u= R + jX , (3.35)
and ˜F and ˜u are the amplitude of the complex force and velocity, respectively.
Note there is a small error in the equation for power in Kinsler et al. [102, p. 276]. See Section D.1.1 for more details.
3.3.7.2 ANSYS Workbench
A model of the previously described system will be created in ANSYS Workbench of a duct with a hemispherical free-field that simulates a plane baffle, as shown in Figure 3.10. Note the change in the location of the origin of the coordinate system between the theoretical model shown in Figure 3.9 and Figure 3.10. It is convenient in the theoretical model to define the piston at z = 0, whereas when creating a geometric model of a sphere in ANSYS, the default location is at the origin and hence the location of the piston is placed at z = −L.
Schematic of the finite element model that will be created in ANSYS Workbench of a circular duct radiating into a plane baffle, which is modeled withFLUID130 infinite acoustic elements on the surface of a hemi-spherical acoustic volume.
Instructions
The ANSYS Workbench archive model radiation open duct.wbpz , which contains the.wbpj project file, is included with this book. The following
in-150 3. Ducts structions provide an overview of the steps required to modify the previous model.
• Start ANSYS Workbench and load the project driven duct pres dist.
wbpj.
• To ensure that the original model is not corrupted, save the project by selectingFile | Save As and type a filename such as radiation open duct.
wbpj.
• Start DesignModeler.
• For this model the starting and finishing locations of the duct will be reversed compared to the previous model, so that the duct finishes at the origin, and a hemisphere that represents the baffled acoustic region is centered at the origin and extends into the +Z axis, as shown in Figure 3.10. In the Tree Outline window, click on the branch for Extrude1. In the row Direction, change it to Reversed and click the Generate icon. In the Tree Outline window click on the icon for XYPlane to show the XY axes and notice that the duct now extends in the −Z direction.
• As the orientation of the duct has been altered, the definitions for theNamed Selections of the inlet and outlets are incorrect and must be fixed. In the Tree Outline window, right-click on the entry for NS_outlet and left-click onEdit Selections. Make sure the Faces selection filter is active, then hold down the <Ctrl> key and left-click on the four faces on the XY plane at the exit of the duct. Left-click on the Apply button and then left-click the Generate icon.
• Repeat these steps to redefine theNamed Selection for NS_inlet as the 4 faces at the far end of the duct at z = −3m.
3.3. Example of a Circular Duct 151
• The next step will be to create a hemisphere to represent the free-field acoustic region which will have its origin at the exit of the duct. In the menu bar, left-click onCreate | Primitives | Sphere. Change the Operation to Add Frozen. Keep the coordinates of the origin as (0,0,0). Make the radius a parameter by clicking on the box next toFD6 and name it sphere_r. Press theGenerate icon. The radius will be set as a function of the wavelength in theParameter Set.
• A second smaller sphere will be created at the exit of the duct that will be used as a transition region for the acoustic finite elements. The finite element mesh in the duct has a swept or “mapped mesh,” and the large spherical region for the free-field will have an inflation mesh. This smaller spherical region at the exit of the duct enables one to have a transition zone between the two mesh regions. Create another sphere at the same location. In the rowFD6, Radius (>0), enter the value 0.05 to make it easier to see the two spheres. Define the radius as a parameter calledsphere_duct_r, which will be set as the radius of the main duct in theParameter Set.
• Click onCreate | Boolean, and change the Operation to Intersect. For the Tool Bodies select the two spheres that were just created. Change Preserve Tool Bodies? to Yes, Sliced. Change the Intersect Result to Union of All Intersections.
• Insert Create | Slice three times to slice all bodies along the XYPlane, ZXPlane, YZPlane.
152 3. Ducts
• The next step is to delete the 8 unwanted bodies of the sphere that are in the −Z axis. Insert a Create | Body Operation, and change the row Type to Delete. Change the Select Mode to Box Select and select the 8 bodies that are the 4 large and 4 small one-eighth spherical bodies comprising the hemisphere that overlaps the duct. Click the Generate icon.
• In theOutline window, in the Parts branch, select all the solid bodies, and right-click and select Form New Part. There should be 1 Part, 12 Bodies listed in the Tree Outline.
• Click on theSave Project icon.
That completes the creation of the solid model. There are two new param-eters that were created and need to be assigned values.
• In theWorkbench Project window, double-click on the Parameter Set box.
• The first parameter that will be defined is the radius of the outer hemisphere.
In the row forP12 sphere_r, in the Value cell, enter 0.7.
• The next parameter that will be defined is the radius of the duct. Click on the rowP13 sphere_duct_r. In the window Properties of Outline B9:P14, click in the box for Expression and type P9/2, which will define it as the duct diameter divided by 2.
3.3. Example of a Circular Duct 153
• Click on theRefresh Project icon, which will update the model with the dimensions that have just been defined.
The next step is to set up the harmonic analysis.
• Start Workbench Mechanical.
• Under the Mesh branch, check that the Sweep Method is applied to the 4 bodies for the duct.
• Check that theEdge Sizing is applied to the Geometry for the 16 edges on the ends of the cylinder. The Number of Divisions should be 6 and the Behavior should be Hard.
• Right-click on the Mesh branch and left-click on Insert | Sizing. Select the 4 large bodies that comprise the hemispherical free-field and in the row Geometry click the Apply button. Click in the box next to Element Size to define it as a parameter (indicated by an icon with the letter P in a box) which will be linked in theParameter Set definitions. Change the Behavior to Hard.
• In theParameter Set, alter the expression for the P14 Body Sizing Element Size to P8*1[m], and click on Refresh Project. This will define the element size as a variable that can be altered depending on the desired number of elements per wavelength, excitation frequency, and speed of sound. Initially this will equate to an element size of 0.142 m.
• Under theMesh branch, right-click Insert | Sizing. Select the 5 edges that
154 3. Ducts comprise the hemisphere on the end of the duct. The edges on the end of the cylinder already have edge divisions defined. Change theType to Number of Divisions, Number of Divisions to 6, and Behavior to Hard.
• The next step is to create an inflation mesh around the region of the exit of the duct. Make sure the Body selection filter is active and right-click on the Mesh branch and select Insert | Inflation. Select the 4 large bodies that comprise the hemisphere for the free-field, and right-click and select Hide All Other Bodies. In the Details of "Inflation" - Inflation window, in theGeometry row select the 4 large bodies for the hemispherical free-field. In the row forBoundary, select the 4 faces for the outer surface of the small hemisphere. TheInflation Option should be Smooth Transition, the Transition Ratio as 0.8, Maximum Layers as 5, Growth Rate as 1.2, and Inflation Algorithm as Pre.
• Right-click on theMesh branch and select Insert | Method. For the Geometry select the 4 small bodies on the end of the duct that comprise a hemisphere.
Change theMethod to Automatic.
• Right-click on theMesh branch and select Insert | Sizing. Select the same 4 bodies as above. Change the Type to Element Size, and in the row for Element Size enter 8e-3.
3.3. Example of a Circular Duct 155
• The next step is to apply a force to the piston face. There are several ways this can be achieved. One way is to apply a force to the face NS inlet and couple all the nodal displacements in the Z axis by inserting an object Conditions | Coupling, which is a beta feature in ANSYS Release 14.5.
Instead of using the beta feature, the way it will be done for this example is to apply a force to the vertex on the piston face and use aCommands (APDL) object to couple the displacement of nodes. This method is also instructive to see how components can be selected in ANSYS Workbench using APDL code. Right-click on theHarmonic Response (A5) branch and select Insert
| Force. Select the vertex on the axis of the duct on the inlet face. Set the Magnitude to 1.e-003. Click in the cell next to Direction and then click on an edge that is along the axis of the duct, so that the red arrow that indicates the direction of the force is pointing into the duct.
• Check that the Acoustic Body is defined for the 4 bodies comprising the duct. TheAcoustic-Structural Coupled Body Options should be Coupled With Unsymmetric Algorithm. The Mass Density should be 1.21, and the Sound Speed should be 343. Note that it is also possible to use the option Coupled With Symmetric Algorithm, provided that all Acoustic Body ob-jects in the model are set to the optionCoupled With Symmetric Algorithm.
• Insert anotherAcoustic Body, and select the 8 bodies on the end of the duct that model the hemispherical free-field region using the Box selection filter.
Change the Mass Density to 1.21, and the Sound Speed to 343. Leave the Acoustic-Structural Coupled Body Options as Uncoupled, and Perfectly Matched Layers (PML) should be Off.
• The next step is to create the absorbing conditions on the exterior of the
156 3. Ducts hemisphere to simulate the free-field. In the ACT Acoustic Extensions toolbar, select Boundary Conditions | Absorbing Elements (Exterior to Enclosure). In the Geometry row, select the 4 faces on the exterior of the hemisphere and then click the Apply button. In the row Radius of Enclosure type the value 0.7, which is the radius of the hemisphere and the same value of the parametersphere_r. This step will lay FLUID130 elements on the surface of the hemisphere that provide the acoustic absorption of the outgoing waves.
• The next step is to enable the fluid–structure interaction at the inlet to the duct so that the displacement degrees of freedom become active and hence the acoustic particle velocity can be determined. In the ACT Acoustics toolbar click on Boundary Conditions | FSI Interface. In the window Details of "Acoustic FSI Interface" change the row Scoping Method to Named Selection and Named Selection to NS_inlet.
• As mentioned earlier, a force is applied to a vertex belonging to the pis-ton, and all the nodes belonging to the piston will have their displacement degree of freedom along the Z axis coupled. The coupling of the nodal dis-placements can be achieved using an APDL code snippet. Right-click on Harmonic Response (A5) and select Insert | Commands.
• Click on the branch forCommands (APDL) and enter the following commands which will couple all the nodes associated with NS_INLET in the UZ axis, which will essentially create a rigid piston face, and motion in the UX and UY axes will be disabled.
1 ! Select all the nodes associated with NS_INLET
2 CMSEL ,S, NS_INLET ,NODE
3 ! Couple all the UZ DOFs to create a rigid piston face.
4 ! The following command will use the NEXT available set number
5 ! for the coupling equations.
6 CP ,NEXT ,UZ ,ALL
7 D,ALL ,UX ,0
8 D,ALL ,UY ,0
9 ALLS
3.3. Example of a Circular Duct 157
• One of the results that we want to obtain is the displacement versus fre-quency of the piston due to the applied force. Once the displacement result is calculated, it is possible to determine the mechanical impedance, which will be compared with the theoretical value. Right-click on Solution (A6) and selectInsert | Frequency Response | Deformation. Click in the row for Geometry and select the vertex on the axis of the cylinder on the inlet face for the piston. Change the row Definition | Orientation to Z Axis.
• Although we have modeled the entire system, we will only analyze 1/4 of the model so that we can reduce the number of nodes and elements in the model. On the triad in the lower right of the screen, click on the -Z axis to change the view of the model. Change the selection filter to Bodies, and Select Mode to Box Select. Select all the bodies in the +X and +Y region, so that only 1/4 of the model is selected. Right-click and select Suppress All Other Bodies.
At a later stage, if you wish to confirm that the results obtained using the 1/4 model are the same as the full model, you can select Unsuppress All Bodies and re-run the analyses.
• The model should comprise 3 bodies.
158 3. Ducts
• Right-click on theMesh branch and select Generate Mesh. Once the meshing has completed, in the Statistic branch in the Details of "Mesh", there will be about 1555 nodes, and 3092 elements (do not be concerned if the statistics of your mesh are not exactly these values). The mesh around the outlet of the duct should have a fine mesh and there should a transition layer of fine to coarse elements.
• Click on Analysis Settings and change the Range Minimum to 0, Range Maximum to 200, and the Solution Intervals to 100. This will result in harmonic analyses conducted at frequency increments ∆f of
∆f = (Range Maximum) − (Range Minimum)
(Solution Intervals) . (3.36) Hence, these settings will provide solutions from 2 Hz to 200 Hz in 2 Hz increments. Note that an analysis at the Range Minimum frequency is not conducted. Under theOutput Controls, make sure everything is set to Yes.
In theAnalysis Data Management, make sure that Save MAPDL db is set to Yes.
• The tree forHarmonic Response (A5) should look like the following figure, and the other entries from the previous analyses can be deleted.
3.3. Example of a Circular Duct 159
• Click onFile | Save Project.
That completes the setup of the model. Click theSolve icon and wait for the computations to complete.
3.3.7.3 Results
Once the analysis has completed, click on the Frequency Response branch underSolution (A6). In the Tabular Data window, click on the cell in the top left corner of the table so that all the entries in the table are highlighted. Right-click and selectCopy Cell. These results for the displacement of the piston can be pasted into MATLAB or a spreadsheet for processing. The results that will be calculated are the mechanical impedance and the mechanical power.
If the displacement results are pasted into MATLAB as a variable ansys_uz, then the velocity ansys_vel is calculated as
ansys_vel = j(2πf) × ansys_uz . (3.37) The mechanical impedanceansys_Z_m0 can be calculated as
ansys_Z_m0 = F
ansys vel= 1 × 10−3 N
ansys vel . (3.38)
The mechanical power can be calculated using Equation (3.33).
The MATLAB code to calculate these parameters is
1 ansys_vel =1i*2* pi* ansys_uz (: ,1).*( ansys_uz (: ,2)+1i* ansys_uz (: ,3));
2 ansys_Z_m0 =1e -3* ones( size ( ansys_vel ))./ ansys_vel ;
3 ansys_power =0 .5 *1e -3ˆ2* real ( ones(size ( ansys_Z_m0 ))./ ansys_Z_m0 );