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- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Operational_Amplifiers%3A_Theory_and_Practice_(Roberge)/12%3A_Advanced_Applications/12.05%3A_FURTHER_EXAMPLESSuccessful design almost always involves combining bits and pieces, a concept here, a topology there, to ultimately arrive at the optimum solution. In this section we will see how some of the ideas in...Successful design almost always involves combining bits and pieces, a concept here, a topology there, to ultimately arrive at the optimum solution. In this section we will see how some of the ideas introduced earlier are combined into relatively more sophisticated configurations. The three examples that are presented are all "real world" in that they reflect actual requirements that the author has encountered recently in his own work.
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Math_Numerics_and_Programming_(for_Mechanical_Engineers)/03%3A_Unit_III_-_Linear_Algebra_1_-_Matrices_Least_Squares_and_Regression/16%3A_Matrices_and_Vectors_-_Definitions_and_Operations/16.04%3A_Further_Concepts_in_Linear_AlgebraLet us consider a \(3 \times 2\) matrix \[A=\left(\begin{array}{ll} 0 & 2 \\ 1 & 0 \\ 0 & 0 \end{array}\right) \text {. }\] The column space of \(A\) is the set of vectors representable as \(A x\), wh...Let us consider a \(3 \times 2\) matrix \[A=\left(\begin{array}{ll} 0 & 2 \\ 1 & 0 \\ 0 & 0 \end{array}\right) \text {. }\] The column space of \(A\) is the set of vectors representable as \(A x\), which are \[A x=\left(\begin{array}{ll} 0 & 2 \\ 1 & 0 \\ 0 & 0 \end{array}\right)\left(\begin{array}{l} x_{1} \\ x_{2} \end{array}\right)=\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right) \cdot x_{1}+\left(\begin{array}{l} 2 \\ 0 \\ 0 \end{array}\right) \cdot x_{2}=\left(\begin{array}{c} 2 x_{2}…
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Math_Numerics_and_Programming_(for_Mechanical_Engineers)/03%3A_Unit_III_-_Linear_Algebra_1_-_Matrices_Least_Squares_and_Regression/15%3A_MotivationIf the magnitude of \(F_{\mathrm{f}}\) dictated by the sum of the drag force \(F_{\text {drag }}\) (a combination of all forces resisting the robot’s motion) and the product of the robot’s mass and ac...If the magnitude of \(F_{\mathrm{f}}\) dictated by the sum of the drag force \(F_{\text {drag }}\) (a combination of all forces resisting the robot’s motion) and the product of the robot’s mass and acceleration is less than the maximum static friction force \(F_{\mathrm{f}, \text { static }}^{\max }\) between the wheels and the ground, the wheels will roll without slipping and the robot will move forward with velocity \(v=\omega r_{\text {wheel }}\).
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Math_Numerics_and_Programming_(for_Mechanical_Engineers)This text introduces elementary programming concepts including variable types, data structures, and flow control. After an introduction to linear algebra and probability, it covers numerical methods r...This text introduces elementary programming concepts including variable types, data structures, and flow control. After an introduction to linear algebra and probability, it covers numerical methods relevant to mechanical engineering, including approximation (interpolation, least squares and statistical regression), integration, solution of linear and nonlinear equations, ordinary differential equations, and deterministic and probabilistic approaches.
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Math_Numerics_and_Programming_(for_Mechanical_Engineers)/02%3A_Unit_II_-_Monte_Carlo_Methods/09%3A_Introduction_to_Random_Variables/9.04%3A_Continuous_Random_VariablesThe mean, \(\mu\), or the expected value, \(E[X]\), of the random variable \(X\) is \[\mu=E[X]=\int_{a}^{b} f(x) x d x .\] The variance, \(\operatorname{Var}(X)\), is a measure of the spread of the va...The mean, \(\mu\), or the expected value, \(E[X]\), of the random variable \(X\) is \[\mu=E[X]=\int_{a}^{b} f(x) x d x .\] The variance, \(\operatorname{Var}(X)\), is a measure of the spread of the values that \(X\) takes about its mean and is defined by \[\operatorname{Var}(X)=E\left[(X-\mu)^{2}\right]=\int_{a}^{b}(x-\mu)^{2} f(x) d x .\] The variance can also be expressed as \[\begin{aligned} \operatorname{Var}(X) &=E\left[(X-\mu)^{2}\right]=\int_{a}^{b}(x-\mu)^{2} f(x) d x \\ &=\int_{a}^{b} …
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Math_Numerics_and_Programming_(for_Mechanical_Engineers)/01%3A_Unit_I_-_(Numerical)_Calculus._Elementary_Programming_Concepts/07%3A_Integration/7.02%3A_Integration_of_Bivariate_FunctionsRecall, the interpolant over \(R_{i}\) is defined by the function value at its centroid, \[(\mathcal{I} f)(\boldsymbol{x})=f\left(\tilde{\boldsymbol{x}}_{i}\right)=f\left(\boldsymbol{x}_{i}^{c}\right)...Recall, the interpolant over \(R_{i}\) is defined by the function value at its centroid, \[(\mathcal{I} f)(\boldsymbol{x})=f\left(\tilde{\boldsymbol{x}}_{i}\right)=f\left(\boldsymbol{x}_{i}^{c}\right), \quad \forall \boldsymbol{x} \in R^{n}\] where the centroid is given by averaging the vertex coordinates, \[\tilde{\boldsymbol{x}}_{i}=\boldsymbol{x}_{i}^{c}=\frac{1}{3} \sum_{i=1}^{3} \boldsymbol{x}_{i}\] The integral is approximated by \[I_{h}=\sum_{i=1}^{N} \iint_{R_{i}}(\mathcal{I} f)(x, y) d…
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Math_Numerics_and_Programming_(for_Mechanical_Engineers)/05%3A_Unit_V_-_(Numerical)_Linear_Algebra_2_-_Solution_of_Linear_Systems/25%3A_Linear_Systems/25.01%3A_Model_Problem-_n__2_Spring-Mass_System_in_EquilibriumWe can then form the scalar \(v^{\mathrm{T}} K v\) as \[\begin{aligned} v^{\mathrm{T}} K v &=v^{\mathrm{T}} \underbrace{\left(\begin{array}{cc} k_{1}+k_{2} & -k_{2} \\ -k_{2} & k_{2} \end{array}\right...We can then form the scalar \(v^{\mathrm{T}} K v\) as \[\begin{aligned} v^{\mathrm{T}} K v &=v^{\mathrm{T}} \underbrace{\left(\begin{array}{cc} k_{1}+k_{2} & -k_{2} \\ -k_{2} & k_{2} \end{array}\right)\left(\begin{array}{c} v_{1} \\ v_{2} \end{array}\right)}_{K v} \\ &=\left(\begin{array}{ll} v_{1} & v_{2} \end{array}\right) \underbrace{}_{\left.\begin{array}{cc} \left(k_{1}+k_{2}\right) v_{1} & -k_{2} v_{2} \\ -k_{2} v_{1} & k_{2} v_{2} \end{array}\right)} \\ &=v_{1}^{2}\left(k_{1}+k_{2}\right…
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Math_Numerics_and_Programming_(for_Mechanical_Engineers)/05%3A_Unit_V_-_(Numerical)_Linear_Algebra_2_-_Solution_of_Linear_Systems/28%3A_Sparse_Matrices_in_Matlab/28.02%3A_Sparse_Gaussian_EliminationIf \(A\) is a mathematically sparse matrix and we wish to solve \(A u=f\) by sparse Gaussian elimination as described in the previous chapter, we need only make sure that \(A\) is declared sparse and ...If \(A\) is a mathematically sparse matrix and we wish to solve \(A u=f\) by sparse Gaussian elimination as described in the previous chapter, we need only make sure that \(A\) is declared sparse and then write \(u=A \backslash f\). (As for the matrix vector product, \(f\) need not be declared sparse and the result \(u\) will not be sparse.) In this case the backslash does more than simply eliminate unnecessary calculations with zero operands: the backslash will permute columns (a reordering) i…
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Math_Numerics_and_Programming_(for_Mechanical_Engineers)/03%3A_Unit_III_-_Linear_Algebra_1_-_Matrices_Least_Squares_and_Regression/19%3A_Regression_-_Statistical_Inference/19.01%3A_Simplest_CaseThe parameter \(s_{\gamma, k, q}\) is related to \(\gamma\)-quantile for the \(F\)-distribution, \(g_{\gamma, k, q}\), by \[s_{\gamma, k, q}=\sqrt{k g_{\gamma, k, q}}\] Note \(g_{\gamma, k, q}\) satis...The parameter \(s_{\gamma, k, q}\) is related to \(\gamma\)-quantile for the \(F\)-distribution, \(g_{\gamma, k, q}\), by \[s_{\gamma, k, q}=\sqrt{k g_{\gamma, k, q}}\] Note \(g_{\gamma, k, q}\) satisfies \[\int_{0}^{g_{\gamma, k, q}} f_{F, k, q}(s) d s=\gamma\] where \(f_{F, k, q}\) is the probability density function of the \(F\)-distribution; we may also express \(g_{\gamma, k, q}\) in terms of the cumulative distribution function of the \(F\)-distribution as \[F_{F, k, q}\left(g_{\gamma, k,…
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Math_Numerics_and_Programming_(for_Mechanical_Engineers)/01%3A_Unit_I_-_(Numerical)_Calculus._Elementary_Programming_Concepts/06%3A_Functions_in_Matlab/6.01%3A_The_Advantage_-_Encapsulation_and_Re-UseSecond, encapsulation: a user can take advantage of the function, and the "function" it performs - from inputs to outputs without knowing how this function has been implemented or what is "inside" the...Second, encapsulation: a user can take advantage of the function, and the "function" it performs - from inputs to outputs without knowing how this function has been implemented or what is "inside" the code; from another perspective, what happens inside the function does not affect the user’s higher level objectives the output is the entire "effect" (we discuss this further below).
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Math_Numerics_and_Programming_(for_Mechanical_Engineers)/02%3A_Unit_II_-_Monte_Carlo_Methods/11%3A_Monte_Carlo-_Areas_and_VolumesWe then calculate the sample mean as \[\bar{w}_{n}=\frac{1}{n} \sum_{j=1}^{n} w_{j}\] and the sample standard deviation as \[s_{n}=\sqrt{\frac{1}{n-1} \sum_{j=1}^{n}\left(w_{j}-\bar{w}_{n}\right)^{2}}...We then calculate the sample mean as \[\bar{w}_{n}=\frac{1}{n} \sum_{j=1}^{n} w_{j}\] and the sample standard deviation as \[s_{n}=\sqrt{\frac{1}{n-1} \sum_{j=1}^{n}\left(w_{j}-\bar{w}_{n}\right)^{2}} .\] (Of course, the \(w_{j}, 1 \leq j \leq n\), are realizations of random variables \(W_{j}, 1 \leq j \leq n, \bar{w}_{n}\) is a realization of a random variable \(\bar{W}_{n}\), and \(s_{n}\) is a realization of a random variable \(S_{n}\).) Not surprisingly, \(\bar{w}_{n}\), which is simply the…
