App Note #7
ME’scope Application Note #7 Modeling a Single Degree-of-Freedom Mass-Spring-Damper System INTRODUCTION NOTE: Steps in this Application Note can be duplicated using any Package that includes the VES-3000 Signal Processing, VES-4000 Modal Analysis and VES-5000 SDM options.
A powerful Drawing Assistant automates creation of all of these display entities. We will use this facility and specialize its output by editing Point, Line and Surface properties in creating our Mass-Spring-Damper model.
In this Application Note the 3D model shown in Figure 1 will be created, mass, spring and damper FEA elements will be added to it, the FEA mode will be calculated and animated on the model. Finally, an FRF will be synthesized from the modal model and its physical significance will be examined.
GETTING STARTED Open ME’scope. Execute: File | Project | New to start a new Proj Execute: File | New | Structure. In the resulting Structure 1.STR window, Execute: File | Options. Click the Units tab and select: Mass Unit = Kilograms (kg) Force Unit = Newtons (N) Length Unit = Millimeters (mm) Click Save as Default and then OK. This applies appropriate and consistent dimensional units to all subsequent project entries. BUILDING THE BASE
Figure 1. Quad View of a Mass-Spring-Damper Structure. GRAPHIC DISPLAY MODEL? In ME’scope, a graphic display model is needed in order to display shapes in animation. This model can be created using the drawing tools in a Structure window. The structure model is usually much easier to construct by using several Substructures, each Substructure representing a portion of the structure. In this app note, 4 substructures will be used to construct the mass-spring-damper structure shown in Figure 1. Each substructure also contains a list of Lines drawn between defined points to generate an outline or wire-frame model. Further, it contains lists of Triangle Surfaces and Quadrilateral Surfaces, used to provide a “solid” look to displays. These surfaces also facilitate dynamic Hidden Line identification/suppression and the creation of stress/strain and acoustic Contours.
Selecting a Cube substructure using Drawing Assistant. The base is a simple rectangular plate. Clearly, we can describe this by locating 8 Points in an X, Y, Z space, drawing 12 Lines between adjacent corners and specifying these lines as the edges of 6 Quadrilateral Surfaces. Let’s use the Drawing Assistant to automate these tasks. Execute: Draw | Drawing Assistant. Scroll through the Substructure list to the Cube.
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Double-click on Cube. A default rendering of a Cube (above) is presented. We will now customize its properties into a blue-green rectangular base plate 100 mm on a side and 2 mm thick. Select the Dimensions tab (right, above) and enter 100 for Width (Y), 2 for Height (Z) and 100 for Length (X). Leave all three Points fields at the default value of 2. This will result in the eight-point geometry desired.
Setting the base Appearance. To more clearly visualize the base substructure, Execute: Display | Surfaces | Filled. Execute: Display | Hidden Lines | Dashed.
Entering Dimensions for the base. Select the Position tab and accept the default values for rotational orientation and Local Origin. Note the 0,0,0 origin is at the bottom of the base’s left-rear corner.
Editing base coordinates and properties of base Points. To view the coordinates and properties of the base Points, Execute: Edit | Object | Points.
Setting the base Position. Select the Appearance tab. Click on the Lines Color box. In the resulting Color dialog, click on black and press OK. Click on Surfaces Color and choose bluegreen. Finally, edit the text under Label from Cube to Base, identifying the substructure by name.
This view allows inspection and editing of the X, Y, Z global coordinates of the defined points. We have no need to change these, but will edit a property of all 8 Points, specifically, the Labels used to attach test measurements or finite-element computations. Double-click on the word, Label. This selects all Points for labeling and opens a dialog box for label text entry to be applied to them. (The option checkbox sets selected labels to the corresponding Point numbers.) Enter 2 and click OK, setting all eight Labels to 2. The purpose of this assignment is to attach the 2nd (of 2) modal coefficients or measured amplitudes to all 8 points.
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Execute: File | Save (from the active Window). Assign a File Name: of Base to this structure in the Save Structure As Window that opens and click Save. The base Substructure is now completely defined and saved as a Structure within the open Project file. MAKING THE MASS The mass substructure is formed and saved in exactly the same manner, using different parameter values. Execute: File | New | Structure Repeat the preceding steps to build and save the mass substructure. Note the differentiating parameter entries required: Dimensions: Width = Length = Height = 45 mm
The piston substructure starts as a cylinder. Radius Start = 0 (mm) Radius End = 9 (mm)
Local Origin Position: X = 27.5, Y = 27.5, Z = 47
Points = 2
Line Color = Black, Surface Color = medium-blue
P Angle Start = 0 (deg)
Substructure Label = Mass
P Angle End = 360 (deg)
Points Label = 1
Points = 12
These parameters produce a 45 mm blue mass cube with black lines centered (in X and Y sense) over the base. The bottom of the mass is parallel to the base and 45 mm above (Z direction) it. Save this substructure with the file name Mass.
Length = 2 (mm) Points = 2 Label = Piston
Positioning the properly dimensioned piston.
Mass substructure Points after proper entry. PRODUCING THE PISTON The damper piston is created by opening a new structure, and then opening the Drawing Assistant and selecting the Cylinder object as shown on the preceding page. Enter the following dimensional parameters (changes from default value in red):
Select the Position tab and note the Rotate control status. The Deg. field contains 45 and the Global (rather than Local) choice is enabled. Click the Y key twice, to rotate the display 90° about the Global Y-axis to the position shown above. Then enter the following Local Origin (mm) coordinates: X = 50, Y = 62.5, Z = 23.5. The piston “disk” is now equidistant between the upper base surface and the lower mass surface and parallel to both. Its axis is at the X direction center of the base and
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mass and 12.5 mm to the right of center in the Y direction.
Execute: Edit | Object | Lines.
Edit the Label text from Cylinder to Piston. To view the coordinates and properties of piston Points, Execute: Edit | Object | Points.
Adding a Line to the piston and editing its properties. Add a bold “shaft” line to the piston drawing. Select: Edit | Add Object. Editing piston Points to add mass attachment. The piston is currently defined by 26 Points, 12 equally spaced along each of two 18 mm diameter concentric circles axially separated by 2 mm and 1 at the center of each circle. We must add a 27th point where the “shaft” mates with the lower surface of the mass. Select: Edit | Add Object.
Position the cursor near the top circle center (point 14) and click. As the cursor is moved, a line from point 14 will follow it. Click on the new point, 27; this adds the new line to the list. Deselect: Edit | Add Object. Make the new line bold by clicking on its Bold column entry. This will change the text from No to Yes and the line will now be drawn with a bold stroke.
Click the cursor above the piston disk to add and roughly position the new point, 27. Refine the coordinates for this point by editing the X coord, Y coord and Z coord entries. A simple way to do this is to copy the coordinates of point 14 (X = 50, Y = 62.5, Z = 23.5) at the center of the top circle and paste them into point 27. Then change the Z coordinate to 47 to match the bottom of the mass.
Execute: Display | Surfaces | Filled.
Deselect: Edit | Add Object. Select all of the Points (double-click on Points). Execute: Draw | Group Selected Objects. This will group all points selected into a new Substructure. A Group Selected Object dialog will open. Enter Piston as the name for this Substructure. This replaces the original substructure labeled Piston with one that includes point 27. Before leaving Points, change the Label for all piston points to 1. This will be used to assign the same measurement degree-of-freedom as used for the mass to all points on the piston. Shift your focus to the Lines connecting the Points.
Viewing Surfaces of piston with all lines visible. Deactivate: Display | Hidden Lines | Invisible. Deactivate: Display | Hidden Lines | Dashed. Select (only) the 12 radial lines on the upper surface (lines 48 – 59), the 12 radial lines on the lower surface (lines 1 – 12) and the 12 vertical lines (36 – 47). Double-click on the word Hide.
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Execute: Display | Hidden Lines | Dashed.
Selecting Line and Surface colors for cup in Appearance. On the Appearance tab, set both the Line and Surface colors to light gray as shown above.
Hiding selected piston Lines. This final appearance retains contrasting lines outlining the perimeter of the piston, but eliminate the distracting lines. Execute: File | Save CREATING THE CUP Open a new Cylinder object as a new Structure and assign it the following Dimensions properties: Radius Start = 0 (mm)
Execute: Edit | Object | Points. Add a new point, 26, with coordinates X = 50, Y = 62.5 and Z = 2 directly beneath point 1. Group this new point into the Cup Substructure. Select the top-center point (point 14). Attempt to execute Edit | Delete. The following error message will appear indicating point 14 is a member of one or more Lines or Surfaces. Click the OK button to confirm the deletion.
Radius End = 10 (mm) Points = 2 P Angle Start = 0 (deg) P Angle End = 360 (deg) Points = 12
Point 14 is now deleted from the model. All Lines that connected to it and all Surfaces that contained it are also deleted.
Length = 20 (mm) Points = 2 Label = Cup Enter the Local Origin (mm) coordinates: X = 50, Y = 62.5, Z = 14.5 Rotate the figure 90° about the Global Y-axis as was done for the piston structure.
Adding and removing Points from the cup. Select all of the points.
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Execute: Draw | Group Selected Objects. Enter Cup as the Substructure name in the Group Selected Object dialog. Double-click on the word Label and enter 2 as the label for all points. Edit the cup Lines to draw the bold “shaft” to the base. Execute: Edit | Object | Lines. Add new line, 49, connecting point 1 to point 26. Make this line black in color and bold in stroke. Edit the cup Surfaces to hide all the quadrilaterals. Setting simple spring Appearance with numbered points.
Execute: Edit | Object | Surface Quads.
Radius Start = 0 (mm)
Execute: Display | Surfaces | Filled.
Radius End = 10 (mm) Execute: Display | Hidden Lines | Dashed.
Points = 2
Double-click on the word Hide, highlighting all 12 quadrilateral surfaces. In the Surface Quads dialog that opens, select Yes and click OK.
P Angle Start = 0 (deg) P Angle End = 360 (deg) Points = 12 Length = 35 (mm) Points = 2 Label = Spring On the Position tab, enter the Local Origin (mm) coordinates: X = 50, Y = 37.5, Z = 7 Rotate the figure 90° about the Global Y-axis as was done for the previous structures. On the Appearance tab:
Hiding selected cup Surfaces. The resulting cup display allows the motion of the piston it contains to be seen while maintaining the “solid” look of the Substructure. Execute: File | Save.
Check: Outline Only Uncheck: Surfaces Set the Lines Color to Orange Press Number Points
Our attention now turns to the last Substructure, the spring. WINDING THE SPRING
These Appearance settings eliminate the generation of Surfaces and draw only a minimum number of interconnecting Lines.
Open a new Cylinder object as a new Structure and assign it the following Dimensions properties:
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Execute: Edit | Object | Points.
App Note #7
Labeling spring Points before editing Z coordinates.
Spring Lines after Points edit and before Line edit.
Note that point-numbering uses the Point Label fields. Points can be also numbered by double-clicking on Label and checking Set Label to Row Number in the Points dialog. Either method yields the same result.
Clearly, the Lines need to be edited! Delete lines 12, 24 and 60 that used to close the three circles of the cylinder. Then add lines from 13 to 14 and from 25 to 27 to complete the helix. Complete the bottom of the spring by adding lines from 39 to 1 and from 1 to 2. Finish the top of the spring by adding lines from 38 to 26 and from 26 to 40.
Delete center point, 26, which is not wanted. Renumber the display as discussed above. Edit the Z coord. entries to map the cylinder into a three-turn helix of 35 mm height, as follows. Center point 1 and outer point 2 are at the bottom of the spring and will remain at Z = 7 mm. Points 3 thru 25 are on the helix and should be changed to increment by 1 mm to values of 8 thru 30. Point 26 is the top center of the spring; set its Z coordinate to 42 mm. Points 27 through 38 increment by 1 mm, from 31 to 42, respectively.
Double-click anywhere in the graphic Window to select a four-view display. (Double-click in any of the four resulting display to make it the single-view display.) Double-click on Bold and check Yes in the resulting dialog box. Press OK.
Now we need to add the two attachment Points that will touch the base and mass. These points are 45 mm apart and on the center axis of the helix. Select: Edit | Add Object. Click above the helix to roughly position point 39. Click below it to approximate the position of point 40. Deselect: Edit | Add Object. Edit the coordinate of point 39 to read: X = 50, Y = 37.5 and Z = 47. Set the coordinates of point 40 to be: X = 50, Y = 37.5 and Z = 2. Clear all Labels, then label points 1 and 40 as 2 and points 26 and 39 as 1. Select all spring points. Execute: Draw | Group Selected Objects. Execute: Edit | Object | Lines.
Four views of finished spring with Bold Lines. Execute: File | Save. Our attention now turns to assembling the total graphic model from the five component Substructures just constructed. This is a straightforward process, employing the well-accepted Windows Copy & Paste metaphor.
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ASSEMBLING THE GRAPHIC MODEL
Copying a Substructure. To Copy a Substructure previously saved as a Structure: Double-click on the file name in the upper-left pane. The selected STR file will open. Execute: Edit | Object | Substructure. Select the desired Substructure. Execute: Edit | Copy to Paste.
Pasting a Substructure into a Structure Execute: File | New | Structure or double-click on an existing Structure in the upper-left pane. The selected STR file will open. Execute: Edit | Paste to Drawing. To assemble your Mass-Spring-Damper model, paste the base into a new structure. Then paste the spring, mass, piston and cup into the same structure. Execute: File | Save and assign the name SDOF. Assembling the Structure using Copy & Paste.
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ADDING DYNAMIC EQUATIONS
Select: Edit | Add Object.
The SDOF graphic model just constructed is completely suitable for displaying experimental measurements. We will now bind dynamic equations to it, using simple elements from the ME’scope finite-element library.
Click on point 110 and then on point 111. A line will be drawn between them.
You will start by defining two additional degrees-offreedom (points) to which these elements will be attached. Specifically, we will add a point at the center of the mass and one at the center of the base.
Attaching and specifying the FE Spring. Edit the resulting table entry to define the degrees-offreedom acted upon by the FE Spring and its stiffness: Set Orient 1 to Local Z Set Orient 2 to Local Z Adding 2 new points for Finite Element attachment.
Set Stiffness to 2 N/mm (2000 N/m or 11.42 lb/in)
Open the SDOF structure, select Points as the display and add points 110 and 111. Make the following entries: Point 110: X = 50, Y = 50, Z=70.5, Label = 1 Point 111: X = 50, Y = 50, Z=1, Label = 2
By direct analogy, attach the FE Damper:
Group these new points into the SDOF Substructure. Proceed to connect an FE Spring, FE Mass and FE Damper to points 110 and 111.
Execute: Edit | Object | FE Dampers. Set Damping to 0.005 Ns/mm (5 Ns/m or 0.0286 lb sec/in)
In similar fashion, attach the FE Mass: Execute: Edit | Object | FE Masses. Set Mass to 0.5 kg (1.10 lb) Anchor the bottom of the FE Spring and FE Damper to ground. Select point 111 (exclusively). Select: Draw | Points | Fixed. Click Yes in the Points -> Fixed dialog Connecting the FE Spring, Mass and Damper elements. Execute: Edit | Object | FE Springs.
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EXAMINING MODEL RESPONSES
Resulting Shape table contains global parameters. Fixing the base end of the FE Spring and FE Damper. Calculate the modal properties of the resulting system.
The global properties results agree quite precisely with the traditional definitions:
C 5 0.07906 4 KM 4 2000 0.5
Execute: Modify | Calculate Element Modes. Click Yes in the Calculate Element Modes dialog. Enter 1 as the maximum number of modes to compute in the Compute Complex Modes dialog. Enter a name for the resulting Shape file or accept the Shape Table 1 default; click OK.
K 1 M 2
2000 10.07 Hz 0 .5
f d f n 1 2 10.03 Hz
To verify the correctness of the orthonormalized or Unit Modal Mass scaled modal magnitude, synthesize an FRF from the Shape table.
Calculating the modes defined by the Finite Elements. A Shape Table will open, summarizing the results. The graphic model can now be animated to display the mode shape.
Synthesizing a Frequency Response Function. Execute: Tools | Synthesize FRFs. In the Synthesize FRFs dialog, select: Block Size = 2048 Starting Frequency = 0 Hz Frequency Step = 0.5 Hz
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Click on 1Z in both the Roving and Reference DOF lists and click OK. Enter FRF synth in the New File dialog and click OK.
Acceleration FRF produced by double differentiation. Magnitude of synthesized Frequency Response Function. Choose the desired component type (Real, Imaginary, Magnitude, Phase, Co/Quad or Nyquist) by using the dedicated Display buttons.
Note the amplitude of the mass asymptote has converged on 2000 mm/Ns2 or 2 m/Ns2, the reciprocal of the 0.5 kg. Hence the mass is properly reflected in the measurement synthesized from the Shape properties. Execute: File | Save. Name this Data Block Acceleration FRF. Close the current displays and reopen the original FRF synth.BLK displacement FRF Data Block.
Format | Vertical Axis
Execute: Transform | Inverse FFT.
and Format | Horizontal Axis permit selecting axis type, span and related parameters. Display | Cursor buttons provide Line, Band or Peak cursor with Cursor Values annotation On or Off.
This transformation results in a time function, the Impulse Response Function (IRF). This time history shows the structure’s reaction to a sharp ( function) spike of unit area and infinitesimal duration.
From the log/log Magnitude plot spanning 0.5 to 100 Hz shown above, note the value of the horizontal stiffness asymptote approaches 1/K = 0.5 mm/N as expected. Note the amplitude at resonance. Its ratio to that of the asymptote is the Quality Factor, Q. Compare the result to the classic definition:
1 6.324 2
(4) Impulse Response from via Fourier transformation.
Hence, this single plot indicates faithful representation of the spring and damper values in the UMM scaled Modal Coefficient. Execute: File | Save. Execute: Tools | Differentiate twice to produce an acceleration FRF (mm/Ns2). Reformat your display to match the following plot.
The IRF exhibits a ringing response at the damped natural frequency, fd. It starts from zero value, reaches a maximum in the first quarter-cycle of oscillation and then decays exponentially. Position the cursor on the 9th positive peak, occurring at about 0.92 seconds. Recognize this peak as marking 9 ¼ cycles of activity.
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The Impulse Response with log Magnitude display. Change the display from Real to Magnitude format and select a logarithmic vertical axis. Use your mind’s eye to draw a straight line through the peak values. Note that the intersection of this line with time = 0 is about 100 times the value marked at 0.92 seconds. Compare these findings with classic 1% Settling SDOF system properties:
0.733 .9207 s f n
0.733 1 2
Return the display format to Real. Compute the system’s Step Response Function by integrating the IRF and dividing by the integration time interval (2 seconds). Execute: Tools | Integrate. Execute: Tools | Math, Scale by Mag & Phs
Step ResponseFunction from integrated IRF. Note that the terminal value of the Step Response is 1/K (0.5 mm/N), as theory predicts.
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