• Journal of applied mathematics & informatics
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• 제5권2호
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• pp.525-537
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• 1998

In this paper we prove the inequality of subharmonic functions between a non-negative subharmonic function u and a smooth function $\upsilon$satisfying ｜$\upsilon(0)$\mid$\leq u(0), $\mid$\nabla \upsilon $\mid$ \leq$\mid$\nabla u $\mid$\leq c\Delta u $, where 0$\leq$c$\leq$1 is a constant. Here $\mu$is the harmonic measure on $\partial$D with respect to 0.

• 대한수학회지
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• 제34권3호
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• pp.543-551
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• 1997

We prove a norm inequality of the form $\left\$\mid$ \upsilon \right\$\mid$ \leq (r - 1) \left\$\mid$ u \right\$\mid$_p, 1 < p < \infty$, between a non-negative subharmonic function u and a smooth function $\upsilon$ satisfying $$\mid$\upsilon(0)$\mid$ \leq u(0), $\mid$\nabla\upsilon$\mid$ \leq \nabla u$\mid$$ and $\mid$\Delta\upsilon$\mid$ \leq \alpha\Delta u$, where $\alpha$ is a constant with $0 \leq \alpha \leq 1$. This inequality extends Burkholder's inequality where $\alpha = 1$.

• Advances in nano research
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• 제12권4호
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• pp.353-363
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• 2022

This paper presents an investigation about superharmonic and subharmonic resonances of a carbon nanotube reinforced composite beam subjected to lateral harmonic load with damping effect based on the modified couple stress theory. As reinforcing phase, three different types of single walled carbon nanotubes (CNTs) distribution are considered through the thickness in polymeric matrix. The governing nonlinear dynamic equation is derived based on the von Kármán nonlinearity with using of Hamilton's principle. The Galerkin's decomposition technique is utilized to discretize the governing nonlinear partial differential equation to nonlinear ordinary differential equation and then is solved by using of multiple time scale method. Effects of different patterns of reinforcement, volume fraction, excitation force and the length scale parameter on the frequency-response curves of the carbon nanotube reinforced composite beam are investigated. The results show that volume fraction and the distribution of CNTs play an important role on superharmonic and subharmonic resonances of the carbon nanotube reinforced composite beams.

• Structural Engineering and Mechanics
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• 제85권4호
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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.

• East Asian mathematical journal
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• 제19권2호
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• pp.165-171
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• 2003

We show that any F-harmonic map from a compact manifold M to N is necessarily constant if N possesses a strictly-convex function, and prove 'Liouville type theorems' for F-harmonic maps. Finally, when the target manifold is the real line, we get a result for F-subharmonic functions.

• 대한기계학회논문집B
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• 제27권8호
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• pp.1023-1032
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• 2003

The effect of local forcing on the separated flow over a backward-facing step is investigated through hot-wire measurements and flow visualization with multi-smoke wires. The boundary layer upstream of the separation point is laminar and the Reynolds number based on the free stream velocity and the step height is 13800. The local forcing is given from a slit located at the step edge and the forcing signal is always defined when the wind tunnel is in operation. In case of single frequency forcing, the streamwise velocity and the reattachment length are measured under forcing with various forcing frequencies. For the range of 0.010〈S $t_{\theta}$〈0.013, the forcing frequency component of the streamwise velocity fluctuation grows exponentially and is saturated at x/h = 0.75 , while its subharmonic component grows following the fundamental and is saturated at x/h = 2.0. However, the saturated value of the subharmonic is much lower than that of the fundamental. It is observed that the vortex formation is inhibited by the forcing at S $t_{\theta}$ = 0.019 . For double frequency forcing, natural instability frequency is adopted as a fundamental frequency and its subharmonic is superposed on it. The fundamental frequency component of the streamwise velocity grows exponentially and is saturated at 0.5 < x/ｈ < 0.75, while its subharmonic component grows following the fundamental and is saturated at x/ｈ= 1.5 . Furthermore, the saturated value of the subharmonic component is much higher than that for the single frequency forcing and is nearly the same or higher than that of the fundamental. It is observed that the subharmonic component does not grow for the narrow range of the initial phase difference. This means that there is a range of the initial phase difference where the vortex parring cannot be enhanced or amplified by double frequency forcing. In addition, this effect of the initial phase difference on the development of the shear layer and the distribution of the reattachment length shows a similar trend. From these observations, it can be inferred that the development of the shear layer and the reattachment length are closely related to the vortex paring.

• Structural Engineering and Mechanics
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• 제42권5호
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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.

• ETRI Journal
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• 제16권4호
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• pp.13-25
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• 1995

We have derived a nonlinear spatiotemporal differential equation for space-charge fields from Kukhtarev’s material equations in a moving coordinate system and obtained the spatial subharmonic instability boundaries by using linear stability analysis. It is also found that there is an analogy between the temporal subharmonic and the spatial subharmonc instabilities in the sense that the governing differential equations describing the instability boundaries are formally identical. The experiments for generating spatial subharmonic waves are performed in a photorefractive $Bi_{12}SiO_{20}$ crystal by using conventional moving grating technique. The threshold detunings are experimentally determined and the results are compared with the theory.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 2005년도 춘계학술대회논문집
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• pp.567-571
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• 2005

The slender rectangular cantilever beam has vef interesting to study dynamic behaviors of the harmonic base excitation of a cantilever beam shows many nonlinear dynamics due to unstability , energy transfer and mode coupling. Nonlinear phenomenon shows superharmonic, subharmonic, super subharmonic and chaotic motions of the cantilever beam. Experimental observation and verification of these phenomenon carry much importance for the theoretical study as well as in it self. In the experimental cantilever beam, the chaotic motions of the beam appear as a pink noise signal in FFT analysis and as a torus structure in the oscilloscope analyzed to eventually give information of chaotic motions of the cantilever beam.

• 한국소음진동공학회논문집
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• 제14권12호
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• pp.1314-1321
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• 2004

The non-linear responses of a slender rectangular cantilever beam subjected to lateral harmonic base-excitation are investigated by the 2-channel FFT analyzer. Both linear and nonlinear behaviors of the cantilever beam are compared with each other. Bending mode, torsional mode, and transverse mode are coupled in such a way that the energy transfer between them are observed. Especially, superharmonic, subharmonic, and chaotic motions which result from the unstable inertia terms in the transverse mode are analyzed by the FFT analyzer The aim is to give the explanations of the route to chaos, i.e., harmonic motion \longrightarrow superharmonic motion \longrightarrow subharmonic motion \longrightarrow chaos.