Jordan University of Science and Technology
Faculty of Engineering
Department of Mechanical Engineering
Course: Graduation project

Project Title: Experimental Modal Analysis Name: Hamzeh Ahmad Alqaisi
I.D.: 20080025119
Instructor: Dr.Yousef Najjar
Supervisor: Dr.Naem Alkhader
Due date: 14/11/2012

EXPERIMENTAL MODAL ANALASYS

ABSTRACT

Experimental modal analysis has grown steadily in popularity since the advent of the digital FFT spectrum analyzer in the early 1970’s. Today, impact testing (or bump testing) has become widespread as a fast and economical means of finding the modes of vibration of a machine or structure.

Contents

TITLE PAGE NO. Nomenclature…………………………………………………………………………………………………………………………………
CHAPTER 1:
Introduction…………………………………………………………………………………………………………………………………..
CHAPTER 2:
Experiment setup………………………………………………………………………………………………………………………..
FRF Calculations…………………………………………………………………………………………………………………………….
CHAPTER 3:
Results…………………………………………………………………………………………………………………………………………..
Conclusion……………………………………………………………………………………………………………………………………..

References……………………………………………………………………………………………………………………………………
Appendices
Computer program……………………………………………………………………………………………………………………….

CHAPTER 1:
Introduction:
Modes are used as a simple and efficient means of characterizing resonant vibration. The majority of structures can be made to resonate. That is, under the proper conditions, a structure can be made to vibrate with excessive, sustained, oscillatory motion.
Resonant vibration is caused by an interaction between the inertial and elastic properties of the materials within a structure.
Resonant vibration is often the cause of, or at least a contributing factor to many of the vibration related problems that occur in structures and operating machinery. To better understand any structural vibration problem, the resonances of a structure need to be identified and quantified.
A common way of doing this is to define the structure’s modal parameters.

What is Experimental Modal Analysis?
In a nutshell, we could say that modal analysis is a process whereby we describe a structure in terms of its natural characteristics which are the frequency, damping and mode shapes its dynamic properties. Well that’s a mouthful so let’s explain what that means. Without getting too technical, I often explain modal analysis in terms of the modes of vibration of a simple plate. This explanation is usually useful for engineers who are new to vibrations and modal analysis.
Let’s consider a freely supported flat plate (Figure 1). Let’s apply a constant force to one corner of the plate. We usually think of a force in a static sense which would cause some static deformation in the plate. But here what I would like to do is to apply a force that varies in a sinusoidal fashion. Let’s consider a fixed frequency of oscillation of the constant force. We will change the rate of oscillation of the frequency but the peak force will always be the same value – only the rate of oscillation of the force will change. We will also measure the response of the plate due to the excitation with an accelerometer attached to one corner of the plate.

Now if we measure the response on the plate we will notice that the amplitude changes as we change the rate of oscillation of the input force (Figure 2). There will be increases as well as decreases in amplitude at different points as we sweep up in time. This seems very odd since we are applying a constant force to the system yet the amplitude varies depending on the rate of oscillation of the input force. But this is exactly what happens – the response amplifies as we apply a force with a rate of oscillation that gets closer and closer to the natural frequency (or resonant frequency) of the system and reaches a maximum when the rate of oscillation is at the resonant frequency of the system. When you think about it, that’s pretty amazing since I am applying the same peak force all the time only the rate of oscillation is changing!

This time data provides very useful information. But if we take the time data and transform it to the frequency domain using the Fast Fourier Transform then we can compute something called the frequency response function (Figure 3).

Now there are some very interesting items to note. We see that there are peaks in this function which occur at the resonant frequencies of the system. And we notice that these peaks occur at frequencies where the time response was observed to have maximum response corresponding to the rate of oscillation of the input excitation.
Now if we overlay the time trace with the frequency trace what we will notice is that the frequency of oscillation at the time at which the time trace reaches its maximum value corresponds to the frequency where peaks in the frequency response function reach a maximum (Figure 4). So you can see that we can use either the time trace to determine the frequency at which maximum amplitude increases occur or the frequency response function to determine where these natural frequencies occur. Clearly the frequency response function is easier to evaluate.

Most people are amazed at how the structure has these natural characteristics. Well, what’s more amazing is that the deformation patterns at these natural frequencies also take on a variety of different shapes depending on which frequency is used for the excitation force.
Now let’s see what happens to the deformation pattern on the structure at each one of these natural frequencies. Let’s place 45 evenly distributed accelerometers on the plate and measure the amplitude of the response of the plate with different excitation frequencies. If we were to dwell at each one of the frequencies – each one of the natural frequencies – we would see a deformation pattern that exists in the structure (Figure 5). The figure shows the deformation patterns that will result when the excitation coincides with one of the natural frequencies of the system. We see that when we dwell at the first natural frequency, there is a first bending deformation pattern in the plate shown in blue (mode 1). When we dwell at the second natural frequency, there is a first twisting deformation pattern in the plate shown in red (mode 2). When we dwell at the third and fourth natural frequencies, the second bending and second twisting deformation patterns are seen in green (mode 3) and magenta (mode 4), respectively. These deformation patterns are referred to as the mode shapes of the structure. (That’s not actually perfectly correct from a pure mathematical standpoint but for the simple discussion here, these deformation patterns are very close to the mode shapes, from a practical standpoint.)

These natural frequencies and mode shapes occur in all structures that we design. Basically, there are characteristics that depend on the weight and stiffness of my structure which determine where these natural frequencies and mode shapes will exist. As a design engineer, I need to identify these frequencies and know how they might affect the response of my structure when a force excites the structure. Understanding the mode shape and how the structure will vibrate when excited helps the design engineer to design better structures. Now there is much more to it all but this is just a very simple explanation of modal analysis.
So, basically, modal analysis is the study of the natural characteristics of structures. Understanding both the natural frequency and mode shape helps to design my structural system for noise and vibration applications. We use modal analysis to help design all types of structures including automotive structures, aircraft structures, spacecraft, computers, tennis rackets, golf clubs . . . the list just goes on and on.
Now we have introduced this measurement called a frequency response function but exactly what is it?

CHAPTER 2:
Experiment setup:
Equipment’s required:
1. An impact hammer with a load cell attached to its head to measure the input force.
2. An accelerometer to measure the response acceleration at a fixed point & direction.
3. A 2 or 4 channel FFT analyzer to compute FRFs.
4. Post-processing modal software for identifying modal parameters and displaying the mode shapes in animation.
Figure 6

FRF MEASUREMENTS
The Frequency Response Function (FRF) is a fundamental measurement that isolates the inherent dynamic properties of a mechanical structure. Experimental modal parameters (frequency, damping, and mode shape) are also obtained from a set of FRF measurements.
The FRF describes the input-output relationship between two points on a structure as a function of frequency, as shown in Figure 3. Since both force and motion are vector quantities, they have directions associated with them. Therefore, an FRF is actually defined between a single input DOF (point & direction), and a single output DOF.
An FRF is a measure of how much displacement, velocity, or acceleration response a structure has at an output DOF, per unit of excitation force at an input DOF. Figure 7 also indicates that an FRF is defined as the ratio of the Fourier transform of an output response ( X(w) ) divided by the Fourier transform of the input force ( F(w) ) that caused the output.

Figure7
Depending on whether the response motion is measured as displacement, velocity, or acceleration, the FRF and its inverse can have a variety of names,
· Compliance:(displacement / force)
· Mobility: (velocity / force)
· Inertance or Receptance: (acceleration / force)
· Dynamic Stiffness: (1 / Compliance)
· Impedance: (1 / Mobility)
· Dynamic Mass: (1 / Inertance)
An FRF is a complex valued function of frequency that is displayed in various formats, as shown in Figure8.

Figure8

Figure 9
FRF CALCULATION
Although the FRF was previously defined as a ratio of the Fourier transforms of an output and input signal, is it actually computed differently in all modern FFT analyzers. This is done to remove random noise and non-linearity’s (distortion) from the FRF estimates.
Tri-Spectrum Averaging
The measurement capability of all multi-channel FFT analyzers is built around a tri-spectrum averaging loop, as shown in Figure10 . This loop assumes that two or more time domain signals are simultaneously sampled. Three spectral estimates, an Auto Power Spectrum (APS) for each channel, and the Cross Power Spectrum (XPS) between the two channels, are calculated in the tri-spectrum averaging loop. After the loop has completed, a variety of other cross channel measurements (including the FRF), are calculated from these three basic spectral estimates.

Figure10
In a multi-channel analyzer, tri-spectrum averaging can be applied to as many signal pairs as desired. Tri-spectrum averaging removes random noise and randomly excited nonlinearity’s from the XPS of each signal pair. This low noise measurement of the effective linear vibration of a structure is particularly useful for experimental modal analysis.

Following tri-spectrum averaging, FRFs can be calculated in several different ways. Noise on the Output (H1) This FRF estimate assumes that random noise and distortion are summing into the output, but not the input of the structure and measurement system. In this case, the FRF is calculated as,
H1 = XPS/ Input APS where XPS denotes the cross power spectrum estimate between the input and output signals, and Input APS denotes the auto power spectrum of the input signal. It can be shown that H1 is a least squared error estimate of the FRF when extraneous noise and randomly excited nonlinearity’s are modeled as Gaussian noise added to the output [2].
Noise on the Input (H2)
This FRF estimator assumes that random noise and distortion are summing into the input, but not the output of the structure and measurement system. For this model, the FRF is calculated as,
H2= OutputAPS/XPS
Likewise, it can be shown that H2 is a least squared error estimate for the FRF when extraneous noise and randomly excited non-linearity’s are modeled as Gaussian noise added to the input. [2].
Noise on the Input & Output (HV) This FRF estimator assumes that random noise and distortion are summing into both the input but and output of the system.
Results:
Modal data are extremely useful information that can assist in the design of almost any structure. The understanding and visualization of mode shapes is invaluable in the design process.
It helps to identify areas of weakness in the design or areas where improvement is needed. The development of a modal model is useful for simulation and design studies. One of these studies is structural dynamic modification.
This is a mathematical process which uses modal data (frequency, damping and mode shapes) to determine the effects of changes in the system characteristics due to physical structural changes. These calculations can be performed without actually having to physically modify the actual structure until a suitable set of design changes is achieved. A schematic of this is shown in Figure 30. There is much more that could be discussed concerning structural dynamic modification but space limitations restrict this.
In addition to structural dynamic modification studies, other simulations can be performed such as force response simulation to predict system response due to applied forces. And another very important aspect of modal testing is the correlation and correction of an analytical model such as a finite element model. These are a few of the more important aspects related to the use of a modal model which are schematically shown in Figure 31.

Conclusion: Modern experimental modal analysis techniques have been reviewed in this report. The three main topics pertaining to modal testing; FRF measurement techniques, excitation techniques, and modal parameter estimation (curve fitting) methods were covered.
FRF based modal testing started in the early 1970’s with the commercial availability of the digital FFT analyzer, and has grown steadily in popularity since then. The modern modal testing techniques presented here are just a brief summary of the accumulation of the past 30 years of progress.