CE-300 Mathcad Tutorial

Transcription

1 CE-00 Mathcad Tutorial The purposes of this tutorial are: (1) to get you acquainted with the Mathcad procedures and synta you will use to solve typical problems in CE-00 and (2) to demonstrate how to set up a reusable Mathcad worksheet -- one that can be easily adapted for problems with different loads, dimensions, etc. To use the tutorial, simply read the instructional material (in blue tet) and, when prompted, type the red tet eactly as shown. The symbol >>> indicates the area where you should click and then enter tet. Purple tet in brackets is used to indicate the use of special keyboard keys, such as [Ctrl] for the Control key, [space] for the space bar, and [Enter] for the Enter key. Eample Problem concrete block has the dimensions shown below. Two ropes are attached to the hook at Point, and forces F 1 and F 2 are applied to the ropes as indicated. The density of concrete is γ = 150 pounds per cubic foot. (a) How much does the block weigh? (b) If F 1 =1100 lb and F 2 =1200 lb, what is the resultant of the two forces? (c) If the block is lifted and suspended from the two ropes, what are the magnitudes of the forces F 1 and F 2? In other words, what values of F 1 and F 2 are required for the two forces and the weight of the block to be in equilibrium? ssume that the directions of the two forces remain unchanged. y F F O 2 Define Variables: 2 16 To make your worksheet as general as possible, define a variable for each piece of information given in the problem statement, then assign each of these variables its appropriate value. First consider the length of the block, which we will call L b, given in the problem statement as 2". To define this variable, click your mouse in the space below to create an "insertion point" (a small red cross), then type L.b:2*in and hit [Enter]. Click and type here >>> L b := 2in

2 (1) When you type the period (.) Mathcad displays all subsequent characters as subscripts. (2) When you type the colon (:) Mathcad displays the assignment operator (:=). This operator is used to assign a value to a variable or to define a mathematical epression. () You can assign units to a number by multiplying (*) the number by the name of the unit (e.g., in, ft, lbf, N, sec). Use of units in Mathcad calculations is not required; however, this is one of the most powerful features of the program, so you'll want to use it whenever possible. See the Help topic "Overview of Units and Dimensions" for more information. Now define variables for the remaining given information by clicking in the space at right and typing the epressions highlighted in red: height of the block h.b:2*in >>> h b := 2in width of the block w.b:16*in >>> w b := 16in Force F 1 F.1:1100*lbf >>> F 1 := 1100lbf Force F 2 F.2:1200*lbf >>> F 2 := 1200lbf To define the density of concrete, you'll need to display the Greek Symbol palette. Click the αβ button on the toolbar above, or click the View menu, then select Toolbars, then Greek. With the Greek palette displayed, click the Gamma symbol (γ) and then complete the definition of the variable as follows: density of concrete γ:150*lbf/ft^ >>> γ := 150 lbf ft (1) In Mathcad, the abbreviation lbf refers to pounds-force. If you use the abbreviation lb, Mathcad interprets it as pounds-mass. Since we will never use pounds-mass in this course, always use lbf for units of force in the U.S. Customary System. (2) Use the carat (^) to indicate eponentiation. () Notice that our variables use mied units--the dimensions of the block are specified in inches, while the density of concrete is defined in terms of feet. Not a problem! Mathcad will take care of all the unit conversions for us automatically. This saves a lot of time and eliminates errors in unit conversions. () If you make a mistake and need to edit one of your epressions, consult the Help topic, "Editing an Epression." (a) Calculate the Weight of the Concrete Block The weight of the block is equal to the density of concrete multiplied by the three dimensions of the block. Let's define a new variable W b as the product of these four terms. gain, use the Greek Symbols palette for Gamma (γ), the density of concrete. weight of block W.b:γ*L.b*h.b*w.b >>> W b := γl b h b w b

3 Note that mathematical epressions are case-sensitive. Mathcad recognizes w b and W b as two different variables. t this point, our calculation is complete, but we don't yet know what the answer is. To evaluate the epression for the weight of the block, type the variable name followed by the equals sign (=), as follows: weight of block W.b= >>> W b = lbf Mathcad calculates the result as N. This is the correct answer, of course, but it is given in SI (metric) units, while we are using U.S. Customary units in this problem. To switch to the correct units, first click on the answer, W b = N to select it. You'll know this math region is selected when you see a black line surrounding it. Notice that, to the immediate right of the N, there is a small solid black rectangle. Click this rectangle, type lbf, and then hit [Enter]. Note that Mathcad automatically converts your answer from Newtons to pounds-force. The result should be lbf. Now let's create the tet heading for Part (b) of the eample problem. To create a heading for this section of your worksheet, click in the open space below to create an insertion point, then type "(b) Resultant Force Click and type here >>> (b) Resultant Force (1) Use of the double quotation mark at the start of this line of tet tells Mathcad that you are creating a tet region, rather than a math region. In some cases, you don't actually need to use the quotation mark. nytime you put a space in a line of tet, Mathcad automatically assumes it is a tet region. But use of the quotation mark is recommended to avoid confusion. Note that anytime you type a single word, without the quotation mark, Mathcad assumes that word is a variable and creates a math region, rather than a tet region. (2) You can change the size, font, and format of tet regions as you would in Microsoft Word. Just highlight the tet you want to change, and click the appropriate button(s) on the toolbar above. The appearance of the heading you just created should match the previous ones in this worksheet--so highlight the entire line of tet by clicking and dragging the mouse across it, then click the B button on the toolbar to make it bold and change the font to 12-point. () To move a tet region or a mathematical epression to a new location, first click the epression to select it. The selected region will be outlined with a solid black line. In this mode you can move the region by putting the mouse pointer over the outer boundary of the region and (once the mouse pointer changes to a hand) dragging it to the desired location. Try it with the tet region you just created above. () To move or delete several regions simultaneously, click a blank area of the worksheet, then drag your mouse over the regions you want to select. When this "multiple selection" technique is used, all of the selected regions will be outlined with a dotted black line. Now you can move them all by dragging the mouse, or delete them all by pressing the [Delete] key. To solve Part (b), we'll first need to resolve F 1 and F 2 into their - and y-components. To do this

4 accurately, it's a good idea to draw a picture of each force first. y y F 1 F 1y F 2y F 2 0 O F 1 These pictures were sketched in PowerPoint and then copy-pasted directly into the Mathcad worksheet. Once in the worksheet, they can be moved (by dragging the picture boundary) and re-sized (by dragging the small black rectangular "handles" on the lower and right-hand edges) as required. Now let's define the two components of each force. -component of F 1 F.1:/5[space]*F.1 >>> F 1 := 5 F 1 y-component of F 1 F.1y:/5[space]*F.1 >>> F 1y := 5 F 1 F 2 -component of F 2 F.2:F.2*cos(0*deg) >>> F 2 := F 2 cos0deg ( ) y-component of F 2 F.2y:F.2*sin(0*deg) >>> F 2y := F 2 sin0deg ( ) (1) Your might have noticed that, as you create any mathematical epression in Mathcad, two blue editing lines (one vertical and one horizontal) follow along as you type. These editing lines indicate which term or terms Mathcad will perform the net operation on. In the first two epressions above, hitting the spacebar epands the editing lines to cover not just the 5, but the entire fraction /5. s a result, when you type the multiplication symbol (*), the entire fraction is multiplied by F 1, not just the 5. Had you omitted the [space], the result would be: You could accomplish the same objective by using parentheses, but the space bar is much quicker and results in a cleaner looking equation. Note that, if necessary, you can hit the space bar multiple times to epand the editing lines to successively larger chunks of the epression. (2) When using trigonometric functions like sine and cosine, the angle must be placed inside parentheses, as you did above. If you do not specify degrees (deg) as the units, Mathcad will assume that the angle is in radians. Now determine the resultant force by adding together the - and y-components, applying the Pythagorean Theorem to calculate the magnitude of the resultant, and using the inverse tangent function (atan) to calculate the direction. We'll follow each variable epression with an evaluation of its value, so we can do a "reality check" on the intermediate calculations.

5 y F Ry F R α F R -component of F R (right is positive) F.R:-F.1+F.2 >>> F R := F 1 + F 2 evaluate F.R= >>> F R = 159.2lbf (In the answer above, don't forget to click on the black rectangle, type lbf, and hit [Enter] to convert Newtons to pounds-force. The result should be lbf. Note that F 1 is preceded by a negative sign, because it points to the left.) y-component of F R (up is positive) F.Ry:F.1y+F.2y >>> F.Ry= >>> F Ry := F 1y + F 2y F Ry = lbf 2 2 magnitude of resultant F.R:\F.R^2[space]+F.Ry^2 >>> F R := F R + F Ry F.R= >>> F R = lbf F Ry direction of resultant α:atan(f.ry/f.r) >>> α := atan F R α= >>> α = deg (1) In the epression for magnitude, note that the backslash (\) creates a square root sign. (2) Use the Greek Symbols palette to enter the angle alpha (α). () The final answer is given below. Highlight the answer by right-clicking the tet region, then select Properties from the pop-up menu. Click Highlight Region, and then choose the yellow color. F R =1270 lb 82.8 degrees thru (c) Calculate Magnitudes of the Forces F 1 and F 2 for Equilibrium To complete this requirement, we will use a Mathcad solve block to solve a system of equations simultaneously. solve block always consists of four major parts: (1) series of statements assigning "guess values" to all of the variables we will be solving for.

6 (2) Given statement, which marks the start of the solve block. () The actual equations to be solved. These equations must be written using a special equals sign, called the "Boolean Equal To" and created by typing [Ctrl]=. On the screen, this equals sign appears bolder than the standard equals sign that we used to evaluate the value of an epression. () The Find statement, which calculates the values of the variables. Guess Values: guess value for F 1 F.1:1*lbf >>> F 1 := 1lbf guess value for F 2 F.2:1*lbf >>> F 2 := 1lbf Given Statement: Equations: Sum of forces in the -direction Sum of forces in the y-direction Find Statement: Given >>> Given -F.1*/5+F.2*cos(0*deg)[Ctrl]=0 >>> F 1 5 F.1*/5+F.2*cos(0*deg)-W.b[Ctrl]=0 >>> F 1 5 nswers Find(F.1,F.2)= >>> Find F 1, F 2 + F 2 cos0deg ( ) = 0 + F 2 sin0deg ( ) W b = 0 ( ) = (1) Ideally, the "guess values" should be roughly equal to the values that we epect the final answers to be. In practice, as you can see above, it is almost always OK to simply assign each variable a "guess value" of 1, with the appropriate units. (2) In creating the special equals sign in the equations, be sure to hold down the [Ctrl] key as you type =. () fter typing the Find statement, be sure to change the units to pounds-force by clicking on the small black rectangle to the right of the answer, typing lbf, and hitting [Enter]. () Note that the variable W b, representing the weight of the block, did not have to be assigned a value in the solve block. Mathcad remembered its value of 100 lbf, which was calculated previously. In Mathcad, a variable retains its value for all subsequent calculations, unless it is eplicitly redefined with a new assignment statement. Final Results: lbf F 1 =118 lb thru F 2 =1218 lb 0 degrees thru

7 The basic Mathcad procedures, synta, and techniques demonstrated in this tutorial will allow you to successfully complete most of the computational requirements in CE-00. few advanced Mathcad applications will be demonstrated later in the course, in conjunction with their specific applications.