Multi-frequency Parametric Excitation Research Articles

Rattle dynamics of noncircular face gear under multifrequency parametric excitation

Abstract. A noncircular face gear (NFG) conjugated with a pinion is a new type of face gear which can transmit variable velocity ratio and in which two time-varying excitations exist, namely the meshing stiffness excitation and instantaneous center excitation. Considering the tooth backlash, static transmission error and multifrequency parametric excitation, a nonlinear dynamic model of the NFG pair is presented. Based on the harmonic balance method and discrete Fourier transformation, a semi-analytic approach for the nonlinear dynamic model is given to analyze the dynamic behaviors of the NFG. Results demonstrate that, with increase in the eccentric ratio, input velocity and error amplitude, the NFG will undergo a non-rattle, unilateral rattle and bilateral rattle state in succession, and a jump phenomenon will appear in the dynamic responses when the rattle state of the gears is transformed from unilateral rattle to bilateral rattle.

A stochastic analysis approach for marine riser’s cross-flow/in-line VIV under heave-induced parametric vibration

ABSTRACT This study proposes a theoretical analysis model to analyse the marine riser's coupled cross-flow/in-line vortex induced vibration (VIV) under heave-induced time varying parametric vibration. Combining with the linear-wave theory, the stochastic phase spectrum method is used to simulate the stochastic multi-frequency parametric excitation in the time-domain to reproduce the real sea conditions. The nonlinearities are considered to capture the geometrical coupling of cross-flow/in-line displacements excited by hydrodynamic lift/drag forces and the wake-riser interaction. A generally good agreement between the proposed approach and other analyses can be observed. A parametric study is conducted, it is found that the drag coefficient is of significance for the riser's VIV, and the lift coefficient is affected by vortex shedding. The wave form, in terms of wave height and wave period, need to be considered with caution in design of a riser to avoid the occurrence of resonance under stochastic parametric excitation.

Stability characteristics of multi-frequency parametrically-excited rotor-oscillator systems

The effect of multi-frequency parametric excitation on the stability of an oscillator is examined in the context of a rotating system where the rotor and the stator have comparable masses and hence the ‘stator’ cannot be considered to be mechanically fixed. The differential equations governing the dynamics of such systems have time-varying terms which are periodic in the rotor frequencies. Floquet theory is employed to numerically develop stability diagrams which distinguish stable from unstable parameter combinations. Reasons for the occurrence of instabilities are explained using the two time scale perturbation method. It was found that the most significant resonance in the single rotor case occurred when the rotor speed equaled the sum of two unforced natural frequencies. In addition, harmonic resonances were found to arise when the rotor speed matched the unforced natural frequencies. Furthermore, the excitation from a second rotor caused instabilities to arise around regimes where multiple combination resonances occurred together. The occurrence of certain experimentally-observed resonances and regimes of low amplitude oscillations were correlated with the model, and the corresponding rotor trajectories were explored. This investigation is useful for identifying operating regimes so as to avoid catastrophic failure in multi-frequency rotating machines such as multi-rotor helicopters and gas turbines with contra-rotating stages.

Dual frequency parametric excitation of a nonlinear, multi degree of freedom mechanical amplifier with electronically modified topology

Mechanical or electromechanical amplifiers can exploit the high-Q and low noise features of mechanical resonance, in particular when parametric excitation is employed. Multi-frequency parametric excitation introduces tunability and is able to project weak input signals on a selected resonance. The present paper addresses multi degree of freedom mechanical amplifiers or resonators whose analysis and features require treatment of the spatial as well as temporal behavior. In some cases, virtual electronic coupling can alter the given topology of the resonator to better amplify specific inputs. An analytical development is followed by a numerical and experimental sensitivity and performance verifications, illustrating the advantages and disadvantages of such topologies.

Recent Advances in Physics of Fluid Parametric Sloshing and Related Problems

Liquid parametric sloshing, known also as Faraday waves, has been a long standing subject of interest. The development of the theory of Faraday waves has witnessed a number of controversies regarding the analytical treatment of sloshing modal equations and modes competition. One of the significant contributions is that the energy is transferred from lower to higher harmonics and the nonlinear coupling generated static components in the temporal Fourier spectrum, leading to a contribution of a nonoscillating permanent sinusoidal deformed surface state. This article presents an overview of different problems of Faraday waves. These include the boundary value problem of liquid parametric sloshing, the influence of damping and surfactants on the stability and response of the free surface, the weakly nonlinear parametric and autoparametric sloshing dynamics, and breaking waves under high parametric excitation level. An overview of the physics of Faraday wave competition together with pattern formation under single-, two-, three-, and multifrequency parametric excitation will be presented. Significant effort was made in order to understand and predict the pattern selection using analytical and numerical tools. Mechanisms for selecting the main frequency responses that are different from the first subharmonic one were identified in the literature. Nontraditional sources of parametric excitation and Faraday waves of ferromagnetic films and ferrofluids will be briefly discussed. Under random parametric excitation and g-jitter, the behavior of Faraday waves is described in terms of stochastic stability modes and spectral density function.

Nonlinear Dynamics of a Travelling Beam Subjected to Multi-Frequency Parametric Excitation

This paper deals with two frequency parametric excitation in presence of internal resonance. The cubic nonlinearity is inserted into the equation of motion by considering the mid-line stretching of the beam. The perturbation method of multiple scales is applied directly to the governing nonlinear fourth order integro-partial differential equation of motion. This leads to a set of first order differential equations known as the reduced equations or normalized reduced equations, which are utilized to determine the additional instability zones, appeared in the trivial state stability plot, the bifurcation and stability of fixed-points, periodic, quasi-periodic, mixed mode and chaotic responses. The transition of system behaviour from stable periodic to unstable chaotic occurs through intermittency route

Instability analyses of a top-tensioned riser under combined vortex and multi-frequency parametric excitations

This study investigates the non-linear dynamic responses of a top-tensioned riser (TTR) under combined parametric and external vortex excitations. In real sea conditions, risers simultaneously encounter waves and flow. A general analysis considers only parametric excitation resulting from platform motions or vortex-shedding excitation due to flow bypassing the risers. However, the instability of a TTR under combined excitations can lead to fatigue damage; therefore it is essential to explore the dynamic characteristics in order to maintain the safety and avoid accidents. A derived TTR model subjected to multi-frequency parametric and vortex shedding excitation has been proposed, and different cases of excitation combinations were compared. The effects of key design parameters on the dynamic properties of a TTR under combined excitation are discussed. The surface distribution of instability for a TTR system under multi-frequency parametric and vortex-shedding excitations was analyzed on the basis of various significant wave height and wave peak period combinations, and extreme sea states were considered to determine the maximum strength of a TTR. Several interesting phenomena were observed. The instability of a TTR system excited by combined excitation could be more unstable than that excited only by parametric excitation or vortex-shedding excitation. It can be concluded that the effect of vortex shedding is predominant in the excited instability of a TTR under combined excitation when the outside sea state is mild, but when the sea states is severe, parametric excitation contributes more to the instability, vortex-shedding excitation only strengthens the consequent instability. More modes and a higher frequency of vibration can be excited when extreme states are more severe.

Parametric Instability of an Axially Moving Belt Subjected to Multifrequency Excitations: Experiments and Analytical Validation

This paper experimentally investigates the parametric instability of an industrial axially moving belt subjected to multifrequency excitation. Based on the equations of motion, an analytical perturbation analysis is achieved to identify instabilities. The second part deals with an experimental setup that subjects a moving belt to multifrequency parametric excitation. A data acquisition technique using optical encoders and based on the angular sampling method is used with success for the first time on a nonsynchronous belt transmission. Transmission error between pulleys, pulley/belt slip, and tension fluctuation are deduced from pulley rotation angle measurements. Experimental results validate the theoretical analysis. Of particular note is that the instability regions are shifted to lower frequencies than the classical ones due to the multifrequency excitation. This experiment also demonstrates nonuniform belt characteristics (longitudinal stiffness and friction coefficient) along the belt length that are unexpected sources of excitation. These variations are shown to be sources of parametric instability.

Nonlinear Vibration of Sheet Metal Plates Under Interacting Parametric and External Excitation During Manufacturing

This study is motivated by the vibrations that plague coating processes used in the manufacturing of coated sheet metal. These vibrations arise from time-dependent tension fluctuations within the sheet metal plate as well as from the eccentricity of the rollers used to transport the plate. The time-dependent tension is observed to be rather broad-band and creates multi-frequency parametric excitation. By contrast, the roller eccentricity is largely single-frequency (synchronized with the roller speed) and creates single-frequency external excitation. The plate and excitation sources are studied herein using a single-degree-of-freedom model with a cubic nonlinearity, subject to combined parametric and external excitation. In our study, we investigate the resonances that arise from the synergistic effects of multi-frequency parametric excitation and single-frequency external excitation. For the simpler case of single-frequency parametric excitation, we observe both sum and difference combination resonances in addition to principal parametric resonance. For the case of multi-frequency parametric excitation, we observe a frequency shift for the parametric resonance that derives from the cubic nonlinearity and external excitation. Moreover, the phase relationships of the external and each parametric excitation source have a significant effect on the resulting response amplitude. We use these analyses to explain the resonance mechanisms observed in experiments conducted on an example sheet metal coating process.

Oscillations of weakly non-linear, self-excited systems under multi-frequency parametric excitation

A class of self-excited mechanical or structural systems subjected to parametric excitation is considered. The systems have an arbitrary number of generalized co-ordinates. The equations of motion include weak quadratic and cubic non-linearities in the stiffness, small “negative” linear damping terms, and small “positive” cubic damping terms. Special cases are van der Pol's equation and Rayleigh's equation. The parametric excitation includes multiple frequencies λ m . By using the method of multiple scales, the following resonances are analyzed: λ s ≈ 2 ω q (principal parametric resonance), λ s ≈ ω q , λ s ± λ t ≈ 2 ω q , and λ s + λ t ≈ ω r − ω q , where the ω n are natural frequencies of the system. Steady state response amplitudes are determined and plotted as functions of a detuning parameter, excitation amplitudes and, when λ s ± λ t ≈ 2 ω q , a measure of the relative values of λ s and λ t .

Stability boundaries of an oscillator under high frequency multicomponent parametric excitation

The method of multiple scales is used to investigate certain of the stability boundaries of a damped single-degree-of-freedom system under a multifrequency parametric excitation having sinusoidal inputs with constant frequency spacing and with initial phase angles. The resonance case studied is that for which the difference between any two excitation frequencies is near to twice the natural frequency. Results obtained are compared with those obtained by the monodromy matrix method which is based on Floquet theory. The results compare closely when the excitation level is small, and agree qualitatively at larger excitation.

Experiments on Phase-Reversed Shock Propagation and Parametric Generation

Insertion of a pressure-release baffle in the path of a shock wave effects a 180° phase change which reverses the distortion process in the reflected waveform. Energy contained in the harmonic components of the original shock then returns to the fundamental with further propagation. The dissipation of energy by absorption at the harmonic frequencies is therefore minimized. This allows one to generate higher sound pressure levels at remote field points and to circumvent the limits of acoustic saturation. This process would also appear to improve parametric conversion efficiency. Theory for this effect is presented in a companion paper by Lockwood. Experiments on these effects were conducted in fresh water with a planar polystyrene reflector in-sonified by a beam of primary frequencies in the neighborhood of 440 kHz, at source levels up to 133 dB re 1 μbar at 1 yd. Measurements of propagation curves and of the amplitude responses at a distant field point were made for single and multifrequency parametric excitation. With some exceptions, the results support the aforementioned contentions. [This work was supported by the U. S. Navy Office of Naval Research.]