USDA FOREST SERVICE

RESEARCH PAPER

FPL211

1973

U. S. Department of Agriculture

Forest Service

Forest Products Laboratory

Madison, Wis.

VIBRATION TRANSMISSIBILITY

CHARACTERISTICS

CORRUGATED OF

FIBERBOARD

ABSTRACT

This reports gives vibration transmissibility data for corrugatedfiberboard under a wide variety of usage conditions. Resonant frequen­cies of loaded corrugated pads widely (35 140 Hz.), but generallyvary are considerably above the fundamental forcingrail and truck shipment. Therefore, for these corrugated fiberboard containers and pads are nor attenuate transportation vibrations.

to frequencies encountered in modes of transportation,

likely to neither amplify

ACKNOWLEDGMENT

The author gratefully acknowledges the assistance of Milo H. Schimming, Engineering Technician, in the development and performance of testing procedures.

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VIBRATION TRANSMISSIBILITY CHARACTERISTICS OF CORRUGATED FIBERBOARD

By

W . D. GODSHALL, Engineer

Forest Products Laboratory 1 Forest Service U. S. Department of Agriculture

INTRODUCTION

Corrugated fiberboard is used universally for the shipment of a wide variety of products,for interior packing, and as a shock absorbing cushion. Design criteria and performance data for these applications have been developed and published by the Forest Products Laboratory and others (6,10-12).2 Little information is currently available concerning the performanceof corrugated fiberboard with respect to vibration.

In transportation, packages are subjected to vibration as well as shock. In 1970, damageclaims for American railroads were in excess of $229 million, and other modes of transportexperienced losses of similar magnitude. Some of this is "concealed" damage, frequentlycaused by vibration, in which no effect to the outer container is evident. Even when the transportation vibration level is low, resonant amplification can cause damage. It is becoming increasingly evident that protection against both shock and vibration must be provided, and that packages designed to provide optimum shock protection may not provideadequate protection against vibration, and vice versa.

It is commonly believed that corrugated fiberboard "soaks up'' vibration, particularly at higher frequencies, but quantitative information is not currently available to define the vibration transmission characteristics of corrugated fiberboard or many other packaging or cushioning materials. Because every corrugated container places one or more layers of corrugated fiberboard between the product and the vibration input, these performancecharacteristics cannot be ignored. Thus, specific design information is needed to most economically and efficiently utilize corrugated fiberboard to protect packaged products from damage by vibration.

Because corrugated fiberboard is manufactured in a number of flute styles and combinations and is used in countless different configurations, it was necessary to limit the scope of this study to representative styles, configurations, and test conditions. Experimental work was restricted to flat single-wall specimens with open ends and with uniform load distribution at standardized atmospheric conditions of 73° F. and 50 percent relative humidity.

Two specimen configurations studied were: Two-layer pads, with the flute orientation of the two layers perpendicular to each other, and five-layer pads, with parallel flute orienta­tion in all layers and random alinement of the flutes vertically with respect to each other. These two configurations are shown in figure 1. The two-layer configurat ion is representa t iveof the bottom of a regular slotted container, the most commonly used box style, and the five-layer configuration is typical of corrugated fiberboard pads used for interior blockingand cushioning.

1Maintained at Madison, Wis., in cooperation with the University of Wisconsin. 2 Underlined numbers in parentheses refer to Literature Cited at the end of this report.

Figure 1.--Specimen configurations: Two-layer pads with flutes perpendicular to each other, and five-layer pads with all flutes parallel. M 139 036

Figure 2 .--Test fixture and configuration. Figure 3.--Test fixture and configuration. "Gravity" configuration with no upper "Compression" configuration with upper pad restraint or pad. M 139 033-3 and restraint. M 139 033-7

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Two packaging configurations were also studied. One configuration consists of the load (simulated packaged item) resting on the specimen pad with no upper pad or restraint as shown in figure 2. This is called the "gravity" configuration in this report, and simulates a "loose" pack or one with no top packing. The other configuration employs a lower and an upper specimen pad and has a fixed restraint at the top (fig. 3). This is referred to as the "compression" configuration, and is representative of a "tight" pack,

Tests were conducted at three loading conditions of 0.1, 0.5, and 1.0 pound of static load weight per square inch of specimen material (weight/area or W/A). This range represents the practical range of loading stresses for corrugated fiberboard and for most other packagingand cushioning materials.

Likewise, tests were conducted at three levels of vibration input which represent the most probable range of vibration likely to be encountered in transportation--0.1 G, 0.5 G, and 1.0 G, measured from zero to peak value. The low value is likely to occur almost continuouslyin most modes of transportation, while the highest value is likely to occur occasionally in transient bursts.

BACKGROUND AND LITERATURE

The basic theory involved in this study is that of vibration transmissibility. The premise is well developed and presented with varying degrees of thoroughness in texts on engineering mechanics (3,5,13). A comprehensive coverage is given by Ruzicka and Derby (9).This theory is extensively used in the design of machinery and aerospace and automotive vehicles. Only limited application has so far been made to packaging materials and design.A brief review of applicable basic theory is given in the appendix.

A proposed standard method of test to determine the vibration transmission characteristics of package cushioning materials has been under development by Committee D-10 on Packaging of the American Society for Testing and Materials. Henny (4) and Zell (14,15) have reported on studies of foamed plastic materials based upon this proposed method. Some limited data on other real materials are given by Mustin (8) but no data have been available until now for wood-base materials such as corrugated fiberboard.

MATERIALS AND SPECIMEN PREPARATION

Four corrugated fiberboard materials were evaluated. Three of the materials were manu­factured from the same set of matched components in A-, B-, and C-flute configurations. The facings were nominal 42-pound basis weight (per 1,000 sq. ft.) linerboard, manufactured from 100 percent kraft pulp, which consisted of approximately 90 percent southern pine and 10 per­cent mixed sweetgum and oak. The corrugating medium was nominal 26-pound basis weight and was manufactured from approximately 90 percent semichemical pulp (aspen and birch) and 10 percent kraft pulp. The facing weights were slightly under 42-pound basis weight, so the combined board had a nominal 175-pound bursting strength. These matched corrugated boards had been evaluated previously for shock absorption properties (6,11,12).

The fourth material evaluated was a nominal 200-pound test C-flute board made of 42-poundbasis weight Fourdrinier kraft facings and 26-pound basis weight semichemical corrugatingmedium. These specimens were cut from boxes from the same lot evaluated in studies on verti­cal dynamic loading (1,2). Thus, the data from this study can be related to the findings of these previous studies.

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The specimen sheets were cut to 6 by 6 inches, using a saw with no set to prevent crushing of the edges. The sheets were randomly assembled into the two- and five-layer pads using small spots of nitrocellulose cement staggered about the specimen area, to minimize anystiffening effect from the adhesive. The specimens were preconditioned at 80" F. and 30 percent relative humidity, and were then conditioned and tested at 73" F. and 50 percentrelative humidity.

TEST METHOD, EQUIPMENT, AND INSTRUMENTATION

The method, equipment, and instrumentation used to determine vibration transmissibility are illustrated in figure 4. The test specimen or specimens and the loading block represent­ing the packaged item are placed in a test fixture which represents the outer container. The loading block is free to move in the vertical axis but is constrained by guides to keepit in position laterally. The test fixture is mounted on a vibration exciter which produces a sinusoidal vibration in the vertical axis at controlled frequencies and acceleration levels. The input vibration acceleration level is measured by an accelerometer on the base of the test fixture, and the vibration transmitted through the test specimen to the loading block is measured by a similar accelerometer mounted within the loading block. Vibration transmissi­bility (T) of the specimen material is computed from the ratio of the response acceleration of the loading block (A r ) to the input acceleration of the test fixture (Ai):

Because the vibration exciter was manually controlled, data were taken point by point,varying the frequency in suitable increments from 5 to 500 Hertz (Hz.). The vibration exciter was set to the desired frequency, the vibration amplitude was increased from zero to the desired level, data were taken and the vibration amplitude was again reduced to zero. This procedure was used to minimize any possible fatigue effects, since many incremental readings had to be taken on each specimen. The resulting point by point data were plotted (fig. 5) to obtain a transmissibility versus frequency relationship for a particularspecimen at some particular configuration and test condition.

Vibration Exciter

The vibration exciter available for this study was an electrodynamic type with a frequency range of 5 to 500 Hz. and a maximum vector force rating of 200 pounds force. The amplitudeand frequency of vibration are manually controlled. The static load on the table which can be supported by the leaf spring support flexures is only about 10 pounds, so supplementarymethods of support must be provided for heavier loads.

Test Fixture

The test fixture had to be light in weight, because of the limited drive and static load capabilities of the vibration exciter. It also had to be stiff enough that any structural resonances would lie well above the frequency range of interest for the specimen materials. These conflicting requirements were satisfactorily met by constructing the test fixture of paper-overlaid plywood, as shown in figure 2. All major resonances lie above 250 Hz. The fixture has a weight of 19.5 pounds, complete with guides and upper restraint platen. The

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Figure 4.--Vibration transmissibility test equipment and instrumentation. M 139 033-2

Figure 5 .--Typical t r a n s m i s s i b i l i t y versus frequency re la t ionsh ip f o r corrugated pad.

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total weight with the maximum load weight of 36 pounds is, therefore, 55.5 pounds. This weight, which exceeds the static load capacity of the vibration exciter, is supported from overhead by a long loop of elastic shock cord. The natural frequency of the armature and fixture on this suspension is about 1 Hz., which is well below the operating range of the vibration exciter.

Two different loading blocks were used. The lightweight block shown in figure 2 was con­structed of aluminum honeycomb sandwich for light weight and high natural frequency. This block has a weight of 3.6 pounds, including the accelerometer, providing a 0.1 pound per square inch load for the 6.0- by 6.0-inch specimen. The other loading block, shown in figure 3, was constructed of laminations of maple and steel bolted together. By varyingthe combinations of maple and steel plates, a wide range of loading weights can be obtained. This loading block was used for static stresses of 0.5 and 1.0 pound per square inch.

Instrumentation

Two identical strain gage accelerometers were used to measure the input and responseacceleration levels. These transducers had a full-scale range of ±5 G, a natural frequencyof 430 Hz., and a linear response from 0 to approximately 260 Hz. The accelerometers were controlled from bridge balance units and their output signals were amplified in d.c. instru­mentation amplifiers and fed to both an optical oscillograph, and an oscilloscope which could be switched to either the input or the response channel. Signal levels were visually observed and manually recorded.

RESULTS AND DISCUSSION

Presentation of Data

The basic data obtained in this study were transmissibility versus frequency plots such as figure 5. Because four materials were tested over a wide range of conditions and configura­tions for a total of 72 separate evaluations, it is impractical to present all the data in this form. From the brief summary of vibration theory given in the appendix, it may be noted that transmissibility versus frequency curves for single-degree-of-freedom systems have a distinctive shape (appendix-fig. 2), which can be characterized by the resonant frequency (fr), the maximum transmissibility (at resonance) (T ), the damping ratio, and the frequency at which r the response changes from amplification to attenuation (f T1). Using these quantities, given in tables 1 and 2, the transmissibility curves can be approximated to a satisfactory degree of accuracy (7). The resonant frequencies are also listed in table 3, in a format that facilitates comparisons between materials and test conditions.

Comments on Testing Procedure

A number of conditions were noted during the testing which deserve some comment. Undoubt­edly some of these conditions are due to the limitations of the available equipment, while others are inherent in the test configuration and actual packaging application.

Because of the limited driving force of the vibration exciter, the dynamic reactions of the test fixture and the loading block had an effect on the input acceleration waveform, causingit to become nonsinusoidal and slightly distorted at some frequencies. While the response of the mass-spring system was generally sinusoidal regardless of the wave shape of the excitingvibration, phase changes occurred at some frequencies causing a nonsymmetrical response wave shape. These phenomena made determination of transmissibility difficult at a few frequencies

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Table 1.--Vibration transmissibility of 200-pound C-flute board

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Table 2.--Vibration transmissibility of 175-pound boards with A, B, and C flutes

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Table 3.--Experimentally determined resonant frequencies of corrugated f iberboard

for each specimen tested. The response signal was unfiltered as no tracking filter was available. Therefore, transmissibility was determined on the basis of the maximum zero to peak amplitudes (either positive or negative) of both the excitation and response waveforms as displayed on the oscilloscope or oscillograph.

Each of the two packaging configurations involved certain special considerations. The "gravity" configuration has no upper restraint and the loading block (simulated packaged item)merely rests on the specimen pad underneath it. When the input vibration level is sufficientlyhigh, and the exciting frequency is in the vicinity of the resonant frequency, amplification occurs. This causes the upward displacement of the loading block to exceed the elastic com­pression of the specimen pad; the loading block "floats," losing contact with the pad. As a result of this unstable condition, the loading block experiences repetitive shocks at random intervals rather than a true vibration at the exciting frequency. Whenever this occurs, the resonant frequency and maximum transmissibility cannot be determined experimentally by the normal test procedure.

Resonant frequency can, unity,

however, be calculated by noting the frequency (f T1)

above resonance where the transmissibility is and by dividing this value by for the unity transmissi­bility frequency is always as shown in appendix-figure 2. Therefore, for the test condi­tions at which instability occurred in the vicinity of resonance, the resonant frequency was determined by this alternate technique. It should also be noted that with this gravity loadingconfiguration, the response waveform is periodic but nonsinusoidal whenever amplification occurs. This is due to the lack of upward restraint and the constant effect of gravity duringthe downward portion of the displacement cycle. A typical response waveform is shown in figure 6.

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Figure 6 .--Typical response waveform for the unrestrained "gravity" configuration.

The "compression" configuration also presents some interesting considerations which confound the testing procedure, but which do exist in actual packaging situations. Restraint is provided at the top of the package and two specimens are used, one below and one above the packaged item. Some initial preload is used to provide a tight pack. However, the static stresses on the lower pad are greater than those on the upper pad due to the weight of the packaged item plus the preload. Because the load deformation characteristics of corrugated fiberboard (and other cushioning materials) are nonlinear, each pad has a different stiffness, which is affected bythe amount of preload. This problem has been analyzed in considerable detail by Zell (15) and Mustin (8). For this study a fixed preload of about 0.1 pound per square inch was used for all tests in the compression configuration. This load was obtained by allowing the upper restraint platen as shown in figure 3 to rest freely on the upper specimen before the platen was firmlybolted in place.

Discussion of Results

The range of resonant frequencies for the two-ply materials in all the various configurations was from 42 to 140 Hz. The range for the five-ply materials was from 35 to 90 Hz. The resonant frequencies for the two-ply pads were generally 20 to 40 Hz. higher than for the five-ply pads at any given loading condition.

Relatively small differences in resonant frequencies were found between A-, B-, and C-flute pads made from matched materials. The differences are so slight that it is not possible to draw positive conclusions, but the trend is for A flute to have the lowest resonant frequency,B flute the highest, and C flute to be between A and B flute.

The differences between the two C-flute materials were greater, with the resonant frequencies of the 200-pound test material consistently being below the resonances of the 175-pound test board in equivalent test configurations. These findings suggest that the vibration transmissi­bility characteristics of corrugated fiberboard are more dependent on the properties of the

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corrugated medium or the liners than on the flute style. The two different corrugatingmediums tested were both nominal 26-pound basis weight semichemical papers, but were made by different mills.

Attempts were made to predict resonant frequencies using spring factors obtained by repeatedcyclic loadings in a universal testing machine. This technique had been used successfully in previous studies of top-loaded corrugated containers (1,2). The spring factors obtained and the resonant frequencies calculated are given in table 4. The predicted resonant frequencies were all lower than the experimentally determined resonant frequencies as shown in figure 7, averaging only 81 percent of the actual values. This is probably due to differences between static and dynamic spring factors. Thus, while this method can be used to predict resonant frequencies of corrugated fiberboard, its accuracy is such that actual vibration transmissibility tests may be needed for precise determination of resonant frequencies.

Figure 7.--Comparison of predicted versus experimentallydetermined resonant frequencies for loaded corrugated pads.

This approach does, however, serve to explain the small variation in resonant frequencies over a wide range of static loading stresses and input vibration levels. The basic equationfor resonant frequency is:

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where g is the gravitation acceleration constant.

For a linear material with a constant value of K, the resonant frequency will decrease with increased static loading, W. The K of corrugated fiberboard increases greatly with an increase in W, in the loading range of interest (table 4). Therefore, the term under the radical tends to remain of the same magnitude, and because the resonant frequency is proportional to the square root of this value, it tends to remain about the same over a wide range of static loading values.

Table 4.--Spring factors otained from repeated loading tests, and calculated resonant frequencies1

The range of static loading stresses normally expected in service (from 0.1 to 1.0 p.s.i.)is well below the flat crush values for corrugated fiberboard. Stress-strain characteristics for the matched 175-pound test boards evaluated are shown in figure 8. Because the typicalloading stress is so much lower than the stress which would cause appreciable permanentdeformation, the board remains resilient and elastic. Thus, the transmissibility and the resonant frequency are not likely to change during a normal usage period. This tendency was partially confirmed by retesting a number of specimens, repeating the original test configura­tion, after approximately 6 months. The results agreed closely with the original data.

Except at the minimum loading stress of 0.1 pound per square inch, relatively little difference in resonant frequency could be attributed to the packaging configuration.Theoretically, the "compression" configuration, with both a lower and upper corrugated padand a fixed restraint at the top, should have a resonant frequency 1.4 times higher than a comparable "gravity" configuration, with only one corrugated pad and no top restraint. The two pads in the "compression" configuration act as springs in series so the resonance equation is

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as opposed t o

Figure 8.--Stress-strain characteristics for matched A-, B-, and C-flute corrugated fiberboards.

M 137 288, M 137 290, M 137 289

for the single ended "gravity" configuration, and a ratio loading of should exist between the two

resonant frequencies. However, for all but the minimum stress, the difference in resonant frequencies is insignificantly small. One possible reason for this discrepancy maybe due to the difference in loading stresses between the upper and lower pads, which was discussed previously.

The input acceleration level was found to have an insignificant effect on resonant frequency and peak transmissibility.

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Peak transmissibilities ranged from 1.5 to 6 with no well defined relationship to any of the test parameters of material, configuration, input acceleration, or static stress. The overall average of peak transmissibilities was 3.12. Using an approximation

where

c/cc = fraction of critical damping or damping ratio, and

Tr = transmissibility ratio at resonance, an estimated average damping ratio of 0.16 is

obtained. This compares with a value of 0.115 determined in a similar manner for top-loaded corrugated fiberboard containers (2).

Attenuation of vibration occurs at frequencies higher than times the resonant frequency. For most test configurations, the input vibration was attenuated to 10 percent or less, at frequencies above 250 Hz.

CONCLUSIONS

Flat open-ended multilayer corrugated pads exhibit vibration transmissibility character­istics similar to simple, lightly damped single-degree-of-freedom systems, with resonant frequencies ranging from 35 to 140 Hz., over a wide variation of loading configurations and test conditions.

Of all the test parameters, only the number of plies of corrugated board produced signifi­cant differences in resonant frequencies. Variations in input vibration acceleration level, loading configuration, flute style, and source of fiberboard components had relatively little effect. Typical resonant frequencies for the two-ply materials were in the vicinity of 100 Hz., while typical resonant frequencies for the five-ply materials were about 60 Hz.

Transmissibilities at resonance ranged from 1.5 over 3.0. The approximate corresponding damping

Predictions of resonant frequencies from static desired. Therefore, it is considered advisable the vibration transmissibility characteristics of

to 6.0 with a typical value of slightlyratio is 0.16 of critical.

test data were not as accurate as might be to conduct actual vibration tests to determine

packaging materials.

The resonant frequencies of loaded corrugated pads are higher than those of most other common cushioning materials. This can be considered beneficial, since they are also generallywell above the fundamental forcing frequencies encountered in rail and truck transportation. Therefore, for these modes of shipment, flat corrugated pads and the bottoms of corrugatedcontainers are not likely to either amplify or attenuate transportation vibrations.

A useful approximation is that frequencies above 200 Hz. will be attenuated with two-layer corrugated fiberboard pads, and frequencies above 120 Hz. will be attenuated for five-layer pads.

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LITERATURE CITED

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

Godshall , W. D . 1968. Effects of vertical dynamic loading on corrugated fiberboard containers. USDA

Forest Serv. Res. Pap. FPL 94. Forest Prod. Lab., Madison, Wis.

1971. Frequency response, damping, and transmissibility characteristics of top-loaded corrugated containers. USDA Forest Serv. Res. Pap. FPL 160. Forest Prod. Lab., Madison, Wis.

Harris, C. M., and Crede, C. E. (editors)1961. Shock and Vibration Handbook. 3 vols. McGraw-Hill Book Co., Inc., New York.

Henny, C., and Leslie, F. 1962. An approach to the solution of shock and vibration isolation problems as applied

to package cushioning materials. Shock and Vibration Bull. 30, part III, pp. 66-75. (AD-273 515)

Jacobsen, L. S., and Ayre, R. S. 1958. Engineering vibrations. McGraw-Hill Book Co., Inc., New York.

Jordan, C. A., and Stern, R. K. 1965. New tests probe cushioning properties of corrugated board. Package Eng. 10(12):

76-94.

Lazarus, M. 1969. How to approximate transmissibility curves. Sound and Vibration 3(6): 25, 26.

Mustin, G. S. 1968. Theory and practice of cushion design, SVM-2. U.S. Dep. of Defense, The Shock

and Vibration Information Center, Washington, D.C.

Ruzicka, J. E., and Derby, T. F. 1971. Influence of damping in vibration isolation, SVM-7. U.S. Dep. of Defense,

The Shock and Vibration Information Center, Washington, D.C.

Stern, R. K. 1964. Package cushioning design handbook. MIL-HDBK-304. U.S. Dep. of Defense.

Naval Publications and Forms Center, 5801 Tabor Avenue, Philadelphia, Pa. 19120.

1968. Tests show corrugated pad's performance as cushioning. Package Eng. 13(2): 71-75.

1968. Flat-crush cushioning capability of corrugated fiberboard pads under repeatedloading. USDA Forest Serv. Res. Note FPL-0183. Forest Prod. Lab., Madison, Wis.

Thomson, William T. 1965. Vibration theory and applications. Prentice-Hall, Inc., Englewood Cliffs, N.J.

Zell, G. 1964. Vibration testing of resilient package cushioning materials. Picatinny Arsenal

Tech. Rep. 3160. Picatinny Arsenal, Dover, N.J. (AD-444 825)

1969. Vibration testing of resilient package cushioning material: Polyethylene foam. Picatinny Arsenal Tech. Rep. 3610. Picatinny Arsenal, Dover, N.J. (AD-701006)

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APPENDIX

BASIC VIBRATION TRANSMISSIBILITY THEORY

The theory of vibration transmissibility may be found in many texts on engineering mechanics (3,5,9,13). The reader is referred to these sources for rigorous derivations of equations and more comprehensive discussions of the subject. The following discussion is intended to present the basic relationships as simply as possible for readers without engineering training.

The responses of mechanical systems to vibration inputs can be analyzed by considering the mechanical systems as one or more mass-spring systems with lumped values of mass (M), spring stiffness (K), and viscous damping (c) (3, 5, 9, 13). Such an idealized system (appendix-fig. 1),

Appendix-figure 1.--Lumped representationparameter of mechanical system.

when momentarily excited, will oscillate at its "natural" frequency (f n ) which is determined by the relationship

where (c/c ) is the ratio of the damping present to the critical damping that would preventc oscillation. Thus, the presence of damping only slightly lowers the natural frequency of the system. However, some energy is dissipated each cycle and the vibration is soon damped out.

When a mass-spring system such as that shown in appendix-figure 1 is continuously excited by a sinusoidal vibration of the base, a different reaction occurs. The mass vibrates at the frequency of the exciting vibration (f) rather than at the natural frequency (f n ), but its amplitude of vibration may be larger or smaller than the exciting vibration depending on the relationship of the exciting frequency to the natural frequency. This ratio of the responseamplitude to the excitation amplitude is called the transmissibility (T) of the system.

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Transmissibility for a linear single-degree-of-freedom system can be expressed by the equation

This relationship, shown graphically in appendix-figure 2, holds true for forces, displacements,and accelerations. Transmissibility is a function of frequency and of the damping in the mechanical system.

When the frequency of the exciting vibration is much lower than the natural frequency of the system, only slight amplification occurs and the transmissibility is essentially unity. As the exciting frequency approaches the natural frequency, amplification of the vibration increases until maximum transmissibility is reached at resonance (T r ), when the exciting frequency is very nearly the same as the natural frequency. For all practical purposes, resonance can be

Appendix-figure 2.--Transmissibility relationship of a viscously damped single-degree-of-freedom system. M 133 489

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considered to occur at the natural frequency. The effect of damping on transmissibility is most apparent at resonance; the greater the damping, the lower the maximum transmissibility. Amplification also occurs for exciting frequencies above the natural frequency, with decreasingtransmissibilities, up to where the transmissibility is unity for all values of damping. For all exciting frequencies higher than the transmissibility is less than 1.0 and the vibration of the mass is attenuated. The effects of damping again assume importance at higherexcitation frequencies, as greater damping produces a lesser degree of isolation.

The preceding analysis was based on an idealized system in which the mass and the base were infinitely rigid, the spring was massless and had linear stiffness and viscous damping, the excitation was simple harmonic motion with no distortion, and vibration occurred in only one degree of freedom. This analysis is applicable to the vast majority of situations involvingactual materials, even though few, if any, of the above assumptions are rigorously met. However, the greater the departure from the idealized model, the greater the caution that must be exercised in the interpretation of the test data.

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U.S. Government Printing Office 754-546/30