Fourier Series To solve the ODE \({y^{\prime\prime}} + a{y^{\prime}} + by = f\left( t \right)\), in which \(f\left( t \right)\) is often the combination of \({e^t}\), \(\sin t\) and \(\cos t\), then any reasonable \(f\left( t \right)\) which is periodic, can be discontinuous but not terribly discontinuous, can be written as Fourier Series. \[f\left( t \right) = {c_0} + \sum\limits_{n = 1}^\infty {{a_n}\cos nt + {b_n}\sin nt} \] Theorem: \(u\left( t \right),v\left( t \right)\) are functions with period \(2\pi \) on the real number, …
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Linear 2nd order ODE with constant coefficients Homogeneous: \[{y^{\prime\prime}} + A{y^{\prime}} + By = 0\] Solution: \[y = {c_1}{y_1} + {c_2}{y_2}\] in which, \({y_1}\) and \({y_2}\) are two independent solutions of the homogeneous ODE. The basic method to solve this ODE is to try \(y = {e^{rt}}\). Plug in and we can get \[{r^2}{e^{rt}} + Ar{e^{rt}} + B{e^{rt}} = 0 \to {r^2} + Ar + B = 0\] Case 1: two different real roots \[y = {c_1}{e^{{r_1}t}} + {c_2}{e^{{r_2}t}}\] Case …
I want to analyze a steel frame with pin connections at both sides or one side at some beams. I found the keyword *Release is very useful and convenient, so I resort to this keyword. I wrote the input file with the same syntax rules, so that it will be more convenient for me to establish more models. However, when I try to run the input file, there are warnings and errors even with the input file generated from a …
Euler equation: \[\left\{ \begin{align} & {{x}_{n+1}}={{x}_{n}}+h \\ & {{y}_{n+1}}={{y}_{n}}+{{A}_{n}}h \\ \end{align} \right.\] \[\left\{ \begin{align} & h=\text{step}\text{size} \\ & {{A}_{n}}=f\left( {{x}_{n}},{{y}_{n}} \right) \\ \end{align} \right.\] Error of solution error = exact solution – approximate solution Method 1: smaller step Euler first-order: \(e\sim {{c}_{1}}h\), proportional to step size Method 2: find a better slop \({{A}_{n}}\) Heun’s method = improved Euler method = modified Euler method = RK2 \[\left\{ \begin{align} & {{x}_{n+1}}={{x}_{n}}+h \\ & {{y}_{n+1}}={{y}_{n}}+h\left( \frac{{{A}_{n}}+{{B}_{n}}}{2} \right) \\ \end{align} \right.\] in which, \({{B}_{n}}=f\left( …
Calculation of transverse shear stiffness Definition The effective transverse shear stiffness of the section of a shear flexible beam is defined in Abaqus as \[{\overline K _{\alpha 3}} = f_p^\alpha {K_{\alpha 3}}\] where \({\overline K _{\alpha 3}}\) is the section shear stiffness in the \(\alpha \)-direction; \({K_{\alpha 3}}\) is the actual shear stiffness of the section having units of force; and \(\alpha = 1,{\kern 1pt} {\kern 1pt} {\kern 1pt} 2\) are the local directions of the cross-section. \(f_p^\alpha \) is a dimensionless factor used …