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Published in Ansel C. Ugural, Mechanical Engineering Design, 2022
The linearly elastic behavior of a beam element is governed according to Equation (4.16c) as d4υ/dx4 = 0. The right-hand side of this equation is 0 because in the formulation of the stiffness matrix equations, we assume no loading between nodes. In the elements where there is a distributed load, the equivalent nodal load components are used. The solution is taken to be a cubic polynomial function of x: υ¯=a1+a2x¯+a3x¯+a3x¯2+a4x¯3
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Published in Ansel C. Ugural, Youngjin Chung, Errol A. Ugural, Mechanical Engineering Design, 2020
Ansel C. Ugural, Youngjin Chung, Errol A. Ugural
The linearly elastic behavior of a beam element is governed according to Equation (4.16c) as d4υ/dx4 = 0. The right-hand side of this equation is 0 because in the formulation of the stiffness matrix equations, we assume no loading between nodes. In the elements where there is a distributed load, the equivalent nodal load components are used. The solution is taken to be a cubic polynomial function of x: υ¯=a1+a2x¯+a3x¯+a3x¯2+a4x¯3
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Published in Ansel C. Ugural, Youngjin Chung, Errol A. Ugural, MECHANICAL DESIGN of Machine Components, 2018
Ansel C. Ugural, Youngjin Chung, Errol A. Ugural
The linearly elastic behavior of a beam element is governed according to Equation 4.16c as d4 υ/dx 4 = 0. The right-hand side of this equation is 0 because in the formulation of the stiffness matrix equations, we assume no loading between nodes. In the elements where there is a distributed load, the equivalent nodal load components are used. The solution is taken to be a cubic polynomial function of x: () υ¯ =a1+a2x¯+a3x¯2+a4x¯3
Effective Elastic Stiffness Tensor and Ultrasonic Velocities for 3D Printed Polycrystals with Pores and Texture
Published in Research in Nondestructive Evaluation, 2022
Ruohan Tan, Yongfeng Song, Xiongbing Li, Shu Cheng, Peijun Ni
Then, the influence of bulk porosity on the ultrasonic velocity of longitudinal wave is analyzed. As seen in Figure 7, the longitudinal wave velocity decreases as porosity increases. When the longitudinal wave velocity decreases by 1%, 2%, 3%, 4%, and 5% from its original value, the corresponding porosities are 1.8%, 3.7%, 5.6%, 7.5%, and 9.5%, respectively. Therefore, using the longitudinal wave velocity, it is simple to evaluate the porosity of 3D-printed 316L stainless steel if it is greater than 1.8%. However, due to the real measurement inaccuracy of longitudinal wave velocity, which might result from incidence angle error, thickness measurement error, inadequate parallelism, or insufficient sampling rate, it is difficult to measure velocity changes of less than 1% now. For cases with porosities less than 1.8%, supplemental ultrasonic attenuation or backscattering measurements are necessary. In addition, a functional fit is used to approximate the velocity-porosity curve. As shown in Figure 7, a cubic polynomial function is chosen, and the approximated function is given by:
An extensive analysis of frequency and transient responses in S and C-shaped gears
Published in Australian Journal of Mechanical Engineering, 2022
S. H. Yahaya, M. S. Salleh, Kenjiro T. Miura, A. Abdullah, A. R. M Warikh, Z. Jano
Said-Ball cubic curve (SBCC) was a parametric form applied throughout this study. Said (1990) revealed that SBCC is a third-degree (cubic) polynomial curve that permits an inflection point and is also highly suitable for G2 (curvature) blending application curves. Moreover, the procedures of controlling shape or preserving curves become more effective because SBCC basis functions consist of two shape parameters, and . Figure 2 shows these parameters’ effectiveness. Meanwhile, Farin (2002) remarked that cubic Bézier curve (CBC) does not have any shape parameter. Therefore, the adjustment of control points is only applicable to its shape control of curve when applying this CBC. Ahmad (2009) reviewed the appearances of SBCC such as positivity curve, convex hull features, and geometric mapping suitability. Other uniqueness of shape parameters in SBCC can also represent several basis functions, namely, cubic Ball, CBC, and cubic Trimmer if the shape values are 2, 3 and 4.
Feedback boundary control of linear hyperbolic equations with stiff source term
Published in International Journal of Control, 2018
The choices of parameters are given by which is supersonic. The eigenvalues of the coefficient matrix are Λ1 = 2.2, Λ2 = 0.2. The conditions (18) are not fulfilled due to the sufficiency. However, the negativity of the boundary terms BCi for i = 1, 2,… , 5 holds true. Therefore, we are still able to obtain the exponential decay of the convergency. The prediction of the decay rate is λ1 = 0.0947. The initial data is given by a cubic polynomial perturbed by a trigonometric function. Figure 5 shows that the Lyapunov function of system (25) with ε = 0.01 is decreasing in an exponential rate 0.44358 that is larger than the theoretical prediction.