# Modal Analysis based on Impact-Synchronous Time Averaging of a Rotor Dynamic System

**ABSTRACT**

Dynamic characteristics of rotor dynamic system, namely natural frequencies, damping and mode shape, vary with rotation speed due to gyroscopic effect. To consider such effect, modal analysis is required to be performed while in operation. However, to generate the frequency response functions, it is not viable to perform data reduction in frequency domain as noise and harmonics excited by rotating forces dominate the spectrum and hide the natural frequencies. The full set of dynamic characteristics is difficult to obtain. A novel technique of Impact-Synchronous Time Averaging is developed to screen out these noise and harmonics. Unlike operational modal analysis, blocks of the time response are triggered by the impulse generated from the impact of the force transducer and are averaged. These blocks are synchronous to the impact rather than to the rotating speed and in so doing, averaging out the running speed components while maintaining the impulse response signature. Fast Fourier Transform then operates on the averaged time series to unveil the natural frequencies of the system. From signal analysis point of view, the force transducer performed a dual task of firstly, generating forced input for cross spectrum and secondly, acting as a trigger for time domain averaging of response. As normal FFT analyzers do not contain such function, virtual instrumentation is used to construct the required capabilities.

KEYWORDS: Modal Analysis, Rotordynamics, Dynamic Characteristics, Gyroscopic Effect, Impact-Synchronous Time Averaging.

**INTRODUCTION**

Efficient operation of rotating machines often requires high rotational rates, which gives rise to concern that the system will resonate at unacceptable large amplitudes and become unstable. A comprehensive study of rotor dynamics is needed to overcome the problem of vibration, resonance and failure of structure in order to create faster and highly productivity of rotating machines (Bucher et al. 2001).

Modal analysis is defined as the study of the dynamic characteristics of a mechanical structure. The three parameters that comprehensively described the dynamic characteristics are natural frequencies, mode shapes and damping. Knowledge of these characteristics provides insights into the dynamic behavior of the structure and its mathematical model, allowing for the prediction of the system response, and the change in structural dynamic behavior with the change in geometrical properties, material properties, and boundary conditions of the structure.

Machines with a non-rotating shaft behave much like structures. However, once the rotor is spinning, the modes are no longer planar. With radially symmetric bearings, the rotor center traces out a circle. The rotor whirls either in the same direction as rotation, or against rotation, resulting in both forward and backward whirl modes. When a gyroscope is spinning rapidly, it has a large amount of a conserved physical quantity called angular momentum. Angular momentum is a special measure of rotational motion that cannot be created or destroyed. It can only be transferred between objects. If a twist is applied to the gyroscope around its axis of rotation, it will either spin faster or slower, depending on which way the gyroscope is twisted. A gyroscopic effect occurs in rotating structures whenever the mode shape has an angular (conical/rocking) component. For forward whirl, as shaft speed increases, the gyroscopic effects essentially act like an increasingly stiff spring on the central disk for the rocking motion. Increasing stiffness acts to increase the natural frequency. For backward whirl, the effect is reversed. Increasing rotor spin speed acts to reduce the effective stiffness, thus reducing the natural frequency. In the case of the cylindrical modes, very little effect of the gyroscopic terms was noted, since the center disk was whirling without any conical motion. Without the conical motion, the gyroscopic effects do not appear. Thus, for the soft bearing case, which has a very cylindrical motion, no effect was observed, while for the stiff bearing case, which has a bulging cylinder and thus conical type motion near the bearings, a slight effect was noted (Swanson et al. 2005).

Dynamic characteristics of rotor-shaft system vary with rotation speed. For comprehensiveness, modal analysis during rotating conditions should be extended from the conventional modal analysis in static condition. Analysis in both conditions shines more light onto the dynamic behavior of the rotor-shaft system with rotation speed. This is crucial in systems that contain long flexible shafts or diametrically large rotors.

However, to generate the frequency response functions, it is not viable to perform data reduction in frequency domain as noise and harmonics excited by rotating forces dominate the spectrum and hide the natural frequencies. The full set of dynamic characteristics is difficult to obtain.

In this paper, technique of Impact-Synchronous Time Averaging is developed to screen out these noise and harmonics. Blocks of the time response are triggered by the impulse generated from the impact of the force transducer and are averaged. These blocks are synchronous to the impact rather than to the rotating speed and in so doing, averaging out the running speed components while maintaining the impulse response signature. Fast Fourier Transform then operates on the averaged time series to unveil the natural frequencies of the system.

**THEORIES**

Structural dynamics models of rotating structures

In our introduction of the various mathematical models of rotating structures, we confine ourselves to models which describe ‘small’ vibrations. Testing and modeling the dynamics of rotating machines is a very broad area and the current text concentrates on a fraction of this discipline. Within this subset of models we further restrict ourselves to linear models formulated in terms of a finite number of degrees of freedom (i.e. to a discretized model) which are an approximation of the governing partial differential equation of the model. Such a system of equations is generally time varying (Yakubovich et al. 1975) as, due to the rotation, some properties may change periodically with time,

A common assumption which is often made is that the structure under test contains only isotropic rotating elements (i.e. it is assumed that both inertia and stiffness properties are not a function of the instantaneous angle of rotation). When the former assumptions hold and when the measurements are performed in an inertial coordinate system, we obtain an additional simplification under which the matrices in (1) are no longer time dependent but are only speed dependent (Lalanne et al. 1990),

For a constant speed of rotation Ω, equation (2) represents a general linear time invariant system and, as a result, commonly available tools can be used to analyse the dynamic behaviour of such a system. The differential equation of motion (2), which represents a rotating structure, is said to be non-self-adjoint. Passive non-rotating structures are generally self-adjoint and so their frequency response and their system matrices are symmetric. The effect of rotation on components gives rise to a non-symmetric (and NSA) equation of motion and to a non-symmetric frequency-response function matrix. Models of self-adjoint structures can be completely expanded in terms of a single set of eigenvectors (modes) and eigenvalues. On the other hand, NSA structures require two sets of eigenvectors with a set of eigenvalues to describe their dynamic behaviour fully (Bucher et al. 2001).

Correlation and Frequency Response Functions

As the name suggests, the correlation function examines whether there is any correlation between signals at two points in time. Clearly, a signal is always perfectly correlated with itself, but is there a correlation between a signal at time t and If a signal is a sine wave with period , then we know that there is an excellent correlation, since the response at this two times will be identical. A purely random signal should be uncorrelated for any time besides zero since the signal changes in a completely unpredictable way (Rahman unpublished, Timosheko et al. 1974).

The auto-correlation of discrete input signal is, by definition, given by

which is the means of the product of q(t) and q(t+), and, where q(t+) is the value of input function q measured at sec after t.

Similarly, the autocorrelation of the output is given by

Seeking correlation between two different signals, the cross-correlation of input x(t) and output y(t) is given by

Cross-correlation would be useful in trying to determine whether a vibration at one point of a structure is being influenced by vibrations at some other point. If the cross correlation is high for some value of , then one could deduce that there is a relationship between the two vibration and that the transit time for the motions to be transmitted from one point to the other is equal to .

The Fourier Transform of the auto-correlation function is called Auto Power Spectral Density Zqq(), for input given by

And for output is given by

The Fourier Transform of the cross-correlation function is called Cross Power Spectral Density Zxq(), given by

We will now link the Power Spectral Densities (APS and XPS) with the Frequency Response Function SXQ(). From Fig.1 the input output relationship can be stated in time domain as

If we multiply both sides by x(t+) and mean them, we obtain

Performing Fourier Transform on both sides we obtain

Hence, on an FFT analyser, we can obtain the FRF by dividing the Auto Power Spectral Density (APS) of output by the Cross Power Spectral Density (XPS) of the input and output.

In the similar fashion in can be shown that

i.e. we can also obtain the FRF by dividing the Cross Power Spectral Density (XPS) by the Auto Power Spectral Density (APS) of the input.

Both equations (11) and (12) can be used in multi-channel FFT analyser to determine the frequency response function for FRF Modal Analysis.

Frequency Averaging

In general, the vibration signal from a rotating machine is not completely deterministic, but has some random noise superimposed on it. Because the noise is unpredictable, it alters the spectrum shape, and in many cases can seriously distort the spectrum. If a series of spectra are averaged together, the noise will gradually assume a smooth shape, and the spectral peaks, due to the deterministic part of the signal, will stand out and their levels will be more accurately represented. However, it should be noted that averaging FFT spectra does not reduce the level of noise but rather smoothens it (Bodre unpublished).

In fundamental modal testing application, frequency averaging is used as the measurement is done while the machine is not in operation. Therefore, spectrum averaging is enough for signal clean up.

**Impact-Synchronous Time Averaging**

Impact-Synchronous Time Averaging incorporates the time synchronous averaging technique to rapidly improve the signal to noise ratio of the response obtained while machine is in operating condition. Unlike frequency domain averaging, the waveform itself is averaged in a time buffer prior to FFT operation, and the sampling of the signal is triggered by impact force input to the analyzer. If the triggering is synchronized with the repetition rate of the impact, the averaging process will gradually eliminate the running speed components, its harmonics and random noise as they are not synchronized with the trigger. Subsequently, the waveform would contain primarily the response of the system due to the impact force.

The frequency response function is obtained from the cross spectrum of time averaged response and the forced input. The frequency response function obtained from such averaging process is shown in Fig. 2, 3 and 4.

Essentially, the force transducer simultaneously performed a dual task; firstly, it generates forced input for cross spectrum and secondly, it acts as a trigger for time domain averaging of the impulse response.

**Windowing Functions**

Rectangular and exponential windows are normally used in modal analysis, namely for impulse and response signals, as shown in Fig. 5. Rectangular windows are used for signals that start and end at zero. Typical example of these signals is obtained from the impulse of force transducer and the response from the accelerometer for a highly damped structure. Here, the signal decays to zero before reaching the end of the time record. However, in low damped structure, it takes a long time for the response to decay to zero. For this, an exponential window is being used to suppress the tail end of the response to zero (Formenti et al. 1999).

The windowing operation of the analyzer can be expressed by the following equation:

It should be mentioned here that when applying exponential window, the damping used in exponential window should be taken into consideration when determining the damping for particular mode as mentioned in the section above.

**EXPERIMENTAL PROCEDURE AND INSTRUMENTATIONS**

The experiment carried out is aimed at studying the effects of:

1) Frequency Averaging and Impact-Synchronous Time Averaging on the time response and transfer functions while the rotor shaft system is in rotating condition

2) Rotating speed on the dynamic characteristics of the rotor shaft system

Step 1: Setting up the measuring-points

A total of 10 points are selected on the RK4 rotor kit to including the motor, shaft and bearing blocks as displayed in Fig. 6 (Point 1 to 10 is defined from right to left). These selected points need to be refined enough to define the geometry of the structure. Selecting the right point is essential to make sure mounting of accelerometer is possible in both vertical and horizontal axis on the RK4 rotor kit. The accelerometer is set at point 5 on the ball bearing outer race. The bearing outer race is attached to the structure by means of fours springs. The outer race assumed the lateral motion of the shaft.

Step 2: Instrument set-up

The impact hammer is connected to channel 1 and accelerometer is connected to channel 2 of UMPC Data Acquisition System. The accelerometer is set at point 5 as the fixed response. The experiment is carried out by fixing the accelerometer and roving the impact hammer from point 1 until point 10 for both non-rotating and rotating conditions.

Step 3: Pre-Processing

Data acquisition is done by using UMPC Data Acquisition System. 5 averages for non-rotating condition and 20 averages are made during rotating condition. The signals are processed by the virtual instruments developed to generate the frequency response functions for various conditions.

Step 4: Post Processing

Post processing is done by using Me’Scope software to extract the modal parameters. Results are viewed to show natural frequencies, mode shapes and damping.

**RESULTS**

**Frequency Response Function (FRF)**

Comparison of Response Time Waveform between Frequency Averaging and Impact-Synchronous Time Averaging Applied during Rotating Condition

The first column in Table 2a and 2b shows the time waveforms of forced input and response for frequency averaging and impact-synchronous time averaging applied during rotating condition, respectively. The focus here is on the response waveforms. In Table 2a, the waveform is in raw form and the signature contained the response of the impact plus components of running speed, its harmonics and noise. Averaging is only performed in the frequency domain, as shown in the second column of Table 2a. In Table 2b, the waveform is time averaged, synchronized by the impact. It is observed that the signature only contained the response from the impact. All non synchronized components appeared to be filtered out. It is only after the averaging process is completed that cross spectrum is performed on the waveform. As a result, a smooth FRF is generated as shown in the second column of Table 2b.

**Comparison of Frequency Response Function between Frequency Averaging and Impact-Synchronous Averaging Applied During Rotating Condition**

Table 3 shows comparison of FRF between frequency averaging and impact-synchronous averaging applied during rotating condition. Performing data reduction in frequency domain the noise and harmonics excited by rotating forces dominate the spectrum and hide the natural frequencies. The full set of dynamic characteristics is difficult to obtain. It is observed that with the application of impact-synchronous time averaging, the noise and harmonics have been well averaged out.

**Comparison of Dynamic Characteristics of Rotor Shaft System for Static and Running Conditions**

Table 4 shows the first two non-rotating mode of RK4 which utilized frequency domain averaging. Table 5 shows the first two rotating mode (at 3000RPM) of RK4 which utilized Impact-Synchronous Time Averaging. Modal analysis of rotor shaft system in static condition is being extended to analysis of the system while in rotating conditions. It can be seen that the frequency response functions of rotor-shaft system vary with rotation speed. Analysis in both conditions shines more light onto the dynamic behavior of the rotor-shaft system with rotation speed.

It is worthwhile to mention here that the second natural frequency (conical mode) would have changed even more if a diametrically larger rotor has been used. A gyroscopic effect occurs in rotating structures whenever the mode shape has an angular (conical/rocking) component. For forward whirl, as shaft speed increases, the gyroscopic effects essentially act like an increasingly stiff spring on the central disk for the rocking motion. Increasing stiffness acts to increase the natural frequency.

**DISCUSSION****Impact-Synchronous Time Averaging**

The effectiveness of Impact-Synchronous Time Averaging modal analysis of rotor shaft system in operating condition is sought. The results, as shown in Table 3, indicated its ability to significantly reduce the running speed component, its harmonics and noise.

Impulsive excitation is a transient excitation. To perform this kind of testing, the structure is excited by an impact hammer and the response of the structure is measured. The input force spectrum exerted on the structure is a combination of the stiffness of the hammer/tip as well as the stiffness of the structure. Basically the input power spectrum is controlled by the length of time of the impact pulse. A long pulse in the time domain, results in a short or narrow frequency spectrum. A short pulse in the time domain, results in a wide frequency spectrum. The hammer itself is equipped with a transducer that enables to record the time history of the input signal. The energy and the frequency contents of the impact hammer can be controlled to a certain extent by the mass of the head of the hammer and the type of tip used on the hammer. A hard tip will for instance increase the bandwidth of the input signal. A soft tip works otherwise. During the study, it is also found that vinyl cover used as the impact hammer tip provide best coherence.

The energy that goes into the system is in general quite low. For this reason, when conducting analysis during running condition, the raw time response is dominated by the rotating forces of the machine. By synchronizing the time series of the response to the impact of the hammer, the non-synchronized rotating forces are filtered out. This outcome of this signal analysis process is, in effect, opposite to the common time averaging process (triggered by a tachometer) which is intended to eliminate the non integer-multiples of the running speed.

**Virtual Instrumentation**

In the technique developed here, the force transducer performed a dual task of firstly, generating forced input for cross spectrum and secondly, acting as a trigger for time domain averaging. The analysis algorithm needs to be reconstructed. The concept of virtual instrumentation is introduced here to overcome the limitation imposed by the conventional FFT analyzers without going into the other extreme of software development.

Virtual instrumentation is the use of customizable software and modular measurement hardware to create user-defined measurement systems, called virtual instruments. Traditional hardware instrumentation systems are made up of pre-defined hardware components, such as digital multi-meters, oscilloscopes and FFT analyzer that are completely specific to their stimulus, analysis, or measurement function. Because of their hard-coded function, these systems are more limited in their versatility than virtual instrumentation systems. The primary difference between hardware instrumentation and virtual instrumentation is that software is used to replace a large amount of hardware. The software enables complex and expensive hardware to be replaced by already purchased computer hardware; e. g. analog to digital converter can act as a hardware complement of a virtual oscilloscope. However, in the analysis of dynamic systems, the user needs a sound knowledge of digital signal analysis to ascertain the reliability of the processed signatures.

**CONCLUSIONS**

1. The study has demonstrated the effectiveness of using Impact-Synchronous Time Averaging in the determination of dynamic characteristics of a rotor dynamic system while in rotating condition.

2. The application of Impact-Synchronous Time Averaging is effective in filtering out the non-synchronous running speed frequency components, its harmonics and noise.

3. Virtual instrumentation provides versatilities normally deprived by conventional FFT analyzers.

**REFERENCES**

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4 Formenti, D., and Macmillan, B. (1999). The Exponential Window. Journal of Sound and Vibration.

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7 Timosheko, S., Yong, D.H., and Weaver, W.J. (1974). Vibration Problems in Engineering. 4th Edition. John Wiley & Sons.

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