• Journal for History of Mathematics
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• v.34 no.5
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• pp.153-164
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• 2021

A functional equation is an equation which is satisfied by a function. Some elementary functional equations can be manipulated with elementary algebraic operations and functional composition only. However to solve such functional equations, somewhat critical and creative thinking ability is required, so that it is educationally worth while teaching functional equations. In this paper, we look at the origin of functional equations, and their characteristics and educational meaning and effects. We carefully suggest the use of the functional equations as a material for school mathematics education.

• Wind and Structures
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• v.25 no.5
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• pp.415-432
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• 2017

The nonlinear aerodynamic instability of a tensioned plane orthotropic membrane structure is theoretically investigated in this paper. The interaction governing equation of wind-structure coupling is established by the Von $K\acute{a}rm\acute{a}n's$ large amplitude theory and the D'Alembert's principle. The aerodynamic force is determined by the potential flow theory of fluid mechanics and the thin airfoil theory of aerodynamics. Then the interaction governing equation is transformed into a second order nonlinear differential equation with constant coefficients by the Bubnov-Galerkin method. The critical wind velocity is obtained by judging the stability of the second order nonlinear differential equation. From the analysis of examples, we can conclude that it's of great significance to consider the orthotropy and geometrical nonlinearity to prevent the aerodynamic instability of plane membrane structures; we should comprehensively consider the effects of various factors on the design of plane membrane structures; and the formula of critical wind velocity obtained in this paper provides a more accurate theoretical solution for the aerodynamic stability of the plane membrane structures than the previous studies.

• Structural Engineering and Mechanics
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• v.76 no.3
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• pp.293-310
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• 2020

This study derives the differential equations of free vertical bending and torsional vibrations for two- and three-pylon suspension bridges using d'Alembert's principle. The respective algorithms for natural vibration frequency and vibration mode are established through the separation of variables. In the case of the three-pylon suspension bridge, the effect of the along-bridge bending vibration of the middle pylon on the vertical bending vibration of the entire bridge is considered. The impact of torsional vibration of the middle pylon about the vertical axis on the torsional vibration of the entire bridge is also analyzed in detail. The feasibility of the proposed method is verified by two engineering examples. A comparative analysis of the results obtained via the proposed and more intricate finite element methods confirmed the former feasibility. Finally, the middle pylon stiffness effect on the vibration frequency of the three-pylon suspension bridge is discussed. It is found that the vibration frequencies of the first- and third-order vertical bending and torsional modes both increase with the middle pylon stiffness. However, the increase amplitudes of third-order bending and torsional modes are relatively small with the middle pylon stiffness increase. Moreover, the second-order bending and torsional frequencies do not change with the middle pylon stiffness.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1995.10a
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• pp.442-445
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• 1995

This paper presents the position tracking control of a laser scanning mirror system in which piezoelectic actuator is incorporated. Using the shear mode of the piezoelectric actuator,angular oscillation of a laser scanning mirror is derived. Torsion bar is rhen designed and attached to the piezoelctric actuator in order to magnify the amplitude generated by the actuator. Finite element modeling and analysis are essntial for designing the piezoelectic actuator. The torsional resonance mode of the piezoelectric actuator is found from the model analysis of the actuator and the mechanical shear is matched with the driving frequency. Transfer function between the electrical excitation and the mechanical shear deformation at resonance frequency is found form the response of the actuator calculated by the finite element analysis and the governing equation of the system is derived from d'Alembert's principle. Tracking control performance for desired trajectory which is, in fact, sinusoidal curve is presented in order to demonstrate the validity of the proposed system.

• Wind and Structures
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• v.26 no.6
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• pp.355-367
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• 2018

This paper studies the aerodynamic stability of a tensioned, geometrically nonlinear orthotropic membrane structure with hyperbolic paraboloid in sag direction. Considering flow separation, the wind field around membrane structure is simulated as the superposition of a uniform flow and a continuous vortex layer. By the potential flow theory in fluid mechanics and the thin airfoil theory in aerodynamics, aerodynamic pressure acting on membrane surface can be determined. And based on the large amplitude theory of membrane and D'Alembert's principle, interaction governing equations of wind-structure are established. Then, under the circumstance of single-mode response, the Bubnov-Galerkin approximate method is applied to transform the complicated interaction governing equations into a system of second-order nonlinear differential equation with constant coefficients. Through judging the frequency characteristic of the system characteristic equation, the critical velocity of divergence instability is determined. Different parameter analysis shows that the orthotropy, geometrical nonlinearity and scantling of structure is significant for preventing destructive aerodynamic instability in membrane structures. Compared to the model without considering flow separation, it's basically consistent about the divergence instability regularities in the flow separation model.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.37 no.1
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• pp.239-246
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• 2017

There are various transversal structures (small dams or drop structures) in median and small streams in Korea. Most of them are concrete structures and it is so hard to exclude low-level water. Unless drainage valves and/or gates would not be installed near bottom of bed, sediment from upstream should be deposited and also contaminants attached to the sediments would devastatingly threaten the water quality and ecosystem. One of countermeasures for such problem is the bypass pipe installed underneath the transversal structure. However, there is still issued whether it would be workable if the gravels and/or stones would roll into and be not excluded. Therefore, in this study, the conditions to exclude the rip stone which enter into the bypass pipe was reviewed. Based on sediment transport phenomenon, the behavior of stones was investigated with the concepts from the critical shear stress of sediment and d'Alembert principle. As final results, the basis condition (${\tau}_c{^*}$) was derived using the Lagrangian description since the stones are in the moving state, not in the stationary state. From hydraulic experiments the relative velocity could be obtained. In order to minimize the scale effect, the extra wide channel of 5.0 m wide and 1.0 m high was constructed and the experimental stones were fully spherical ones. Experimental results showed that the ratio of flow velocity to spherical particle velocity was measured between 0.5 and 0.7, and this result was substituted into the suggested equation to identify the critical condition wether the stones were excluded. Regimes about the exclusion of stone in bypass pipe were divided into three types according to particle Reynolds number ($Re_p$) and dimensionless critical shear force (${\tau}_c{^*}$) - exclusion section, probabilistic exclusion section, no exclusion section. Results from this study would be useful and essential information for bypass pipe design in transveral structures.