1. Vibrationdata 1 Unit 15 SDOF Response to Base Input in the Frequency Domain

2. Vibrationdata 2 Introduction n Steady-state response of an SDOF System n Base Input: PSD – stationary with normal distribution

3. Vibrationdata 3 Miles Equation n Miles Equation is the simple method for calculating the response of an SDOF to a PSD n Assume white noise, flat PSD from zero to infinity Hz n As a rule-of-thumb, it can be used if PSD if flat within + 1 octave of the natural frequency

4. Vibrationdata 4 Miles Equation The Miles equation is a simplified method of calculating the response of a single- degree-of-freedom system to a random vibration base input, where the input is in the form of a power spectral density. The Miles equation is P is the power spectral density level at the natural frequency f n is the natural frequency where is the overall response Q is the amplification factor

5. Vibrationdata 5 SDOF System, Base Excitation The natural frequency fn is The amplification factor Q is The damping coefficient C is

6. Vibrationdata 6 SDOF Free Body Diagram The equation of motion was previously derived in Webinar 2.

7. Vibrationdata 7 Sine Transmissibility Function Either Laplace or Fourier transforms may be used to derive the steady state transmissibility function for the absolute response. After many steps, the resulting magnitude function is where where f is the base excitation frequency and fn is the natural frequency.

8. Vibrationdata 8 The base excitation frequency is f. The natural frequency is fn. Frequency Ratio (f / fn)

9. Vibrationdata Transmissibility Curve Characteristics 9 The transmissibility curves have several important features: 1. The response amplitude is independent of Q for f << fn. 2. The response is approximately equal to the input for f << fn. 3. Resonance occurs when f  fn. 4. The peak transmissibility is approximately equal to Q for f = fn and Q > 2. 5. The transmissibility ratio is 1.0 for f =  2 fn regardless of Q. 6. Isolation is achieved for f >> fn.

10. Vibrationdata Exercises 10 vibrationdata > Miscellaneous Functions > SDOF Response: Steady-State Sine Force or Acceleration Input Practice some sample calculations for the sine acceleration base input using your own parameters. Try resonant excitation and then +/- one octave separation between the excitation and natural frequencies. How does the response vary with Q for fn=100 Hz & f =141.4 Hz ?

11. Vibrationdata “Better than Miles Equation” 11  Determine the response of a single-degree-of-freedom system subjected to base excitation, where the excitation is in the form of a power spectral density  The “Better than Miles Equation” is a.k.a. the “General Method”

12. Vibrationdata Miles Equation & General Method 12 n The Miles equation was given in a previous unit n Again, the Miles equation assumes that the base input is white noise, with a frequency content from 0 to infinity Hertz n Measured power spectral density functions, however, often contain distinct spectral peaks superimposed on broadband random noise n The Miles equation can produce erroneous results for these functions n This obstacle is overcome by the "general method" n The general method allows the base input power spectral density to vary with frequency n It then calculates the response at each frequency n The overall response is then calculated from the responses at the individual frequencies

13. Vibrationdata General Method 13 The general method thus gives a more accurate response value than the Miles equation. The base excitation frequency is f i and the natural frequency is f n The base input PSD is

14. Vibrationdata Navmat P-9492 Base Input 14 PSD Overall Level = 6.06 GRMS Frequency (Hz) Accel (G^2/Hz) Frequency (Hz) Accel (G^2/Hz) 200.01 800.04 3500.04 20000.007

15. Vibrationdata 15 Apply Navmat P-9492 as Base Input fn = 200 Hz, Q=10, duration = 60 sec Use: vibrationdata > power spectral density > SDOF Response to Base Input

16. Vibrationdata 16  4.5-sigma is maximum expected peak from Rayleigh distribution  Miles equation also gives 11.2 GRMS for the response  Relative displacement is the key metric for circuit board fatigue per D. Steinberg (future webinar) SDOF Acceleration Response = 11.2 GRMS = 33.5 G 3-sigma = 49.9 G 4.5-sigma SDOF Pseudo Velocity Response = 3.42 inch/sec RMS = 10.2 inch/sec 3-sigma = 15.3 inch/sec 4.5-sigma SDOF Relative Displacement Response = 0.00272 inch RMS = 0.00816 inch 3-sigma = 0.0121 inch 4.5-sigma

17. Vibrationdata Pseudo Velocity  The "pseudo velocity" is an approximation of the relative velocity  The peak pseudo velocity PV is equal to the peak relative displacement Z multiplied by the angular natural frequency  Pseudo velocity is more important in shock analysis than for random vibration  Pseudo velocity is proportional to stress per H. Gaberson (future webinar topic)  MIL-STD-810E states that military-quality equipment does not tend to exhibit shock failures below a shock response spectrum velocity of 100 inches/sec (254 cm/sec)  Previous example had peak velocity of 15.3 inch/sec (4.47-sigma) for random vibration 17

18. Vibrationdata 18 Peak is ~ 100 x Input at 200 Hz Q^2 =100 Only works for SDOF system response Half-power bandwidth method is more reliable for determine Q.

19. Vibrationdata 19 Peak Design Levels for Equivalent Static Load Author Design or Test Equation Qualifying Statements Himelblau, et al 33 However, the response may be non-linear and non-Gaussian Fackler 33 3  is the usual assumption for the equivalent peak sinusoidal level Luhrs 33 Theoretically, any large acceleration may occur NASA 3  for STS Payloads 2  for ELV Payloads Minimum Probability Level Requirements McDonnell Douglas 44 Equivalent Static Load Scharton & Pankow 55 See Appendix C DiMaggio, Sako, Rubin nn See Appendices B and D for the equation to calculate n via the Rayleigh distribution AhlinCn See Appendix E for equation to calculate Cn

20. Vibrationdata 20 Rayleigh Peak Response Formula Maximum Peak fn is the natural frequency T is the duration ln is the natural logarithm function is the standard deviation of the oscillator response Consider a single-degree-of-freedom system with the index n. The maximum response can be estimated by the following equations. a.k.a. crest factor

21. Vibrationdata Conclusions 21  The General Method is better than the Miles equation because it allows the base input to vary with frequency  For SDOF System (fn=200 Hz, Q=10) subjected to NAVMAT base input… We obtained the same response results in the time domain in Webinar 14 using synthesized time history!  Response peaks may be higher than 3-sigma  High response peaks need to be accounted for in fatigue analyses (future webinar topic)

22. Vibrationdata Homework 22  Repeat the exercises in the previous slides  Read T. Irvine, Equivalent Static Loads for Random Vibration, Rev N, Vibrationdata 2012 T. Irvine, The Steady-state Response of Single-degree-of-freedom System to a Harmonic Base Excitation, Vibrationdata, 2004 T. Irvine, The Steady-state Relative Displacement Response to Base Excitation, Vibrationdata, 2004