Frequency Of Harmonic Excitation Research Articles

التحليل العددي وتطبيقات الاهتزاز المزدوج في العارضات ذات الرقائق المركبة الغير متماثلة

This paper presents a comprehensive numerical analysis and application of extension-bending-shear-torsion coupled vibration in asymmetric laminated composite beams, in which the exact closed-form solutions were previously obtained to investigate the steady-state dynamic response of such beams under various harmonic loading and boundary conditions. Key objectives include predicting natural frequencies and corresponding mode shapes, as well as establishing quasi-static responses for different harmonic excitation frequencies. Numerical examples are used to validate the accuracy and efficiency of the proposed solutions by comparing static and dynamic results with those from existing literature and finite element analysis. The study reveals excellent agreement between the closed-form solutions and previously published results, emphasizing the utility of the developed approach for both static and dynamic coupled vibration analysis in asymmetric laminated beams. The results offer valuable insights into the dynamic behavior of asymmetric composite beams, with potential applications in engineering structures subjected to complex loading conditions. Keywords: Coupled vibration, steady state response, asymmetric laminated beam, harmonic loading.

Stochastic dynamics analysis for unilateral vibro-impact systems under combined excitation

Vibro-impact system, as an important type of non-smooth system, exhibits intricately nonlinear characteristics. Inevitably, the vibro-impact system will encounter random excitations, but the conventional methods ar7e not eligible for simultaneous determination of its transient responses and reliabilities. Commonly, existing methods of applying non-smooth transformation tend to ignore the essential non-smooth characteristics of vibro-impact system. To this end, this paper proposes a unified framework based on direct probability integral method (DPIM) to simultaneously determine stochastic dynamic responses and reliabilities of unilateral vibro-impact systems under combined harmonic and random excitation without non-smooth transformation, and captures their complicated dynamical behaviors. Firstly, the impact velocity dependent coefficient of restitution is introduced to establish the motion equation of vibro-impact system. Secondly, the probability density integral equation (PDIE) for the unilateral vibro-impact system is derived from the perspective of probability conservation. Then, the PDIE and governing differential equation of the system is solved in a decoupled and efficient way. Moreover, the first-passage reliability is assessed by introducing extreme value mapping of the stochastic dynamic response. Numerical results of three typical examples using the proposed framework are compared with those using Monte Carlo simulation (MCS), quasi-MCS and from the reference, which highlights the advantages of DPIM in computing the stochastic responses and reliabilities of vibro-impact system under random excitations and random parameters. The stationary probability density functions exhibit periodic fluctuations under combined harmonic and stochastic excitation. Specially, the noise intensity and frequency of harmonic excitation pose the great influence on the reliabilities of systems.

Nonlinear vibration and acoustic radiation of an internally resonant buckled beam

Although the nonlinear vibration of buckled beam has been thoroughly studied, its acoustic radiation does not get much attention. This paper deals with the nonlinear vibro-acoustic problem of an internally resonant buckled beam immersed in an infinite fluid. A finite element method based on an arbitrary Lagrangian-Eulerian framework is adopted to analyze the nonlinear interaction between the large-deformed buckled beam and the finite-amplitude acoustic waves in the fluid. The effects of acoustic pressure excitation, external harmonic excitation, and internal resonance on the vibro-acoustic responses of the buckled beam are discussed in the first and second primary resonances. It is found that the amplitude of the acoustic pressure can be comparable to the amplitude of the external harmonic force. This leads to an increase in the critical harmonic excitation amplitude for the occurrence of quasi-periodic responses and an increase in the second mode frequency component of quasi-periodic responses compared to the case neglecting the acoustic pressure excitation on the beam. Saturation phenomena are discovered in the second mode frequency components of beam displacement and acoustic pressure. Due to the 2:1 internal resonance, different time-frequency characteristics, quadrupole acoustic directivity patterns with low radiation efficiency, and dipole acoustic directivity patterns with high radiation efficiency can be observed at different harmonic excitation amplitudes and frequencies. Understanding these phenomena will help design loudspeakers and recognize sound sources.

Optimal and Quasi-Optimal Automatic Tuning of Vibration Neutralizers

Vibration neutralizers are single-degree-of-freedom devices affixed to vibrating structures in order to reduce the response at a specific troublesome harmonic excitation frequency. As this frequency may vary over time, it becomes imperative to track and adjust the neutralizer to maintain the optimal performance. Recent years have witnessed the emergence of adaptive tunable vibration neutralizers, offering real-time adjustment capabilities through external actions. Thanks to real-time control algorithms, these devices enable the automatic mitigation of vibration levels in mechanical structures. A particularly successful algorithm for the automatic tuning of these devices leverages the phase angle between the base acceleration and the neutralizer’s mass. This study critically examines the justification for employing such an algorithm and scrutinizes its optimal applicability limits, particularly in the context of viscous and structurally damped systems. The findings reveal that this algorithm accurately approximates optimum tuning for systems with low damping. Moreover, from an engineering perspective, the algorithm remains acceptable even for heavily damped structures. Through a focused and comprehensive analysis, this paper provides valuable insights into the efficacy and limitations of the phase-angle-based tuning algorithm, contributing to the advancement of adaptive vibration control strategies in smart structures.

Experimental study on characteristics of liquid dynamics in a cylindrical water tank under harmonic excitation

AbstractAs the offshore engineering industry continued to develop, issues such as damage to storage structures due to liquid shaking became increasingly prevalent. This paper explored the dynamic response of liquid within a rigid cylindrical water tank subjected to harmonic excitation through vibration table tests, analyses, and numerical simulations. Using the Laplace equation and the Bessel function of the first kind, the analytical formula of liquid dynamic response was derived. A numerical model was established by ANSYS software to verify the accuracy of the analytical solution. At the same time, the response characteristics of hydrodynamic pressure and liquid level wave height were studied by vibration table tests. In this experiment, a rigid cylindrical water tank with varying depths of liquid was subjected to harmonic excitations utilizing a vibration table. In most cases, the experimental results agreed well with the analytical formula. However, under conditions of relatively severe liquid sloshing, there was a discrepancy in peak values between the experiment and the analytical formula. The experimental data indicated that both the hydrodynamic pressure and liquid level wave height increase with an increase in harmonic excitation frequency and amplitude.

An asymmetric bistable vibro-impact DEG for enhanced ultra-low-frequency vibration energy harvesting

In order to enhance the performance of bistable energy harvesters with dielectric elastomers (DEs) under ultra-low frequency vibrations, this paper proposes an asymmetric bistable vibro-impact dielectric elastomer generator (ABVI DEG), which mainly consists of a vibro-impact (VI) DEG, two identical pre-compressed springs, two pairs of undeformed springs, and a base. Two kinds of dynamics models of the proposed energy harvester under ultra-low frequency harmonic excitations are developed, and the instantaneous impact model is selected due to its computational efficiency and accuracy. The instantaneous impact model is extended by a series of impact experiments and validated by comparing the dynamical responses of the proposed energy harvester obtained from the two kinds of dynamics models. On this basis, the dynamical behavior and the energy harvesting (EH) performance of the proposed energy harvester under ultra-low frequency harmonic excitations are studied for the initial conditions of the system, different bistable potential wells, and various parameters, which include the pre-stretched ratio of the membranes and the distance between the two membranes. A further comparative study demonstrates the superiority of the proposed energy harvester under the ultra-low frequency harmonic excitation. This work not only provides an insight into dynamical behavior of the proposed ABVI DEG, but helps guide the design and optimization of the proposed EH system in the ultra-low frequency vibration environment.

Modelling and vibration response of a magnetically suspended flexible rotor considering base motion

Active magnetic bearings can provide no-friction support, high operating speed, and active vibration control for rotating machinery. The magnetically suspended rotor will inevitably be excited by the base motion when working on the moving carrier, causing the rotor vibration, and even instability. The main purpose of this study is to model a magnetically suspended flexible rotor under base motion and predict the vibration response. A general model of the magnetically suspended flexible rotor under base motion is established based on the finite element method and Lagrange's equation, considering the rotor shrinkage fit and AMB sensor/actuator being not collocated. The accuracy of the rotor model in free-free state is verified in terms of modal frequencies and mode shapes through the modal test. The vibration response of magnetically suspended rotor is experimentally validated using an AMB-rotor test rig mounted on the shaker. The time-domain response of the rotor under base single harmonic and shock excitation in the experiments is close to that predicted by numerical simulation. Finally, the influence of the frequency and amplitude of the base single harmonic excitation, the amplitude and pulse width of the base shock excitation, and the AMB controller parameters on the vibration response of the magnetically suspended rotor are investigated.

Harvesting weak vibration energy by amplified inertial force and multi-stable buckling piezoelectric structure

One of the main challenges in vibration energy harvesting is to improve the efficiency for low-frequency and weak ambient vibration. A desired harvester should own a broadband working frequency and large-amplitude under weak excitation. In this paper, a novel harvester is proposed, which can enhance the energy harvesting performance for weak ambient vibration by amplified inertial force and multi-stability. This harvester is constituted of an inertial mass, two piezoelectric buckled beams and two linkages. Under external excitation, the inertial force produced by inertial mass is amplified and acts on the piezoelectric buckling beams via linkages, by which the amplified inertial force can drive the piezoelectric buckling beams to execute inter-well oscillation densely and generate high output power over a broadband frequency. Furthermore, the harvester possesses a super-harmonic vibration characteristic and can realize the frequency up-conversion effect, so the low-frequency excitation can excite the high-frequency vibration of piezoelectric beam and increase the output. The influence of the beam’s buckling level is studied through potential energy analysis. The output performance and nonlinear dynamic characteristics are studied under different types of excitations, i.e., the frequency sweeping, harmonic and random excitations. The validation experiment was carried out. The results prove that the harvester owns the super-harmonic characteristic and can execute snap-through motion under weak random excitation. Thus it can produce a larger output for weak excitation.

VIBRATION TRANSMISSIBILITY OF THE COFFEE FRUIT-PEDUNCLE SYSTEM: A FORCED VIBRATION STUDY OF HIGH FREQUENCY AIMING MECHANICAL HARVESTING

Semi-mechanized and mechanized harvesting use machines that promote the transference of vibrational energy and impact to achieve the detachment of coffee fruits. The aim of this study was to evaluate the vibration transmissibility in coffee fruit-peduncle systems, using high-speed cameras, submitted to high frequency harmonic excitation in different combinations between frequency and amplitude of vibration, identifying working ranges suitable to perform selective harvesting. Vibration transmissibility was determined for the coffee fruit-peduncle systems, for the maturation stages unripe and ripe that were subjected to a sinusoidal harmonic displacement, in which the input parameters were frequency (35, 45 and 55 Hz) and peak-to-peak amplitude (3.5, 5.0 and 6.5 mm). An experiment was used to study the effect of frequency and amplitude on vibration transmissibility in a completely randomized design in a factorial scheme 3 x 3 x 2, with three replications. The frequency of 35 Hz, associated with the amplitudes 3.5-6.5 mm, was the one that most influenced the results of vibration transmissibility. For the frequency of 55 Hz and amplitude of 6.5 mm, in the ripe maturation stage, the vibration transmissibility was higher than 1.0, which could be a suitable combination for selective coffee harvesting.

Operational modal analysis under harmonic excitation using Ramanujan subspace projection and stochastic subspace identification

The presence of deterministic harmonic excitation, such as that induced by rotating machinery, violates the classical operational modal analysis (OMA) assumption that the system output is strictly ergodic. This paper proves that the presence of harmonic excitation has no effect on the identification of structural modes except in the case where some of the harmonic excitation frequencies are close to the structural frequencies. Therefore, the harmonic component must be removed from the mixed random and harmonic system output before further processing. This paper proposes a Ramanujan subspace projection (RSP) method for harmonic removal, which is realized by projecting the raw system output onto the complex conjugate division (CCD) of the Ramanujan subspace. The novelty of this study is that it reveals the relationship between the frequencies defined in the period and frequency domains, allowing the RSP method to directly extract the harmonic component from the specific CCD without any frequency domain analysis. In addition, an energy indicator is proposed to select the underlying CCDs with the most robust harmonic feature. Using the indicator, one only needs to project the raw system output onto a subset of the CCDs rather than all of them, thereby significantly reducing the computational effort. After removing the harmonic components from the raw output, the remainder can be fed into the covariance-driven stochastic subspace identification (Cov-SSI) method for OMA. The numerical, experimental and field test results show that the proposed RSP method is not only resistant to random noise but also capable of precisely extracting the weak harmonic component from the raw output with less computation. Furthermore, the modal parameters of the structures subjected to mixed random and harmonic excitation can be accurately identified by combining the Cov-SSI and RSP methods.

Quasi-Periodic Solutions of a Damped Nonlinear Quasi-Periodic Mathieu Equation by the Incremental Harmonic Balance Method With Two Time Scales

Abstract Quasi-periodic (QP) solutions of a damped nonlinear QP Mathieu equation with cubic nonlinearity are investigated by using the incremental harmonic balance (IHB) method with two time scales. The damped nonlinear QP Mathieu equation contains two incommensurate harmonic excitation frequencies, one is a small frequency while the other nearly equals twice the linear natural frequency. It is found that Fourier spectra of QP solutions of the equation consist of uniformly spaced sidebands due to cubic nonlinearity. The IHB method with two time scales, which relates to the two excitation frequencies, is adopted to trace solution curves of the equation in an automatical way and find all frequencies of solutions and their corresponding amplitudes. Effects of parametric excitation are studied in detail. Based on approximation of QP solutions by periodic solutions with a large period, Floquet theory is used to study the stability of QP solutions. Three types of QP solutions can be obtained from the IHB method, which agree very well with results from numerical integration. However, the perturbation method using the double-step method of multiple scales (MMS) obtains only one type of QP solutions since the ratio of the small frequency to the linear natural frequency of the first reduced-modulation equation is nearly 1 in the second perturbation procedure, while the other two types of QP solutions from the IHB method with two time scales do not need the ratio. Furthermore, the results from the double-step MMS are different from those numerical integration and the IHB method with two time scales.

Time-varying dynamic analysis of a plate entering the finite subsonic airflow field

Devices entering rigid wall constrained finite airflow field are commonly seen in engineering practice. The devices gradually move from the infinite airflow field to the finite airflow field, which makes the device and the outside flow field a complicated time-varying fluid–structure interaction system. During the entry process, the stiffness, damping and mass of the devices change constantly over time, which makes the dynamic behaviors of the system show complex time-varying properties. Hence, it is needed to study the stability and vibration characteristics of the devices entering the finite airflow field. In this study, based on the von Karman nonlinear plate theory and the linear potential flow theory, the governing equation of the plate entering the finite airflow field is established by employing the Hamilton’s principle. By using the generalized eigenvalue method to linear system, the variations of the natural frequencies of the plate in the process of entering the finite airflow field are investigated, from which the stability change rules of the system are obtained. The vibration properties of the system are studied by analyzing the displacement time response of the plate in the process of entering the finite flow field. From the results, the different system parameters such as entry position, entry velocity, aspect ratio, external harmonic excitation amplitude and external harmonic excitation frequency have significant impacts on the stability and vibration properties of the plate entering the finite airflow field.

Online Frequency Estimation on a Building-like Structure Using a Nonlinear Flexible Dynamic Vibration Absorber

The online frequency estimation of forced harmonic vibrations on a building-like structure, using a nonlinear flexible vibration absorber in a cantilever beam configuration, is addressed in this article. Algebraic formulae to compute online the harmonic excitation frequency on the nonlinear vibrating mechanical system using solely available measurement signals of position, velocity, or acceleration are presented. Fast algebraic frequency estimation can, thus, be implemented to tune online a semi-active dynamic vibration absorber to obtain a high attenuation level of undesirable vibrations affecting the main mechanical system. A semi-active vibration absorber can be tuned for application where variations of the excitation frequency can be expected. Adaptive vibration absorption for forced harmonic vibration suppression for operational scenarios with variable excitation frequency can be then performed. Analytical, numerical, and experimental results to demonstrate the effectiveness and efficiency of the operating frequency estimation, as well as the acceptable attenuation level achieved by the tunable flexible vibration absorber, are presented. The algebraic parametric estimation approach can be extended to add capabilities of variable frequency vibration suppression for several configurations of dynamic vibration absorbers.

Non-probabilistic reliability-based robust design of micro-scale topology optimization (NRRD-MTO) for structural vibro-acoustic problem under harmonic excitation and natural frequency constraints

Non-probabilistic reliability-based robust design of micro-scale topology optimization (NRRD-MTO) for structural vibro-acoustic problem under harmonic excitation and natural frequency constraints

Quantitative prediction of rattle noise: An experimentally validated approach using the harmonic balance method

Rattle noise in a car’s interior must be avoided to prevent customer complaints. Virtually predicting such disturbing sounds without expensive prototyping has become more important in recent years to reduce development costs. For this reason, methods for simulating the acoustic behavior of rattling systems are gaining attention. This research validates the harmonic balance method for a quantitative prediction of rattle noise. For this purpose, a test rig is developed showing representative rattling and non-rattling configurations. The test rig consists of an oscillating beam-like structure with a lumped mass at its tip. When amplitudes are large enough, the beam serves as an impulsive excitation of a plastic plate. The nonlinear dynamic system response is measured using a 3D scanning laser Doppler vibrometer and the harmonic balance method is used to calculate the system’s response, which predicts the first and higher order harmonics with high degree of accuracy. Furthermore, measurements and simulations show that solutions for the system’s response are not necessarily unique for all predefined harmonic excitation frequencies. Finally, a motivation is given for why higher harmonics have to be taken into account to adequately predict the acoustic behavior.

Vibration Control of Time-Varying Delay under Complex Excitation.

The existing available research outcomes on vibration attenuation control for time-delay feedback indicate that, for the delay dynamic vibration absorber with fixed time-delay control parameters, under harmonic excitation, a good vibration attenuation control effect occurs on the vibration of the main system. However, the effect is not obvious for complex excitation. Aiming at the above problems, in a short time interval, a harmonic excitation with the same displacement size as the complex excitation was established. Then, by calculating its equivalent amplitude and equivalent frequency, a harmonic equivalent method for complex excitation was proposed in this paper. The time-delay parameters were adjusted according to the equivalent frequency of harmonic equivalent excitation in real time; therefore, a good vibration attenuation control effect was obtained through the delay dynamic vibration absorber in the discrete time interval. In this paper, research on a time-varying delay dynamic vibration absorber was conducted by taking the two-degree-of-freedom vibration system with a delay dynamic vibration absorber as an example. The simulation results show that the proposed control method can reduce the vibration of the main system by about 30% compared with the passive vibration absorber. This can obviously improve the performance of the time-delay dynamic vibration absorber. It provides a new technical idea for the design of vehicle active frame system.

A method to distinguish harmonic frequencies and remove the harmonic effect in operational modal analysis of rotating structures

Operational modal analysis (OMA) is used to identify the modal parameters of a structure under operational conditions, which reflects the real boundary conditions and state-dependent parameters. For rotating structures under operational conditions, in addition to broadband excitations, the harmonic excitation is also present and even dominates the responses in some cases. Since only the responses are recorded in OMA, the first problem lies in separating the resonant frequencies from the harmonic excitation frequencies. Moreover, if the frequency of the harmonic excitation is close to, or coincides with a resonant frequency of the system, classical modal analysis procedures will fail to identify the modal parameters accurately. These issues are addressed in this paper and a method is presented for distinguishing harmonic frequencies and removing the harmonic effect using correlation functions of the responses. Firstly, the correlation functions are deduced for the system under the combination of broadband and harmonic excitations, yielding the sum of free decay response of the system and sinusoids with the same frequencies as the harmonic excitations. Then, the principle of the method is presented using a numerical study on a two degree-of-freedom system. Both the noise resistance and method comparison studies are conducted in the simulation case. Subsequently, the effectiveness of the method is demonstrated through experiments on a light-damped beam, where cases of different harmonic intensity and different harmonic frequencies are considered. Lastly, the proposed method is applied to a rotating blade to extract modal parameters in its operational state.

A novel nonlinear mechanical oscillator and its application in vibration isolation and energy harvesting

The nonlinear design of a new vibration structure is an essential part of the development of vibration isolation and energy harvesting technologies, and is one of the most effective technical means to improve the efficiency of low-frequency excitation. Due to structural constraints, high-efficiency vibration isolation and energy harvesting from ultra-low frequency or low intensity excitation has been a theoretical bottleneck and technical challenge in this field. In this study, this paper firstly proposes the theory and method of vibration isolation and energy harvesting based on the high-order quasi-zero stiffness (HQZS) mechanism, which is expected to break through the above bottleneck. Secondly, a detailed study of the HQZS oscillator with electromagnetic transduction under both stochastic and harmonic excitations is emphasized. The HQZS oscillator can not only achieve arbitrarily small stiffness near the equilibrium position through geometrically nonlinear parameter design, but also achieve vibration isolation or high-energy interwell oscillation under ultra-low frequency or low intensity excitations. Finally, compared with quad-stable and quasi-zero stiffness (QZS) systems, the results show that the HQZS oscillator has better vibration isolation and energy harvesting performance for ultra-lower stochastic excitation intensity or harmonic excitation frequency. Also, the initial isolation frequencies and maximum harvested power frequencies of the HQZS oscillator are lower than that of the QZS oscillator. The results will not only help to deepen the understanding of HQZS systems, but also provide new theoretical support and technical approaches for vibration isolation and energy harvesting under ultra-low frequency or low intensity excitation.

Bending-torsion coupling bursting oscillation of a sandwich conical panel under parametric excitation

Regarding the dynamic behavior of the functionally graded materials (FGM) conical panels under the static and harmonic in-plane loading, most studies focus on parametric resonances and dynamic instability which determine the stability regions. Studies on the bursting oscillation of the conical panels under the lower frequency excitation are quite rare. In this paper, we present a novel nonautonomous system of the cantilever functionally graded materials sandwich conical panels with a slowly varying parametric excitation and study the bending-torsion coupling bursting oscillations. The face sheets and core layer are the metal ceramic composite layer and homogeneous metal layer, respectively. In two surfaces layers, the component materials are gradually changed in the transverse direction and the temperature dependent properties show the form of the power law distribution. The excitations in the meridional direction include the static preload and lower frequency harmonic excitation. The mode functions of the first-order bending mode and the first-order torsion mode are obtained in terms of Chebyshev polynomials with the help of energy principle. The bending-torsion coupling nonautonomous nonlinear oscillation system is derived by Lagrange equations. Under the slowly varying in-plane load, the equilibrium solutions of the system are obtained by treating them as a generalized autonomous system and they will be no longer stable through the pitchfork bifurcation with the symmetry breaking. The bursting oscillations of the novel dynamic system are analyzed by numerical methods. It is interesting to find that the mechanisms of bursting oscillation are associated with the pitchfork bifurcation with the symmetry breaking.

Inter-turn Fault Identification of Surface-Mounted Permanent Magnet Synchronous Motor Based on Inverter Harmonics

Inter-turn short-circuit faults can lead to further faults in motors. This makes monitoring and identifying such faults particularly important. However, because of interference in their working environment, fault signals can be weak and difficult to detect in permanent magnet synchronous motors. This paper proposes a method for overcoming this by extracting the inverter harmonics as an excitation source and then extracting characteristic of fault measurements from the negative sequence voltage. First of all, a model of permanent magnet synchronous motor faults is established and a fault negative sequence voltage is introduced to calculate the fault indicators. Then the high frequency harmonic excitation in the voltage is extracted. This is injected into the original voltage signal and the high frequency negative sequence component is separated and detected by a second-order generalized integrator. Simulation results show that the proposed method can effectively identify inter-turn short-circuit faults in permanent magnet synchronous motors while remaining highly resistant to interference. The method is especially effective when the severity of the fault is relatively small and the torque is relatively large.