Superharmonic Resonance Research Articles

Nonlinear dynamic modeling for analysis of large spacecraft with extendible appendages

A nonlinear dynamic model of a large spacecraft with extendible ultra-flexible appendages under the perturbation of gravity gradient that can capture large deformation is proposed in this paper. The proposed model is derived accurately based on the Euler-Bernoulli beam theory and Hamilton's principle with non-Cartesian deformation variables (the stretch and transverse deformation). To address the problem of velocity residual, the piecewise-actual (PWA) extension strategy is presented for the first time. Under the proposed extension strategy, the time varying dynamic characteristics of the natural frequency are investigated by numerical methods. Moreover, the influences of the parameters on the stability of the appendages are studied using the eigenvalue method. Finally, to show the accuracy of the proposed nonlinear model, the dynamic response of the proposed model is compared with those of the previous models for different attitude angle moving states. All results are consistent with each other in the case of small deformation, which verifies the correctness of the proposed nonlinear model. The natural frequencies of stretch and transverse deformations decrease gradually with time due to the axial extension of the flexible beam. The stability analysis demonstrates the rapid extension leads to the dynamic instability of the system. Furthermore, the super-harmonic resonance phenomenon is observed when the attitude angle keeps Sun-facing by the proposed model which indicates that the proposed model overcomes the limitations of nonlinear models with Cartesian variables in large deformation.

Dynamic characteristics analysis and the identification signal of the horizontal tail drive shaft system with the ballistic impact damage of a helicopter

The ballistic impact damage (BID) will change the dynamic characteristics of the horizontal tail drive shaft system (HTDSS) and will directly affect the flight safety of the helicopter. Aiming at the problem of how the BID affects the dynamic characteristics of the HTDSS and how to identify the BID in time, the dynamic characteristics of the HTDSS with the BID are studied in this paper. The BID is simplified as the ideal geometric damage, and the calculation method of the BID is given. The finite element dynamic equation of the HTDSS with the BID is established, and the effect mechanism of the ballistic impact parameters on the dynamic characteristics is revealed. The result shows: the BID causes the mass loss and the stiffness asymmetric of the ballistic impact position of the TDS; the eccentric excitation introduced by the BID leads to the obvious increase of the system response, and the stiffness asymmetry leads to the super-harmonic resonance of the system. The obvious increase of system response and the appearance of 2Ω frequency component can be used as the identification signal of the BID. Finally, the experiment was carried out, which verified the correctness of the established dynamic model, and explained the reliability of the proposed identification signal of the BID.

Thermoelastic resonance analysis of BNNT-reinforced nanocomposite beams heated by internal thermal source

The nanocomposite structures due to excellent chemical and thermal resistant have considerable potential for use in an aggressive environment with high corrosion and temperature gradient. This study analyzes subharmonic (Ω/ωL=2 and 3) and superharmonic (Ω/ωL=1/2 and 1/3) resonance behavior of boron nitride nanotube-reinforced composite (BNNT-RC) beams under harmonic excitation and internal thermal source. The thermal analysis is carried out for determining the temperature change using the two-dimensional heat conduction equation with the assumption of an arbitrary variation in the thickness direction. In the framework of adjacent equilibrium configurations and Galerkin method, the modified couple stress Timoshenko beam equations are reduced to one nonlinear decoupled equation. Next, closed-form relations of vibration amplitude (a) and detuning parameter (σ∼) are developed for subharmonic and superharmonic excitations using the multiple-scale perturbation technique. The nonlinear frequency and resonance responses are verified by comparison with the existing ones in the literature. Finally, parametric studies are presented to discuss the effects of length scale parameter, temperature change, energy rate of thermal source, detuning parameter, and geometric ratios. The snap-through buckling behavior is observed in the curves of ωNL/ω∗-ΔT and different unstable paths exist after the occurrence of instability. The thermoelastic effect can increase the stability range of harmonic load amplitude for various values of detuning parameter and it is more pronounced for the temperature variation induced by an internal thermal source.

Parametric resonance of multi-frequency excited MEMS based on homotopy analysis method

The nonlinear dynamics and control of micro electromechanical systems (MEMS) is an important topic at present and homotopy analysis method (HAM) is an effective semi-numerical and semi-analytical method for solving strongly nonlinear problems. In this paper, the parametric resonance of MEMS with multi-frequency excitation is studied by HAM. Firstly, the differential equation of electrostatically driven microbeam is processed by Taylor expansion, and it is transformed into the parametric motion model with multi-frequency excitation. Then, the approximate solution and amplitude–frequency response equation of the system are obtained by HAM, and compared with the numerical solution. Finally, the influence of direct current (DC) and alternating current (AC) on principal parametric resonance and superharmonic resonance is discussed. The results show that HAM is an effective method to analyze the parametric vibration of multi-frequency excitation system, and the amplitude–frequency response curve of microbeam about parametric motion depends on the time scale in high DC and high AC state. This study effectively extends the application of HAM in parametric resonance, which is of great significance to the study of nonlinear vibration of MEMS.

Modelling and nonlinear analysis of the wave-induced vibrations of a single hybrid riser conveying two-phase flow

This paper analyses the nonlinear dynamic response of a single hybrid riser pipe to wave excitations. Detailed nonlinear equations of motion are derived for the riser pipe as it conveys two-phase flow. Hence its dynamics are described by two coupled nonlinear equations, relating to both longitudinal and transverse displacements. Wave excitations forcing the system are imposed as oscillating inline flow. In addition, considering that the riser is a structure that is vibrating in a moving fluid, the inline force is modelled with the extended Morison equation. These equations are solved by the method of multiple scale perturbation. Theoretical results show that longitudinal and transverse responses are coupled at some specified natural frequencies. Similarly, the riser pipe is also prone to primary and super-harmonic resonance excitations at some wave frequencies. This presents a complex nonlinear multi-frequency excitation problem identical to the numerical studies conducted on a typical single hybrid riser pipe based on the frequency response curves. The uncoupled primary resonance behaviour of the transverse displacements for all the void fractions studied exhibited a hardening nonlinear behaviour of non-unique responses and jump phenomenon. However, a distinctive feature of the harmonics was observed where the curves separate into two to form an isolated branch of periodic solutions. These are known as Isola. Also, the presence of internal resonance results in the transfer of energy between the planar axis resulting in undesirable axial resonance peaks which are destabilized by the presence of the nonlinear anti-resonance effect.

Nonlinearity modulation in a mode-localized mass sensor based on electrostatically coupled resonators under primary and superharmonic resonances

A general model of a mode-localized mass sensor incorporating two weakly coupled clamped-clamped microbeams under electrostatic excitation is presented, and a reduced-order model considering quadratic and cubic nonlinearities is established. The multiple time scales method is used to solve the dynamic characteristics of the coupled resonators under primary resonance, simultaneous superharmonic and primary excitations, and one-third superharmonic resonance respectively, and to analyze the contribution of each harmonic excitation term. It is shown that the sensor can display softening, hardening, and linear behaviors by tuning the overall nonlinear coefficient in three different excitation scenarios. Furthermore, the conditions for restoring linear behavior with the highest possible amplitude without any hysteresis under different excitations are obtained. Finally, the mass sensitivities represented by the relative shift of amplitude ratio are calculated for all the resulting dynamic behaviors. The results show that the sensitivity is highest, for the hardening behavior in the in-phase mode and for the softening behavior in the out-of-phase mode. Interestingly, the sensitivities of the linear behavior obtained by nonlinearity modulation are the same for the two vibration modes, which is improve the output stability. Consequently, the sensor resolution can be significantly enhanced below the pull-in instability, while avoiding noise mixing.

Crack Fault Diagnosis and Location Method for a Dual-Disk Hollow Shaft Rotor System Based on the Radial Basis Function Network and Pattern Recognition Neural Network

The crack fault is one of the most common faults in the rotor system, and researchers have paid close attention to its fault diagnosis. However, most studies focus on discussing the dynamic response characteristics caused by the crack rather than estimating the crack depth and position based on the obtained vibration signals. In this paper, a novel crack fault diagnosis and location method for a dual-disk hollow shaft rotor system based on the Radial basis function (RBF) network and Pattern recognition neural network (PRNN) is presented. Firstly, a rotor system model with a breathing crack suitable for a short-thick hollow shaft rotor is established based on the finite element method, where the crack’s periodic opening and closing pattern and different degrees of crack depth are considered. Then, the dynamic response is obtained by the harmonic balance method. By adjusting the crack parameters, the dynamic characteristics related to the crack depth and position are analyzed through the amplitude-frequency responses and waterfall plots. The analysis results show that the first critical speed, first subcritical speed, first critical speed amplitude, and super-harmonic resonance peak at the first subcritical speed can be utilized for the crack fault diagnosis. Based on this, the RBF network and PRNN are adopted to determine the depth and approximate location of the crack respectively by taking the above dynamic characteristics as input. Test results show that the proposed method has high fault diagnosis accuracy. This research proposes a crack detection method adequate for the hollow shaft rotor system, where the crack depth and position are both unknown.

Responses of a Strongly Forced Mathieu Equation—Part 1: Cyclic Loading

AbstractThis work concerns the response of a damped Mathieu equation with hard cyclic excitation at the same frequency as the parametric excitation. A second-order perturbation analysis using the method of multiple scales unfolds resonances and stability. Superharmonic and subharmonic resonances are analyzed and the effect of different parameters on the responses are examined. While superharmonic resonances of order two have been captured by a first-order analysis, the second-order analysis improves the prediction of the peak frequency. Superharmonic resonances of order three are captured only by the second-order analysis. The order-two superharmonic resonance amplitude is of order ε0, and the order-three superharmonic amplitude is of order ε. As the parametric excitation level increases, the superharmonic resonance amplitudes increase. An nth-order multiple-scales analysis will indicate conditions of superharmonic resonances of order n + 1. At the subharmonic of order one-half, there is no steady-state resonance, but known subharmonic instability is unfolded consistently. Analytical expressions for resonant responses are presented and compared with numerical results for specific system parameters. The behavior of this system could be relevant to applications such as large wind-turbine blades and parametric resonators.

Nonlinear Vibration of a Multi-Degree-of-Freedom Gear Transmission System With Multipiecewise Linear Functions

AbstractPeriodic and period-doubling vibrations in a gear transmission system subjected to forced excitations with multipiecewise linear functions are investigated in this work by using the incremental harmonic balance (IHB) method. The nonlinear ordinary differential equations that govern vibration of the gear transmission system are formulated by employing the Newton's second law. Analytical results reveal abundant interesting phenomena, including jumps, bifurcations, softening-spring behaviors, and primary, super-harmonic, and subharmonic resonances, which have not been exhibited in existing investigations on nonlinear vibrations of gear transmission systems. Nonlinear phenomena and resonances of the gear transmission system are revealed by considering different numbers of degrees-of-freedom of the system. The bifurcations containing saddle-node, period-doubling, and Hopf bifurcations are observed in frequency response curves. The period-doubling phenomena are characterized by phase-plane diagrams and Fourier spectra. Further, analytical results obtained by the IHB method match very well with those from numerical integration.

Responses of a Strongly Forced Mathieu Equation—Part 2: Constant Loading

AbstractThe present study deals with the response of a damped Mathieu equation with hard constant external loading. A second-order perturbation analysis using the method of multiple scales (MMS) unfolds resonances and stability. Non-resonant and low-frequency quasi-static responses are examined. Under constant loading, primary resonances are captured with a first-order analysis, but are accurately described with the second-order analysis. The response magnitude is of order ϵ0, where ϵ is the small bookkeeping parameter, but can become arbitrarily large due to a small denominator as the Mathieu system approaches the primary instability wedge. A superharmonic resonance of order two is unfolded with the second-order MMS. The magnitude of this response is of order ϵ and grows with the strength of parametric excitation squared. An nth-order multiple scales analysis under hard constant loading will indicate conditions of superharmonic resonances of order n. Subharmonic resonances do not produce a non-zero steady-state harmonic, but have the instability property known to the regular Mathieu equation. Analytical expressions for predicting the magnitude of responses are presented and compared with numerical results for a specific set of system parameters. In all cases, the second-order analysis accommodates slow time-scale effects, which enable responses of order ϵ or ϵ0. The behavior of this system could be relevant to applications such as large wind-turbine blades and parametric amplifiers.

Experimental and theoretical investigation of superharmonic resonances in a planar oscillator under angular base excitation

Experimental and theoretical investigation of superharmonic resonances in a planar oscillator under angular base excitation

Investigation of analytical solution of super harmonic resonance of rotor system in permanent magnet synchronous motors considering mixed eccentricity

The unbalanced magnetic pull results in the nonlinearity of the system for the permanent magnet synchronous motor rotor having mixed eccentricity. The mass unbalance and static eccentricity form multi-frequency excitation, which results in the coupling effect of forward and backward whirling motions on the steady-state of super harmonic resonance. The analysis is difficult due to the phases of complex amplitudes of two motions which is no longer time invariant compared with the main resonance. To solve this problem, an analysis method of complex amplitudes is presented in this paper. The rotor is considered a Jeffcott rotor, and multi-scale method is applied to this system. A mathematical analysis method of the complex amplitude phases is employed for their time variant characteristics. The results show that both phases are linear time variant and their variation rates are both equal in magnitude to the difference between twice the excitation frequency and natural frequency of the generating system. The characteristics lead to the frequency shift of the two motions by the difference value from the natural frequency of the generating system. The numerical analysis reveals that the phases are independent of the initial condition and hence the complex amplitudes. The analysis solves two problems: the analytical solution of the rotor system carrying phases information can be expressed, which is in good agreement with the numerical result, and the solution construction is analyzed easily, which indicates the contribution of every component of the analytical solution. The phase characteristics analysis is the prerequisite for the frequency characteristics which can be easily carried out, and the coupling mechanism of the two motions is explained.

Magneto-Aero-Elastic Superharmonic and Subharmonic Resonances, Bifurcations and Chaos of Conductive Spinning Circular Plates

This paper focuses on the magneto-aero-elastic nonlinear resonances, bifurcations and chaos for superharmonic and subharmonic responses of conductive circular plates spinning in air and magnetic fields. The electromagnetic forces acting on in-plane are presented by using electromagnetic field theory, and a rotational dissipative damping aerodynamic model is employed to characterize the aerodynamic forces acting on the plates. Considering the effects of the geometric nonlinearity and centrifugal tensioning, the magneto-aero-elastic governing equations of spinning conductive circular plates are derived by the Hamiltonian principle. Based on the mode shapes with Bessel functions, the transverse nonlinear vibration differential equations of spinning plates with clamped-edge are derived by the Galerkin method. Amplitude-frequency equations for subharmonic and superharmonic responses are obtained by using the approach of multiple scales, respectively. Numerical calculations indicate the influences of important parameters (e.g. magnetic field intensity, excitation amplitude and rotational speed) on amplitude frequency characteristics. Moreover, it is demonstrated by the numerical results that system responses are sensitive to the variation of these important parameters, where the complex intermittency, period-doubling and chaotic dynamical behaviors exist.

Complicated nonlinear oscillations caused by maneuvering of a flexible spacecraft equipped with hinged solar arrays

We present a general methodology for the nonlinear dynamical modeling of a three-axis stabilized spacecraft equipped with flexible solar arrays. The large-span multi-panel solar arrays are modeled as flexible thin plates that are connected to the rigid central body of the spacecraft by means of nonlinear flexible hinges. We construct a low-dimensional yet accurate dynamical model by using a Galerkin expansion in terms of global modes of the system that we compute first by using the Rayleigh–Ritz method. The hinged connections between the rigid and flexible parts of the system are imposed employing Lagrange multipliers. The model is used to study the spacecraft response triggered by various maneuvering scenarios. We in particular focus on the coupling between vibrations of the flexible components and the rigid motion of the spacecraft induced by hinge nonlinearities during these orbital and attitude maneuvering operations. In all cases considered, it is found that four global modes are sufficient to accurately compute the system’s response. We also observe other complicated nonlinear dynamical phenomena, such as hysteresis and superharmonic resonance that may be of concern in spacecraft design. Our modeling approach can be applied to other multibody systems directly.

Analysis of super-harmonic resonance and periodic motion transition of fractional nonlinear vibration isolation system

The precision instruments and equipment are often utilized in low-frequency and micro-amplitude vibration systems, in which many vibration isolators of rubber materials are widely used. Ignoring the low-frequency amplitude will result in errors in the fatigue life design of the vibration isolators and predicting the dynamic response of each frequency band accurately becomes necessary. However, integer-order models cannot describe the frequency dependence of rubber materials, while the fractional-order models can describe it instead. On the other hand, the elastic restoring force is strongly nonlinear under large deformation, and the vibration of the nonlinear system contains multiple harmonic components. In order to solve those issues, the fractional nonlinear Nishimura model is used to characterize the constitutive relation of vibration isolators such as air springs, which are made of carbon black filled natural rubber. The high-order harmonic balance method is used to obtain the steady-state response of the vibration system, while the fourth-order Runge–Kutta method is applied to simulate the dynamic response of the system in the low-frequency region, and the Lyapunov exponent is used to determine the stability of the system. Furthermore, the influence of parameters on the amplitude–frequency characteristics of different frequency bands is also studied, and a method to solve the optimal damping coefficient is proposed based on the primary resonance amplitude–frequency curves. The results show that there is a diversity of periodic motions in the process of adjacent super-harmonic resonance transition. Numerical simulations also demonstrate that multi-periodic motions coexist in the system. The motion transition law between the polymorphic coexistence region and its adjacent regions is summarized.

Dynamical modeling and parametric analysis of an electret-based wave energy converter

Carbon neutrality and the Internet of Underwater Things have injected new impetus into the development of wave energy converters (WECs) albeit most of WEC projects have ended in failure due to low cost-effectiveness. Hereupon we present the dynamical modeling and parametric analysis of a novel electret-based WEC that has a distinct power take-off system compared to the conventional ones and has exhibited high power density in our previous study. Linear wave theory and Morison equation are applied for wave field and forces, respectively. WEC motions parallel with incident waves are modeled, and the electrical subsystem of the generator is treated as a current source and a capacitor. With these simplifications, the governing equations of the nonlinear and non-smooth electromechanical system are obtained and then numerically solved using an event-driven algorithm. The new power take-off system is experimentally validated. We find that the power output varies nonlinearly with wave parameters under regular wave excitation but linearly under irregular ones, and that superharmonic resonance can significantly enhance energy conversion performance at relatively low frequencies, based on which the optimum parameters are obtained. A peak average power of around 2 mW (at a matched load resistance) is generated from each generator encapsulated in the WEC of a diameter of 15 cm. The findings shed light on the characteristics of electret-based WECs and provide support for the development of wave energy farm; moreover, the methodologies of modeling and analysis can be extended to the design and optimization of the scaleups of the proposed electret-based WEC.

Study on Vibration Suppression of an Inclined Cable with a Nonlinear Energy Sink Under the Axial Excitation

The nonlinear energy sink (NES) has been verified to have a broadband damping effect in many studies. In this paper, the in-plane vibration of an inclined cable attached with an NES is considered. First, nonlinear motion equation of the cable under an axial harmonic excitation (parametric excitation) is derived on the basis of Hamilton’s principle. The ordinary differential equations (ODEs) of the system are derived by Galerkin method and solved by fourth-order Runge–Kutta method. In this way, the suppression effects of the NES on primary resonance, 1/2-order sub-harmonic resonance and second-order super-harmonic resonance of the cable are investigated when the cable is subjected to a parametric excitation. Then, by optimizing the parameters of the NES individually, the corresponding results are compared with those of the uncontrolled system and the cable with a tuned mass damper (TMD). Meanwhile, the robustness of the NES against changes in the amplitude of axial excitation is also studied. The results demonstrate the high-efficiency vibration suppression of the NES and the vibration suppression effect of the optimized NES on the cable shows better performance in terms of multi-modality compared with the optimized TMD.

Nonlinear vibration analysis of meshing gear wheels considering the teeth elasticity

In this study, the nonlinear vibration of a meshing gear pair with the teeth elasticity is investigated. The elastic elements between the teeth and the body of each gear wheel are introduced to partially incorporate the teeth elasticity in the model. This model was originally considered by the second author as a rigid-elastic modeling approach of meshing gear wheels, which is well-suited for multibody systems. The contact analysis of the meshing teeth is performed using the Kelvin-Voigt method. By deriving the equations of motion for the mentioned model, the results of the numerical solution are presented. The method of multiple scales is utilized here for the first time to analytically analyze the nonlinear vibration of a gear system in which the teeth elasticity is partially incorporated by considering identical elastic elements between the teeth and the gear body. In this study, the effect of system parameters on the frequency response including the primary, sub-harmonic, and super-harmonic resonances is investigated. The results of the analytical solution show that the effect of external excitation amplitude and intensity on the system response amplitude is the same in the primary and sub-harmonic resonances. The results obtained numerically and analytically for the model have an acceptable agreement.

Nonlinear vibration of the axle box front cover of high-speed train and its effect on connecting bolts

In order to explore the causes for failure of connecting bolts between the axle box front cover and axle box body on high-speed train, field test was carried out firstly in this work. The results showed that the wheel had 20th-order polygonization and the front cover vibrated drastically with the dominant frequencies of 290 Hz and 1160 Hz at the speed of 300 km/h, which were 1/2 and 2 times the rotational frequency 580 Hz of 20th-order polygonal wheel, respectively. Secondly, bench test was carried out to reproduce the unique and drastic vibration phenomenon of the field test, it was proved that the wheel polygonization took responsibility for the drastic vibration of the front cover. OMA (Operational Modal Analysis) and theoretical analysis were made next to certify existence of nonlinear vibration (subharmonic resonance of order 1/2 and superharmonic resonance of order 2) of the front cover under 20th-order polygonization. Finally, the cause of bolt failure and dynamic characteristic of the front cover resulted from different wheel polygon order was studied, results showed that relative vibration existed between axle box body and front cover. The relative vibration enlarged when the front cover was in nonlinear vibration under 20th-order wheel polygonization, which definitely caused a larger dynamic transverse load for the bolts and accelerated its vibration fatigue.

Vibration characteristics of slotted disc spring series vibration isolation unit

There have been many pieces of research on the vibration isolation theory of the single slotted disc spring. However, the theory of the series slotted disc spring and its bandwidth optimization is rarely mentioned. Therefore, by considering the effect of boundary friction of series slotted disc spring units, we optimized the general formula for the load–deflection relationship of a slotted disc spring. The accuracy of the optimization was verified by simulation. Furthermore, it expands on the application range of the general formula, making it suitable for both single and series slotted disc springs. The nonlinear dynamic model of the slotted disc spring was established, and the multiscale method was used to study system parameters under strong excitation conditions. System parameters affected the response characteristics of the primary, superharmonic, and subharmonic resonance of series slotted disc spring units. Results showed that reducing the nonlinear stiffness and increasing the viscous damping coefficient and friction reaction coefficient are beneficial to improving the vibration isolation performance. We further deduced that the effective bandwidth of vibration isolation could be increased by adding the mass of the block, slotted depth, and number of slotted disc spring units. Therefore, this research provides theoretical guidance that is significant for engineering applications.