Evaluating Vibration Environments Using the Shock …

1 18 sound AND Vibration /APRIL 2003 The Shock response spectra computed directly from the timehistories of simulated stationary Vibration data are comparedto the expected values for the maximum response peaks com-puted from the autospectra of the data. The data are assumedto be random with a Gaussian probability density function andthe results reveal excellent agreement when that is truly thecase. However, for the complex Vibration environment gener-ated by a repetitive Shock machine, the Shock response spec-tra computed directly from the time history data are higherthan the expected values. This indicates that for the sameautospectrum, the damage potential of a repetitive shockmachine is greater than that for a truly random Vibration . Thesame conclusion undoubtedly applies to many other complexbut not random Vibration Environments such as those pro-duced by reciprocating Shock response spectrum (SRS) is broadly defined as thepeak response of a simple oscillator (single degree-of-freedomsystem)

2 To an excitation as a function of the natural frequencyof the It was originally introduced to evaluate thedamage potential of mechanical transients, but can also be usedto evaluate the damage potential of stationary random vibra-tions as measured by the peak value for the response of a simpleoscillator exposed to the Vibration over a finite ,3 Thislatter application of the SRS directly competes with the use ofstatistical procedures to predict the peak value for the responseof a simple oscillator under the assumption that the excitationto the oscillator is a stationary random Since theSRS does not require the excitation to be random, a direct com-parison of SRS results to a statistical computation of the maxi-mum value of the oscillator response can be used to evaluatethe randomness of the excitation. Such a comparison has beenused to detect transients in otherwise stationary random vibra-tion Of interest here is the use of such a comparisonto detect differences between the damage potential of randomvibrations versus the complex (sometimes called quasi-ran-dom) vibrations produced by pneumatic hammer-type vibra-tion test machines, commonly referred to as repetitive Computation of Shock Response SpectrumThe computation of the Shock response spectrum (SRS) fora stationary random Vibration involves the determination of themaximum value for the response of a lightly damped, linearoscillator to the excitation, as illustrated in Figure 1a.

3 Assum-ing an acceleration excitation x(t) produces an accelerationresponse y(t), the frequency response function of the simpleoscillator is given by6where fn is the undamped natural frequency and is the damp-ing ratio of the oscillator. Assuming the acceleration excitationx(t) is random with an autospectrum Gxx(f), the response y(t)of the oscillator will be random with a narrow bandwidth, asshown in Figure 1b, and will have a standard deviation ap-proximated by6If it is further assumed the autospectrum of the excitation isrelatively uniform at frequencies near the natural frequency fn,then Equation 2 is closely approximated by6It should be mentioned that the standard deviation of the os-cillator response is often computed in terms of velocity ratherthan acceleration,2,3 because the stress produced by the reso-nant response of a structure is proportional to Forthe application at hand, however, only comparisons of relativevalues are of interest, so acceleration units are used since theyare more familiar to most Shock and Vibration is recommended that a conservative maximum value forthe response of the oscillator to a stationary random excitationbe estimated by2.

4 3wherefn= undamped natural frequency of the oscillatorT= duration of the excitation y= standard deviation of the oscillator response, as de-fined in Equation 2P(T)= probability that the value Ym will be exceeded dur-the exposure duration TFor design purposes, a probability of P(T) = (5%) iscommonly assumed in Equation ,3 However, to estimate anSRS, the expected value of Ym is needed since it correspondsto the average value of the SRS as normally computed. The ex-pected value and standard deviation for the maximum responseof the oscillator is estimated by4where all terms are as defined in Equation 4. It follows that thenormalized random error (coefficient of variation) for an esti-mate of the maximum response of the oscillator is given bywhere the hat (^) denotes estimate of. Note from Equation 7that the normalized random error for estimates of the maximumresponse value will be less than r < (10%) for fnT > results in Equations 4 through 7 involve two criticalassumptions:1.

5 The response y(t) of the oscillator has a normal (Gaussian)probability density function. As long as the excitation x(t)is random and the oscillator response is linear, even if x(t)is not Gaussian, this assumption is often acceptable becausethe narrow bandwidth filtering of the oscillator suppressesdeviations from the Gaussian form in the response y(t).82. The peak values of the oscillator response are statisticallyindependent. It is clear from the relatively smooth variationsin the envelope for the peak values of the oscillator responsein Figure 1b that the peak values are not statistically inde- Evaluating Vibration EnvironmentsUsing the Shock Response SpectrumGeorge R. Henderson, GHI Systems, Inc., San Pedro, CaliforniaAllan G. Piersol, Piersol Engineering Company, Woodland Hills, California(1)(2)(3)(4)(5)(6)(7)Hfjfffjfx ynnn()=+ +12122 yxyxxHfGfdf=()() 02 YPTfTPTPTmyn() =() () 21 ln; EYfTfTfTmynnn[]=()+() () 20577221 ln ln; ln.

6 YfTfTmynn[]=()()12821. ln; ln rmmmnYYEYfT .. =[][]=()+12820 5772 ln yxxnnGf f=()+ 144219 sound AND Vibration /APRIL 2003pendent. However, computer simulation studies indicate thestatistical independence assumption is acceptable for valuesof Ym/ y > , which corresponds to fnT > 250 for a Gaussianrandom of Estimated and Computed ResultsComputations were performed on data produced by twosources: (a) a computer generated random signal simulating aGaussian random Vibration , and (b) the signal from an accel-erometer mounted on the table of a commercial repetitive Shock (RS) machine. In both cases, an autospectrum (PSD) was com-puted Using a conventional fast Fourier transform (FFT) basedPSD analysis algorithm,6 and a Shock response spectrum (SRS)was computed Using the Ramp-Invariant Method. 9 All analy-ses were performed over T = sec of data that were digitizedusing a sampling rate of 25,000 samples per sec. The lower fre-quency limit for the analyses was fixed to 40 Hz to comply withthe second assumption after Equation 7.

7 The upper frequencylimit was fixed at 2500 Hz (10% of the sampling rate) to restrictthe magnitude error in the SRS values to less than 5%10 as wellas to suppress an inherent bias error in the Damping ratios of 5% ( = corresponding to Q= 10) and 1% ( = corresponding to Q = 50) were used forall standard deviation and SRS computations. The frequencyresolution for the analyses was 10 Hz for the PSD values and1/12 octave band for the SRS computations. However, all PSDand SRS results are presented at 1/3-octave band center fre-quencies for Data. The simulated random Vibration data weregenerated with a PSD of Gxx(f) = g2/Hz over the frequencyrange from 40 to 2500 Hz. The directly computed SRS for thesimulated random Vibration data, the expected value for themaximum response given by Equation 5 and the P(T) = given by Equation 4, all computed with 5% damping, arecompared in Figure 2. Note in Figure 2 that the directly com-puted SRS values are in good agreement, on average, with thepredicted values of Equation 5 and are just enveloped by theP(T) = values of Equation 4, as would be expected with19 SRS values.

8 It is clear from these results that Equations 3and 4, Using the standard deviation computed from Equation2, provide accurate results for truly random Machine Data. The probability density function for thetable Vibration produced by the repetitive Shock (RS) machineused for these studies is shown in Figure 3. Note that the prob-ability density function for the table motion in terms of accel-eration values deviates substantially from the Gaussian results are consistent with the findings in Henderson12for a different RS machine and, further, would be intuitivelyanticipated for a Vibration response that is produced by a se-quence of transients rather than a stationary random , it should be mentioned that these particular RS ma-chines are both of an early design. Studies of an RS machineof more recent design revealed a table motion that is muchcloser to the Gaussian PSD values of the Vibration generated by the RS machineat the 1/3-octave band center frequencies used for the compu-tations in Equations 2 through 5 are shown in Figure 4.

9 Thedirectly computed SRS values for the RS machine vibrationdata, the expected value of the maximum response given byEquation 5 and the P(T) = value given by Equation 4, allcomputed with 5% damping, are compared in Figure 5. A simi-lar comparison of the values computed with 1% damping areshown in Figure 6. From the results in Figure 5, it is seen thatthe directly computed SRS values with 5% damping exceed thepredicted values of Equations 4 and 5 at most frequencies bymargins of up to 3:1. It is clear that Equations 4 and 5, usingthe standard deviation computed from Equation 2, do not pro-vide accurate results for this particular RS machine is, a component exposed to the excitation of the RS ma-chine would have a substantially higher peak response thanpredicted by either Equation 4 or 5. On the other hand, the re-sults with 1% damping in Figure 6 also reveal higher computedSRS values than predicted by Equations 4 and 5 at most fre-quencies, but by smaller results in Figures 5 and 6 might be explained as Shock machines produce what is essentially a com-plex periodic Vibration with natural and sometimes intention-ally introduced random modulations of both magnitude andfrequency.

10 Hence, the Vibration does have a limited randomcharacter. The half-power point bandwidth for a simple oscil-lator is approximated by6 Bhp 2 fn where the bandwidth ofthe oscillator is directly proportional to the damping ratio. Thenarrow bandwidth filtering operation of the simple oscillatorsessentially invokes the Central Limit Theorem and thus sup-presses deviations from the Gaussian form where the narrowerthe bandwidth, the greater the suppression of It follows that the narrower bandwidth for the1% damping produces a more Gaussian response that makesEquations 2 through 4 more accurate. This conclusion is con-Figure 1. Illustration of simple oscillator: (a) schematic diagram. (b) timehistory response to random 2. Predicted versus computed Shock response spectrum forGaussian random 3. Probability density function for the table Vibration of a repeti-tive Shock 4. Power spectral density function for the table Vibration of arepetitive Shock (t)t(b)y(t)x(t)mkc(a)1041031021011001011 02 Shock Response Spectrum, gNatural Frequency, HzExpected value computed from autospectrumP(T)