Amplitude Frequency Response Curves Research Articles

Application and characterization of digital Sallen–Key filter in nuclear spectrum smoothing

In the nuclear spectrum analysis processing, spectrum smoothing can remove the statistical fluctuation in the spectrum, which is beneficial for peak detection and peak area calculation. In this work, a spectrum smoothing algorithm is proposed based on digital Sallen–Key filter, which contains four parameters (m, n, k, D). The amplitude–frequency response curve of Sallen–Key filter is deduced and the filtering performance is analyzed. Meanwhile, the effects of the four parameters on the shape of the smoothed spectrum are explored: D affects the counts and peak areas of the spectrum, and the peak area can be corrected by the peak area correction function S’. The parameters of m, n and k affect the peak position after smoothing, making the peak position shift to the right, and the peak position correction function P’ can be used to correct the peak position, when n¿2, the spectrum data appear negative after smoothing, when k¿2, the smoothed spectrum broadening degree is greater than 20%. Smoothness (R), noise smoothing factor (NSF), spectrum count ratio before and after smoothing (PER), and comprehensive evaluation factor (Q) are used to evaluate the smoothing effect of the algorithm. The parameters of the algorithm are optimally selected: about the gamma spectrum of 137Cs and 60Co, the optimal parameters are m=1.5 n=2 k=2 D=1, about the characteristic X-ray spectrum of Fe and quasi-geological sample (TiMnFeNiCuZn), the optimal parameters are m=1.1 n=1.1 k=1.3 D=1. Based on Sallen–Key smoothing method, Fourier transform method, Gaussian function method, wavelet transformation method, center of gravity method and least squares method, the gamma spectrum of 137Cs is smoothed and denoised in this paper. The results show that the Sallen–Key method has better spectrum denoising effect (R=0.6056) and comprehensive performance indicators (Q=0.6104), which can be further applied for the smoothing of nuclear spectrum data.

Ultra-low frequency vibration energy harvesting of piezoelectric vibration systems with an adjustable device

Energy harvesting from external Ultra-low frequency vibrations is currently attracting a lot of attention. However, for the same system, it is difficult to satisfy the need for more efficient energy harvesting in a variety of external environments. Accordingly, an adjustable gearing device connected to the energy harvesting system is used herein for adapting energy harvesting to a wide range of external environments. The device uses four symmetrical inclined springs. One end of the springs is connected to a gear and the other end to a mass element. The geometrical parameters of the system can be varied by rotating the gears, thus varying the dynamic stiffness of the system. The kinetic equations of the system are derived and their nonlinear terms are Taylor-expanded. The amplitude-frequency response curve of the system is obtained by solving using the harmonic balance method. Multi-color plots of RMS values of induced voltage and electric power are obtained using the fourth-order Runge-Kutta method. The results show that the device has excellent adjustable performance. Adjusting the rotation angle of the gears can change the soft and hard characteristics of the system's dynamic stiffness and the intrinsic frequency, thus improving the system's energy harvesting performance in different environments.

Instrumentation of the Nigeria network of seismological stations

“Living in fear of Nigeria biggest Earthquake” is a sub-heading of the Punch daily newspaper of Nigeria, dated, 21st of August, 2016, reported the earthquake/tremor recently witnessed in the ancient town of Saki, accompanied by a series of aftershock events that lasted for about three (3) months (March/May, 2016) southwest Nigeria[1]. Similarly, roughly five (5) years later, another series of earthquakes/tremors occurred again in Saki town, which was reported by an online news vendor named “Ripples Nigeria” dated September 8, 2021, under the sub-heading of “earth tremor rocks Saki in Oyo state”[2]. However, these events were not captured nor recorded by any of the functional seismological stations in Nigeria. The nearest seismological station located at the Obafemi Awolowo University (OAU) Ile-Ife, also failed to capture the events. We therefore seek to examine the instrumentation of the Nigerian National Network of Seismographic Stations. This is necessary to understand the functionalities and the capabilities of the deployed seismometers in each of the seismic stations. Therefore, we evaluated the bandpass limit of the seismic wave frequency for the respective seismic stations in Nigeria through the computation of the amplitude-frequency response curve, phase response curve, and count to cm/sec.

Nonlinear vibration control of magnetic bearing system considering positive and negative stiffness

The nonlinear dynamics are studied for an active magnetic bearing (AMB) system under the low-frequency sinusoidal excitation, which can affect the operational stability of a high-speed magnetic bearing. A single-degree-of-freedom nonlinear dynamics model of magnetic bearing was developed with a c-type active magnetic bearing as the research target. The characteristics of the magnetic bearing system show positive and negative stiffness for different feedback gains. The amplitude–frequency response curves of the vibration characteristics of the positive stiffness magnetic bearing system were obtained by using the method of multiple scales. The influence of each parameter on the resonance amplitude and nonlinearity of the magnetic bearing system is understood by analyzing the amplitude–frequency response curve. The bifurcation diagram for the rotor with rotor mass, and control parameters is derived from the numerical simulation of the negative stiffness magnetic bearing system. The parameter intervals that cause single-cycle vibration, double-cycle vibration, and chaotic state of the magnetic bearing rotor are discovered by analyzing the bifurcation diagrams of each parameter. The simulation results show that the magnetic bearing system exhibits complex nonlinear dynamic characteristics when it is a negative stiffness system.

Nonlinear vibrations and control of PFG-GR laminated composite cantilever rectangular variable cross-section plate with NPPF controller

This paper studies the nonlinear vibrations and control of the piezoelectric functionally graded graphene-reinforced laminated composite cantilever rectangular (PFG-GRLCCR) variable cross-section plate with the NPPF controller under the simultaneous resonant cases. The nonlinear dynamic governing equations of motion for the PFG-GRLCCR variable cross-section plate are obtained by using the extended Halpin-Tsai mode, linear constitutive equations of the piezoelectric materials, classical laminated plate theory, von Karman large deformation theory, and Hamilton principle. Considering Galerkin method, the nonlinear ordinary differential governing equations are obtained. The nonlinear positive position feedback (NPPF) controllers are designed according to the similarity principle. The multiple scale method is used to investigate the nonlinear vibration characteristics of the stable motion for the PFG-GRLCCR variable cross-section plate with the NPPF controllers. The amplitude-frequency response curves, force-amplitude response curves, time histories, phase portraits, and bifurcation diagrams are given to investigate the nonlinear dynamic characteristics of the PFG-GRLCCR variable cross-section plate with the NPPF controller. The results demonstrate that the NPPF controller can suppress the nonlinear vibrations of the structure well, and the effectiveness of the controllers is best when the NPPF controller has a similar nonlinear characteristic with the structure. The complex nonlinear vibration characteristics appear for the PFG-GRLCCR variable cross-section plate with the NPPF controller.

Primary resonance characteristics of a cylindrical bubble based on the multi-scale method

This paper describes a primary resonance theoretical model for a cylindrical bubble under acoustic excitation. Based on the multi-scale method, an analytical solution of the bubble–wall equation with second-order accuracy is obtained and numerically verified. The oscillation characteristics in the time domain and the frequency response characteristics of the oscillations under primary resonance are analyzed with different amplitudes and frequencies of acoustic excitation and the equilibrium radius of the bubble. This study yields the following primary findings: (1) For the cylindrical bubbles, the primary resonance of the bubble exists in unstable regions. Nonlinear behaviors such as jumps, hysteresis, and multivalued solutions may be widely present. (2) As the amplitude of the acoustic excitation and the bubble equilibrium radius increase, the backbone of the amplitude–frequency response curve bends to the left and the unstable region gradually expands. (3) When the dimensionless amplitude of the acoustic excitation is less than 0.005 and the bubble equilibrium radius is less than 1.0 × 10−5 m, the unstable region of resonance disappears.

Nonlinear dynamics of graphene-reinforced aluminum matrix composite aero-engine blade in thermal environment

This paper analyzes the nonlinear dynamic behaviors of the graphene-reinforced aluminum-based aero-engine blade under the primary resonance using the amplitude-frequency response curves, bifurcation diagram, Lyapunov exponent and Poincaré map. The lateral aerodynamic load, longitudinal aerodynamic load, centrifugal force and temperature are investigated for the graphene-reinforced aluminum-based aero-engine blade. Three kinds of configurations for the functionally graphene-reinforced gradient are considered. The effective Young's modulus of the composite material is given by Halpin-Tsai mode. The rule of the mixture gives other material properties. The results demonstrate that the bifurcation diagrams of the modal amplitude for the blade with the lateral aerodynamic load, longitudinal aerodynamic load and temperature have the periodic-chaos dynamic evolution process under the disturbance of the engine rotation speed. When the disturbance amplitude reaches a specific value, a new round of the periodic-chaos dynamic evolution will be continued. The results also clearly indicate that the aero-engine blade has the rich and complex nonlinear dynamic behaviors which include the hyperchaos, chaos, almost periodic, typical period-doubling bifurcation and anti-period-doubling bifurcation vibrations. This research is more helpful for the engineers to understand the nonlinear dynamic behavior of the blade.

Vibration Suppression Study of Parallel Combined Damping Nonlinear Energy Sink on Piecewise Linear Systems

The Nonlinear Energy Sink (NES) is a passive control device capable of achieving targeted energy transfer and effectively suppressing system vibrations. This paper conducts dynamic modeling and vibration suppression analysis of parallel combined damping nonlinear energy sinks in piecewise linear systems. Initially, the system's amplitude-frequency response curve is obtained using the harmonic balance method, validated by the Runge-Kutta method for accuracy. Subsequently, an analysis is performed on the vibration reduction capabilities of the parallel combined damping nonlinear energy sink and the influence of parameters on its vibration suppression effectiveness. Finally, the study investigates changes in the main system's maximum response amplitude with varying damping under different stiffness conditions and determines optimal parameter values based on trend analysis. Research findings indicate that the vibration reduction performance of the parallel combined damping nonlinear energy sink surpasses that of a single connected damping nonlinear energy sink. Furthermore, after parameter optimization, the main system achieves superior vibration reduction effects.

Buckling Stability and Vibration Characteristics of Functionally Graded Graphene Platelet-Reinforced Composites Multi-Arc Concave Honeycomb Sandwich Panels Under Thermal Loading

The utilization of functionally graded graphene platelet-reinforced composites (FG-GPLRC) and honeycomb sandwich constructions is prevalent in the field of engineering. Therefore, it is crucial to investigate the buckling and vibration properties of these materials under heat circumstances. This paper specifically focuses on the vibration and thermal buckling characteristics of a novel multi-arc concave honeycomb sandwich plate (MACHSP) equipped with an adjustable positive-negative Poisson’s ratio. The governing equations of motion for the system are derived based on the first-order shear deformation theory in conjunction with Hamilton’s principle. The formulation of the system’s characteristic equation involves the establishment of displacement functions that satisfy distinct boundary conditions. Using computational programming, natural frequencies, mode shapes, and critical buckling temperatures of MACHSP with three different FG-GPLRC distributions under thermal influences are determined. Subsequently, these results are compared with outcomes from finite element simulations and findings from previously published literature. Additionally, the amplitude-frequency response characteristics of the structure under harmonic excitation are analyzed, followed by a discussion of the impact of structural parameters on its buckling and vibration characteristics. The results indicate that the established theoretical model for predicting the buckling and vibration characteristics of MACHSP with FG-GPLRC distribution aligns well with simulation models and prior research. In contrast to solid plates, MACHSP not only achieves a substantial reduction in weight but also demonstrates a significant increase in natural frequency. Furthermore, graphene platelets have the potential to enhance the structure’s natural frequency values and critical buckling temperatures. Lastly, various structural parameters exert a notable influence on the structure’s performance.

Full-scale resonant bending fatigue testing of casing joints under bending moment load

The quality and reliability of oil casings play a pivotal role in the safety and efficiency of offshore oil and gas exploration. This study conducted research on resonant bending fatigue of 9-5/8-inch casing joints and established a systematic theoretical model for resonant bending fatigue based on the Euler–Bernoulli beam equation. The relationships between parameters affected by frequency and mode shape functions were deduced and solved to analyze the influence of wall thicknesses, length and weights at both ends of the specimen on the natural frequency of the resonant bending fatigue system. A full-scale fatigue test was then designed and conducted to investigate the circumferential stress distribution, reveal the stress amplitude–frequency response characteristics of the specimen, and explore resonance response behaviors under different excitation forces. This was done to establish a method for evaluating the oil casing life and obtain fatigue life characteristic curves of the specimen. A finite element model was employed to analyze the internal stress distribution of the coupling under bending moment load, considering the influence of material properties, thread lead angle and non-linear contact on casing joints. Finally, the fatigue life algorithm was summarized and compared by integrating the stress simulation results with the material properties of the specimen, elucidating the fatigue damage evolution process of casing joints. This research provides a significant engineering reference value for structural design optimization, fatigue testing and life analysis of oil casing threaded joints.

Investigation on the 1:2:3 internal resonance of high dimensional nonlinear dynamic system for FGGP reinforced rotating pre-twisted blade under the aerodynamic forces

The study of internal resonances is one of the main research areas in nonlinear dynamics. The energy transfer behaviors between multiple vibrational modes caused by internal resonances lead to complex nonlinear vibration behaviors. In this paper, the rotating blade is modeled as a functionally graded graphene platelet (FGGP) reinforced rotating pre-twisted plate. The nonlinear vibration of a high dimensional nonlinear dynamic system for FGGP reinforced rotating pre-twisted blade under the aerodynamic forces with 1:2:3 internal resonance is investigated. The axial excitations on the blade are primarily caused by axial aerodynamic force in the tip clearance and contact forces at the top of the blade. The subsonic airflow is treated as a transverse excitation and is calculated using the vortex lattice method. Rayleigh-Ritz method is used to obtain the mode shapes of the rotating composite pre-twisted blade. The three-degree-of-freedom ordinary differential equations of the rotating FGGP reinforced pre-twisted plate are derived by using Lagrange equation. Based on the multiscale method, the averaged equations of the rotating FGGP reinforced pre-twisted plate in the case of primary resonance and 1:2:3 internal resonance under axial and transverse excitations are acquired. We present a discussion on the amplitude-frequency response, force-amplitude response, bifurcations, and chaotic motions of a rotating FGGP reinforced pre-twisted cantilever plate under both axial and transverse excitations. A crescent-shaped independent resonance region is found in the amplitude-frequency response curves. The system exhibits complex nonlinear vibration behavior as axial or transverse excitation amplitude changes.

Combined resonance of graphene platelets reinforced metal foams cylindrical shells with spinning motion under nonlinear forced vibration

The recent studies on dynamical behavior of spinning cylindrical shells are mainly concerned on the principal and internal resonances, however, there is a lack of study on coupled resonance of spinning graphene platelets reinforced metal foams (GPLRMF) cylindrical shells. To remedy this defect, the present work attempts to investigate the combined resonances of spinning cylindrical shells under multi-source excitations. Considering Reddy's higher order shear deformation theory (HSDT), the highly dimensional nonlinear governing motion equations is derived and then discretized through Galerkin truncation. Afterwards, the method of varying amplitudes (MVA) is applied to obtain the steady-state approximate solution, in which the jump phenomenon and bifurcation behavior are discussed. Additionally, the dynamic model is confirmed via comparing with existing literature. Eventually, numerical analyses indicate the internal stable/unstable loops may appear unexpectedly, and the existence of the loop mainly depends on the variations of initial phase angle, external excitation amplitude, damping coefficient and so on. In addition, the amplitude frequency response curve of nonlinear vibration can be adjusted and supported through parametric resonance, and different from internal resonance, the multi-jump phenomenon of combined resonance will take place in the spinning cylindrical shells.

Nonlinear vibrations of FG-GRLCC rectangular variable cross-section plate subjected to transverse and parametric excitations in thermal environment

This paper investigates the linear and nonlinear vibration responses for the functionally graded graphene-reinforced laminated composite cantilever (FG-GRLCC) rectangular variable cross-section plate subjected to the transverse and parametric excitations under the thermal environment. Halpin-Tsai model is used to calculate the material properties of the graphene-reinforced structure. Using the classical laminated plate theory, von Karman large deformation theory, Hamilton principle and Galerkin method, the dynamic model is given for the FG-GRLCC rectangular variable cross-section plate. The natural frequencies and mode shapes are analyzed for the FG-GRLCC rectangular variable cross-section plate under the thermal environment through using Rayleigh-Ritz method. The averaged equations of the system are obtained based on the multiple scale perturbation (MSP) under the primary, 1/2 sub-harmonic and 1:1 internal resonances. The comparisons between the theoretical algorithm and finite element method are proposed to illustrate the accuracy of the present model. The results about the frequency veering phenomenon and mode shape interaction have been illustrated. The amplitude-frequency response curves and force-amplitude response curves are depicted for the FG-GRLCC rectangular variable cross-section plate. The nonlinear and chaotic vibrations for the FG-GRLCC rectangular variable cross-section plate is studied by using the bifurcation diagrams and max Lyapunov exponents.

Comparison of Amplitude-Frequency Response Characteristics Between DCM and Arctan Algorithms in ϕ-OTDR

In phase-sensitive optical time-domain reflectometry (ϕ-OTDR) systems, DCM and Arctan demodulation algorithms are typically used to recover the information on vibration events along the fiber. The amplitude-frequency response characteristics of the DCM and Arctan algorithms are investigated and compared using a 3×3 coupler demodulation technique. We first testify the demodulation capability of the DCM and Arctan algorithms through experiments in ϕ-OTDR. Then the amplitude-frequency response ranges of the two algorithms are analyzed according to the demodulated phase error distribution from both the experiment and simulation. It is shown that the Arctan algorithm has a wider amplitude-frequency range than the DCM algorithm. The phase error generation mechanism is also investigated and we find that, in a noise-free condition, the phase errors still appear with use of the DCM algorithm, which cannot be explained by the general perspective that the phase errors arise from the sensitivity of the differential operator to noise. For both algorithms the phase error is related to the amplitude (D) and frequency (f) of the vibration signal, and pulse repetition frequency <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(f_pulse)$</tex-math></inline-formula> . When the phase error tolerance is less than 0.1, the expression of the amplitude-frequency response characteristic of DCM algorithm is 5.7 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(D+1)f\leq f_pulse$</tex-math></inline-formula> , whereas that of the Arctan algorithm is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2Df\leq f_pulse$</tex-math></inline-formula> .

Experimental and numerical investigations on nonlinear snap-through vibrations of an asymmetrically composite laminated bistable thin plate simple supported at four corners

The nonlinear and snap-through vibrations of the bistable asymmetric composite laminated thin square plate with four corners simple support (FCSS) are investigated through the experimental testing and numerical simulations. Four corners simply support boundary conditions are designed to capture the nonlinear dynamic snap-through phenomena of the bistable asymmetric composite laminated thin square plate. An electromechanical shaker is utilized to excite the FCSS bistable thin square plate. Under the appropriate excitation amplitude and frequency, the FCSS bistable thin square plate is able to generate large vibrations around two stable states, namely, the snap-through vibrations. The vibration signals are acquired by using a laser displacement sensor. The complicated nonlinear dynamics are experimentally investigated for focusing on the nonlinear snap-through vibrations. Analyzing the vibration experiment, the various and interesting dynamic responses of the FCSS bistable asymmetric composite laminated thin square plate are obtained through using the amplitude-frequency response curves, force-amplitude response curves and bifurcation diagrams. Based on the dynamic model, numerical simulations are utilized to analyze the periodic and chaotic snap-through vibrations of the FCSS bistable asymmetric composite laminated thin square plate. The snap-through phenomena are successfully captured. The excitation amplitude and frequency of the snap-through regions are summarized. Comparing the experimental and numerical critical parameters of the snap-through regions, the results of the experiment and numerical simulations are a good qualitative agreement.

Bifurcation and chaos of Li-doped 6,6,12-graphyne impacted by hydrogen atoms: A new way to explore hydrogen storage

Hydrogen energy, as a green new energy source, has great significance for environmental protection due to its widespread application. This article uses nonlinear dynamics theory to study the problem of hydrogen energy storage firstly, which is different from traditional methods by using simulation and experimental methods. In this paper, the bifurcation and chaos of Li-doped 6,6,12-graphyne impacted by hydrogen atoms are studied. The perturbation of 6,6,12-graphene by Li atoms is treated as the Gauss white noise. The incremental harmonic balance method (IHB) is used for drawing the amplitude-frequency response curve of the system under simple harmonic excitation. Theoretical analysis shows that the inhomogeneity of 6,6,12-graphyne stiffness can affect the system stability, which decides the actual hydrogen storage capacity of graphyne. And then simulation and experiment results confirm the results of theoretical analysis. The maximum increase hydrogen storage efficiency can be more than 10% by selecting appropriate distribution of Li atoms.The research results of this article have a certain promoting effect on the industrial application of hydrogen energy.

Effect of negative stiffness nonlinearity on the vibration control effectiveness of tuned negative stiffness inerter damper

Recently, tuned negative stiffness inerter damper (TNSID) integrating the advantages of both negative stiffness and inerter elements has been proposed and verified to be effective for structural vibration control. In previous studies, the stiffness of the negative stiffness element was normally regarded as a constant for simplicity. However, it is well-known that negative stiffness is generally accompanied by nonlinearity, which may lead to undesirable nonlinear vibrations and unstable responses. Therefore, in this paper, the influence of the nonlinearity of negative stiffness, including weakening and strengthening nonlinear negative stiffness, on the performances of two representative TNSIDs with different configurations is investigated. The working mechanisms and optimal parameters of the TNSIDs with simplified linear negative stiffness are introduced firstly. Following that, based on the harmonic balance method, the dynamic responses of the two TNSIDs with different nonlinear negative stiffness are attained. The unstable problems and performance degradation caused by the nonlinearity of negative stiffness are analysed. Parametric studies are further carried out to evaluate the effects of the nonlinear coefficient, excitation amplitude, internal frequency ratio, and damping ratio on the performances of the two nonlinear TNSIDs. Moreover, to eliminate the impact of the negative stiffness nonlinearity, a new structural parameter optimization is conducted by minimizing the maximum displacement amplitude-frequency response of the TNSIDs. The results indicate that the nonlinearity of negative stiffness could lead to the bend and jump of the displacement amplitude-frequency response curves of TNSIDs, as well as superharmonic resonances, especially when the performance of TNSIDs is sensitive to the variation of negative stiffness. Besides, the unstable areas and the maximum displacement amplitude-frequency responses of TNSIDs are dependent on the excitation amplitude and nonlinear coefficient. To avoid the unstable vibration phenomenon and maintain effective vibration control performance, TNSID-P2 is recommended to be equipped with a weakening negative stiffness under both moderate and strong excitations, while TNSID-P1 should be designed with a weakening negative stiffness and work under a moderate excitation due to its performance being highly sensitive to the variation of negative stiffness.

H∞ Optimization of Three-Element-Type Dynamic Vibration Absorber with Inerter and Negative Stiffness Based on the Particle Swarm Algorithm.

Dynamic vibration absorbers (DVAs) are extensively used in the prevention of building and bridge vibrations, as well as in vehicle suspension and other fields, due to their excellent damping performance. The reliable optimization of DVA parameters is key to improve their performance. In this paper, an H∞ optimization problem of a novel three-element-type DVA model including an inerter device and a grounded negative stiffness spring is studied by combining a traditional theory and an intelligent algorithm. Firstly, to ensure the system's stability, the specific analytical expressions of the optimal tuning frequency ratio, stiffness ratio, and approximate damping ratio with regard to the mass ratio and inerter-mass ratio are determined through fixed-point theory, which provides an iterative range for algorithm optimization. Secondly, the particle swarm optimization (PSO) algorithm is used to further optimize the four parameters of DVA simultaneously. The effects of the traditional fixed-point theory and the intelligent PSO algorithm are comprehensively compared and analyzed. The results verify that the effect of the coupling of the traditional theory and the intelligent algorithm is better than that of fixed-point theory alone and can make the two resonance peaks on the amplitude-frequency response curves almost equal, which is difficult to achieve using fixed-point theory alone. Finally, we compare the proposed model with other DVA models under harmonic and random excitation. By comparing the amplitude-frequency curves, stroke lengths, mean square responses, time history diagrams, variances and decrease ratios, it is clear that the established DVA has a good vibration absorption effect. The research results provide theoretical and algorithm support for designing more effective DVA models of the same type in engineering applications.

Nonlinear Oscillations of a Composite Stepped Piezoelectric Cantilever Plate with Aerodynamic Force and External Excitation

Axially moving wing aircraft can better adapt to the flight environment, improve flight performance, reduce flight resistance, and improve flight distance. This paper simplifies the fully unfolded axially moving wing into a stepped cantilever plate model, analyzes the structural nonlinearity of the system, and studies the influence of aerodynamic nonlinearity on system vibration. The model is affected by aerodynamic forces, piezoelectric excitation, and in-plane excitation. Due to Hamilton’s principle of least action, the mathematical model is established based on Reddy’s higher-order shear deformation theory, and using Galerkin’s method, the governing dimensionless partial differential equations of the system are simplified to two nonlinear ordinary differential equations, and then a study of the influence of the various engineering parameters on the nonlinear oscillations and frequency responses of this model is conducted by the method of multiple scales. It was found that the different values of a5, a6, b6 and b8 can change the shape of the amplitude–frequency response curve and size of the plate, while different symbols a7 and b7 can change the rigidity of the model. The excitations greatly impact the nonlinear dynamic responses of the plate.

Nonlinear resonant responses of hyperelastic cylindrical shells with initial geometric imperfections

The resonant responses of the hyperelastic cylindrical shell with the initial geometric imperfections are investigated by considering both geometric and material nonlinearities. The hyperelastic cylindrical shell is subjected a radial harmonic excitation. Based on Donnell's theory, hyperelastic constitutive relations and Lagrange equation, the differential governing equations of motion are derived for the hyperelastic cylindrical shell with the initial geometric imperfections. Using the multiple scale method, the perturbation analysis for the governing equations of motion is conducted under the 2:1 internal resonance condition. The amplitude-frequency and force-amplitude response curves are obtained for the hyperelastic cylindrical shell. The effect of different parameters on the linear frequencies, amplitude-frequency response curves, force-amplitude response curves and chaotic responses of the hyperelastic cylindrical shell are discussed. The numerical results indicate that the existence of the geometric imperfections leads to the increase of the linear frequencies of the hyperelastic cylindrical shell. The amplitude-frequency response curves have the typical double-jumping phenomena and the response amplitudes increase with the increase of the external excitation amplitudes. The imperfection amplitudes, imperfection forms and structure parameters have the obvious influences on the resonant peaks evolutions. The vibrations of the first-order and second-order modes have the synchronization phenomena. With the changes of the parameters, the dynamic responses of the hyperelastic cylindrical shell with the imperfections change the period to chaotic vibrations alternately.