Using Microsoft Excel Built-in Functions and Matrix Operations. EGN 1006 Introduction to the Engineering Profession

Transcription

1 Using Microsoft Ecel Built-in Functions and Matri Operations EGN 006 Introduction to the Engineering Profession

2 Ecel Embedded Functions Ecel has a wide variety of Built-in Functions: Mathematical Financial Statistical Logical Database Conversion User-defined *** EGN 006 Introduction to the Engineering Profession

3 EGN 006 Introduction to the Engineering Profession Ecel Embedded Functions These functions allow us to : Perform more comple operations Combine data for parametric calculations Manipulate the contents of the datasheet Search for values in the datasheet

4 Ecel Embedded Functions t V Eample: Open Ecel and start from an empty datasheet and enter the following data: EGN 006 Introduction to the Engineering Profession

5 EGN 006 Introduction to the Engineering Profession Ecel Embedded Functions Enter the following formula for an oscillating particle position at any time t with a frequency ω=0.75 ( t ) = sin( ω t ) By clicking in the f button or entering: =sin(0.75*a) on cell B t V

6 Ecel Embedded Functions Plot the position (t) as: t EGN 006 Introduction to the Engineering Profession

7 EGN 006 Introduction to the Engineering Profession Ecel Embedded Functions Enter the following formula for the particle velocity at any time t with a frequency ω=0.75 V ( t ) = ω cos( ω t ) By clicking in the f button or entering: =0.75*cos(0.75*A) on cell C t V

8 Ecel Embedded Functions Plot the velocity V(t) as: V t V EGN 006 Introduction to the Engineering Profession

9 EGN 006 Introduction to the Engineering Profession Ecel Embedded Functions We can also perform multi-dimensional calculations: Assume that the temperature of the surface of an electronic board is given by the function: T (, y) = e 0.( + y) [ sin( ) cos( y ) ]

10 Ecel Embedded Functions Enter the following data for the position (,y): y EGN 006 Introduction to the Engineering Profession

11 EGN 006 Introduction to the Engineering Profession Ecel Embedded Functions Use the formula for the surface temperature on cell B as: = EXP(0.*(B$+$A))*SIN(B$-)*COS($A-) y

12 Ecel Embedded Functions Use a surface graph to plot T(,y) as: EGN 006 Introduction to the Engineering Profession

13 EGN 006 Introduction to the Engineering Profession Matri Operations A Matri is a collection of independent values ordered in a row-column format: 0 0 (5) The above Matri is said to be (5) or by 5 because it has rows and 5 columns. The first number is the first dimension or the number of rows. The second number is the second dimension or the number of columns.

14 EGN 006 Introduction to the Engineering Profession Matri Operations When a Matri has just one () column (N) is said to be a vector. The following is a () vector: 0 ( ) Matrices are very useful in the solution of systems of multiple linear equations arising from many problems: Electricity, Heat Transfer, Fluid Mechanics, Optics, etc.

15 Matri Operations The fundamental Matri operations are:. Addition and Subtraction. Multiplication by a Scalar. Transpose. Multiplication of Two Matrices 5. Determinant 6. Inversion EGN 006 Introduction to the Engineering Profession

16 EGN 006 Introduction to the Engineering Profession Matri Operations. Addition and Subtraction: To add or subtract two matrices they both must have the same eact dimensions. The result contains the addition or subtraction of corresponding elements. In Ecel, simply enter the matrices, add or subtract the first element of each matri into a new cell, and copy the cell to form the new matri: [A] () [C]=[A]+[B] - 6 () [B] () - -

17 EGN 006 Introduction to the Engineering Profession Matri Operations. Multiplication by a Scalar: The resulting matri of a scalar-matri multiplication has the same dimensions as the original matri with all its elements multiplied by the scalar. In Ecel, simply enter the Matri and the Scalar, multiply the first element of the matri times the scalar (with absolute address) into a new cell, and copy the cell to form the new matri: Scalar 5 [A] - [C]=Scalar [A] () - ()

18 EGN 006 Introduction to the Engineering Profession Matri Operations. Transpose: The transpose of a matri positions the rows on the column locations and the columns on the row locations. The result is a Matri with the opposite dimensions as the original one ( 5 5). In Ecel, use the built- in-function =transpose(). Remember to use [ctrlshift-enter] when entering the results because the =transpose() function will occupy multiple cells: [A] - 5 transpose[a] 0 (5) - (5)

19 EGN 006 Introduction to the Engineering Profession Matri Operations. Multiplication of two Matrices: To multiply two matrices the number of columns of the first matri must equal the number of rows of the second. The resulting matri will have as many rows as the first and as many columns as the second. In Ecel, use the built-in-function =mmult(,). Remember to use [ctrl-shift-enter] when entering the results because the =mmult(,) function will occupy multiple cells: [A] - () [C]=[A][B] [B] - - () 0 ()

20 Matri Operations Another Multiplication eample: [A] [c]=[a][b] 0 (55) - 0 (5) [b] (5) 0 EGN 006 Introduction to the Engineering Profession

21 EGN 006 Introduction to the Engineering Profession Matri Operations 5. Determinant: Only the determinants of square matrices can be obtained. The determinant of a singular matri is zero (0). In Ecel, use the builtin-function =mdeterm(). [A] - - () determinant[ A] -0

22 EGN 006 Introduction to the Engineering Profession Matri Operations 6. Inversion: Only the inverse of square matrices can be obtained. The inverse of a matri has the same dimensions as the original one. In Ecel, use the built-in-function =minverse(). Remember to use [ctrl-shift-enter] when entering the results because the =minverse() function will occupy multiple cells: [A] inverse[a] (55) - 0 (55)

23 Introduction to the Engineering Profession Solution of systems of multiple linear equations: If a system of linear equations is well-posed (same number of equations as unknowns and no equation is the combination of one or more of the others) a Matri-Vector Analogy can be found to facilitate the solution of the system. Given the Matri Operations EGN 006 Introduction to the Engineering Profession facilitate the solution of the system. Given the following system of five (5) equations and five (5) unknowns: = + + = = + = + = + +

24 Introduction to the Engineering Profession Where the unknowns 5 can represent electric intensity, energy, temperature, flow velocities, etc., depending on the application. An analog Matri-Vector system can be derived as: Matri Operations EGN 006 Introduction to the Engineering Profession = 0 5 Or simplified as: [ ] } { } { b A =

25 EGN 006 Introduction to the Engineering Profession Matri Operations The solution of the system is given by: In Ecel: [ A] { b} { } = [A] - - {b} (55) (5) inverse[a] {}=inv[a]{b}.8 (55) (5)