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## Unit 15

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**Unit 15**SDOF Response to Base Input in the Frequency Domain**Introduction**• Steady-state response of an SDOF System • Base Input: PSD – stationary with normal distribution**Miles Equation**• Miles Equation is the simple method for calculating the response of an SDOF to a PSD • Assume white noise, flat PSD from zero to infinity Hz • As a rule-of-thumb, it can be used if PSD if flat within + 1 octave of the natural frequency**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 where is the overall response Q is the amplification factor P is the power spectral density level at the natural frequency f nis the natural frequency**SDOF System, Base Excitation**The natural frequency fn is The damping coefficient C is The amplification factor Q is**SDOF Free Body Diagram**The equation of motion was previously derived in Webinar 2.**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.**Frequency Ratio (f / fn)**The base excitation frequency is f. The natural frequency is fn.**Transmissibility Curve Characteristics**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.**Exercises**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 ?**“Better than Miles Equation”**• 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”**Miles Equation & General Method**• The Miles equation was given in a previous unit • Again, the Miles equation assumes that the base input is white noise, with a frequency content from 0 to infinity Hertz • Measured power spectral density functions, however, often contain distinct spectral peaks superimposed on broadband random noise • The Miles equation can produce erroneous results for these functions • This obstacle is overcome by the "general method" • The general method allows the base input power spectral density to vary with frequency • It then calculates the response at each frequency • The overall response is then calculated from the responses at the individual frequencies**General Method**The general method thus gives a more accurate response value than the Miles equation. The base excitation frequency is fi and the natural frequency is fnThe base input PSD is**Navmat P-9492 Base Input**PSD Overall Level = 6.06 GRMS Accel (G^2/Hz) Frequency (Hz)**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**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 • 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)**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**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.**Rayleigh Peak Response Formula**Consider a single-degree-of-freedom system with the index n. The maximum response can be estimated by the following equations. Maximum Peak a.k.a. crest factor**Conclusions**• 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)**Homework**• 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