• Transactions of the Korean Society of Machine Tool Engineers
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• v.10 no.1
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• pp.43-50
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• 2001

Subharmonic resonances of order 1/2 of a continuous rotating shaft with distributed mass are discussed. The restoring force of the shaft exhibits geometric stiffening nonlinearity due to the extension of the shaft center line. It is assumed that a distributed lateral force, such as the gravity, acts on the rotor. The possibility of the occurrence of subharmonic resonances, the shapes of resonance curves, and internal resonance phenomena are investigate.

• Structural Engineering and Mechanics
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• v.42 no.5
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• pp.677-699
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• 2012

This article studies the secondary resonances of a clamped-clamped microresonator under combined electrostatic and piezoelectric actuations. The electrostatic actuation is induced by applying the AC-DC voltage between the microbeam and the electrode plate that lies at the opposite side of the microbeam. The piezoelectric actuation is induced by applying the DC voltage between upper and lower sides of piezoelectric layer. It is assumed that the neutral axis of bending is stretched when the microbeam is deflected. The drift effect of piezoelectric layer (the phenomenon where there is a slow increase of the free strain after the application of a DC field) is neglected. The equations of motion are solved by using the multiple scale perturbation method. The system possesses a subharmonic resonance of order one-half and a superharmonic resonance of order two. It is shown that using the DC piezoelectric actuation, the sensitivity of AC-DC electrostatically actuated microresonator under subharmonic and superharmonic resonances may be tuned. In addition, it is shown that the tuning domain of the microbeam under combined electrostatic and piezoelectric actuations at subharmonic and superharmonic conditions is larger than the tuning domain of microbeam under only the electrostatic actuation.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 1994.10a
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• pp.35-39
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• 1994

One of the important characteristics of the response of nonlinear systems is the existence of subharmonic resonances. When some conditions in parameter space are satisfied. It is possible even in the presence of damping for a periodically excited nonlinear system to possess a response which is the combination of a contribution at the excitation frequency and a component at the system natural frequency. The system natural frequency being a submultiple of the excitation frequency implies that the resulting response is a subharmonic oscillation. In general, there also co-exists, for the system, a response at the excitation frequency, and initial conditions determine which of the steady-state responses is achieved in an experiment or a numerical simulation. In single-degree-of-freedom systems with harmonic excitation, depending on the type of the nonlinearity, e.g., cubic or quadratic the frequency of subharmonic response is respectively, one-third or one-half of that of the excitation frequency. Although subharmonic resonance is one of the principal characteristics of a nonlinear system the subharmonic responses of structures in the presence of internal resonances have been studied very rarely. In this work, we consider subharmonic responses in the two-mode approximation of the plate equations. It is assumed that the two modes are in one-to-one internal resonance. Constant and periodic steady-state solutions of the averaged equations are studied. Finally, the results of direct time integration of the original equations of motion are presented and compared with those obtained from the averaged equations.

• Structural Engineering and Mechanics
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• v.85 no.4
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• pp.545-562
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• 2023

In this work, superharmonic and subharmonic resonance of rotating stiffened FGM truncated conical shells exposed to harmonic excitation in a thermal environment is investigated. Utilizing classical shell theory considering Coriolis acceleration and the centrifugal force, the governing equations are extracted. Non-linear model is formulated employing the von Kármán non-linear relations. In this study, to model the stiffener effects the smeared stiffened technique is utilized. The non-linear partial differential equations are discretized into non-linear ordinary differential equations by applying Galerkin's method. The method of multiple scales is utilized to examine the non-linear superharmonic and subharmonic resonances behavior of the conical shells. In this regard, the effects of the rotating speed of the shell on the frequency response plot are investigated. Also, the effects of different semi-vertex angles, force amplitude, volume-fraction index, and temperature variations on the frequency-response graph are examined for different rotating speeds of the stiffened FGM truncated conical shells.

• Journal of Aerospace System Engineering
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• v.13 no.5
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• pp.9-15
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• 2019

In this paper, dynamic analysis of a nonlinear active vibration absorber is conducted with a time delay acceleration feedback to suppress the vibration of a nonlinear single degree of freedom primary system. The primary system consisting of linear and nonlinear cubic springs, mass, and damper is subjected to the multi-harmonic hard excitation with a parametric excitation. It is proposed to reduce the vibration of the primary system and the absorber by using a lead zirconate titanate (PZT) stack actuator in series with a spring in the absorber which configures as an active vibration absorber. The method of multiple scales (MMS) is used to obtain the approximate solution of the system under the internal resonance, subharmonic, superharmonic, and principal parametric resonance conditions simultaneously. Frequency and time responses of the system are investigated considering a delay in the feedback for the various parameters of the absorber configuration and controlling force.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.22 no.11
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• pp.1610-1624
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• 1998

An axisymmetric jet is forced with two helical fundamental waves of identical frequency spinning in opposite directions and an additional axisymmetric sub harmonic wave. The subharmonic component rapidly grows downstream from subharmonic resonance with the fundamental, significantly depending on the initial phase difference. The variations of the subharmonic amplitude with the initial phase difference show cusp-like shapes. The amplification of the sub harmonic results in 'vortex pairing of helical modes'. Furthermore, azimuthal variation of the amplification induces an asymmetric jet cross-section. When the initial subharmonics is imposed with an initial phase difference close to a critical value, the jet-cross section evolves into a three-lobed shape. One lobe is generated by the enhanced vortex pairing and the other two lobes are generated by the delayed vortex pairing. Thus, it is confirmed that the initial phase difference between the fundamental and the subharmonic plays an important role in controlling the jet cross-section.

• International Journal of Fluid Machinery and Systems
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• v.9 no.1
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• pp.1-16
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• 2016

Behaviors of surges appearing near the stall stagnation boundaries in various fashions in systems of a single-stage compressor and flow-path systems were studied analytically and were tried to put to order. Deep surges, which enclose the stall point in the pressure-mass flow plane, tend to have either near-resonant surge frequencies or subharmonic ones. The subharmonic surge is a multiple-loop one containing, for example, in a (1/2) subharmonic one, a deep surge loop and a mild surge loop, the latter of which does not enclose the stall point, staying only within the stalled zone. Both loops have nearly equal time periods, respectively, resulting in a (1/2) subharmonic surge frequency as a whole. The subharmonic surges are found to appear in a narrow zone neighboring the stall stagnation boundary. In other words, they tend to appear in the final stage of the stall stagnation process. It should be emphasized further that the stall stagnation initiates fundamentally at the situation where a volume-modified reduced resonant-surge frequency becomes coincident with that for the stagnation boundary conditions, where the reduced frequency is defined by the acoustical resonance frequency in the flow-path system, the delivery flow-path length and the compressor tip speed, modified by the sectional area ratio and the effect of the stalling pressure ratio. The real surge frequency turns from the resonant frequency to either near-resonant one or subharmonic one, and finally to stagnation condition, for the large-amplitude conditions, caused by the non-linear self-excitation mechanism of the surge.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.21 no.9
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• pp.813-819
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• 2011

This paper predicted nonlinear dynamic responses of a cantilevered carbon nanotube(CNT) resonator incorporating the electrostatic forces and van der Waals interactions between the CNT cantilever and ground plane. The structural model of CNT includes geometric and inertial nonlinearities to investigate various phenomena of nonlinear responses of the CNT due to the electrostatic excitation. In order to solve this problem, we used Galerkin's approximation and the numerical integration techniques. As a result, the CNT nano-resonator shows the softening effect through saddle-node bifurcation near primary resonance frequency with increasing the applied AC and DC voltages. Also we can predict nonlinear secondary resonances such as superharmonic and subharmonic resonances. The superharmonic resonance of the nano-resonator is influenced by applied AC voltage. The period-doubling bifurcation leads to the subharmonic resonance which occurs when the nano-resonator is actuated by electrostatic forces as parametric excitation.

• Journal of KSNVE
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• v.7 no.1
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• pp.55-60
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• 1997

In order to examine the validity of an asymptotic solution obtained from the method of multiple scales, we investigate a third-order subharmonic resonance response of a bar constrained by a nonlinear spring to a harmonic excitation. The motion of the bar is governed by a linear partial differential equation with a nonlinear boundary condition. The nonlinear boundary value problem is solved by using the finite difference method. The numerical solution is compared with the asymptotic solution.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2006.05a
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• pp.62-65
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• 2006

In this paper, nonlinear nonplanar vibration of a flexible rectangular cantilever beam is analyzed when one-to-one resonance occurs to the beam. The planar and nonplanar motions of the beam are analyzed in time and frequency domains. In frequency domain, FFT analyzer is used to perform autospectrum and cepstrum analyses for nonlinear response of the beam. In time domain, an oscilloscope is used to investigate the phase difference between the planar and nonplanar motions and to perform Torus analysis in the phase space. Through those analyzing process, the main frequencies of superharmonic, subharmonic, and super-subharmonic motions are investigated in the nonplanar motion due to one-to-one resonance. Analyzing the phase difference between the planar and nonplanar motions, it is observed that the phase difference varies in time.