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The impulse excitation technique-also known as the impulse excitation of vibration, resonant vibration, impact acoustic resonance, ping test and eigen frequency methods

^{1}-has been receiving increased attention as a precise and repeatable way to measure the dynamic elastic properties of materials

^{2}and detect defects. It is one of the quickest and easiest methods available for these applications. Furthermore, with recent advances in computer software and hardware, and similar advances in programming languages and their integrated development environments, nondestructive testing instruments designed for use with the impulse excitation technique can now be developed at much lower costs than traditional analog instruments.

Given these benefits, it's not surprising that such instruments are finding widespread use in the high-tech ceramic industry as research and quality control tools.

## The Principle of Impulse Excitation

The impulse excitation technique is described in detail in ASTM standards E1867 and C1259. Basically, a small hammer or impulse tool strikes a solid and creates a standing wave. An impulse-excited body can vibrate in several different modes, such as flexural or transverse (bending), torsional (twisting), and longitudinal, and each mode has a fundamental resonant frequency and higher frequency overtones associated with it.An impulse-excited solid might be expected to vibrate at several resonant frequencies simultaneously. However, for simple shapes, such as beams and plates, it is possible to support them and excite them in such a way that only one mode of vibration is prevalent (see Figure 1). The sample is struck at one of the antinodes (in this case, the center of the top face), and the frequency of vibration is then measured at another antinode (the left end) using a microphone and frequency analyzer or frequency counter. Other sensors, such as contact piezoelectric probes and laser vibrometers, can also be used.

^{3,4}and for discs.

^{5,6,7}Solutions for plates

^{8,9}and shells

^{10,11}are available elsewhere. For bars of any uniform cross-section, and for the flexural mode of vibration, Equation 1 can be expanded to:

^{12,13,14}(see Equation 2)where f

_{n}is the flexural resonant frequency; L is the length of the sample;

*k*is the radius of gyration (t/√12 or D/4 for a rectangular prism or cylinder respectively, where t = the dimension in the direction of vibration and D = diameter);

*m*is a constant dependent on the order of frequency ~p(2

_{n}*n*+1)/2; and T

_{n}is the correction factor to take into account shear deformation. The subscript

*n*denotes the order of frequency (1 = fundamental, 2 = first overtone, etc.).

T_{n} is a function of the slenderness ratio, *L/k,* and Poisson's ratio. *T _{n}* approaches unity for long, thin bars (

*L/k*>400). For standard test bars (rectangular or circular cross section),

*T*is well known

_{n}^{13,14}and precise for the fundamental (

*n*= 1) and is given in the ASTM standards. For more complicated shapes, finite element analysis is often used.

For beams, the Young's modulus is derived from the flexural or longitudinal frequency, and the shear modulus is derived from the torsional frequency. For isotropic elastic solids, any of the five elastic constants can be calculated from any other two. As a result, all five elastic constants can be obtained from just the Young's modulus and shear modulus. Two frequency measurements are usually required from two different modes of vibration. However, since the correction factor *T _{n}* in Equation 2 is a function of the Poisson's ratio, as well as the order of the frequency,

^{15,16}it is possible (at least in principle) to derive the Poisson's ratio from the overtones for a single mode of vibration. Therefore, all of the elastic constants can be obtained from one mode of vibration. This has implications where only one mode of vibration can be excited easily-such as with cylinders, where the torsional mode of vibration is difficult to excite-and for high-temperature testing, where a simplified design can be used to reduce costs and speed testing.

## Some Practical Uses

The impulse excitation technique is primarily used in research and development applications and for quality control/quality assurance in manufacturing.In research, the requirement is generally to measure the elastic constants of the materials being studied. In this case, ASTM standards such as E1876 and C1259 can be followed, and any elastic material of almost any size can be measured. The main limitation regarding size is determined by the highest frequency that can be measured by the frequency analyzer, and by the precision of the dimension measurements. For example, a very small or very thick piece will tend to have a very high frequency and might not be measurable, depending on the frequency range of the instrument. Additionally, very small samples will need to be measured to high precision to reduce the propagation of errors in Equation 2. One can expect errors of about 1% in Young's modulus calculations in most cases, even though the resonant frequencies can be measured to tolerances as high as +0.06%.^{17} Nevertheless, according to some independent research at Oak Ridge National Laboratories by Lara-Curzio *et al*,^{2} impulse excitation gives higher precision and repeatability compared to static methods such as nanoindentation and four-point bending for isotropic materials.

In quality control and quality assurance applications, a precise measure of the elastic constants might also be desired; however, there is usually more interest in monitoring product quality directly. For samples of similar sizes and bulk densities, one or more characteristic resonant frequencies* can be used as a good indicator of quality. This has the advantage that the dimensions and mass of the sample being tested need not be measured. Lower control limits for the characteristic frequency can be defined, which allows defective parts to be easily identified and rejected. Furthermore, the power spectra and waveforms displayed by sophisticated impulse excitation instruments can be obtained on control samples and compared to those of the parts being tested. Unusual features, such as extra peaks in the power spectrum, would indicate a substandard part.

In both cases, the analysis is quick and easy. The operator simply sets up the supports for the sample(s), obtains the dimensions and mass, and measures the resonant frequencies. Generally, between one and four resonant frequencies need to be measured (one is typical), each requiring just a few seconds to complete. This compares to the several minutes required to obtain such measurements using static methods. The dimension, mass and frequency data are then recorded in a spreadsheet, and the instrument uses that information to automatically calculate the elastic constants.

*Sometimes the square of a characteristic resonant frequency is used as a quality indicator.

## R&D Experiments

To demonstrate the capabilities of the impulse excitation method, a commercial nondestructive impulse excitation testing system^{§}was used to measure the resonant frequencies of some ceramic solids. The instrument analyzes the resonant frequencies by using a fast Fourier transform algorithm and displays the waveform and power spectrum of the vibrations. Although the instrument has built-in algorithms for Equation 2, a Microsoft

^{®}Excel

^{®}spreadsheet was used here for convenience.

Under the research category, several thin samples (approximately 500 æ) of a NiO/8YSZ anode support material used in a solid oxide fuel cell (SOFC) were received from Zbigniew Rak, Ph.D., senior scientist at the Energy Research Centre of the Netherlands (ECN). The samples were in the form of six rectangular bars (samples 1F-, ~50 mm x 20 mm) and one whole plate (sample 4F, ~130 mm x 130 mm). The aim of this work was to determine if the nondestructive tester could be used to measure the Young's moduli of these very thin and porous (~38-40%) samples, which would be difficult to measure accurately by any other means. Note that the bars were test specimens cut from a whole plate.

The results for the NiO/8YSZ anodes are shown in Table 1. For sample 4F, the "flexural" frequency is the value for the third mode of vibration, and the "torsional" frequency is the value for the second mode of vibration for a square plate.^{9} The Young's modulus of sample 4F was calculated from frequency equations for square plates derived from solutions by Hurlebaus.^{9} Similar results were obtained using equations provided by Leissa,^{8} and Leissa and Narita.^{11}

^{§}Buzz-o-sonic, supplied by BuzzMac Software LLC, Glendale, Wis.

Sample 4F had a significantly lower Young's modulus than expected from its bulk density. This may be due to processing variables. Also, 4F was much larger than the bars and likely to contain more defects.

Even though the samples were only 500 æ thick and had up to 40% porosity, the elastic moduli were easily measured, and a correlation to bulk density was obtained. The tests were performed in a matter of seconds. Although the bars were machined from larger pieces, the production specimens could have been measured directly.

## QC/QA Experiments

A typical goal in ceramic manufacturing is to identify weak parts nondestructively. Fortunately, in many cases the impulse excitation technique can be used to do just that.To demonstrate this application of the technique, six square ceramic tiles were tested.^{†} Two of the tiles (Control A and Control B) were control samples, believed to be good-quality parts, while the other four samples (Samples 1-4) were taken from a low-quality batch.

The basic assumptions used in the impulse excitation technique are that the samples are isotropic, homogenous and linearly elastic. Some of these assumptions will not hold where defects are present. Thus, for the square tiles tested here, one might expect that all of the difference in resonant frequencies along the major axes would be a good indicator of product quality-the larger the difference, the more defective or heterogeneous the part. For a homogeneous composition, the difference in resonant frequencies will be due to defects only. Consequently, the longitudinal resonant frequencies of the six tiles were measured along both main axes. The difference in each frequency, expressed as a ratio of the minimum, was then calculated as shown in Figure 3.

^{-1}). In general, Q

^{-1}increases with the number or severity of defects.

^{18}(see Equation 3)where

*A*is amplitude at time (

*t*);

*A*is the initial amplitude following impulse excitation;

_{0}*f*is the resonant frequency of interest; and z is the damping ratio, which determines system damping capacity. For a given damping ratio, higher frequencies decay more rapidly. Some analog instruments take advantage of this by allowing the overtones to decay and the fundamental to be more easily measured. More sophisticated instruments are designed to measure the power spectra so that all of the vibrations can be analyzed before the overtones decay.

_{n}^{-1}) can be obtained from the damping ratio by the equation: (see Equation 4) where d, the ratio of successive amplitudes, is called the logarithmic decrement.

^{19}The internal frictions of the samples analyzed in this example are shown in Figure 5. Once again, Sample 1 appears to be defective.

As illustrated by Figures 3-6, Sample 1 is shown to be defective. To confirm this analysis, a proprietary strength test was used. Sample 1 was found to have approximately half the strength of the other samples, even though it was not visibly cracked.

## Quick and Easy Analysis

As manufacturers search for ways to become more competitive, quality will play an increasingly important role. As demonstrated in the examples discussed in this article, tools such as the impulse excitation technique can be used to quickly and accurately analyze difficult samples such as thin, porous NiO/8YSZ anode materials and other high-tech ceramic components.## Author's Acknowledgement

The Energy Research Centre of the Netherlands (ECN) is gratefully acknowledged for its support of the work described in this article.**For more information about the impulse excitation technique, contact BuzzMac Software LLC, 6720 North Sidney Place #104, Glendale, WI 53209; (414) 352-5419; fax (253) 540-9798; e-mail paul@buzzmac.com ; or visit http://www.buzzmac.com .**

## References

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3. E. Goens, "Euber die Bestimmung des Elastizitätsmoduls von Stäben mit Hilfe von Biengungsschwingungen," Annalen der Physik, B. Folge, Band 11 649-678 (1931)

4. S. P. Timoshenko, "On the Transverse Vibrations of Bars of Uniform Cross Section," Phil. Mag. Ser., 6 [43] 125-131 (1922)

5. G. Martincek, "The Determination of Poisson's Ratio and The Dynamic Modulus of Elasticity from the Frequencies of Natural Vibration in Thick Circular Plates," J. Sound Vib., 2 [2] 116-127 (1965).

6. J. C. Glandus, F. Platon, and P. Boch, "Measurement of the Elastic Moduli of Ceramics", Materials in Engineering Applications, 1 243-247 (June 1979)

7. J. C. Glandus, "Rupture fragile et resistance aux choces thermiques d cèramiques à usages mècaniques," Ph.D. Dissertation, University of Limoges (1981)

8. Arthur W. Leissa, "Vibration of Plates," Acoustical Society of America, Published in 1993; Originally issued by NASA in 1973 (published 1969)

9. S. Hurlebaus, "Nondestructive Evaluation of Composite Laminates," NDT.net 4 [3] (1999)

10. Arthur W. Leissa, "Vibration of Shells," Acoustical Society of America, Published in 1993; Originally issued by NASA in 1973

11. Arthur W. Leissa and Y. Narita, "Vibrations of Completely Free Shallow Shells of Rectangular Planform," J. Sound & Vib. 96 [2] 207-218 (1984)

12. G. Pickett, "Equations for Computing Elastic Constants from Flexural and Torsional Resonant Frequencies of Vibration of Prisms and Cylinders," Proceedings ASTM, 45 846-865 (1945)

13. S. Spinner, T. W. Reichard, and W. E. Tefft, "A Comparison of Experimental and Theoretical Relations Between Young's Modulus and the Flexural and Longitudinal Resonance Frequencies of Uniform Bars," J. Res. of the National Bureau of Standards-A. Physics and Chemistry, 64 [2] (1960)

14. A. Spinner, and W. E. Tefft, "A Method for Determining Mechanical Resonance Frequencies and for Calculating Elastic Moduli from these Frequencies," Proceedings ASTM, 61 1221-1238 (1961)

15. X. Q. Wang, "The Measurement of Elastic Moduli by Flexural Vibration Testing," Ninth International Conference on Sound and Vibration (Orlando, Florida, USA), 8-11 July (2002)

16. Kolluru S. V., Popovics, J. S., and Shah, S. P., "Determining Elastic Properties of Concrete Using Vibrational Resonance Frequencies of Standard Test Cylinders," Cement, Concrete, and Aggregates, CCAGDP, 22 [2] 81-89 (2000)

17. Smith, John S., Wyrick, Michael D., and Poole, Jon M., "An Evaluation of Three Techniques for Determining the Young's Modulus of Mechanically Alloyed Materials," Dynamic Elastic Modulus Measurement in Materials, ASTM STP 1045, Alan Wolfenden, ed., ASTM, Philadelphia, PA, 1990

18. C. Y. Wei and S. N. Kukureka, "Evaluation of Damping and Elastic Properties of Composite Structures by the Resonance Technique," by J. Mat. Sci., 35 3785-3792 (2000)

19. E. Schreiber, O. Anderson, and N. Soga, "Dynamic Resonance Method for Measuring the Elastic Moduli of Solids, Elastic Constants and their Measurement," p121, McGraw Hill (1973)