16 SOUND AND VIBRATION/OCTOBER 2002

Base(Shaker & Fixture)

Mass

Y0

Y

η

M

k

K = k (1 + i η)

This article presents common-sense guidelines for single-axis sweep sine and random vibration testing. Several actualcase histories are included as well to demonstrate these tech-niques.

Single-axis shakers (mostly electrodynamic1) are widelyemployed for modal analysis, failure simulation and environ-mental testing in many research and design projects. In particu-lar they are used for automotive and aerospace applications and(in combination with laser-vibrometers) for testing miniaturehigh-technology MEMS sensors. A test engineer should havea good theoretical background to help prevent new designsfrom encountering long and costly feedbacks for design correc-tion actions. Demonstration skills and teamwork with designand manufacturing engineers are also desirable. Many theoreti-cal and experimental ‘details’ on shaker testing are not clearlycovered, although they save time and assure high quality. Thisarticle presents helpful guidelines for sweep sine and randomvibration testing, general relationships between natural andresonant frequencies and fatigue failure testing. Simple (butuseful) equations are derived and applied to predict or inter-pret test parameters and results. The practical examples pro-vided here are based on real case histories with names andproject/test details changed.

Maximum Shaker AccelerationPhil, a young dynamic testing engineer, was engaged in mea-

suring the fundamental resonant frequencies of two aluminumcondensers for an automotive air conditioning system. Hebolted the condensers, with no rubber vibration isolators, sym-metrically to an aluminum fixture as shown in Figure 1. Heinstalled accelerometers on the top of each specimen andstarted a sweep sine test in the vertical direction at an excita-tion of 4 g pk and sweep rate of 0.5 octave/min. The sweep sinefrequency range of 80-160 Hz was recommended by Jay, a fi-nite element modeling engineer, who estimated the condenser’snatural frequency to be 113 Hz. The acceleration seemed highfor an experimental modal analysis but Phil did it on purposeto later employ the results on a fatigue test at an accelerationof 4 g pk. From an equation found in a manual, Phil calculatedthe maximum shaker acceleration

where the combined mass of the armature and fixture was 24lb and the mass of one condenser was 8 lb. The manual stated,however, that the dynamic response of the fixture and test itemmay reduce the calculated rating but did not provide details.To play it safe, Phil introduced a “safety factor” of 5. But evenin this case, the calculated value (50 g pk/5 = 10 g pk) was wellabove 4 g pk; the test setup seemed right. Nevertheless, thesweep sine test was automatically stopped within 1 min by thevibration controller because of an overload. It happened repeat-edly and Phil addressed the problem to his supervisor Robertwho restarted the test at an acceleration of 1 g pk and obtainedexperimental data for both condensers. The resonant frequen-cies measured 102 and 104 Hz (below the predicted value byapproximately 10%) but the peak transmissibilities proved veryhigh – 48 and 49. To explain what happened, Robert sketcheda single-degree-of-freedom model2 incorporating a mass, baseand parallel spring and dashpot as shown in Figure 2. The baseplayed the role of the shaker and fixture; the mass, spring and

dashpot simulated the test unit parameters. Assuming that thebase moved harmonically with a displacement , sothe differential equation for the displacement of themass took the form:

where ω = 2πf was the angular frequency of vibration and K =k (1+iη) was the complex spring constant combining the springconstant k and loss factor η. In the ideal case of viscous damp-ing (Figure 3), the loss factor equals

where the quantities c and cc = 2 Mk are the coefficients ofviscous damping and the critical damping, respectively; ωn =2πfn = k M/ is the angular natural frequency and fn is thenatural frequency.

In general, the loss factor may be governed by multiple en-ergy dissipation mechanisms. For the all-metal structures, theloss factor includes two main components: (1) the internal lossfactor that appears in most solid materials with alternatingstresses due to hysteresis and (2) the so-called structural lossfactor depending on vibration energy absorption at junctures,edges and adjacent structures. It is also true for building ele-

Helpful Guidelines forSingle-Axis Shaker TestingRoman Vinokur, Westlake Village, California

(1)(2)

(3)

MY K Y Y�� + − =( )0 0

η ωω

= 2ccc n

gmax = =+

Shaker Force RatingTotal Accelerated Mass

2000 lbf24 (( )2 8

50× =

=40 lb

g pk

Y yei t= ω

Figure 1. Test setup for 1-DOF shaker test on two condensers symmetri-cally installed on the fixture.

Accelerometer Specimen

Bracket

A B C

Fixture

Specimen

Expansion Head(part of Fixture)

Shaker

Figure 2. Single-degree-of-freedom vibration system. The damping el-ement is a dashpot but in Equation (2), a more general case of damp-ing is considered.

Y y ei t0 0= ω

17DYNAMIC TESTING REFERENCE ISSUE

ments: walls, window glazing, ceilings, etc.3,4 The internal fric-tion component is typically constant and the structural com-ponent normally decreases with frequency so the total loss fac-tor tends to reduce with frequency (Figure 3). It is noteworthyin the literature, that damping in vibratory systems is com-monly considered viscous for mathematical simplification andall other damping mechanisms (hysteresis, etc.) are modeledby an equivalent damping component obtained from equalenergy considerations. It must be emphasized again that thedamping component in Equation (2) may simulate multipledissipation mechanisms. The solution of Equation (2) is

From Equation (4), one can find the transmissibility

where a = –yω2 and a0 = –y0ω2 are the accelerations of the massand base, respectively. The transmissibility (Figure 4) is themodulus of the frequency response function defined as theFourier transform of the output divided by the Fourier trans-form of the input. The frequency response function should notbe confused with the transfer function defined as the Laplacetransform of the output divided by the Laplace transform of theinput. These terms are often used interchangeably and aresometimes a source of misunderstanding. The other importantresult following from Equation (4) is the amplitude of the forcedeveloped by the spring

From Equation (6), the maximum spring force equals

where Tpeak = T(fr) if the resonant frequency fr belongs to thetest frequency range (fr ≈ fn provided that η << 1) and Tpeak ≈ 1if the resonant frequency is well above the test frequencies.Using Equation (7), the maximum shaker acceleration limit canbe expressed as

where Mj and Tpeak j are respectively the mass and peak trans-missibility of the single-degree-of-freedom structural compo-nents moved by the shaker (j = 1, . . .). Considering the arma-ture and fixture like the systems with Tpeak ≈ 1 and substitutingthe numeric data into Equation (8), Robert calculated the maxi-mum shaker acceleration as

Thus, the shaker acceleration on Phil’s test notably exceededthe maximum shaker acceleration. Indeed, test specimens andfixtures are not ideal single-degree-of-freedom structures withthe same transmissibility at all points. Thus, the average trans-missibilities should be utilized in Equation (8). For Phil’s test

specimen (Figure 1), the transmissibility is maximum at pointC (in the central position) and close to 1 at point A (near oneof the corner brackets bolted to the fixture). The transmissibil-ity at point B (between points A and C) should be close to theaverage transmissibility used in Equation (8). In any case, Equa-tion (8) is more accurate than Equation (1) if the transmissi-bilities can be obtained through the preceding shaker test at alower acceleration or evaluated. It is noteworthy that Phil nevermeasured such a high transmissibility in his prior experience.He commonly attached the specimens to the fixture throughspecial rubber isolators with loss factors of 0.15-0.25 (peaktransmissibilities ranged between 4 and 7). However, for all-aluminum or all-steel structures at relatively high frequencies,the loss factors may be as low as 0.001-0.010 in order of mag-nitude. For this test, the average peak transmissibility was 48.5and according to Equation (5), that corresponds to a loss fac-tor of 0.02. Structures with high peak transmissibilities mayendure severe internal stresses at their resonant frequenciesthat can result in fatigue failure. It makes sense to increase vi-bration energy absorption in the test fixtures to attenuate in-herent high-frequency resonances. Fixtures made of magne-sium are superior because the internal loss factor formagnesium is notably higher than aluminum or steel.5 How-ever, in practice this is useful only at frequencies where thehysteresis outperforms the structural loss component.

Natural and Resonant FrequenciesThe transmissibility of a real vibration system occurs not at

its undamped natural frequency but at a somewhat differentfrequency called the resonant frequency or the frequency ofmaximum forced amplitude.2 It is noteworthy that in books onvibration theory, the relationship between the two is analyzedonly for the case of ideal viscous damping. For small values ofdamping, the frequencies are very close together, but from timeto time they become a matter of dispute. It happened againwhen Jay learned that his prediction fell 10% above the aver-age resonant frequency measured. In Jay’s opinion, it was justa normal discrepancy between the natural (computed) and

(5)

(6)

(7)F Ma Tpeakmax = 0

gM Tj j

j

max = ∑Shaker Force Rating

peak (8)

(9)

y yi

ff

in

= +

−

+0 2

1

1

η

η

Tyy

aa

ffn

= = = +

−

+0 0

2

2 2

2

1

1

η

η

gmax [( ) ( ) ( )].=

× + × + ×≈ <2000 lbf

lb g pk g pk

24 1 8 48 8 492 5 4

F K y yk i y

nffn

i

Ma

= − =+

−

+

=

+

( )( )

01 2

0

2 12

01 2

1

η ω

ω η

η

−−

+

=

ffn

Ma T2 2

2

0

η

(4)

Figure 4. The transmissibility of the 1-DOF vibration system from Fig-ure 2. The dimensionless frequency equals f/fr.

Figure 3. The loss factor dependence on frequency for two differentdamping mechanisms.

Frequency

Lo

ss F

act

or

Viscous Damping

Damping in all-metaland building structures

Dimensionless Frequency

Tra

nsm

issi

bili

ty

5

4

3

2

1

00 0.5 1.0 1.5 2.0

Tpeak

18 SOUND AND VIBRATION/OCTOBER 2002

resonant (measured) frequencies rather than a deficiency of hiscomputer model. To explain the real tendencies, Robert sug-gested that in Equation (5),

|ε|<< 1, and η << 1, and transformed Equation (5) to a simplerform

As follows from this equation, the peak transmissibilityshould be attained at a minimum of the function z(ε) = 4ε2 +η2. The necessary condition for the minimum is

with the only solution

Thus, the ratio of the resonant and natural frequencies is

where both loss factor η and its derivative are calculated at thefrequency f = fn. Several important conclusions result fromEquation (9):1. If the loss factor, at frequencies equal or close to the natural

frequency, is constant

then the resonant frequency coincides with the natural fre-quency (fr = fn).

2. If, in the range mentioned, the loss factor grows with fre-quency

then the resonant frequency is below the natural frequency(fr < fn).

3. If, in the range mentioned, the loss factor decreases with fre-quency

then the resonant frequency exceeds the natural frequency(fr > fn).

4. In case of ideal viscous damping, the loss factor grows withfrequency. Using Equations (3) and (9) one obtains

This equation clearly indicates that the relative differencebetween the resonant and natural frequencies is very low forsmall values of damping (for instance at η = 0.1, it comes toonly 0.25%).

5. As mentioned previously for a number of solid structures,the loss factor tends to reduce with frequency (Figure 3). Inthis case, the resonant frequency is expected to exceed thenatural frequency.Jay checked the calculation, agreed and got back to his FEA

model. For product development, computer modeling is mostlya good guess until the experiment makes the product speak foritself. Without a thorough theoretical analysis, neither experi-mental nor computer modeling can be fully successful.

Random Excitation TestPhil’s new project was a single-axis vibration fatigue test on

the small auxiliary radiators (2 lb each) for trucks. The radia-

Figure 6. The auto spectral density g2/Hz of the acceleration of theshaker and sample.

Figure 5. Test setup in the “slip-table” mode.

tors were designed to be bolted to the truck’s frame through twoaluminum brackets with no rubber isolators. Two small radia-tors were attached to the shaker similar to their “in-vehicle”installation. As shown in Figure 5, the shaker was arranged inthe “slip-table” mode to excite the radiators in their most vul-nerable state. First of all, Phil performed a sweep sine test tomeasure the resonant frequency (31 Hz) and peak transmissi-bility (15). The fatigue failure test was to be performed withrandom excitation to closely simulate real environmental con-ditions. The project leader provided Phil with the averagedauto spectral density (ASD) of the acceleration measured on thetruck frame during a special road test. The curve looked cum-bersome, but in a rough approximation it can be depicted likea horizontal segment at a level of 0.01 g2/Hz between 5 and 41Hz and zero at all other frequencies (Figure 6). In this case, thetotal acceleration calculated was

For random vibration, the rms (root-mean-square) accelera-tion is indeed the standard deviation of the acceleration. Peakvalues may numerically exceed the rms value at least by a fac-tor of 3. The peak acceleration from the road test report rangedup to 3 g. To implement accelerated life testing, shaker accel-eration should be increased to appropriately excite stresses inthe specimen. Phil reasonably suggested that the most signifi-cant stresses develop at the resonant frequency (31 Hz) and cal-culated the total numbers of cycles for the product lifetime (10years) and accelerated test duration (24 hours). Using a fatiguestrength dependence on the total number of cycles,6 he calcu-lated an increased excitation factor of 3. So, the auto spectraldensity of the shaker acceleration should be increased to S2 =0.09 g2/Hz in the same frequency range (5-41 Hz). The calcu-lated total acceleration in this case was 1.8 g rms. The next im-portant step was to check that the shaker total displacementdid not exceed the maximum peak-to-peak (pk-pk) displace-ment. The shaker was designed for a maximum displacementof 2 in. pk-pk, but it was recommended not to exceed 75% ofthat value for long tests. The maximum permissible displace-

Fixture

ShakerBrackets

Small Radiators

SlipTable

a grms020 01 41 5 0 6= × −( ) =. / .Hz Hz g rms

100

10

1

0.1

0.01

0.001

Frequency, Hz1 10 100

On shaker – approximated

On shaker – accelerated test

On sample – accelerated test

Auto

Spect

ral D

ensi

ty

T ≈+1

4 2 2ε η

dzd

ddε

ε η ηε

= + =8 2 0

ε εη η

ε= = −r

dd4

ff

dd f fr

n

n≈ − ( )1

4

η η/

ffn

= +1 ε

ffr

n= −1

4

2η

dd f fn

η/( )

= 0

dd f fn

η/( )

> 0

dd f fn

η/( )

< 0

19DYNAMIC TESTING REFERENCE ISSUE

leader approved the proposal and it was implemented with nodelay. The shaker test on the new samples was promptly com-pleted, the results proved very satisfactory, and mass produc-tion moved forward.

SummaryThis article, written in the form of interrelated case stories

based on the author’s experience, presents some feasible tech-niques for single-axis shaker testing, an important and some-times critical part of research, design and manufacturingprojects. The simple (but effective) equations can help vibra-tion engineers in creating sweep sine and random tests, inter-preting the experimental data and developing improvementproposals. The relationship between resonant and natural fre-quencies, commonly described only in the case of viscousdamping, may be of more general interest. In addition to ex-perimental and theoretical facts, this article illustrates the ben-efits of creative engineering teamwork.

References1. Lang, G. F., and Snyder, D., “Understanding the Physics of Electro-

dynamic Shaker Performance,” Sound and Vibration, Oct. 2001, pp.24- 33.

2. Den Hartog, J. P., Mechanical Vibrations, Dover Publications, Inc. edi-tion, 1985.

3. Vinokur, R., “Influence of the Edge Conditions on the Sound Insula-tion of a Thin Finite Panel,” Soviet Physics – Acoustics (USA), 1981,Vol. 26, No. 1, pp. 72-73.

4. Craik, R., “Damping of Building Structures,” Applied Acoustics,1981, Vol. 14, No. 5, pp. 347-359.

5. Vibration Test Equipment, Fixture Design Notes by John RaymondIV, Unholtz-Dickie Corporation.

6. Linderberg, Michael R., Mechanical Engineering Reference Manual,9th edition, recommended by the National Society of ProfessionalEngineers, 1994.

The author can be reached at rvinokr@aol.com.

ment was considered to equal 1.5 in. or 38 mm pk-pk. Philknew a simple equation

interrelating the peak displacement and acceleration on sineshaker tests. But what if the acceleration and displacement arerandom functions?

Robert helped again. As follows from the previous equation,the auto spectral density for the displacement should be

where Sa2 is the auto spectral density for the acceleration. Af-

ter that, the standard deviation for the displacement was cal-culated as

where L and U equals 5 and 41 Hz, respectively. The pk-pkvalue may numerically exceed the rms value more than 6 timesbut even with a factor of 9, the shaker pk-pk displacement wasbelow the permissible value of 38 mm. As for the maximumshaker acceleration limit in random mode, it should not be aproblem for this test based on Robert’s experience.

Design ImprovementPhil started the vibration fatigue test in random mode. It

proved relatively short. Within 25 minutes, all the specimenssuffered the same critical failure – broken brackets. Phil boltednew specimens to the fixture through the rubber washers forvibration energy absorption but with no significant improve-ment. For effective vibration energy isolation, the rubber iso-lators should be somewhat loose. When compressed, they arenot very effective. A design modification seemed unavoidableto fix the problem. It was not good news for the company be-cause the small radiators should have already been at the pro-duction stage and the manufacturing engineers were waitingfor the shaker test results as the final ‘go’ condition. Fortu-nately, Phil found the no-delay and no-cost engineering solu-tion.

During the test, Phil observed the auto spectral density of thesignals measured by the accelerometers on the test units. Dueto a notable peak caused by the resonance (Figure 6), the rmsacceleration on the units proved as high as 10 g rms. It seemedbeneficial to move the resonant frequency beyond the effectivefrequency range of 5-41 Hz. At an engineering meeting, he sug-gested that steel brackets of the same shape be used in place ofthe aluminum ones. It proved very convenient because therewas no need to change the existing tooling equipment. How-ever, Jay had a concern. The ultimate tensile strength of thecheap steel recommended by Phil was just twice that of the alu-minum material in the existing brackets. It may not be suffi-cient to resist fatigue failure and would have to be experimen-tally confirmed. However, Phil was positive that the selectionof the material was right because: (1) available tooling may failbending a stronger steel; (2) it should be a no-cost modifica-tion in order to save the company’s money; and (3) most im-portantly with the steel brackets, the natural frequency is welloutside the effective frequency range. Since the brackets playthe role of springs, the natural frequency of the radiator withthe steel brackets was evaluated as

where fn Al is the measured natural frequency of the radiatorwith the aluminum brackets, EAl is the Young’s modulus of alu-minum, and ESteel is the Young’s modulus of steel. The project

Sf

Sy a2

2

2

21

2= −

( )π

(10)

ya

f0

022

= −( )π

D S f dfS df

f

S

L Uy

L

U

= = = − =

−

∞

∫ ∫2

02 4 2 3 3

2 3

4 4 3

1 1

0 3

4 3

1

5

1

( )

.

π π

π

4413 93

2g rms/Hz mm rms≈ .

f fEEn n Steel AlSteel

AlHz

GPa70 GPa

Hz= ≈ ≈31200

52