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Solving Single Algebraic Equations
Published in David E. Clough, Steven C. Chapra, Introduction to Engineering and Scientific Computing with Python, 2023
David E. Clough, Steven C. Chapra
A transcendental equation is one which “transcends” algebra by including terms such as the exponential and trigonometric functions. For example, f(x)=e−0.5xsin(3x−1)
Tem and Quasitem Modes: Basic Planar Transmission Lines
Published in Edward F. Kuester, Theory of Waveguides and Transmission Lines, 2020
A coplanar stripline as in Figure 8.23 is to have Zc = 50 Ω, while εeff = 4. If εr = 17, obtain values of w/h and s/h such that these line parameters are achieved. A transcendental equation has to be solved numerically to obtain the solution.
Low-Dimensional Semiconductors
Published in Jyoti Prasad Banerjee, Suranjana Banerjee, Physics of Semiconductors and Nanostructures, 2019
Jyoti Prasad Banerjee, Suranjana Banerjee
The transcendental equation, (5.27) cannot be solved analytically but can be solved, using graphical or numerical technique. The parameter θ1 is a function of effective electron mass, height, and width of the quantum well.
Semi-analytical solutions for seismic responses of tunnel tube in pier-supported submerged floating tunnel under vertical excitation
Published in Ships and Offshore Structures, 2022
Renfei He, Yong Yuan, Zexu Fan, Chong Li, Weiguo He
To make Equation (18) have nonzero solutions, the determinant of A(b) is set to zero: Equation (19) is a transcendental equation with multiple roots. By substituting Equation (12) into Equation (19) and solving the resulting equation, the natural circular frequencies (ω1, ω2, … , ωn) can be obtained. Due to the complexity of Equation (19), deriving the analytical solutions for the natural circular frequencies is difficult. The semi-analytical solutions can be acquired numerically with computer software, such as MATLAB (Yu et al. 2018). Thereafter, substituting the solutions into Equation (12) and then Equation (18), the corresponding D vectors can be calculated. Note that D has infinitely many solutions and can be determined by setting the amplitude of the mode shape function to 1. Then, the nth mode shape function of the tunnel tube can be integrated by the nth mode shape function of each component:
Hygrothermoelastic response of a finite hollow circular cylinder
Published in Waves in Random and Complex Media, 2022
To obtain the equation for temperature and moisture function, we first define the Hankel transform and its inverse transform over the variable in the range as where the kernel of the transform is Here, are the roots of the transcendental equation Second, we define the finite Fourier transform and its inverse transform over the variable in the range defined as where and are the positive roots of the transcendental equation Now, applying Laplace, Hankel, and Fourier transform and their inversions (defined above) to Equations (5) and (6) and making use of the transformed boundary and initial conditions (9)–(18), one obtains the temperature and the moisture distribution function for the uncoupled effect expressed as follows: where
Effect of Time-Varying Delay on Stability of Real-Time Hybrid Simulation with Multiple Experimental Substructures
Published in Journal of Earthquake Engineering, 2022
Liang Huang, Cheng Chen, Menghui Chen, Tong Guo
The system is stable if and only if all the characteristic roots of Eq. (20b) are located in the left half plane of the Laplace domain. When the characteristic roots have purely imaginary solution, the system is critically stable and the time delay is defined as the critical delay marked as , ... However, implementing this approach faces three challenges including: (1) Eq. (20b) is a transcendental equation due to the term , which often has no closed-form solutions. (2) When the system has multiple sources of delay (i.e. ), the stability boundary of Eq. (20b) may not have analytical expression. (3) The complexity of solving Eq. (20b) increases almost exponentially with the increase of the number of DOFs of system. Due to these challenges, previous researches are often limited to simple cases for the characteristic roots analysis method.