Internal Resonance Conditions Research Articles

Nonlinear behaviors and chaos control of a fluid-conveying piezoelectric pipe in supercritical regime

The study investigated the steady state response of a piezoelectric pipe conveying fluid under external and internal resonances in the supercritical regime of fluid velocity. Internal resonance conditions between the first and second frequency modes were studied at a specific flow velocity in the supercritical regime. The Galerkin method and the multiple scale method were used to extract the vibration amplitude versus the excitation frequency and amplitude. The steady state solutions lost stability through saddle-node and Hopf bifurcations, leading to periodic, double periodic, multi-periodic, and chaotic behaviors in the forced and frequency response curves. Time response, FFT, phase portrait, and Lyapunov exponents were presented to predict the unstable regions of the piezo-pipe system between the Hopf points. The Wolf algorithm was utilized to evaluate the Lyapunov exponents. Additionally, a fuzzy terminal sliding mode controller was designed in the chaotic region of the piezoelectric pipe conveying fluid, and its parameters were optimized using the genetic algorithm. The controller effectively stabilized the chaotic motion of the system, demonstrating better performance than the sliding mode control in numerical results.

External and internal resonances of thin-walled curved beams under three-directional moving harmonic loads

This paper deals with the internal and external resonances of simply supported curved beams subjected to three-directional harmonic moving loads. Firstly, based on the differential transformation methods and the Glerkin methods, the analytical solutions are newly proposed for vertical, torsional, radial and axial response of thin-walled curved beams under three-directional harmonic moving loads. Then, in the proposed analytical solutions, this study attains the conditions for the external resonance for in-plane and out-of-plane vibrations by letting the denominator of the proposed analytical solutions equal to zero. Besides, the semi-analytical solution for the internal resonance of curved beams is first proposed in this paper by the coincidence of both two mentioned-above resonant frequencies of beams. The internal resonance will occur when the beam is designed with the critical length for the given cross-section. The derived condition for the internal resonance can offer a basis for structural design to avoid the resonant appearance. All the analytical solutions are in good agreement with the results addressed in existing literature and numerical ones obtained from the finite element method (FEM). The results reveal that the corresponding motion of curved beams diverges, reaching a maximum value during the duration of the harmonic moving loads, when in the external resonance. Moreover, under the internal resonance condition, the curved beams are in repetitive transition between the out-of-plane mode and the in-plane mode without any external energy input. Additionally, in this scenario, the excitation frequency can coincide with the two resonance frequencies of beams resulting in simultaneous occurrences of out-of-plane and in-plane resonances on curved beams. The aim of this study highlights the potential resonant appearances on beams and provide reference in the design phase to avoid the internal resonance appearance and ensure the safety of structures.

“Cumulative effect” of second harmonic Lamb waves in a lossy plate

The second harmonic Lamb waves have high sensitivity to microstructural defects in materials and are therefore promising for incipient damage detection and monitoring of thin-walled structures. Existing studies have shown that the second harmonic Lamb waves can be cumulative with increasing propagation distance under the internal resonance conditions, which is conducive to nonlinear wave measurements in view of structural health monitoring. However, when propagating in a lossy structure with damping, the cumulative properties of the second harmonic Lamb waves are affected by energy dissipation and thus need to be re-examined. In this paper, a method for predicting the cumulative characteristics of second harmonic Lamb waves in damped plates is proposed. Instead of using material damping parameters which are difficult to obtain in practice, the proposed method relies on the attenuation patterns of Lamb waves at fundamental and double frequencies while taking into account the influence of the wave beam divergence. The proposed methodology is validated by finite element simulations and experiments. The results show that the cumulative second harmonic Lamb waves in the damped plate tend to increase and then decrease, and a “sweet” zone of relatively large amplitude can be predicted using the proposed method. The elucidation of the cumulative characteristics of the second harmonic Lamb waves provides guidance for effective system design for structural damage detection and monitoring applications.

Effects of internal resonance on the nonlinear acoustic radiation induced by axial and bending vibration of hyperelastic cantilever beams

This paper provides a numerical nonlinear vibro-acoustic study of a slender hyperelastic cantilever beam with 2:1 internal resonance condition. The model of beam includes geometric and material nonlinearities, while the model of acoustics includes the Arbitrary Lagrangian-Eulerian frame and the perfectly matched layers. The finite element method and the time integration method are used to discretize the structure and acoustic field. A loosely coupled serially staggered procedure is used to compute the variable values of structure and acoustic field. The results show that the transition from the axial mode to the bending mode of beam can be excited by the axial mode excitation. This leads to the alteration of acoustic directivity pattern and frequency components. Besides, due to the weak or strong coupling effects of axial and bending modes of beam, the symmetric or asymmetric acoustic directivity patterns can be witnessed in different frequency components.

One-To-One-To-One Internal Resonance Analysis of a Double-Cable-Stayed Flexible Beam-Structure Mounted with a Manipulator System

This paper deals with the theoretical studies to investigate the in-plane eigenvalue problems and the one-to-one-to-one internal resonance behaviors of a double-cable-stayed flexible beam-structure mounted with a manipulator system (DCFBMS) under constant rotational speed. A more precise and rigorous dynamic model with proper boundary conditions has been developed in which geometric nonlinearity and coupling nonlinearity of cables and beam are considered while the manipulator has been modeled as a rigid link. The in-plane eigenvalue problem of the model considering the boundary and continuous conditions has been evaluated and graphically demonstrated. Then, the ordinary differential equations (ODEs) are obtained by Galerkin’s method and multiple scales method has been employed to further analyze the vibration attributes of steady-state responses and their stability under internal resonance condition. The nonlinear response, stability and bifurcations for one-to-one-to-one internal resonance have also been investigated by varying system parameters. Analytically obtained results have been verified by Runge–Kutta method and found to be in good agreement. The present theoretical results deliver a useful insight into the nonlinear dynamic behavior and operational stability of DCFBMS.

Achieve hybrid energy harvesting based on the internal resonance of a two-degree-of-freedom cantilever beam

Taking a cantilever beam with two-degree-of-freedom (DOF) as an example, this paper studies the energy transfer mechanism between the vibration DOF and the rotation DOF, proposes a hybrid energy harvesting scheme, establishes the dynamic equation and solves it. Through the simulation of the analytical equation, it is found that under the condition of internal resonance of two DOFs, the two nonlinear peaks on the rotation DOF are very conducive to energy harvesting in the wide frequency domain. Furthermore, the influence of excitation amplitude, electromechanical coupling coefficient, time domain constant and other parameters on the resonance frequency and amplitude response of the energy harvesting system is studied.

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.

Shape Memory Alloy-Spring Pendulum for Vibration Suppression of Wind-Induced Transmission Tower-Line Systems

Wind-induced vibration suppression of transmission line structures is critical for structural reliability and serviceability. In this study, a shape memory alloy-spring pendulum (SMA-SP) is proposed for a wind-induced tower-line system to improve the reliability and serviceability of the control unit. A numerical calculation method of the transmission tower line structure coupled with the SMA-SP is established with special emphasis on the solid elements of the damper. Furthermore, the effectiveness and reliability of vibration suppression on wind-induced transmission line structures are evaluated and parametric analyses are conducted; the results reveal the influences of key parameters on the performance of SMA-SP and provide design recommendations for applications. In this study, it is demonstrated that the SMA-SP provides more stable control effects on the wind-induced vibration of transmission line structures than a conventional spring pendulum (SP), and the proposed numerical method effectively demonstrates the nonlinear mechanism of the damper. The parametric analyses show that SMA-SP exhibits low sensitivity levels to variations in the tuning frequency ratio and achieves optimal reduction effects after satisfying the internal resonance condition.

Mixing of Non-Collinear Lamb Wave Pulses in Plates with Material Nonlinearity

Guided waves have been extensively studied in the past few years, and more recently nonlinear guided waves have attracted significant research interest for their potential for early damage detection and material state characterization. Combined harmonic generation due to wave mixing can offer some advantages over second harmonic generation. However, studies focused on Lamb wave mixing are still very limited, and have mainly focused on collinear wave mixing and used plane wave assumption. In this paper, numerical simulations and experiments are conducted to understand the interaction of mixing non-collinear Lamb wave pulses with non-planar wavefronts. The results demonstrate that the generated secondary wave is cumulative under internal resonance conditions and the sum-frequency component of the combined harmonics is useful for characterizing material nonlinearities.

Analytical analysis of nonlinear internal resonance bandgap of pendulum-type metamaterial

In this paper, a pendulum-type metamaterial (PTM) is designed with a pendulum bob hinged at the primary mass. Considering the effect of geometric nonlinearity, the nonlinear dynamic equations of PTM unit cell are presented with the aid of the Bloch theorem. The analytical formula of dispersion equation is deduced to discuss the nonlinear internal resonance bandgap using the harmonic balance method. The obtained bandgap of the metamaterial is in good agreement with the numerical simulation result. The nonlinear geometric influence of pendulum on PTM bandwidth is significant. The bandgaps under different internal resonances are substantially different from each other due to the nonlinear effects. The upper boundaries of the frequency bandgap under 1:1/2 and 1:1/3 internal resonance rise nonlinearly to higher than those under linear and 1:1 internal resonance conditions. It shows that the impact of 1:1/2 and 1:1/3 internal resonance on the bandgap could be more obvious as the strong nonlinearity is taken into consideration.

Nonlinear dynamics of friction-induced regenerative chatter in internal turning with process damping forces

Stability in internal turning operation depends on several factors such as cutting conditions, tool overhang, associated tool dynamics, regeneration, friction and process damping. In the present work, an improved friction-induced regenerative chatter mechanism is introduced for internal turning operations with the help of a two-degree of freedom (DOF) cutting tool model by considering structural nonlinearities and process damping effects. The model is initially validated with the existing linear stability analysis theory via the natural parameter continuation method. The stability lobes obtained from the proposed model predict the stable regions with good accuracy. Furthermore, the nonlinear behaviour of the cutting tool with Stribeck and regenerative effects under internal resonance and primary resonance conditions is investigated. The nonlinear dynamic responses and the cutting stability are predicted using the higher-order method of multiple time scales (MMTS). The effect of nonlinearities on the frequency response and stability is studied in detail and the system parameters for stable cutting operations are identified. In order to validate the cutting states of the stability diagrams, internal turning experiments are performed on AISI 1020 carbon steel workpieces and the stability conditions are found to be in good agreement.

Three-to-One Internal Resonance of L-Shaped Multi-Beam Structure with Nonlinear Joints

In this paper, a reduced-order analytical model for an L-shaped multi-beam structure with nonlinear joints is presented to investigate the nonlinear responses of the system with three-to-one internal resonances conditions. Firstly, the global mode shapes are used to obtain an explicit set of nonlinear ordinary differential equations of motion for the system. Then, the first two natural frequencies of the system are calculated to determine the specific tip mass that results in three-to-one internal resonance. Subsequently, an approximation of the analytical solution of the dynamic model with two-degree-of-freedom is derived by using the multi-scale method. The accuracy of the approximation solution is verified by comparing it with the numerical solution obtained from the original motion equations. Based on the nonlinear dynamical model obtained by this paper, the frequency response curves are given to investigate the nonlinear dynamic characteristic of the L-shaped multi-beam structure with nonlinear joints. The results show that the nonlinear stiffness of the joints has a great influence on the nonlinear response of the system with three-to-one internal resonance conditions.

Mitigation of Structural Vibrations of MDOF Oscillators by Modal Coupling Due to Hysteretic Dampers

In civil engineering, structural elements characterized by hysteresis are often encountered, such as materials with limited elastic fields, microsliding friction and elastomeric absorbers. Hysteretic nonlinearities produce a wide variety of dynamical phenomena, such as significant modal coupling, bifurcations and superabundant modes. This paper investigates nonlinear modal interactions in the dynamic response of a two-degree-of-freedom system (2DOF) with hysteretic elements. These phenomena are notably important in internal resonance conditions, where modal interactions produce strong modifications in the response with possible beneficial effects. In specific conditions, the transfer of energy between the two modes leads to a notable reduction in the maximum response amplitude; the exploitation of this feature to achieve vibration mitigation of the forced response is the main goal of the paper. Two configurations are investigated: the hysteretic element at the top (vibration damper) and the hysteretic element at the base (isolator). In both cases, several internal resonance conditions occur since, by increasing the excitation intensity, the frequencies of the hysteretic system change, as well as their ratio. Qualitative similar results are obtained, characterized by a transfer of energy between the two modes. For both configurations, the usefulness of exploiting these nonlinear phenomena in vibration mitigation has been shown.

Traditional and non-traditional active nonlinear vibration absorber with time delay combination feedback for hard excitation

In this work, vibration suppression of a base excited nonlinear spring–mass primary system subjected to simultaneous external hard harmonic and parametric excitations is carried out by using a modified traditional and non-traditional active nonlinear vibration absorber (ANVA). The ANVA consists of a mass, time delayed damper and a PZT stack actuator in series with a linear and nonlinear spring. The ANVA utilizes various combinations of time delayed displacement, velocity and acceleration feedback gains of the primary system for vibration reduction. The governing coupled nonlinear equations of motion for the system with weighted modal matrix approach have been derived and solved by the method of multiple scales (MMS) for simultaneous superharmonic, principal parametric and primary resonance with 1:1 internal resonance conditions. The reduced equations obtained from MMS are solved using Newton’s method, which is then compared with the numerical methods, showing good agreement. The trivial state instability regions are also analyzed for various feedback gains. The parametric analyses are carried out for various system parameters, viz., variation in the nonlinear stiffness, time delay in the damper, changes in the amplitude of excitations, time delay in the feedbacks, stiffness of the absorber and various combinations of feedback gains through different feedbacks. From these parametric analyses, stable and unstable regions of operating frequencies are obtained at which the system response amplitude is minimum for a wide range of operating frequencies. Further, it is shown that 100% vibration reduction of the system can be achieved for a certain range of operating frequencies.

Torsional behavior of a non‐eccentric structure considering the effect of shear deformations in vertical members

AbstractTorsional moments can occur in a column owing to the geometric nonlinearity when the top of the column undergoes a large displacement in two horizontal directions. Periodically generated torsional moments can excite the torsional mode of a building owing to the internal resonance. For high‐rise buildings, there is a risk that long‐period seismic motions can cause a non‐negligible torsional response due to this phenomenon. To investigate this risk, this study focuses on non‐eccentric single‐story and two‐story shear‐type structural models with an aggregate column, which is assumed to be an elastic Timoshenko beam. We derived the equations of motion for the models under horizontal ground motions and confirmed that the equations were identical to those of the models with an aggregate column of an elastic Euler beam. We then performed time‐history response analyses of the models satisfying the internal resonance conditions under the input ground motions of harmonic waves, Gaussian white noise acceleration, acceleration records of previous earthquakes, and a long‐period design wave. The results showed that the maximum horizontal acceleration at the corner of the top floor slab could be increased by more than 60% of the acceleration at the center of the slab in the worst case.

Numerical study of coupled slosh modes in a 3D vessel subjected to multi-directional excitations

Present study explores liquid sloshing in a three-dimensional (3D) rectangular tank subjected to surge and coupled surge, sway and heave excitations. A number of applications of practical interest in nuclear systems such as, spent fuel storage tanks, fast reactor vessels, pool type coolant etc will be benefited from such analysis. Effect of harmonic excitation is investigated emphasizing finite liquid depth when the internal resonance condition 1:1 is satisfied between the natural frequencies of dominant modes (1,0), (0,1) and (0,2). The numerical model was validated against available in–house experimental shake table studies. Temporal snapshots of velocity magnitudes and free surface time history at different locations are presented when the tank is excited under two different loading conditions. The amplification of diagonal slosh modes were observed under planar excitation. This analysis assumes significance, as the attendant hydrodynamic forces could eventually cause fatigue failure of the tank structure. Non-linear square-like, swirling and wave breaking features were observed, when the tank was subjected to earthquake excitation.

An Internal Resonance Based Method for Reducing Torsional Vibration of a Gear System

The gear system is one of the most widely-used transmission systems due to its accurate transmission ratio and high efficiency. However, torsional vibration may severely degrade transmission performance and shorten the gear lifespan. In view of this, a nonlinear interaction principle suitable for vibration energy transfer is researched, and an internal resonance based method is put forward to reduce the torsional vibration of the gear system. According to the coupling relationship between the gear torsional vibration mode and the vibration absorber mode, the 1:1 internal resonance condition is analyzed by the multiple scale method and the sufficient and necessary conditions for establishing internal resonance are obtained. Through stability analysis, the vibration energy transfer channel based on internal resonance is successfully established, by which vibration energy can be transferred to and dissipated by the vibration absorber. Based on numerical and virtual prototyping simulations, vibration reduction performances are examined, including effectiveness, damping characteristics and robustness. The research results show that the proposed internal resonance based method can effectively reduce the torsional vibration of the gear system.

The role of the hysteretic restoring force on modal interactions in nonlinear dynamics

Hysteretic behaviour characterizes elements of a wide class of mechanical systems: the dependence of their restoring force on the deformation history has a great influence on the dynamic response. For this reason, an increasing interest has been registered in the study of typical phenomena of nonlinear dynamics, also because hysteresis belongs to the class of strong nonlinearities. The paper is devoted to the analysis of the modification of the response of two degrees-of-freedom chain systems due to the characteristics of the restoring force of a hysteretic element; attention is given to its dissipation characteristic, comparing the cases of full and reduced hysteresis. With increasing excitation amplitude, the strong hysteretic nonlinearity modifies the nonlinear frequencies and in turn their ratio, easily leading to internal resonance conditions. For a system close to a 3:1 resonance condition, the modification of the frequency response curves (frcs) for increasing excitation intensity is illustrated and compared with the response of similar systems not in internal resonance. The general trend of the phenomena is slightly qualitatively influenced by the dissipation property, whereas the quantitative differences are notable. The only evident difference is the presence of a frequency range of coexisting solutions in the case of reduced hysteresis. In both cases, after a bifurcation a novel mode arises around the first resonance, with similar frequency and different shape. The case of reduced hysteresis makes it possible to better investigate the evolution of nonlinear modes of a system, close to the Hamiltonian system embedded by the actual dissipative one. The occurrence of this novel mode, peculiar of systems with strong nonlinearities, is responsible of a substantial transfer of energy between the two modes in internal resonance conditions. Finally, an example of quasiperiodic oscillations is presented to show one of the other possible scenarios which arise even in the presence of significant dissipation.

Nonlinear modal interaction between longitudinal and bending vibrations of a beam resonator under periodic thermal loading

The paper investigates the nonlinear modal interaction of longitudinal and bending vibrations of a beam resonator under periodic thermal loading. The mode of parametric oscillations is investigated under conditions of internal multiple resonance between some flexural and longitudinal forms of free oscillations of the resonator. The possibility of generation in the system of the longitudinal-bending mode was found, the frequency of the slow envelope of which essentially depends on the parameter of the internal frequency detuning, which is directly related to the magnitude of external disturbances subject to high-precision measurement.

Nonlinear dynamics of drill string in horizontal well under axial load and supercritical regime of flowing fluid

In this article, the nonlinear vibration of the spinning pipe conveying fluid in a borehole drilling is investigated under the axial compression load, nonlinear fluid forces, internal and external resonances. The spinning pipe is modeled as a simply supported rotor with imbalance mass and eccentricity. When the flowing fluid is in supercritical regime the pipe oscillates around an equilibrium position in the working frequency range of external excitation. Two to one internal resonance condition between forward and backward natural frequencies of the pipe is found at specific values of flow velocity and rotating speed. The multiple scales method is occupied to solve the governing ordinary differential equations and solvability condition is applied to get the modulation equations. The Runge–Kutta integration and the arc-length continuation method are used for directly solving ordinary differential equations. Bifurcation diagrams are plotted and time responses, phase portraits and Fast Fourier transformation of amplitudes are achieved. The Wolf algorithm is used to determine the Lyapunov Exponents. Periodic, multi-periodic, quasi-periodic, double jump and chaotic behaviors can be seen in simulation results.