Dynamics and Vibration

Tutorial

Startup

To use Dynamics and Vibration Analysis (DVA), you must first start TK Solver. Once in TK, select Dynamics & Vibration from the Applications Menu. The DVA Menu will appear.

The menu structure mimics the organization of Blevins' book—chapters, tables, and cases. You will use the mouse to move through the menu. As you highlight a chapter folder, a description box in the middle portion of the screen provides details. If you double-click on a chapter folder, the menu opens the folder to show the tables available. Descriptions are also available for each of the tables. Again, if you double-click on a Table folder, the menu opens the folder to display the available cases.

Problem 1

As an example, you'll use the Menuing System to load in the formulas required to analyze the natural frequency of an iron steam pipe running between two buildings. This comes from Chapter 8, Table 1, Case 7 in Formulas for Natural Frequency and Mode Shape.

In the Dynamics & Vibration Analysis menu, double-click on the Chapter 8 folder. Next, double-click on the Table 1 folder. Finally, double-click on Case 7, Clamped-Clamped to make your selection. Click on the OK button to load your selection.

A dialog box will appear asking you to Load or Include. Respond with the Load option at this time. If you have already entered data into TK, a different dialog box will appear asking if you want to Reset, Merge or Include the selection. If you received this message, select Reset.

Once your selection is loaded, TK Solver displays a Variable Sheet along with icons for other sheets.

The first portion of the Variable Sheet identifies the case you loaded. The rest of the sheet displays the variables you will use in entering inputs and getting results. To move the highlight within a TK Solver sheet, you can use the mouse or, as long as the sheet is active, the arrow keys.

Move to the Input column and enter the following values for the corresponding variables:

Given: Initial mode number (iI) 1 Final mode number (i) 3 Span (L) 2 Modulus of elasticity (E) 2E11 Area Moment of Inertia (I) 2.898E-6Mass per unit length (m) 22.064 Find: Frequency of initial mode (fiI) Frequency of final mode (fi)

The Variable Sheet should look like the following:

To solve the problem, press the Solve key, F9; click on the Solve button or select Solve from the Commands Menu. The solution is displayed as shown below.

The solution can also be viewed in tabular form. Double-click on the Table Sheet icon to open the sheet. There are two tables available, Freq and Shape. Highlight the name Freq and click the right mouse button. This opens the subsheet for Freq. Subsheets are used to specify the details for the various objects in TK. In general, whenever you want to learn or specify more about an object, you open the subsheet. In this case, the subsheet tells you which lists are included in the table.

You are more interested in actually viewing the data, so just click the right mouse button again to open the Interactive Table Freq.

There is another table in this model called Shape. It reports the mode shapes for 200 locations along the beam. You can view this output by opening the Interactive Table named Shape.

A plot of the mode shapes was automatically generated when the model was solved. Press F7; click the Plot button or select Display Plot from the Commands Menu to view this plot.

Problem 2

This problem will show you how to work in different units and merge cases together to look up material properties.

You will work with a flat circular plate, clamped on the outer edge with a point load in the center. Use the Menuing System to reset TK and load Chapter 11, Table 1, Case 12.

The Variable Sheet appears as shown.

The plate is 18 inches in diameter and 0.25 inches in thickness with a 500 lbm mass in the center. The model is set up in metric units but it is easy to work in other units. The variable a represents the radius of the plate. Go to the Unit column for this variable and enter in. Do the same for the variable h. Change the unit for M to lbm. Now, move to the Input column and enter the values specified above (a = 9, h = .25, M = 500).

Entries in the Unit column are case-sensitive. If you enter LBM instead of lbm for the unit of the variable M, TK displays a "?" in front of the value of the variable. This is an indication that TK does not know the conversion to the unit specified.

Check the Unit Sheet for a complete listing of units and conversion factors.

The plate is made of high-carbon steel. To merge in a material table to return the values of the modulus of elasticity, Poisson's ratio, and density: open the Dynamics & Vibration Menu and select Chapter 16, Table 2, Case 1.

When prompted, choose the Merge option.

The Variable Sheet will be updated with the new variables appearing at the bottom.

Enter 2 as the input for the variable matnum and 20 as the input for the variable T, the temperature. Solve and TK provides a complete solution. The values of E, nu, and mu were evaluated from Chapter 16, Table 1 and then used in Chapter 11, Table 1, Case 12.

Problem 3

TK Solver includes several features you can use to enhance the Dynamics & Vibrations application to suit your own needs. To point out these features, select Dynamics & Vibration from TK's Applications Menu and load Chapter 6, Table 2, Case 3.

Choose the Reset option when prompted.

This model computes frequencies and mode shapes for a system of two unequal masses and two unequal springs.

Open the Blevins text to page 48 to see the formulas used in the model. Double-click on the Rule Sheet icon to see the TK Solver representation of the formulas. There are eight equations.

The first two rules were added to make the frequency equations easier to read and understand. The rest of the equations were taken straight from the book. Return to the Variable Sheet to see how this model works.

Enter the following inputs.

k1 = 1e6 k2 = 2.5e6 M1 = 50 M2 = 10

Here's the resulting Variable Sheet.

Solve. Given the four inputs and the two extra rules, there are actually six independent equations and six unknowns. Now, consider the possibility that the second mass, M2, might actually vary anywhere from 2 to 50 pounds. You can use TK's list solving feature to investigate how changes in M2 will affect other variables such as the frequencies, f1 and f2.

Place an L in the Status column for the variables M2, f1, and f2. This associates lists with each of these variables. Next, select the List Fill option from the Commands Menu. When the dialog box appears, fill the list M2 with values starting at 2 and going to 50 in steps of 2.

When you press F10, click the List Solve icon , or select List Solve from the Commands Menu, TK will repeatedly solve the model using each of the M2 values. The resulting f1 and f2 values will be stored in the corresponding lists.

The resulting lists are stored in the List Sheet but, to view the relationships, create a plot and table. Open the Plot Sheet by double-clicking on its icon. Name the plot Freq and then click the right mouse button to open the subsheet for Freq. Enter M2 as the X-axis list name. Cursor down and enter f1 and f2 as the Y-axis list names. (Remember, TK is case-sensitive.) To further identify the two curves, change the Symbol Count values to 5 and the Character entries to 1 and 2, respectively.

Press F7 to view the plot.

Next, open the Table Sheet. Name the table Freq and open the subsheet by clicking on the right mouse button. Enter M2, f1, and f2 as the three lists to include in the table. Click again on the right mouse button to see the Interactive Table with the inputs and outputs.

You will use TK's list solving feature often in investigating relationships between variables.

Problem 4

Suppose you want to design the mass-spring system from Problem 3 so that the f1 and f2 frequency values differ by a specified factor, say 4. You can add an equation to the Rule Sheet to take care of this. Make the Rule Sheet active. The location of this new rule does not matter, so you can add the following equation to the end of the sheet:

f2 = 4*f1

At this point, if you try solving, TK reports an error, Inconsistent. This is because you added a new constraint without removing any of the inputs on the Variable Sheet. There are seven independent equations and only six unknowns.

Make the Variable Sheet active. Make the second mass, M2, the new unknown by blanking its input value. Press F9 to solve and... huh? No solution. Both the frequency equations have at least two unknowns and TK's Direct Solver cannot solve them. This is a job for TK's Iterative Solver!

Enter a G in the Status column of the variable M2 to indicate that this variable will get an initial guess value.

Now when you solve, TK will then iterate on this value until the inconsistency is no longer noticeable. Solve to try it.

In general, you can add equations to any of the models in this application. Save these enhanced models to separate files because you should not overwrite the existing files.