies of fluid-loaded plates. In this method, the coupled finite element method–boundary element method is used for the free flexural vibration analysis of plates with arbitrary fluid loading effects and arbitrary elastic boundary conditions, and the genetic algorithm method is combined with the finite element method–boundary element method for searching the optimal values of plate’s boundary parameters. By using this method, multiple natural frequencies of a given fluid-loaded plate can be optimized simultaneously to different target values. The coupled finite element method–boundary element method is first validated by comparing with earlier published results. The proposed optimization method is then applied to the optimal boundary condition design of four different cases. The results show natural frequencies of a fluid-loaded plate are sensitive to its boundary conditions. The possibility of optimizing the natural frequencies of a fluid-loaded plate by modifying boundary conditions is demonstrated, as well as the effectiveness of the proposed method as a structural optimization tool. According to the authors’ knowledge, this study is the first attempt of optimizing fluid-loaded plate natural frequencies by considering arbitrary boundary conditions as optimization variables.

Dynamic Analysis of a Rod Vibro-Impact System with Intermediate Supports

Abstract The two-mass resonant vibro-impact module is presented as the rod system with cylindrical intermediate supports. The corresponding design diagram is constructed. Based on the finite element method, the frequency of free oscillations is defined for the corresponding location of the intermediate supports. A stress-strain state of the elastic element is considered. The stiffness of the intermediate supports is defined by solving the contact problem between the cylindrical rod supports and the flat spring. The dynamics of the vibro-impact rod system with multiple natural frequencies is analyzed taking into account the contact stiffness of the intermediate supports. The determination of contact and equivalent stresses occurring during the operation of the vibro-impact rod system is performed.

Natural frequency assignment for mass-chain systems with inerters

Abstract This paper studies the problem of natural frequency assignment for mass-chain systems with inerters. This is the problem to determine whether an arbitrary set of positive numbers may be assigned as the natural frequencies of a chain of n masses in which each element has fixed mass and is connected to its neighbour by a parallel combination of a spring and inerter. It is proved that mass-chain systems with inerters may have multiple natural frequencies, which is different from conventional mass-chain systems (without inerters) whose natural frequencies are always simple. It is shown that arbitrary assignment of natural frequencies including multiplicities is not possible with the choice of n inerters and n springs. In particular, it is shown that an eigenvalue of multiplicity m may occur only if n ⩾ 2 m - 1 . However, it is proved that n - 1 inerters and n springs are necessary and sufficient to freely assign an arbitrary set of distinct positive numbers as the natural frequencies of an n-degree-of-freedom mass-chain system.