• Title, Summary, Keyword: equation of a plane

### Vibration Analysis of an Axially Moving Membrane with In-plane/Out-of-plane Deformations (면내/면외변형을 고려한 이송되는 박막의 진동해석)

• Shin Changho;Chung Jintai
• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.14 no.9
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• pp.910-918
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• 2004
• The vibration analysis of an axially moving membrane are investigated when the membrane has the two sets of in-plane boundary conditions, which are free and fixed constraints in the lateral direction. Since the in-plane stiffness is much higher than the out-of-plane stiffness, it is assumed during deriving the equations of motion that the in-plane motion is in a steady state. Under this assumption, the equation of out-of-plane motion is derived, which is a linear partial differential equation influenced by the in-plane stress distributions. After discretizing the equation by using the Galerkin method, the natural frequencies and mode shapes are computed. In particular, we put a focus on analyzing the effects of the in-plane boundary conditions on the natural frequencies and mode shapes of the moving membrane.

### Vibration Analysis of an Axially Moving Membrane with In-Plane/out-of-Plane Deformations (면내/면외변형을 고려한 이송되는 박막의 진동해석)

• 신창호;정진태
• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• pp.164-168
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• 2004
• The vibration analysis of an axially moving membrane are investigated when the membrane has the two sets of in-plane boundary conditions, which are free and fixed constraints in the lateral direction. Since the in-plane stiffness is much higher than the out-of-plane stiffness, it is assumed during deriving the equations of motion that the in-plane motion is in a steady state. Under this assumption. the equation of out-of\ulcornerplane motion is derived, which is a linear partial differential equation influenced by the in-plane stress distributions. After discretizing the equation by using the Galerkin method, the natural frequencies and mode shapes are computed. In particular, we put a focus on analyzing the effects of the in-plane boundary conditions on the natural frequencies and mode shapes of the moving membrane.

### Elastic Analysis of a Half-Plane Containing an Inclusion and a Void Using Mixed Volume and Boundary Integral Equation Method (혼합 체적-경계 적분방정식법을 이용한, 함유체와 공동을 포함한 반무한 고체에서의 탄성해석)

• Lee, Jung-Ki;Yoon, Koo-Young
• Transactions of the Korean Society of Mechanical Engineers A
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• v.32 no.12
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• pp.1072-1087
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• 2008
• A mixed volume and boundary integral equation method (Mixed VIEM-BIEM) is used to calculate the plane elastostatic field in an isotropic elastic half-plane containing an isotropic or anisotropic inclusion and a void subject to remote loading parallel to the traction-free boundary. A detailed analysis of stress field at the interface between the isotropic matrix and the isotropic or orthotropic inclusion is carried out for different values of the distance between the center of the inclusion and the traction-free surface boundary in an isotropic elastic half-plane containing three different geometries of an isotropic or orthotropic inclusion and a void. The method is shown to be very accurate and effective for investigating the local stresses in an isotropic elastic half-plane containing multiple isotropic or anisotropic inclusions and multiple voids.

### Elastic Wave Propagation in Monoclinic System Due to Harmonic Line Load

• Kim, Yong-Yun
• The Journal of the Acoustical Society of Korea
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• v.17 no.2E
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• pp.47-52
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• 1998
• An analysis of dynamic responses is carried out on monoclinic anisotropic system due to a buried harmonic line source. The load is in the form of a normal stress acting along an arbitrary axis on the plane of symmetry within the orthotropic materials: In case that the line load is acting along the symmetry axis normal to the plane of symmetry, plane wave equation is coupled with verital shear wave and longitudinal wave. However, if the line load is acting along an arbitrary axis normal to the plane of symmetry, plane wave equation is coupled with vertical shear wave, longitudinal wave and horizontal shear wave. We first considered the equation of motion in a reference coordinate system, where the line load is coincident with a symmetry axis of the orthotropic material. Then the equation of motion is transformed into one with respect to general coordinate system with azimuthal angle by using transformation tensor. Plane wave solutions of monoclinic systems are derived for infinite media. Finally complete solutions for the plane harmonic wave are obtained by calculating the inverse of the integral transforms, in which bulk wave poles are avoided by deforming the contour of the integration to the complex plane. Numerical results for examples of orthotropic material belonging to monoclinic symmetry are demonstrated.

### Elastic Analysis of a Half-Plane Containing Multiple Inclusions Using Volume Integral Equation Method (체적 적분방정식법을 이용한, 다수의 함유체를 포함한 반무한 고체에서의 탄성해석)

• Lee, Jung-Ki;Ku, Duck-Young
• Transactions of the Korean Society of Mechanical Engineers A
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• v.32 no.2
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• pp.148-161
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• 2008
• A volume integral equation method (VIEM) is used to calculate the plane elastostatic field in an isotropic elastic half-plane containing multiple isotropic or anisotropic inclusions subject to remote loading. A detailed analysis of stress field at the interface between the matrix and the central inclusion in the first column of square packing is carried out for different values of the distance between the center of the central inclusion in the first column of square packing of inclusions and the traction-free surface boundary in an isotropic elastic half-plane containing multiple isotropic or anisotropic inclusions. The method is shown to be very accurate and effective for investigating the local stresses in an isotropic elastic half-plane containing multiple isotropic or anisotropic inclusions.

### Nonlinear wind-induced instability of orthotropic plane membrane structures

• Liu, Changjiang;Ji, Feng;Zheng, Zhoulian;Wu, Yuyou;Guo, Jianjun
• 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.

### Elastic analysis in a half-plane using volume integral equation method (반무한 공간에서의 체적 적분방정식법을 이용한 탄성해석)

• Ku, D.Y.;Lee, J.K.
• Proceedings of the Korean Society of Precision Engineering Conference
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• pp.81-82
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• 2007

### Conformable solution of fractional vibration problem of plate subjected to in-plane loads

• Fadodun, Odunayo O.;Malomo, Babafemi O.;Layeni, Olawanle P.;Akinola, Adegbola P.
• Wind and Structures
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• v.28 no.6
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• pp.347-354
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• 2019
• This study provides an approximate analytical solution to the fractional vibration problem of thin plate governing anomalous motion of plate subjected to in-plane loads. The method of variable separable is employed to transform the fractional partial differential equations under consideration into a fractional ordinary differential equation in temporal variable and a bi-harmonic plate equation in spatial variable. The technique of conformable fractional derivative is utilized to solve the resulting fractional differential equation and the approach of finite sine integral transform method is used to solve the accompanying bi-harmonic plate equation. The deflection field which measures the transverse displacement of the plate is expressed in terms of product of Bessel and trigonometric functions via the temporal and spatial variables respectively. The obtained solution reduces to the solution of the free vibration problem of thin plate in literature. This work shows that conformable fractional derivative is an efficient mathematical tool for tracking analytical solution of fractional partial differential equation governing anomalous vibration of thin plates.

### Elastic Analysis of Unbounded Solids Using a Mixed Numerical Method (혼합 수치해석 방법을 이용한 무한고체의 탄성해석)

• Lee , Jung-Ki;Heo, Kang-Il
• Proceedings of the KSME Conference
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• pp.341-348
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• 2001
• A Mixed Volume and Boundary Integral Equation Method is applied for the effective analysis of plane elastostatic problems in unbounded solids containing general anisotropic inclusions and voids or isotropic inclusions. It should be noted that this newly developed numerical method does not require the Green's function for anisotropic inclusions to solve this class of problems since only Green's function for the unbounded isotropic matrix is involved in their formulation for the analysis. This new method can also be applied to general two-dimensional elastodynamic and elastostatic problems with arbitrary shapes and number of anisotropic inclusions and voids or isotropic inclusions. Through the analysis of plane elastostatic problems in unbounded isotropic matrix with orthotropic inclusions and voids or isotropic inclusions, it will be established that this new method is very accurate and effective for solving plane elastic problems in unbounded solids containing general anisotropic inclusions and voids or isotropic inclusions.

### Vibration Analysis of a Flexible Spinning Disk Considering the effect of Misalignment (회전축 정렬불량을 고려한 유연회전디스크의 진동해석)

• Jeong, Jin-Tae;Heo, Jin-Uk;Choe, Gi-Yeong
• Transactions of the Korean Society of Mechanical Engineers A
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• v.26 no.5
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• pp.952-959
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• 2002
• The natural frequencies of a flexible spinning disk misaligned with the axis of rotation are studied in an analytic manner. The effects of misalignment on the natural frequency need to be investigated, because the misalignment between the axis of symmetry and the axis of relation cannot be avoided in the removable disks such as CD-R, CD-RW or DVD disks. Assuming that the in -plane displacements are in steady state and the out-of-plane displacement is in dynamic state, the equations of motion are derived for the misaligned spinning disk. After the exact solutions are obtained fur the steady -state in-plane displacements, they are plugged into the equation for the dynamic-state out-of-plane motion. The resultant equation is a linear equation for the out -of-plane displacement, which is discretized by the Galerkin method. Based on the discretized dquations, the effects of the misalignment are analyzed on the vibration characteristics of the spinning disk, i.e., the natural frequencies and the critical speed.