App Note #11
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2/25/2014
ME’scope Application Note #11 Using a Tuned Absorber to Suppress Vibration INTRODUCTION NOTE: The steps in this Application Note can be duplicated using any Package that includes the VES-5000 SDM option.
plitude resonance is replaced by two resonances on either side of it. More importantly, the vibration level at the troublesome resonant frequency is greatly reduced because of the suppression of the resonance at that frequency.
Tuned Absorbers are used in many industrial applications to suppress the level of troublesome resonant vibration. A Tuned Absorber is modeled as a mass attached to the problem structure with a spring & damper, as shown in Figure 1. Constructing a physical Tuned Absorber that behaves as a mass-spring-damper system is a separate challenge. Nevertheless, ME’scope is useful for examining the effects of different absorber parameters and locations before fabricating an actual absorber. The key idea behind the use of the Tuned Absorber is that the mass and spring stiffness are chosen so that the Tuned Absorber vibrates at the same frequency as the troublesome resonance. Also, the Tuned Absorber is attached to the structure at an anti-node (high amplitude point) of the troublesome resonance, so the absorber will absorb the vibration energy instead of the resonance. The best way to examine the effect of a Tuned Absorber is to compare FRFs at an active point on the structure before and after the absorber has been attached to it.
Figure 2. Driving Point FRFs Overlaid. How It Works Adding a Tuned Absorber to a structure involves solving a sub-structuring problem. To solve this problem, three parts are needed;
The mode shapes of the unmodified structure.
The rigid body mode shape of the Tuned Absorber mass.
The spring & damper required to attach the Tuned Absorber to the structure.
Use of the Tuned Absorber will be illustrated by attaching an absorber to an SDOF system. The accuracy of the results of adding the Tuned Absorber will be checked in two different ways;
Comparing the modes of the SDOF model with the Tuned Absorber attached with the modes of a 2-DOF model.
Comparing the modal frequencies with those calculated from an analytical handbook formula.
By comparing with the modes of a 2-DOF model, and analytical frequencies from a handbook. Figure 1. Tuned Absorber & SDOF Model.
CONSTRUCTING THE SDOF MODEL
Figure 2 shows the typical effect of a Tuned Absorber on a structure. After the absorber is applied, a single high am-
In this example, an SDOF (mass-spring-damper) model will be created, and its single flexible body mode of vibration
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App Note #11
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calculated. Then a tuned absorber will be added to the SDOF model, and the answer compared with the modes of a 2-DOF model.
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Press the up X button twice, to rotate the Line by 90 degrees about the Global X-axis, as shown in Figure 3.
Building the SDOF Model
Open the App Note 11-Using a Tuned Absorber to Suppress Vibration Project from the ME’copeVES\Application Notes folder.
Execute File | New | Structure (empty) Structure window.
Enter “SDOF” in the dialog box that opens, and click on OK.
to open a new
To set up the units for the SDOF model,
Right click in the graphics area of STR: SDOF, and execute Structure Options. Figure 3. Centerline of the Model. Adding the Mass, Spring & Damper Next, the mass, spring & damper will be added to the stick model in Figure 3.
Execute FEA | FEA Properties in the STR: SDOF window.
Click on the Mass tab.
Execute Edit | Add to add a new property to the tab.
Enter 1 kg into the spreadsheet, as shown below.
Click on the Stiffness tab.
Execute Edit | Add to add a new property to the tab.
Enter 394,942 N/m into the spreadsheet.
Click on the Damping tab.
Execute Edit | Add to add a new property to the tab.
Enter 25.12 N/(m/sec) into the spreadsheet.
On the Units tab, choose the appropriate Mass, Force, and Length units shown above, and click on OK.
To begin drawing the structure model,
Right click in the graphics area of STR: SDOF, and execute Drawing Assistant.
The Drawing Assistant tabs will be displayed.
On the Substructure tab, double click on the Editable Line substructure.
A Line substructure will be added to the Structure window and additional Drawing Assistant tabs will be displayed.
On the Dimensions tab, make the following entries: Length (m) = 1
To add the FEA Mass to the stick model,
Points = 2
The Line substructure will be drawn with two end-points and a length of 1 meter along the Y-axis. Next, the Line will be changed to lie along the (vertical) Z-axis instead of the Y-axis.
Right click in the graphics area of STR: SDOF, and execute Object List | FEA Masses.
Right click in the graphics area, and execute Add Objects.
Click near the Point at the top of the Line, to add the FEA Mass element to the Point.
On the Position tab, verify that 45 Degrees and Global are selected in the Rotate section.
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App Note #11
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Right click in the graphics area, and execute Add Objects again to terminate the Add operation.
Notice also that the mass is added as a new row to the masses spreadsheet on the right side of the window.
Click on the FEA Property cell in the masses spreadsheet, and select Mass 1 from the list.
To add the FEA Spring to the stick model,
Right click in the graphics area of STR: SDOF, and execute Object List | FEA Springs.
Right click in the graphics area, and execute Add Objects.
Click near the Point at the top of the Line, and then near the Point at the bottom of the Line, to add the FEA Spring element to the Point.
Right click in the graphics area, and execute Add Objects again to terminate the Add operation.
Make the selections in the following dialog box, and click on OK.
Notice that the spring is also added as a new row in the springs spreadsheet on the right side of the window.
Click on the FEA Property cell in the springs spreadsheet, and select Spring 1 from the list.
Repeat the steps above that were used to add the FEA Spring, but this time add an FEA Damper to the stick model.
To verify that all of the FEA Objects have been correctly added to the SDOF model,
The final dialog box will report that all 6 DOFs of the selected Point have been fixed. Now the model is read for calculating the SDOF modes.
Execute FEA | Calculate FEA Modes in the STR: SDOF window.
Click on Yes in the dialog box that opens, enter parameters into the next dialog box as shown below, and click on OK.
Execute FEA | FEA Objects List in STR: SDOF
Calculating the SDOF Modes Before the modes of the SDOF model can be calculated, the ground point of the model must be fixed.
Right click in the graphics area of STR: SDOF, and execute Object List | Points.
Hold down Ctrl, and click near the bottom Point on the Line to select it.
Right click in the graphics area of STR: SDOF, and execute Animation Equations | Fix DOFs
Click on No in the following dialog box that opens
After the modes have been calculated,
Click on the New File button in the dialog that opens, and enter “SDOF mode” into the next dialog box.
A new Shape Table window will open listing the modes of the SDOF model, as shown in Figure 4.
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Enter “SDOF FRF” into the next dialog box, and click OK.
Figure 4. SDOF Modes. Animating the Mode Shapes To display the modes of the SDOF model in animation:
Right click in the spreadsheet area of SHP: SDOF Modes, and execute Tools | Create Animation Equations.
Select STR: SDOF in the dialog box that opens, and click through the remaining dialog boxes.
Right click in the spreadsheet area of SHP: SDOF Modes, and execute Tools | Animate Shapes.
As you click on each Shape button in SHP: SDOF Modes to animate its shape, notice the following,
The first two modes, with near zero frequencies, have rigid body mode shapes.
The third mode is the flexible body mode of the mass on the spring.
A Data Block window will open showing the driving Point FRF with a 100 Hz peak. This is the expected (displacement/force) driving point FRF for an SDOF.
Since the two rigid body modes are of no interest in this example, they will be deleted from the Shape Table.
Right click in the spreadsheet area of SHP: SDOF Modes, and execute Tools | Animate Shapes again to terminate the animation.
Select the first two modes, execute Edit | Delete Selected Shapes, and click on Yes to delete the modes. Figure 5. Driving Point FRF at the Mass.
Synthesizing a Driving Point FRF NOTE: A Driving Point FRF is any FRF where the Roving DOF is the same as the Reference DOF. To synthesize the driving Point FRF at the mass Point,
Right click in the spreadsheet area of SHP: SDOF Modes, and execute Tools | Synthesize FRFs.
The FRF Synthesis dialog box will open.
ADDING A TUNED ABSORBER TO THE SDOF MODEL A Tuned Absorber will be attached to the SDOF model, and we will look at its effects on the Driving Point FRF at DOF 2Z. First, the FEA Objects of the SDOF model must be hidden to prevent them from being used again during the Tuned Absorber calculation.
Right click in the graphics area of STR: SDOF, and execute Object List | FEA Masses.
Enter 200 Hz for the Ending Frequency, select 2Z for both DOFs as shown below, and click on OK.
Click on the Visible cell in the masses spreadsheet to hide the FEA mass.
Select Displacement in the next dialog box, and click on OK.
Repeat the two steps above to hide the FEA Spring and FEA Damper.
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Edit the Tuned Absorber mass Point coordinates to (0, 0, 2) in the Points spreadsheet to move it away from the SDOF mass.
NOTE: A Tuned Absorber can only be attached to a selected Point, and will only take effect in the direction of a selected shape DOF.
New Mode Shapes in Animation
To display the new mode shapes in animation:
Right click in the graphics area of STR: SDOF, and execute Object List | Points.
Hold down Ctrl, and click near the top (mass) Point on the Line to select it.
Execute the SDM | Add Tuned Absorber command. The Tuned Absorber dialog box will open.
Select the SHP: SDOF Mode, enter a 1 kg mass, 100 Hz frequency, and 2% damping into the dialog box, as shown below, and click on Calculate New Modes.
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Right click in the spreadsheet area of SHP: SDOF with Absorber, and execute Tools | Create Animation Equations.
Select STR: SDOF in the dialog box that opens, and click through the remaining dialog boxes.
Right click in the spreadsheet area of SHP: SDOF with Absorber, and execute Tools | Animate Shapes.
As you click on each Shape button in SHP: SDOF with Absorber to animate its shape, notice the following,
The Tuned Absorber mass moves in-phase with the SDOF mass at 61.8 Hz.
The Tuned Absorber mass moves out-of-phase with the SDOF mass at 161.7 Hz.
Synthesizing the Driving Point FRF at 2Z To synthesize the driving Point FRF at the mass Point,
Right click in the spreadsheet area of SHP: SDOF with Absorber, and execute Tools | Synthesize FRFs.
The FRF Synthesis dialog box will open.
Click on the New File button in the dialog box that opens, enter “SDOF with Absorber” into the next dialog box, and click OK.
The SHP: SDOF with Absorber Shape Table will open, with new modes in it, as shown in Figure 6.
Enter 200 Hz for the Ending Frequency, select 2Z for both DOFs, and click on OK.
Select Displacement in the next dialog box, and click on OK.
Select BLK: SDOF FRF in the next dialog box, and click on Add To.
The BLK: SDOF FRF Data Block window will now contain two FRF Traces.
Right click in the graphics area of BLK: SDOF FRF, execute Format | Cascade, and select 2 from the list.
The two FRFs will be displayed as shown in Figure 7.
Figure 6. SDOF with Tuned Absorber Modes. The new mode shapes include two rigid body (near zero frequency) modes, and flexible body modes at 61.8 & 161.7 Hz. Notice also that a new FEA mass, FEA Spring & FEA Damper were added to the SDOF model, and the absorber mass, spring, & damper properties were added to their respective spreadsheets in the FEA | FEA Properties window.
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The Calculate New Modes button in the Add Tuned Absorber box solved a substructuring problem. It used the mode of the unmodified SDOF system plus the rigid body mode of the Tuned Absorber, and then connecting the two SubStructures together using the Absorber spring and damper. Both methods yielded the same result. To compare the Tuned Absorber frequencies with analytically derived results, the reference book “Formulas for Natural Frequency and Mode Shape” by Robert D. Blevins, was used. It contains the following formulas (in Table 6.2, page 48) for a two mass, two spring system, 1
Figure 7. Driving Point FRFs Overlaid.
3 5 1 2 1 3 2 2
Now it is clear that the Tuned Absorber replaced the SDOF 100 Hz mode with two modes on either side of it (61.8 & 161.75 Hz.). The response of the structure at DOF 2Z and 100 Hz has been reduced by about 55 dB. This is shown by the cursor values in Figure 7.
3 5 1 2 2 3 2 2
1
If 100 Hz were the only frequency at which there was a vibration problem, then the Tuned Absorber did its job. However, a higher amplitude resonance at 62 Hz has also been created, which could cause an even greater problem than the one we solved at 100 Hz.
The springs spreadsheet reveals that the springs both have a stiffness of 394,942 N/m. Using this and a mass of 1 Kg in the above formulas gives,
Checking the Results
This is good agreement even though these formulas are for undamped natural frequencies, whereas ME’scope calculated the damped natural frequencies.
The accuracy of the results of adding the Tuned Absorber can be checked in two different ways;
Un-hide all on the FEA Objects on the model and calculate it modes
Compare the modal frequencies with those calculated from an analytical handbook formula.
To calculate the modes of the 2-DOF model,
Make sure that both FEA Masses, FEA Springs & FEA Dampers are Visible by clicking on that property in their respective spreadsheets.
Execute FEA | Calculate FEA Modes
A new Shape Table will open, containing the same two modes as obtained from the Add Tuned Absorber command (Figure 6.). You can animate these mode shapes and compare them with the Tuned Absorber modes.
1 61.82 Hz ,
2
k m
1
k m
1
2
2 2
2 163.84 Hz
EXAMPLE #2: ADDING A TUNED ABSORBER TO A PLATE STRUCTURE In this example, we will look at the effects of adding a Tuned Absorber to the center of the plate structure shown in Figure 1. The experimental mode shapes of the plate were obtained by impact testing the plate, and then curve fitting a set of FRF measurements. The FRF measurements, mode shapes, and structure model are all contained in the PLTMODES Project in the Examples subdirectory on disk. To open the PLTMODES project, Select the ME’scope\Examples subdirectory in the lower pane of the Project Panel in the ME’scope window. Double click on PLTMODES.PRJ in the middle pane of the Project Panel. The Structure, Data Block, and Shape Table windows for this Project will open in the Work Area.
An Explanation You might ask, “What’s the difference?” The Modify | Calculate Element Modes command calculated the modes (eigenvalues and eigenvectors) of 2-DOF model directly by solving an eigensolution problem.
Execute the Windows | Arrange Windows command in the ME’scope window to display them in tiled format. Execute the Draw | Animate command to display the mode shapes in animation.
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Execute Display | Point Labels to display the Point numbers. Notice that Point #13 is a nodal point (little or no motion) for all of the shapes except Shapes 2 & 3. Suppose that these are the modes of a platform and that a sensitive instrument is to be mounted at Point #13. Therefore, we would like to investigate the effects of placing a Tuned Absorber at or near Point #13 (or any other anti-nodal Point) to suppress the structural response.
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and 752 Hz modes have been replaced by a 359 Hz and a 606 Hz mode. There is also a 6th mode at 9849 Hz, which is unrealistic. This mode is the result of using a truncated modal model with the SDM algorithm.
Suppose we designed the Tuned Absorber to suppress the 423 Hz mode. The only other parameter that needs to be specified is the weight (mass) of the Tuned Absorber. Let’s choose a 1-pound mass. Execute File | Options in the Structure window, and select units of Pounds force, Pounds mass, and inches.
Figure 9. New Modes of the Plate with Absorber. Comparing FRFs
Select Point #13 on the Plate model. Execute the Modify | Add Tuned Absorber command. The Tuned Absorber dialog box will open.
Finally, to compare driving Point FRFs at DOF 13Z before and after the Tuned Absorber is added, Execute the Tools | Synthesize FRFs command in the PLTMODES.SHP window. Enter an Ending Frequency of 1500 Hz, select 13Z for both DOFs, and click on OK.
Enter the parameters shown in Figure 8. Press the Calculate New Modes button, and click on OK. A Shape Table window will open with the new modes in it.
Repeat the above step for the Shape Table with the new modes. Paste one Data Block into the other, and overlay the two Traces. Figure 10 shows the two Traces overlaid. Clearly, the vibration response at 423 Hz has been substantially reduced by the tuned absorber.
Figure 8. Adding Tuned Absorber to Plate Structure. Execute the Draw | Animate command in the Structure window, and select the new Shape Table from the Animation Source list on the Toolbar.
Figure 10. Driving Point FRFs Overlaid.
Notice that the 339, 813, and 978 Hz modes were not affected by the Tuned Absorber. On the other hand, both the 423
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