When a steel bar is hit by a hammer, a clear sound can be heard because the steel bar vibrates at its resonant frequency. If the bar is oscillated at this resonant frequency, it will be found that the vibration amplitude of the bar becomes very large. Therefore, when a machine is designed, it is important to know the resonant frequency of the machine. The analysis to obtain the resonant frequency and the vibration mode of an elastic body is called ‘mode analysis’.
The finite element method (FEM) analysis makes it possible to obtain the vibration mode for a various complex shape of an elastic body. In this chapter, vibration modes and resonant frequencies of a straight beam, a HDD suspension, a one axis moving table and a high-speed spindle using elastic hinges are solved by the ANSYS software.
Mode analysis; Vibration mode; Resonant frequency; Type of elements
When a steel bar is hit by a hammer, a clear sound can be heard because the steel bar vibrates at its resonant frequency. If the bar is oscillated at this resonant frequency, it will be found that the vibration amplitude of the bar becomes very large. Therefore, when a machine is designed, it is important to know the resonant frequency of the machine. The analysis to obtain the resonant frequency and the vibration mode of an elastic body is called ‘mode analysis’.
It is said that there are two methods for mode analysis. One is the theoretical analysis and the other is the finite element method (FEM). Theoretical analysis is usually used for a simple shape of an elastic body, such as a flat plate and a straight bar but theoretical analysis cannot give us the vibration mode for the complex shape of an elastic body. FEM analysis can obtain the vibration mode for it.
In this chapter, three examples are presented, which use beam elements, shell elements and solid elements, respectively, as shown in Fig. 4.1.
Obtain the lowest three vibration modes and resonant frequencies in the y direction of the straight steel bar shown in Fig. 4.2 when the bar can move only in the y direction.
Thickness of the bar is 0.005 m; width of the bar is 0.01 m; length of the bar is 0.09 m. Material of the bar is steel with Young's modulus, E = 206 GPa, and Poisson's ratio ν = 0.3. Density ρ = 7.8 × 103 kg/m3.
Boundary condition: All freedoms are constrained at the left end.
Before mode analysis is attempted using ANSYS, an analytical solution for resonant frequencies will be obtained to confirm the validity of ANSYS solution. The analytical solution of resonant frequencies for a cantilever beam in y direction is given by,
where length of the cantilever beam, L = 0.09 m, cross section area of the cantilever beam, A = 5 × 10− 5 m2,
Young's modulus E =206 GPa.
The area moment of inertia of the cross section of the beam is
Mass per unit width M = ρAL/L = ρA = 7.8 × 103 kg/m3 × 5 × 10− 5 m2 = 0.39 kg/m.
λ1 = 1.875, λ2 = 4.694, λ3 = 7.855.
For that set of data the following solutions is obtained: f1 = 512.5 Hz; f2 = 3212 Hz; f3 = 8994 Hz.
Fig. 4.3 shows the vibration modes and the positions of nodes obtained by Eq. (4.1).
In FEM analysis, it is very important to select a proper element type which influences the accuracy of solution, working time for model construction and CPU time. In this example, the two-dimensional elastic beam, as shown in Fig. 4.4, is selected for the following reasons:
A two-dimensional elastic beam has three degrees of freedom at each node (i, j), which are translatory deformations in the x and y directions and rotational deformation around the z axis. This beam can be subject to extension or compression bending due to its length and the magnitude of the area moment of inertia of its cross section.
From the ANSYS Main Menu, select Preprocessor → Element Type → Add/Edit/Delete.
Then the Element Types window, as shown in Fig. 4.5, is opened.
From the ANSYS Main Menu, select Preprocessor → Sections → Beam → Common Sections.
This section describes the procedure of defining the material properties of the beam element.
From the ANSYS Main Menu, select Preprocessor → Material Props → Material Models.
Next, define the value of density of material.
To draw a cantilever beam for analysis, the method of using keypoints is described.
From the ANSYS Main Menu, select Preprocessor → Modeling → Create → Keypoints → In Active CS.
The Create Keypoints in Active Coordinate System window, Fig. 4.12, opens.
By implementing the following steps, a line between two keypoints is created.
From the ANSYS Main Menu, select Preprocessor → Modeling → Create → Lines → Lines → Straight Line.
The window Create Straight Line, shown in Fig. 4.15, is opened.
From the ANSYS Main Menu, select Preprocessor → Meshing → Size Cntrls → Manual Size → Lines → All Lines.
The window Element Sizes on All Selected Lines, shown in Fig. 4.17, is opened.
From the ANSYS Graphic Window, the preview of the divided line is available, as shown in Fig. 4.18, but the line is not really divided at this stage.
From the ANSYS Main Menu, select Preprocessor → Meshing → Mesh → Lines.
The Mesh Lines window, shown in Fig. 4.19, opens.
The left end of nodes is fixed in order to constrain the left end of the cantilever beam.
From the ANSYS Main Menu, select Solution → Define Loads → Apply → Structural → Displacement → On Nodes.
The Apply U, ROT on Nodes window, shown in Fig. 4.20, opens.
In order to obtain the vibration modes only in the y direction, the following boundary conditions are set.
From the ANSYS Main Menu, select Solution → Define Loads → Apply → Structural → Displacement → On Lines.
The window Apply U, ROT on Nodes, shown in Fig. 4.24, opens.
The following steps are used to define the type of analysis.
From the ANSYS Main Menu, select Solution → Analysis Type → New Analysis.
The New Analysis window, shown in Fig. 4.31, opens.
In order to define the number of modes to extract, the following procedure is used.
From the ANSYS Main Menu, select Solution → Analysis Type → Analysis Options.
The Modal Analysis window, shown in Fig. 4.32, opens.
From the ANSYS Main Menu, select Solution → Solve → Current LS.
The Solve Current Load Step window, shown in Fig. 4.34, opens.
From the ANSYS Main Menu, select General Postproc → Read Results → First Set.
From the ANSYS Main Menu, select General Postproc → Plot Results → Deformed Shape.
The window Plot Deformed Shape, shown in Fig. 4.37, opens.
From the ANSYS Main Menu, select General Postproc → Read Results → Next Set.
Follow the same steps outlined in Section 4.2.5.2 and calculate the results for the second and third modes of vibration. The results are plotted in Figs 4.39 and 4.40.
A suspension of HDD has many resonant frequencies with various vibration modes and it is said that the vibration mode with large radial displacement causes the tracking error. So the suspension has to be operated with frequencies of less than this resonant frequency.
Obtain the resonant frequencies and determine the vibration mode with large radial displacement of the HDD suspension as shown in Fig. 4.41.
Material: Steel.
Young's modulus, E = 206 GPa, Poisson's ratio ν = 0.3.
Density ρ = 7.8 × 103 kg/m3.
Boundary condition: All freedoms are constrained at the edge of a hole formed in the suspension.
In this example, the three-dimensional elastic shell is selected for calculations as shown in Fig. 4.41. Shell element is very suitable for analysing the characteristics of thin material.
From the ANSYS Main Menu, select Preprocessor → Element Type → Add/Edit/Delete.
Then the Element Types window, as shown in Fig. 4.42, is opened.
This section describes the procedure of defining the material properties of shell element.
From the ANSYS Main Menu, select Preprocessor → Material Props → Material Models.
Next, define the value of density of material.
From the ANSYS Main Menu, select Preprocessor → Sections → Shell → Lay-up → Add/Edit.
To draw a suspension for analysis, the method using keypoints on the window are described in this section.
From the ANSYS Main Menu, select Preprocessor → Modeling → Create → Keypoints → In Active CS.
Table 4.1
KP No. | X | Y | Z | KP No. | X | Y | Z |
---|---|---|---|---|---|---|---|
1 | 0 | − 0.6e − 3 | 9 | 8.3e − 3 | − 0.8e − 3 | ||
2 | 8.3e − 3 | − 2.15e − 3 | 10 | 10.5e − 3 | − 0.8e − 3 | ||
3 | 8.3e − 3 | − 1.4e − 3 | 11 | 10.5e − 3 | 0.8e − 3 | ||
4 | 13.5e − 3 | − 1.4e − 3 | 12 | 8.3e − 3 | 0.8e − 3 | ||
5 | 13.5e − 3 | 1.4e − 3 | 13 | 0 | − 0.6e − 3 | 0.3e − 3 | |
6 | 8.3e − 3 | 1.4e − 3 | 14 | 8.3e − 3 | − 2.15e − 3 | 0.3e − 3 | |
7 | 8.3e − 3 | 2.15e − 3 | 15 | 8.3e − 3 | 2.15e − 3 | 0.3e − 3 | |
8 | 0 | 0.6e − 3 | 16 | 0 | 0.6e − 3 | 0.3e − 3 |
Areas are created from keypoints by performing the following steps.
From the ANSYS Main Menu, select Preprocessor → Modeling → Create → Area → Arbitrary → Through KPs.
Table 4.2
Area No. | Keypoint Number |
---|---|
1 | 9, 10, 11, 12 |
2 | 1, 2, 3, 4, 5, 6, 7, 8 |
3 | 1, 2, 14, 13 |
4 | 8, 7, 15, 16 |
The Solid Circle Area window, shown in Fig. 4.53, opens. Input [D] the values of 12.0e-3, 0, and 0.6e-3 to the X, Y, and Radius boxes as shown in Fig. 4.53, respectively, and click the [E] OK button. Then the solid circle is made in the drawing of suspension as shown in Fig. 4.54.
In order to make a spring region and the fixed region of the suspension, the rectangular and circle areas in the suspension are subtracted by Boolean operation.
From the ANSYS Main Menu, select Preprocessor → Modeling → Operate → Booleans → Subtract → Areas.
From the ANSYS Main Menu, select Preprocessor → Modeling → Operate → Booleans → Glue → Areas.
The Glue Areas window, shown in Fig. 4.59, opens. Click the [E] Pick All button.
From the ANSYS Main Menu, select Preprocessor → Meshing → Size Cntrls → Manual Size → Areas → All Areas.
The window Element Sizes on All Selected Areas, shown in Fig. 4.60, opens.
From the ANSYS Main Menu, select Preprocessor → Meshing → Mesh → Areas → Free.
The Mesh Lines window, shown in Fig. 4.61, opens.
The suspension is fixed at the edge of the circle.
From the ANSYS Main Menu, select Solution → Define Loads → Apply → Structural → Displacement → On Lines.
The window Apply U, ROT on Nodes, shown in Fig. 4.63, opens.
The following steps are performed to define the type of analysis.
From the ANSYS Main Menu, select Solution → Analysis Type → New Analysis.
The New Analysis window, shown in Fig. 4.67, opens.
In order to define the number of modes to extract, the following steps are performed.
From the ANSYS Main Menu, select Solution → Analysis Type → Analysis Options.
The Modal Analysis window, shown in Fig. 4.68, opens.
From the ANSYS Main Menu, select Solution → Solve → Current LS.
From the ANSYS Main Menu, select General Postproc → Read Results → First Set.
From the ANSYS Main Menu, select General Postproc → Plot Results → Deformed Shape.
The Plot Deformed Shape window, shown in Fig. 4.70, opens.
From the ANSYS Main Menu, select General Postproc → Read Results → Next Set.
Perform the same steps indicated in Section 4.2.4.2 and calculated results from the second mode to the sixth mode of vibration are displayed on the windows as shown in Figs 4.72–4.76.
A one-axis table using elastic hinges has been often used in various precision equipment, and the position of a table is usually controlled at nanometer order accuracy using a piezoelectric actuator or a voice coil motor. Therefore, it is necessary to confirm the resonant frequency in order to determine the controllable frequency region.
Obtain the resonant frequency of a one-axis moving table using elastic hinges when the bottom of the table is fixed and a piezoelectric actuator is selected as an actuator.
Material: Steel, thickness of the table: 5 mm.
Young's modulus, E = 206 GPa, Poisson's ratio ν = 0.3.
Density ρ = 7.8 × 103 kg/m3.
Boundary condition: All freedoms are constrained at the bottom of the table and the region A indicated in Fig. 4.77, where a piezoelectric actuator is glued.
In this example, the solid element is selected to analyse the resonant frequency of the moving table.
From the ANSYS Main Menu, select Preprocessor → Element Type → Add/Edit/Delete.
Then the Element Types window, as shown in Fig. 4.78, opens.
This section describes the procedure of defining the material properties of solid element.
From the ANSYS Main Menu, select Preprocessor → Material Props → Material Models.
Next, define the value of density of material.
To draw the moving table for analysis, the method using keypoints on the window are described in this section.
From the ANSYS Main Menu, select Preprocessor → Modeling → Keypoints → In Active CS.
The window Create Keypoints in Active Coordinate System, shown in Fig. 4.83, opens.
Areas are created from keypoints by carrying out the following steps.
From the ANSYS Main Menu, select Preprocessor → Modeling → Create → Areas → Arbitrary → Through KPs.
The window Create Area thru KPs, shown in Fig. 4.85, opens.
From the ANSYS Main Menu, select Preprocessor → Modeling → Operate → Booleans → Subtract → Areas.
From the ANSYS Main Menu, select Preprocessor → Modeling → Create → Areas → Arbitrary → Through KPs.
From the ANSYS Main Menu, select Preprocessor → Modeling → Operate → Booleans → Add → Areas.
The window Create Area thru KPs, shown in Fig. 4.89, opens.
From the ANSYS Main Menu, select Preprocessor → Modeling → Create → Area → Circle → Solid Circle.
The Solid Circle Area window, shown in Fig. 4.91, opens.
Table 4.4
No. | X | Y | Radius |
---|---|---|---|
1 | 0 | 0.0122 | 0.0022 |
2 | 0.005 | 0.0122 | |
3 | 0.035 | 0.0122 | |
4 | 0.04 | 0.0122 | |
5 | 0 | 0.0428 | |
6 | 0.005 | 0.0428 | |
7 | 0.035 | 0.0428 | |
8 | 0.04 | 0.0428 | |
9 | 0.0072 | 0.015 | |
10 | 0.0072 | 0.02 |
From the ANSYS Main Menu, select Preprocessor → Modeling → Operate → Booleans → Subtract → Areas.
From the ANSYS Main Menu, select Preprocessor → Meshing → Mesh Tool.
The Mesh Tool window, shown in Fig. 4.94, opens.
Next, by performing the following steps, the thickness of 5 mm and the mesh size are determined for the drawing of the table.
From the ANSYS Main Menu, select Preprocessor → Modeling → Operate → Extrude → Elem Ext Opts.
The window Element Extrusion Options, shown in Fig. 4.100, opens.
From the ANSYS Main Menu, select Preprocessor → Modeling → Operate → Extrude → Areas → By XYZ Offset.
The window Extrude Areas by Offset, shown in Fig. 4.101, opens.
The table is fixed at both the bottom and the region A of the table.
From the ANSYS Main Menu, select Solution → Define Loads → Apply → Structural → Displacement → On Areas.
The Apply U, ROT on Areas window, shown in Fig. 4.104, opens.
The following steps are performed to define the type of analysis.
From the ANSYS Main Menu, select Solution → Analysis Type → New Analysis.
The New Analysis window, shown in Fig. 4.108, opens.
In order to define the number of modes to extract, the following steps are performed.
From the ANSYS Main Menu, select Solution → Analysis Type → Analysis Options.
The Modal Analysis window, shown in Fig. 4.109, opens.
From the ANSYS Main Menu, select Solution → Solve → Current LS.
The Solve Current Load Step window opens.
From the ANSYS Main Menu, select General Postproc → Read Results → First Set.
From the ANSYS Main Menu, select General Postproc → Plot Results → Deformed Shape.
The Plot Deformed Shape window, shown in Fig. 4.111, opens.
From the ANSYS Main Menu, select General Postproc → Read Results → Next Set.
Perform the same steps indicated in Section 4.4.4.2, and the calculated results for the higher modes of vibration are displayed on the windows as shown in Figs 4.113 and 4.114.
In order to easily judge the vibration mode shape, the animation of mode shape can be used.
From the Utility Menu, select PlotCtrls → Animate → Mode Shape.
The Animate Mode Shape window, shown in Fig. 4.115, opens.