Normalize so that L = 1



A Method for Converting a Sine Tone to a Narrowband PSD

Revision G

By Tom Irvine

Email: tom@

October 9, 2013

______________________________________________________________________________

Variables

|m |Mass |

|c |Viscous damping coefficient |

|k |Stiffness |

|[pic] |Acceleration of mass |

|[pic] |Base acceleration |

|N |Overall GRMS response to narrowband base input |

|S |Sine base input peak amplitude (G peak) |

|Q |Amplification Factor |

|( |Standard deviation scale factor |

Introduction

Consider a single-degree-of-freedom system subject to base excitation.

Figure 1.

Assume a case where the base input is a sine tone which must be converted to a narrowband PSD for either analysis or test purposes. The conversion will be made in terms of the acceleration response of the mass to each input.

The conversion formula is

[pic] (1)

[pic] (2)

Equation (2) does not immediately give a corresponding base input PSD level. The matching PSD is derived in a separate calculation.

Specifically, the N response level is calculated for a given narrowband PSD input using the method in Reference 1.

If the test item’s natural frequency is unknown, then it is taken as the narrowband center frequency, which is the sine input frequency. Otherwise, the known natural frequency may be used.

N can initially be calculated for an input PSD of 1 G^2/Hz. Then the PSD can be scaled accordingly to satisfy equation 2 for a given amplification factor Q, sine amplitude S and coefficient (.

Set [pic]according to Reference 2.

[pic] (3)

This means that the sine response will be equal to the random response 1.9-sigma level. Note that the 1-sigma level is equal the RMS level assuming a zero mean value.

The recommended bandwidth for the PSD is one-twelfth octave.

One-twelfth octave appears to be a reasonable bandwidth which would allow a corresponding synthesized time history to have a normal distribution with a kurtosis of 3.0, if proper care is taken. A wider bandwidth may also be used if desired.

Example

An example is given in Appendix A. The example shows that equation (3) is satisfactory in terms of a resulting narrowband PSD which envelops the sine tone in terms of both peak response and fatigue damage.

References

1. T. Irvine, An Introduction to the Vibration Response Spectrum, Revision D, Vibrationdata, 2009.

2. T. Irvine, Extending Steinberg’s Fatigue Analysis of Electronics Equipment Methodology to a Full Relative Displacement vs. Cycles Curve, Revision C, Vibrationdata, 2013.

3. David O. Smallwood, An Improved Recursive Formula for Calculating Shock Response Spectra, Shock and Vibration Bulletin, No. 51, May 1981.

4. T. Irvine, Derivation of the Filter Coefficients for the Ramp Invariant Method as Applied to Base Excitation of a Single-degree-of-Freedom System, Revision B, Vibrationdata, 2013.

5. ASTM E 1049-85 (2005) Rainflow Counting Method, 1987.

6. Dave Steinberg, Vibration Analysis for Electronic Equipment, Second Edition, Wiley-Interscience, New York, 1988.

7. T. Irvine, Miner’s Cumulative Damage via Rainflow Cycle Counting, Revision F, Vibrationdata, 2013.

8. Test Methods and Control, Martin Marietta, M-67-45 (Rev 4), Denver, Colorado, January 1989. See Paragraph 8.20.4.3.

APPENDIX A

Example

An SDOF system is to be subjected to an 18 G, 100 Hz sine tone for 60 seconds. Derive an equivalent narrowband PSD with the same duration.

Set the band center frequency equal to 100 Hz. Set Q=10.

Set the bandwidth equal to one-twelfth octave such that the band limits are 97.15 and 102.93 Hz.

The overall response level is 94.7 GRMS per equation (3).

The corresponding PSD level is 17 G^2/Hz for this band as calculated using Matlab script: sine_to_narrowband.m.

Coordinates

The coordinate points of the narrowband PSD are

|Freq (Hz) |Accel (G^2/Hz) |

|97.15 |17 |

|102.93 |17 |

Time History Synthesis

[pic]Figure A-1.

A 60-second time history is synthesized for the narrowband PSD as shown in Figure A-1.

The overall level is 9.9 GRMS.

The kurtosis is 3.0.

The histogram has an approximately normal distribution as shown in Figure A-2.

[pic]

Figure A-2.

[pic]

Figure A-3.

The resulting PSD is shown in Figure A-3. Both curves have an overall level of 9.9 GRMS.

Peak Response Analysis

[pic]

Figure A-4.

The shock response spectra for the narrowband PSD and the sine tone are shown in Figure A-4. The synthesis yields a higher peak acceleration beginning at 73 Hz.

Any system with a natural frequency below, say, 50 Hz would be considered as isolated.

Fatigue Analysis

The relative displacement response was calculated for each base input using a natural frequency of 100 Hz and Q=10, using the method in References 3 and 4.

Next, a rainflow cycle count was performed on each relative response using the method in Reference 5.

A damage index D was calculated using

[pic] (A-1)

where

|[pic] |is the response amplitude from the rainflow analysis |

|[pic] |is the corresponding number of cycles |

|b |is the fatigue exponent |

The damage index is only intended to compare the effects of the two base input types.

The fatigue exponent is assumed to be b > 6.4.

The fatigue results for the example are

|Base Input |Fatigue Damage, fn=100 Hz, Q=10 |

|Type |b=5.0 |b=6.0 |b=6.4 |b=8.0 |b=10.0 |b=12.0 |

|Sine |1.01 |0.18 |0.089 |0.0055 |0.00017 |5.3e-06 |

|Narrowband PSD |0.73 |0.17 |0.093 |0.0099 |0.00067 |4.9e-05 |

The Narrowband PSD is thus has a greater fatigue damage potential that the Sine input for

b > 6.4.

Furthermore, there is an implicit assumption in the above table that the test item’s natural frequency is the same as the sine tone frequency. In many cases the item’s natural frequency may be unknown.

Note that Reference 6 gives a value of b=6.4 for electronic equipment for both sine and random vibration. Un-notched aluminum samples tend to have a value of b ( 9 or 10, as shown in Reference 7 for example.

Add Narrowband to Broadband PSD

Now assume that the narrowband PSD is to be added to a broadband PSD. Note that the G^2/Hz values can be simply added.

Assume that the broadband random level is 0.1 G^2/Hz from 20 to 2000 Hz.[1]

The coordinate points of the combined PSD are

|Freq (Hz) |Accel (G^2/Hz) |

|20 |0.1 |

|97.14 |0.1 |

|97.15 |17.1 |

|102.93 |17.1 |

|102.94 |0.1 |

|2000 |0.1 |

This table could then be inserted into a finite element model. The frequency step in the finite element analysis should be fine enough to pick up a number of points within the narrowband. An appropriate value would be, say, 0.5 Hz or lower.

The table could also be input to a vibration control computer for a shaker table test with the same concern for the frequency resolution.

APPENDIX B

Trade Study

Recall the formula for the overall GRMS response to narrowband base input

[pic] (B-1)

Continue with the example from Appendix A with a variable coefficient (.

The response levels are

|[pic] |Response GRMS |Base Input PSD (G^2/Hz) |

|1.9 |94.7 |17.0 |

|1.8 |100.0 |18.9 |

|1.7 |105.9 |21.2 |

|1.6 |112.5 |23.9 |

|1.5 |120.0 |27.2 |

Note that some references use smaller fatigue exponents which yield more conservative, higher PSD base input levels. Reference 8, for example, gives a value of b=4.0 for “Electrical Black Boxes.”

The fatigue damage results are given in the following table. The bold font indicates cases where the Narrowband PSD exceeds the Sine in terms of fatigue damage.

|Base Input |Fatigue Damage, fn=100 Hz, Q=10 |

|Type |b=4.0 |b=4.5 |b=5.0 |b=5.5 |b=6.0 |b=6.4 |

|Sine |5.75 |2.41 |1.01 |0.42 |0.18 |0.089 |

|Narrowband PSD, ( = 1.9 |3.43 |1.57 |0.73 |0.35 |0.17 |0.093 |

|Narrowband PSD, ( = 1.8 |4.33 |2.04 |0.98 |0.48 |0.24 |0.14 |

|Narrowband PSD, ( = 1.7 |5.40 |2.61 |1.29 |0.65 |0.33 |0.19 |

|Narrowband PSD, ( = 1.6 |6.88 |3.43 |1.74 |0.90 |0.47 |0.28 |

Thus, an ( = 1.6 covers the conservative b=4.0 case from Reference 8.

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[1] A typical PSD would have a starting ramp to limit low-frequency displacement, but this is omitted for simplicity.

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