# BASICS OF STRUCTURAL VIBRATION TESTING AND ANALYSIS

Contents: 1. Section One | 2. Section Two | 3. Section Three | 4. Section Four

## Section 1

Introduction
Structural vibration testing and analysis contributes to progress in many industries, including aerospace, auto-making, manufacturing, wood and paper production, power generation, defense, consumer electronics, telecommunications and transportation. The most common application is identification and suppression of unwanted vibration to improve product quality.

This application note provides an introduction to the basic concepts of structural vibration. It presents the fundamentals and definitions in terms of the basic concepts. It also discusses practical applications and provides real world examples.

This paper covers the following topics:

• Basic terminology
• Models of single and multiple degree of freedom
• Continuous structure models
• Measurement techniques and instrumentation
• Vibration suppression methods
• Modal analysis
• Operating deflection shape analysis

Basic Terminology of Structural Vibration
The term vibration describes repetitive motion that can be measured and observed in a structure. Unwanted vibration can cause fatigue or degrade the performance of the structure. Therefore it is desirable to eliminate or reduce the effects of vibration. In other cases, vibration is unavoidable or even desirable. In this case, the goal may be to understand the effect on the structure, or to control or modify the vibration, or to isolate it from the structure and minimize structural response.

Vibration analysis is divided into sub-categories such as free vs. forced vibration, sinusoidal vs. random vibration, and linear vs. rotational vibration.

Free vibration is the natural response of a structure to some impact or displacement. The response is completely determined by the properties of the structure, and its vibration can be understood by examining the structure's mechanical properties. For example, when you pluck a string of a guitar, it vibrates at the tuned frequency and generates the desired sound. The frequency of the tone is a function of the tension in the string and is not related to the plucking technique.

Forced vibration is the response of a structure to a continuous forcing function that causes the structure to vibrate at the frequency of the excitation. For example, the rear view mirror on a car will always vibrate at the frequency associated with the engine's RPMs. In forced vibration, there is a deterministic relationship between the amplitude of the corresponding vibration level and the forcing function. The relationship is dictated by the characteristics of the structure.

Sinusoidal vibration is a special type of vibration. The structure is excited by a forcing function that is a pure tone with a single frequency. Sinusoidal vibration is not very common in nature, but it provides an excellent engineering tool that enables us to understand complex vibrations by breaking them down into simple, one-tone vibrations. The motion of any point on the structure can be described as a sinusoidal function of time as shown in Figure 1 (top).

Random vibration is very common in nature. The vibration a driver feels when driving a car results from a complex combination of sources, including rough road surface, engine vibration, wind buffeting the car's exterior, etc. Instead of trying to quantify each of these effects, they are commonly described by using statistical parameters. Random vibration quantifies the average vibration level over time across a frequency spectrum. Figure 1 (bottom) shows a typical random vibration versus time plot.

Figure 1. Sinusoidal vibration (top) and random vibration (bottom).

Rotating imbalance is another common source of vibration. The rotation of an unbalanced machine part can cause the whole rotating machine to vibrate. The imbalance generates the forcing function that affects the structure. Examples include a washing machine, an automobile engine, shaft system, steam or gas turbines, and computer disk drive. Rotational vibration is usually harmful and unwanted, and the way to eliminate or minimize it is to properly balance the rotating part of the machine.

The common element in all these categories of vibration is that the structure responds with some repeti­tive motion that is related to its mechanical properties. By understanding the basic structural models, measurement and analysis techniques, it is possible to successfully characterize and treat vibration in structures.

Time and Frequency Analysis
Structural vibration can be measured by using electronic sensors that convert vibration motion into electrical signals. By analyzing the electrical signals, the nature of the vibration can be understood. Signal analy­sis is generally divided into time and frequency domains; each domain provides a different view and insight into the nature of the vibration.

Time domain analysis starts by analyzing the signal as a function of time. An oscilloscope, data acquisition device, or dynamic signal analyzer can be used to acquire the signal. Figure 2 illustrates a structure model, such as a single-story building, responding to an impact of vibra­tion that is measured at point A and plotted versus time. The dashed lines indicate the motion of the structure as it vibrates about its equilibrium point.

Figure 2. Mechanical structure responds with vibration plotted versus time.

The plot of vibration versus time provides information that helps characterize the behavior of the structure. Its behavior can be characterized by measuring the maximum vibration (or peak) level, or finding the period (time between zero crossings), or estimating the decay rate (the amount of time for the envelope to decay to near zero). These characteristic parameters are the typical results of time domain analysis.

Frequency analysis also provides valuable information about structural vibration. Any time history signal can be transformed into frequency domain. The most common mathematical technique for transforming time signals into frequency domain is called Fourier Transform, named after the French Mathematician Jean Baptiste Fourier. The mathematical processing involved is complex, but today's dynamic signal analyzers race through it automatically in real-time.

Fourier Transform theory states that any periodic signal can be represented by a series of pure sine tones. Figure 3 illustrates how a square wave can be constructed by adding up a series of sine waves; each of the sine waves has a frequency that is a multiple of the fundamental frequency of the square wave. The amplitude and phase of each sine tone must be carefully chosen to get just the right waveform shape. When using a limited number of sine waves as in Figure 3, the result resembles a square wave, but the composite waveform is still ragged.

Figure 3. A square wave can be constructed by adding pure sine tones.

As more and more sine waves are added in Figure 4, the result looks more and more like a square-wave.

Figure 4. As more sine tones are added the square waveform shape is improved.

In Figures 3 and 4, the third graph shows the amplitude of each sine tones. In Figure 3, there are three sine tones, and they are represented by three peaks in the third plot. The frequency of each tone is represented by the location of each peak on the frequency coordinate in the horizontal axis. The amplitude of each sine tone is represented by the height of each peak on the vertical axis. In Figure 4, there are more peaks as there are more sine tones added together to form the square wave. This third plot can be interpreted as the Fourier Transform of the square wave.

In structural analysis, usually time waveforms are measured and their Fourier Transforms are computed. The Fast Fourier Transform (FFT) is a computationally optimized version of the Fourier Transform. The third plot in Figure 4 also shows the measurement of the square wave with a signal analyzer that computes its Fast Fourier Transform. With testing experience, structural vibration can be understood by studying frequency domain spectrum.

The Decibel dB Scale
Vibration data is often displayed in a logarithmic scale called Decibel (dB) scale. This scale is useful because vibration levels can vary from very small to very large values. When plotting the whole data range on most linear scales, the small signals become virtually invisible. The dB scale solves this problem because it compresses large numbers and expands small num­bers. A dB value can be computed from a linear value per following equation,

where xref is a reference number that depends on the type of measurement. Comparing the motion of a mass to the motion of the base, base measurement is used as the reference in the denominator and the mass as the measurement in the numerator.

In dB scale, if the numerator and denominator are equal, the level is zero dB. A level of +6 dB means the numerator is a factor of two times the reference value, and +20 dB means the numerator is a factor of 10 times the reference.

Figure 5 shows an FFT spectrum in linear scale on the top, and dB scale on the bottom. Notice that the peak near 200 Hz is nearly indistinguishable on the linear scale, but very pronounced with the dB scale.

Figure 5. The dB scale allows us to see both large and small num­bers on the same scale as shown for the FFT with linear scale on the top and dB scale on the bottom.

## Section 2

Structural Vibration

Structural vibration can be complex, so let's start with a simple model to derive some basic concepts and build up to more advanced models. The simplest vibration model is the single-degree-of-freedom, or a mass-spring-damper model. It consists of a simple mass (M) that is suspended by an ideal spring with a known stiffness (K), and a dashpot damper from a fixed support. A dashpot damper is like a shock absorber in a car. It produces an opposing force that is proportional to the velocity of the mass.

Figure 6. Simple Mass-Spring-Damper Vibration Model

In this model, the factors that affect vibration are completely characterized by the parameters M, K and C. Knowing these values, the structural response to excitation can be predicted exactly.

The mass is a measure of the density and amount of the material. A marble has a small mass and a bowling ball has a relatively larger mass. The stiffness is a measure of how much force the spring will pull when stretched by a given amount. A rubber band has a small stiffness and a car leaf spring has a relatively large stiffness. A sports car with a tight suspension has more damping than a touring car with a soft suspension. When the touring car hits a bump, it oscillates up and down for a longer time than the sports car. Different materials have different damping qualities. Rubber, for example, has much more damping than steel.

If the mass is displaced by pulling down and releasing, the mass will respond with motion similar to Figure 7. The mass will oscillate about the equilibrium point and after every oscillation, the maximum displacement will decrease due to the damper element, till the motion becomes so small that it is undetectable. Eventually the mass element will stop moving.

Figure 7. Free Vibration of Mass-Spring-Damper Model

Figure 7 shows that the time between every oscillation is same. The time plot crosses zero at regular intervals. The time for the displacement to cross zero with a positive slope to the next zero crossing with positive slope is named the period. It is also related to the frequency of oscillation.

Frequency can be computed by dividing one by the period value. For the mass-spring-damper model subject to free vibration, the frequency of oscillation is completely determined by the parameters M, K, and C. It is called the natural frequency denoted by the symbol fn. Frequency is measured in cycles per second with units of Hertz (Hz). Assuming the damping is small, then the mathematical relationship is given by

A larger stiffness will result in a higher fn, and a larger mass will result in a lower fn.

Figure 7 also reveals something about damping. Theory tells us that the amplitude of each oscillation will diminish at a predictable rate. The rate is related to the damping factor C. Usually damping is described in terms of the damping ratio ζ. That ratio is related to C by

The damping ratio can vary from zero to infinity. When it is small (less than about 0.1), the system is lightly damped. When excited, it will oscillate, or ring, for a long time as shown in Figure 8. When the damping ratio is large, the system is 'over damped'. It will not oscillate at all and it may take a long time to return to its equilibrium position. When the damping ratio = 1, the system is 'critically damped' and will return to the equilibrium position in the shortest possible time.

Figure 8. Free vibration of mass-spring-damper for three cases: A - under damped, B - critically damped, C - over damped.

Vibration theory also describes how the mass-spring-damper model will respond to forced vibration. Imagine that the base of the mass-spring-damper is not fixed, but instead forced to move up and down a small distance. If this base motion is sinusoidal, it is straightforward to predict how the mass will respond to the forced vibration. The mass will begin to move in sinusoidal motion at the same frequency as the base; the amplitude of the mass displacement will vary depending on the frequency of the base motion. This is shown in Figure 9 where the mass displacement is larger than the base displacement. There will also be a phase difference between the base and the mass displacement.

Figure 9. Mass-Spring-Damper Model responds to forced vibration with change in amplitude and phase: A-mass vibration, B-base displacement.

The relative amplitude and phase of the mass displacement will vary with the frequency of the base excitation. Varying the base frequency and recording the corresponding mass displacement amplitude (divided by the base displacement amplitude) and the phase, results in a plot of the vibration amplitude and phase versus excitation frequency.

Figure 10 shows the results for three different values of the damping ratio (ξ). In all cases, the amplitude ratio (Magnitude) is 1 (or zero dB for dB scale) for low frequencies. This means that the amplitudes are equal at low frequencies. As the frequency increases, the magnitude rises to some maximum value. The frequency of this peak is fn, the natural frequency. At frequencies above fn, the magnitude falls of at a constant rate. This plot is known as a Bode Diagram. Note the horizontal axis is of log scale with unit Hz. Note that the peak is higher for light damping. For critical damping (red curve) and over damping (not shown), the magnitude does not increase above 1 (or 0 dB).

Figure 10. Bode Diagram of the vibration of the mass-spring­-damper system with A: ζ = 0.1, B: ζ = 0.5, C: ζ = 1

The lower plot in Figure 10 shows the phase relation between the base and the mass for different frequencies. At low frequencies below fn, the phase is zero degrees meaning that the mass is in phase with the base. When the base vibration frequencies coincide with fn (at resonance), the phase is 90 degrees. At high frequencies, the phase is 180 degrees; the mass is out of phase with the base and they move in opposite directions. Note that different damping ratios affect the slope of the phase change.

The Q Factor is a common term used to represent how underdamped an structure is. Vibration theory shows that the damping ratio is related to the sharpness of the peak of the magnitude plot. The damping ratio, ζ, can be determined by computing Q, defined as the resonant frequency divided by the half power bandwidth around the peak at fn. Q factor is computed as:

where fn is the resonant frequency at the peak in Hz; f1 and f2 are the half power points measured -3 dB down from the peak as shown in Figure 11.

Another way to measure damping is to simply record the peak amplitude ratio. However this should not be confused with the Q factor as they are not equivalent.

Figure 11. The Q factor is computed by dividing the peak frequen­cy by the half power bandwidth.

The single-degree-of-freedom, mass-spring-damper model is an over simplification of most real structures. The concepts and terminologies introduced in analyzing how free and forced vibration affect this simple model also apply to analyzing more complex structures.

Multi-Degree of Freedom Model
The first model introduced in this paper is a single-story building. This model can be extended to a two or three story building resulting in a two or three degree of freedom system. The two story building model can be rep­resented by interconnecting simple mass-spring-damper systems as shown in Figure 12.

Figure 12. A two story building can be modeled as a two degree of freedom model and simplified into two coupled mass-spring-damper systems.

The coupled mass-spring-damper system will have two resonant frequencies. This system is characterized by a Bode Diagram similar to Figure 13. The damping of each resonance can be determined using the Q factor technique. Note that the first resonance is more light­ly damped compared to the second resonance judging from the sharpness of the peaks.

Figure 13. Bode diagram of 2 mass-spring-damper system.

In the models considered so far, mass is lumped into one point. A continuous structure such as a beam or string where the mass is distributed over volume requires another type of model. Figure 14 illustrates a beam structure pinned at both ends so that it can rotate but cannot translate. Under excitation, the beam will deform, vibrate and deform per different shapes depend­ing on the frequency of the excitation as well as mount­ing method (boundary conditions) of the beam ends. The beam will have a first res­onant frequency at which all its points will move in uni­son; at the first resonant frequency, the beam will take the shape shown to the right in Figure 14 labeled First Mode Shape. At a higher frequency, the beam will have a second resonant frequency and mode shape, and a third, and fourth, etc. Theoretically there are an infinite number of resonant frequencies and mode shapes.

However at higher frequencies, the structure acts like a low-pass filter and the vibration levels get smaller and smaller. The higher modes are harder to be excited and have relatively less effect on the overall vibration of the structure.

Figure 14. A beam fixed at both ends is an example of a continu­ous structure model.

Figure 15 shows an experimentally measured Bode Diagram for a 1 x 0.25 inch cross section and 8 inch long steel beam. There are many resonant frequencies, begin­ning as low as 265 Hz and many higher ones across the frequency range. Around every resonant peak, its phase goes through a 180 degree phase shift. Notice that when the resonances are well separated per frequen­cy axis, the frequency response function of each resonance is similar to that of a simple spring-mass-damper system.

Figure 15 A typical Bode Diagram for a beam.

This overview of structural vibration analysis shows that the field has many complexities and considerable depth. Fortunately for technical and non-technical people alike, the fundamental phenomena and concepts apply to models no matter it is simple or complex, and they can be represented by either single-degree-of-freedom model, multiple-degree-of-freedom model, or continuous structure model.

Vibration Measurements
Vibration Sensors
Structural vibration is commonly measured with electronic sensors called accelerometers. These sensors convert an acceleration signal to an electronic voltage signal that can then be measured, analyzed and recorded with electronic hardware. There are many types of accelerometers. Some common one requires a power supply connected by a cable to the accelerometer as shown in Figure 16. Some accelerometers have internal circuitry that accepts the DC power from the analyzer's ADC channel. The dynamic signal analyzer includes a calibration setting parameter for each transducer that allows the voltage signal to be converted into the measurement of acceleration, i.e., g or m/s2.

Figure 16. Typical instrumentation for accelerometer.

Manufacturers calibrate each accelerometer and supply a sensitivity value. For example, a 100 mV/g nominal sensitivity accelerometer will have a calibration value of 102.3 mV/g. Measurement accuracies depend on using the correct sensitivity value in the signal analyzer and on using an accelerometer with the right range of sensitivity for the application. A high sensitivity sensor with 1000 mV/g sensitivity may not be appropriate for an application of high acceleration level. In this case the too high voltage from the sensor will saturate the input channel circuitry on the signal analyzer. If the acceleration level is very low, a small sensitivity accelerometer, such as 10 mV/g one, may produce a signal that is too weak to measure and affect the accuracy of measurement.

Sensitivity also has an important impact on the signal to noise ratio. Signal to noise ratio is the ratio of the signal level divided by the noise floor level and typically measured in dB scale as:

All sensors and measurement hardware are subject to electronic noise. Even when the structure is not vibrate, electronic noise from the measurement elements may still shows some small acceleration level. This is due to the sensor cables picking up electronic noise from stray signals in the air, from noise in the power supply or from internal noise in the analyzer electronics. High quality hardware is designed to minimize the internal noise making low signal measurements possible. The signal to noise ratio limits the lowest measurement that can be made.

For example, if the noise floor is 1 mV, then with a typical 100 mV/g accelerometer, the smallest level that can be read, can be computed as:

1 mV divided by 100mV⁄g = 0.01 g

If a 20 g acceleration is measured with this accelerometer, then the signal to noise ratio can be computed as:

In this case, the analyzer will always show a level of at least 0.01 g because of the noise floor. It is a good practice to not trust a measurement with a signal to noise ratio that is below 3:1 or 4:1. When the signal to noise ratio is too low, one solution is to use an accelerometer with a higher sensitivity.

For example, with 1 mV of noise, the smallest level that can be read by a 1000 mV/g accelerometer can be computed as:

1 mV divided by 1000mV⁄g = 0.001 g

Some accelerometer power supplies include a gain setting that multiplies the signal by 1, 10, or 100. Unfortunately, this gain setting also amplifies the noise that is picked up by the wires and inherent in the sensor and power supply. In most cases, increasing the power supply gain will not solve signal to noise issues.

## Section 3

Excitation Methods

In some applications, vibration measurements are made during normal operation of a structure or machine. For example, an automobile can be instrumented with many accelerometers and driven on road or a test track while vibration signals are measured and analyzed. Many other cases require a more tuned excitation to yield reproducible and predictable results. The two most common methods are the impact hammer and electrodynamic shaker.

An impact hammer, as shown in Figure 17, is a specialized measurement tool that produces short dura­tion vibration levels by striking the structure at certain point. The hammer incorporates a sensor (called a force sensor) that produces a voltage signal proportional to the force of impact. This enables precise measurement of the excitation force. An impact hammer is often used for modal analysis of structures where use of a shaker is not convenient; examples are in the field or with very large structures. Different impact tip materials allow tailoring of the frequency content of the impact force. For low frequency measurements, a soft rubber tip concentrates the excita­tion energy in a low narrow frequency range. A hard metal tip gives good excitation energy content out to high frequencies.

Figure 17. Impact hammer instrumented with a force sensor to meas­ure the excitation force and different hardness tips

For laboratory vibration measurements, modal shakers are the instruments of choice. Modal shakers are rated by the force they produce. Modal Shakers vary in size and force from several pound force to hundred pound force.

Figure 18. Modal shakers are used in laboratory measurements and vary in size from small to large.

Modal Shaker is connected to structures by means of a thin metal rod called a stinger in general. Force sensor is mounted on the structure, and then connected through the stinger to the modal shaker. There is a type of sensor called impedance head, which is a combination of force sensor and accelerometer in one. Using this sensor, both the driving force and acceleration level at the driving point on the structure can be measured simultaneously.

A dynamic signal analyzer incorporates a source type of signal, which is amplified and sent to the modal shaker to excite the structure under test.

Dynamic Signal Analyzers
The most common equipment for analyzing vibration signals is a computer based data acquisition system called a dynamic signal analyzer (DSA) (see Figure 19). The first generation DSAs used analog tracking filters to measure frequency response. Modern analyzers use digital technology and are far faster and more versatile. Using disk drives to store large volumes of data for post processing, they can record all sorts of data including time, frequency, amplitude and statistic data.

Figure 19. Crystal Instruments Dynamic Signal Analyzer family

A modern DSA consists of many electronic modules as shown in Figure 20. First, the analyzer measures electronic signals with an analog front end that may include special signal conditioning such as sensor power supply, TEDS (transducer electronic data sheets that read the calibration and other information from a chip embedded in the sensor), adjustable voltage gains settings and analog filters. Next, the sys­tem converts the analog signal to a digital format via an analog to digital converter (ADC). After the signal is digitized, the system processes it with a digital signal processor (DSP), which is a mini-computer opti­mized to do rapid mathematical calculations. The DSP performs all required calculations, including additional filtering, computation of time and frequency measure­ments, and management of multiple channel signal measurements.

Most modern signal analyzers connect with a PC for the setup, display ad reporting functions. The DSP interfaces with the PC; a software user interface displays results on the PC screen. Some older model of analyzers did not have a PC interface; they include buttons and a display screen on the analyzer chassis. The PC interface speeds and simplifies test setup and reporting.

Figure 20. Signal analyzer architecture

Quantization: Analog to Digital Conversion and Effective Bits
One measure of DSA quality is the bit count of the analog to digital conversion (Demler, 1991). When an analog signal is converted into a digital signal, it undergoes quantization, which means that a perfectly smooth analog signal is converted into a signal represented by stair steps as shown in Figure 21. To accurately represent an analog signal, the stair steps of the digital signal should be as small as possible. The step size depends on the number of "bits" in the ADC and the voltage range of the analog input. For example, if the voltage range is 10 volts and the ADC uses 16 bits, the smallest step size can be computed as,

Step size = 10 volts divided by 216 = 0.15 millivolts

A 24-bit ADC, reduces the step size to

Step size = 10 volts divided by 224 = 0.0006 millivolts

The 24-bit ADC has a step size that is 256 times smaller than the step size of the 16-bit ADC. Twenty four bit is the highest bit count available in modern DSAs. A 24-bit ADC can accurately measure large signals and small signals at the same time. ADCs with lower bit counts require the user to change to lower voltage range when measuring a low level signal in order to achieve the similar measurement accuracy.

Figure 21. When an analog signal is converted to digital the smooth analog signal becomes a stair step signal with the size of the step depending on the bit count of the ADC.

The bit count is also related to dynamic range of the DSA, another important measurement quality consideration. Dynamic range shows the ratio of the largest signal to the smallest signal that a DSA can accurately measure; dynamic range is reported in the dB scale. A typical 24-bit ADC with good low-noise performance will have 110 - 120 dB of dynamic range. Dynamic range is also affected by the noise floor of the device.

Measurement Types: time vs. frequency, FFT, PSD, FRF, Coherence
Most analyzers carries out time and frequency measurements. Time measurements include capturing tran­sient signals, streaming long duration events to the computer disk drive, and statistic measures. The sampling rate, another measure of DSA quality, is related to time measurements. High speed ADCs can attain sampling rates of up to 100,000 samples per second (100 kHz). Some analyzers incorporate multiplexers that use one ADC sampling at a high rate and switch it between the different input channels. For example, if the DSA has 8 channels and a multiplexing 100 kHz ADC, it will only sample at 12.5 kHz per channel. High-quality DSAs do not use multiplexers and provide high sampling rates regardless of the number of input channels.

Most DSAs can compute a variety of frequency measurements including Fast Fourier Transform, Power Spectral Density, Frequency Response Functions, Coherence and many more. The DSP computes these signals from digitized time data. Time data is digitized and sampled into the DSP block by block. A block is a fixed number of data points in the digital time record. Most frequency functions are computed from one block of data at a time.

Fast Fourier Transform (FFT) is the discrete Fourier Transform of a block of time signal. It represents the frequency spectrum of the time signal. It is a complex signal meaning that it has both magnitude and phase information and is normally displayed in a Bode Diagram. Figure 22 shows the FFT of a square wave measured by a Crystal Instruments DSA Analyzer. The horizontal axis shows frequency ranging from zero to 225 Hz; the vertical axis is m/s2 ranging from 0 to 14 m/s2. A square wave is composed of many pure sine waves indicated by the discrete peaks at even intervals in the FFT.