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11.23 A particle of mass m is attached to one end of a spring and allowed to come to rest in an equilibrium position. If an external force, f(t), is then applied to the particle, its motion is described by the 2 y equation m d 2 + a dy + k y = f (t ), y '(0) = 0, y(0) = 0 where a is a damping constant and k is the spring s dt dt stiffness constant. Assuming m = 1, a = 2, k = 1, and f (t ) = e t, describe the motion of the spring. Then determine the motion of the spring if an impulse of 1 lb-sec is applied after 1 sec. Plot both graphs on one set of axes.
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m = 1; a = 2; k = 1; equation = m y''[t]+ a y'[t]+ k y[t] Exp[ t]; (* without impulse *) temp = LaplaceTransform[equation, t, s]; temp2 = Solve[temp, LaplaceTransform[y[t], t, s]] /. {y' 0, y 0}; temp3 = temp2[[1, 1, 2]]; Print["Solution Without Impulse (dashed)"] f1[t_]= InverseLaplaceTransform[temp3, s, t] g1 = Plot[f1[t], {t, 0, 10}, PlotStyle Dashing[{.01}]; equation = m y''[t]+ a y'[t]+ k y[t] Exp[ t]+ DiracDelta[t 1] (* with impulse *) temp = LaplaceTransform[equation, t, s]; temp2 = Solve[temp, LaplaceTransform[y[t], t, s]] /. {y' 0, y 0}; temp3 = temp2[[1, 1, 2]]; Print["Solution With Impulse (solid)"] f2[t_] = InverseLaplaceTransform[temp3, s, t] g2 = Plot[f2[t], {t, 0, 10}]; Solution Without Impulse (dashed) 1 t t2 2 Solution With Impulse (solid) 1 t t2 + 1 t( 1 + t) UnitStep[ 1 + t] 2 Show[g1, g2 ,PlotRange All]
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12.1 Vectors and Matrices
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Vectors and matrices are represented as lists ( 3) in Mathematica. A vector is a simple list and a matrix is a list of vectors. The elements of a vector or a matrix may be entered manually as a list or, more conveniently, by the use of built-in commands. Vectors
Table[expression, {i, n}] constructs an n-dimensional vector whose elements are the values of expression for i = 1, 2, 3,..., n. Array[f, n] generates an n-dimensional vector whose elements are f, f, ... , f[n]. f is a function of one variable.
Matrices
Table[expression, {i, m}, {j, n}] constructs an m n matrix whose elements are the values of expression for (i, j) = (1, 1) ,...,(m,n). Array[f, {m, n}] generates an m n matrix whose elements are f[1,1], ... , f[m,n]. f is a function of two variables. DiagonalMatrix[list] creates a diagonal matrix whose diagonal entries are the elements of the one-dimensional array list. IdentityMatrix[n] creates an n n identity matrix.
Although matrices are represented as lists, they may be viewed as matrices by using the MatrixForm command.
MatrixForm[list] prints the elements of the two-dimensional array list in a rectangular arrangement enclosed by brackets. If list is a simple (one-dimensional) array, MatrixForm prints it as a column vector, i.e., an n 1 matrix.
Using //MatrixForm to the right of list is equivalent to MatrixForm[list] and is a bit more convenient. Care must be taken, however, not to use //MatrixForm in the definition of the matrix. This command is for display purposes only. (See Problem 12.3.)
EXAMPLE 1
m = {{1, 1}, {1, 2}}; m //MatrixForm 1 1 1 2
Linear Algebra
Additionally, a matrix can be introduced via the menu Insert Table/Matrix New . . . (On a PC, a matrix can also be inserted by right-clicking the mouse and selecting Insert Table/Matrix . . .)
This produces a grid as shown. Once the grid has been set up, you can conveniently enter the numbers, using the [TAB] key to go from cell to cell. Options for filling with 0s and 1s are particularly convenient for large, sparse matrices.
EXAMPLE 2 A matrix is a list in Mathematica. Even if it is input via the Create Table/Matrix menu, it is repre-
sented internally as a list of depth 2. 1 2 3 m = 4 5 6 7 8 9 {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}
EXAMPLE 3 A vector can be represented as a simple list or as an n 1 matrix. Either way, MatrixForm will print it as a column vector.