SDOF Response to Base Input in the Frequency Domain

1
Unit 15

• SDOF Response to Base Input in the Frequency
  Domain

2
Introduction

• Steady-state response of an SDOF System
• Base Inputs
• Pure Sine
• PSD stationary with normal distribution

3
SDOF System, Base Excitation
The natural frequency fn is
The damping coefficient C is
The amplification factor Q is
4
SDOF Free Body Diagram
The equation of motion was previously derived in
Webinar 2.
5
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.
6
Frequency Ratio (f / fn)
The base excitation frequency is f. The
natural frequency is fn.
7
Transmissibility Curve Characteristics

• The transmissibility curves have several
  important features
• 1. The response amplitude is independent of Q for
  f ltlt fn.
• 2. The response is approximately equal to the
  input for f ltlt fn.
• 3. Resonance occurs when f ? fn.
• 4. The peak transmissibility is approximately
  equal to Q for f fn and Q gt 2.
• 5. The transmissibility ratio is 1.0 for f ?2
  fn regardless of Q.
• 6. Isolation is achieved for f gtgt fn.

8
Exercises
vibrationdata gt Miscellaneous Functions gt 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 fn100 Hz f 141.4 Hz ?
9
Better than Miles Equation

• Determine the response of a single-degree-of-freed
  om 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

10
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

11
General Method
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
12
Navmat P-9492 Base Input
PSD Overall Level 6.06 GRMS
Accel (G2/Hz)
Frequency (Hz) Accel (G2/Hz)
20 0.01
80 0.04
350 0.04
2000 0.007
Frequency (Hz)
13
Apply Navmat P-9492 as Base Input
fn 200 Hz, Q10, duration 60 sec
Use vibrationdata gt power spectral density
gt SDOF Response to Base Input
14
SDOF Acceleration Response 11.2 GRMS
33.5 G 3-sigma 49.9 G 4.47-sigma
SDOF Pseudo Velocity Response 3.42
inch/sec RMS 10.2 inch/sec 3-sigma
15.3 inch/sec 4.47-sigma SDOF Relative
Displacement Response 0.00272 inch RMS
0.00816 inch 3-sigma 0.0121 inch
4.47-sigma

• 4.47-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)

15
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

16
Peak is 100 x Input at 200 Hz Q2 100
Only works for SDOF system response Half-power
bandwidth method is more reliable for determine Q.
17
Peak Design Levels for Equivalent Static Load
Author Design or Test Equation Qualifying Statements
Himelblau, et al 3s However, the response may be non-linear and non-Gaussian
Fackler 3s 3s is the usual assumption for the equivalent peak sinusoidal level
Luhrs 3s Theoretically, any large acceleration may occur
NASA 3s for STS Payloads 2s for ELV Payloads Minimum Probability Level Requirements
McDonnell Douglas 4s Equivalent Static Load
Scharton Pankow 5s See Appendix C
DiMaggio, Sako, Rubin ns See Appendices B and D for the equation to calculate n via the Rayleigh distribution
Ahlin Cn See Appendix E for equation to calculate Cn
18
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
fn is the natural frequency
T is the duration
ln is the natural logarithm function
is the standard deviation of the oscillator response
19
Conclusions

• The General Method is better than the Miles
  equation because it allows the base input to vary
  with frequency
• For SDOF System (fn200 Hz, Q10) 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)

20
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