webinar_15_SDOF_frequency_domain

Unit 15
Vibrationdata
SDOF Response to Base Input in the
Frequency Domain
1
Introduction


Vibrationdata
Steady-state response of an SDOF System
Base Inputs:
Pure Sine
PSD – stationary with normal distribution
2
SDOF System, Base Excitation
Vibrationdata
The natural frequency fn is
fn 
1
2
k
m
The amplification factor Q is
The damping coefficient C is
C  2 k m
Q  1 /( 2 )
3
SDOF Free Body Diagram
Vibrationdata
The equation of motion was previously derived in Webinar 2.
z  2ξ ωnz  ωn2z  y
x  z  y
4
Vibrationdata
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
x

y
1 (2) 2
1   2
2 2
2
where
  f / fn
where f is the base excitation frequency and fn is the natural frequency.
5
TRANSMISSIBILITY MAGNITUDE
SDOF SYSTEM SUBJECTED TO BASE EXCITATION
20
Q = 10
Q=2
Q=1
Transmissibility (G out / G in )
10
Vibrationdata
1
0.1
0.1
1
10
Frequency Ratio ( f / fn )
Frequency
Ratio (f / fn)
The base excitation frequency is f.
The natural frequency is fn.
6
Transmissibility Curve Characteristics
Vibrationdata
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.
7
Exercises
Vibrationdata
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 ?
8
“Better than Miles Equation”
Vibrationdata

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”
9
Miles Equation & General Method



Vibrationdata
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
10
Vibrationdata
General Method
The general method thus gives a more accurate response value than the Miles
equation.

1 ( 2 i ) 2 
x GRMS f n ,    
ŶAPSD ( f i ) f i ,
2


i 1   1 i 2    2  i 2 


N
i  fi / f n
The base excitation frequency is f i and the natural frequency is f n
The base input PSD is ŶA PSD (fi )
11
Vibrationdata
Navmat P-9492 Base Input
PSD
Level
GRMS
PSD Overall
OVERALL
LEVEL==6.06
6.06 GRMS
ACCEL (G /Hz)
0.1
2
Accel
(G^2/Hz)
0.01
0.001
20
100
1000
Frequency
(Hz)
Accel
(G^2/Hz)
20
0.01
80
0.04
350
0.04
2000
0.007
2000
FREQUENCY
(Hz) (Hz)
Frequency
12
Apply Navmat P-9492 as Base Input
Vibrationdata
fn = 200 Hz, Q=10,
duration = 60 sec
Use:
vibrationdata > power spectral density > SDOF Response to Base Input
13
SDOF Acceleration Response
=
=
=
11.2 GRMS
33.5 G 3-sigma
49.9 G 4.47-sigma
Vibrationdata
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)
14
Pseudo Velocity
Vibrationdata

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
PV  n Z

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
15
Vibrationdata
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.
16
Peak Design Levels for Equivalent Static Load
Author
Design or Test
Equation
Qualifying Statements
Vibrationdata
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
NASA
3s for
STS Payloads
2s for
ELV Payloads
McDonnell
Douglas
4s
Scharton & Pankow
5s
DiMaggio, Sako, Rubin
ns
Ahlin
Cn
Theoretically, any large acceleration may occur
Minimum Probability Level Requirements
Equivalent Static Load
See Appendix C
See Appendices B and D for the equation to calculate n via
the Rayleigh distribution
See Appendix E for equation to calculate Cn
17
Vibrationdata
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.
cn 
2 ln fn T 
Cn  cn 
Maximum Peak
fn
T
ln
sn
0.5772
cn
 Cn sn
a.k.a. crest factor
is the natural frequency
is the duration
is the natural logarithm function
is the standard deviation of the oscillator response
18
Conclusions
Vibrationdata

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)
19
Homework

Repeat the exercises in the previous slides

Read
Vibrationdata
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
20