1.4.7 Vibration of a rotating cantilever plate

Product: ABAQUS/Standard  

This example is intended to provide basic verification of the centrifugal load stiffness effect present in vibration problems when the structure is undergoing small vibrations in a rotating coordinate frame. The most common example of such applications is the study of the vibrations of components of rotating machines, such as the blades on turbines and compressors. In such cases two effects that are not present in vibration problems in fixed coordinate systems become important: the initial stressing of the structure caused by the centrifugal loading and the “load stiffness” effect caused by the line of action of the centrifugal load changing if the vibration causes motion in the plane normal to the axis of rotation. In most conventional designs of rotating machines the initial stress effect is a stiffening effect, and the load stiffness effect is a softening effect. In the vibration of blades on turbines or compressors the load stiffness effect is significant only for long blades on small wheels, such as the fan blades on modern high bypass jet engines for aircraft: see Hibbitt (1979). The purpose of this example is to illustrate this effect and verify the capability in ABAQUS for such vibration studies.

Problem description

Analysis

The analysis is done in a series of steps. Step 1 extracts the lowest mode of the system at rest (no rotation of the wheel) using the *FREQUENCY procedure. In this example only the lowest frequency is required: in a practical case several frequencies would probably be needed.

Step 2 is a *STATIC procedure in which the centrifugal load, corresponding to a rotational speed of the system of 25 revolutions/second, is applied using the *DLOAD option. This centrifugal load is applied using both the CENT and CENTRIF load options. The *DLOAD magnitude must be given as with the CENT option and as with the CENTRIF option. The CENTRIF option uses the density defined with the *DENSITY option; therefore, it uses the actual mass matrix of the element in the load calculation, which means that a lumped mass matrix is used for first-order elements and a consistent mass matrix is used for second-order elements. The CENT option always uses a consistent mass matrix. The NLGEOM parameter is used on the *STEP option to indicate that geometric nonlinearity is required, which causes ABAQUS to include the initial stress and load stiffness effects and implies a nonlinear analysis.

Step 3 uses the *FREQUENCY procedure to obtain the lowest frequency at this rotational speed. Step 4 is a *STATIC step to increase the *DLOAD to a rotational speed of 50 revolutions/second, Step 5 obtains the lowest eigenmode at this speed, Step 6 increases the speed to 75 revolutions/second, and Step 7 obtains the lowest eigenmode at this speed.

Substructure analysis

This example is suitable for demonstrating the substructure preload capability in ABAQUS. With this option it is possible to create a finite element mesh, load it using a nonlinear procedure, and create a substructure using the current stiffness after the loading. If the entire wheel had to be modeled with all the rotating blades, the model could be simplified by using this option. The blade would be modeled as a substructure, the centrifugal force applied, and the stiffness formed including the “load stiffness.” The substructure could then be rotated and used for all the blades attached to the wheel.

Preloading is obtained by preceding a *SUBSTRUCTURE GENERATE step with one or several analysis steps. The substructure stiffness is formed from the final loading condition of the preceding general analysis step. Four substructures are generated for each analysis. The first is generated without any preloading. The remaining three substructures are generated after a centrifugal load has been applied so that each includes the load stiffness associated with a different rotational speed. Furthermore, when the substructures are used, the NLGEOM parameter is immaterial in the *FREQUENCY step, since the load stiffness is included in the substructure stiffness matrix and is, thus, included in the frequency extraction whether NLGEOM is used or not.

Results and discussion

Input files

References

Table

Table 1.4.7–1 Spinning beam frequencies (Hz).

 Vibration in the planeVibration normal to
 of the rotation axisthe rotation axis
 (Case A)(Case B)
Rotary speed (cycles/sec)0255075255075
Rayleigh quotient23.6841.7472.10104.2733.4251.9572.44
B2123.4441.2071.00102.2932.9450.8670.24
B2323.6841.7271.94103.7333.4051.7371.66
B3123.4441.2071.00102.2932.9450.8670.24
B3323.6841.8372.14103.9833.5452.0072.02
S8R23.8941.9072.13103.9133.6351.9971.91
S8R523.8141.8272.05103.8233.5351.8771.79
Substructure23.8241.8872.33104.5833.5651.9872.04
C3D8I24.2341.9071.93103.6633.8252.1572.25
C3D1025.1442.7072.88104.9134.6253.0373.34
C3D10M24.8242.4072.51104.4534.3252.7072.96
C3D2024.5342.4572.87105.0234.3053.0173.51
C3D20R24.2842.2572.54104.3834.0652.5572.60


Figure

Figure 1.4.7–1 Plate and wheel geometry.