Finite element analysis, together with Lagrange method, was used to model the behavior of the system. A disc connected to the rotor was used to create an imbalance in the rotor bearing system.
Problem statement
Depending on the program used, the data can then be viewed in the available formats. Vibration based monitoring is a wide field in itself and is narrowed down to only unbalance and misalignment in the system.
Research project objectives
Impact of a solution
Research publications
Dissertation overview
The experimental test setup was used to perform the same tests to confirm the results of the simulation. The finite element model was analyzed to see whether faults can be effectively identified from the simulation results.
Chapter summary
The same was done for the experimental results obtained from the rotor holder test rig. This chapter was used to connect all aspects of the dissertation and to give an overview of what has been done to achieve the objectives that were stated in this chapter.
Background
Infrared thermography, tribology, visual inspection and ultrasonic monitoring are just a few examples of these monitoring systems. When portable monitoring systems and data collectors became available, machine fault diagnosis experienced an explosive growth.
Condition based monitoring
Further analysis of the collected data can also show trends, existing condition, type of failure developing or existing, time to failure, and type of failure that caused the failure. This includes monitoring the condition of the machine being monitored, estimating the expected time to failure and planning the appropriate maintenance time.
Causes of vibration
Forced vibration
The Jeffcott rotor represents a massless elastic shaft supported by bearings at its ends and carrying a disk. The amplitude of the force transmitted to the bearings can be reduced by reducing unbalance, increasing viscous damping and avoiding operation near critical speeds [5].
Self excited vibration
This is when the shaft is actually bent, then when it rotates it tries to correct the bow and this causes vibration. It is known that rotor inertia and gyroscopic action have an influence on the natural frequency of the rotor including reverse rotation [5].
How to measure vibrations
Transducer selection and location
Display formats
The frequency domain refers to the display or analysis of the vibration data as a function of frequency. The time domain vibration signal is typically processed in the frequency domain by applying a Fourier transform, usually in the form of a Fast Fourier Transform (FFT) algorithm.
Data logging
The main advantage of this format is that the repetitive nature of the vibration signal is clearly shown as peaks in the frequency spectrum at the frequencies where the repetition occurs. However, the disadvantage of frequency domain analysis is that a considerable amount of information can be lost during the transformation process.
Dynamic response of an unbalanced rotor
Software such as LabVIEW Signal Express can be used to view and manipulate the data to obtain valuable information. CondMaster is another computer-based program used to analyze and graph machine vibration signatures.
Misalignment of rotor bearing system
Flexible coupling
This model also considers the coupling mass as two discs at the end axle nodes where the coupling connects. The coupling used for this study was modified to reflect a flexible coupling rather than a rigid coupling.
Chapter summary
The flexible coupling model considered for this research has a similar rotational force to that proposed by Lees. The test rig was designed according to the specification and objectives as stated in Chapter 1.
Test rig
- Shaft design
- Bearings
- Flexible coupling
- Motor and control
- Test rig stand
- Isolating foot pads
- Rig safety
The test stand is the medium used to connect the rotor bearing system to the ground. There are two reasons why pad insulation was essential to the test rig.
Design process
It was also important to properly level the test rig to ensure that gravity acted perpendicular to the system. The rotor bearing test rig was also positioned so that the operating panel was easily accessible for an emergency shutdown.
Testing equipment
Accelerometer
The transducer chosen to measure the vibrations was a piezoelectric accelerometer, which works with compression. Improper mounting can cause higher g levels than the transducer can handle and cause damage.
Leonova hand held
This transducer has a magnetic base and needed at least 30 mm of flat surface to be mounted on, which can be seen in Figure 3.3-1. The transducer used for this research was an SLD 144 which had a nominal sensitivity of 10mV/m/s2 and a linear frequency range of 2Hz-10kHz.
Software
Testing procedure
Pre-testing procedure
When a round was created, the short-term memory had to be set to the time signal and the FFT. Now the adjustment had to be set to the desired round to be tested.
Testing
One was the temperature of the bearings, and the other was the vibration at startup. After performing the test, the data had to be stored in the short-term memory of the Leon's infinity.
Post-testing procedure
During testing, care had to be taken that the rotor daer test rig was not bumped or the mass of the system was not changed.
Chapter summary
Shaft finite element analysis
Shape function
By looking at any point on the rotor element, it was seen that the rotational and translational displacements (x,y,θx,θy) were spatial functions of the axial distance and time. It was used to relate the displacement of the nodes from the local to the global coordinate system and also explain the movement of the rotor.
Energy equations
The general formulation of the kinetic energy of the shaft arose from an extension of the disk equation. Where - was the sum of the classical mass matrix and the influence of the secondary effect of the rotational inertia of the shaft and was symmetrical.
![Figure 4.1-5: Nelson and Crandall’s [21]model for flexible coupling](https://thumb-ap.123doks.com/thumbv2/pubpdfnet/10638903.0/61.892.276.651.608.825/figure-4-nelson-crandall-21-model-flexible-coupling.webp)
Fault modelling
Unbalance in rotor disk
Misalignment in coupling
The forces present in the coupling due to the angular misalignment are shown in Figure 4.2-3 by FL for the force on the left and FR for the force on the right. Then the force in the x and y direction for the right side is given by:.
Assembly process
Equation of motion
Lagrange equations were constructed for each component, but adjustments were required to relate these to the standard Lagrange equation. Non-rotating and rotating systems are modeled exactly the same, but the rotating system has an additional gyroscopic force.
Matrix assembly
It was also noted that the system was linear and this is because the bearings used in this research were assumed to be linear. When added to the forcing vector, it was not added to the coupling position, but to the bearings, shown in Figure 4.3-5.
Boundary and initial conditions
Numerical analysis
Runge-Kutta
Runge-Kutta was chosen and used for this study only because Runge-Kutta is a more stable method than Heun's. The Runge-Kutta method analyzes the derivatives obtained from the system four times for each given time step.
Matlab
Now that the finite element model was built, it had to be solved using Runga-Kutta. So before it can be solved, the ordinary differential equations would have to be squared.
Chapter summary
The errors introduced into the system were time dependent and this meant that it had to be added to the Lagrange M-file. It was also shown how to assemble the necessary matrices and how to add the errors to the system.
Experimental results
- Base line
- Unbalance response
- Parallel misalignment
- Angular misalignment
- Unbalance response
- Parallel misalignment
- Angular misalignment
Two types of parallel misalignment tests were performed to get a complete picture of the effect this has on the finite element model. This test was performed to demonstrate the effect that angle deviations had on the finite element model when the size of the angle was varied.
Chapter summary
The results will be discussed in more detail in this chapter, which will give insight into how to identify the error present in the system. This will be done for both numerical and experimental results, and then both will be compared.
Experimental results
This meant that if the mass of the imbalance doubled, so did the force on the system. Comparing the baseline with the 0.5, 1 and 2 degree angular misalignment tests, it was seen that there was little or no change in the frequency of the 3x system.
Numerical simulation
This forcing function was seen to introduce a disturbance into the 1x system frequency with no effect on the 2x and 3x system frequencies. This showed that the severity of an angular deviation can be determined by taking note of the 2x system frequency.
Comparing numerical to simulation
This shows that the severity of the parallel mismatch can be determined by considering 3x the system frequency. There was a direct correlation between the parallel mismatch and the peak values of the dominant system frequencies.
Chapter summary
By comparing the peak values of experiment and simulation, it was seen that there was a difference. This was due to the same reason as for the imbalance along with the linearized coupling model.
Research conclusion
Parallel mismatch was successfully detected by increasing system frequencies by 2x and 3x, with 3x being the higher of the two. These factors only contributed to the difference in the maximum values of the system frequencies.
Future research
This was due to the linearization of the bearings and coupling model and also the fact that the experimental test rig can never be perfectly balanced and aligned. The peak values are important in determining the severity of the error, but the residual technique can eliminate these errors.
Motor details
This chapter gives the specifications of all the components used in the rotor daer test rig. It also shows a detailed shaft design done and assembly drawings used to manufacture the rotor daer test rig.
Shaft design
Kt = concentration factor for bending Kts= concentration factor for torsion Su = Ultimate tensile strength Sy = Yield strength. To keep the joint interchangeable, the minimum shaft diameter has also been made 15mm.
Rotex flexible coupling
Bearing dimensions
Test rig drawings
Disk derivation
Shaft derivation
If the shaft is subjected to a constant axial force, then there is a second contribution to the strain energy. It is necessary to express the strain energy as a function of u and w – components of displacement in (to avoid periodic terms being explicitly a function of time).
Bearing derivation
Matlab code used for simulation
Baseline alignment report
Parallel alignment reports
Angular alignment reports
Baseline varying speed graphs
Unbalance graphs
Parallel misalignment graphs
Angular misalignment graphs
Comparison graph of finite element models
Unbalance graphs
Parallel misalignment graphs
Angular misalignment graphs
