1

SINGLE-DEGREE-OF-FREEDOM SYSTEM RESPONSE TO SINE-ON-RANDOM VIBRATION

By Tom Irvine Email: tomirvine@aol.com August 13, 2008 Introduction Consider a single degree-of-freedom system.

Figure 1.

where

Assume that the amplification factor is Q=10. Allow the natural frequency to be an independent variable. Three natural frequencies are considered: 100, 200, 400 Hz. The system is subjected to simultaneous sine and random base excitation per the specification in Figure 2. Calculate the SDOF response in both the time and frequency domains.

m = mass c = viscous damping coefficient k = stiffness x = absolute displacement of the mass y = base input displacement

m

k c

&&x

&&y

2

Specification

0.001

0.01

0.1

1

100 100020 200 20000.1

1

10

100

15

Sine 15 G peak (Right Scale)PSD 6.1 GRMS (Left Scale)

FREQUENCY (Hz)

AC

CEL

(G2 /H

z)

ACC

EL (G

)

SINE-ON-RANDOM SPECIFICATION COMBINED OVERALL LEVEL = 12.2 GRMS

Figure 2.

Table 1. PSD, 6.1 GRMS, 60 seconds

Frequency (Hz)

Accel (G^2/Hz)

20 0.01 80 0.04 350 0.04 2000 0.007

3

Time Domain Synthesis

-250

-200

-150

-100

-50

0

50

100

150

200

250

0 5 10 15 20 25 30

TIME (SEC)

ACC

EL

(G)

SYNTHESIZED TIME HISTORY SINE-ON-RANDOM SPECIFICATIONOVERALL LEVEL = 12.2 GRMS MIN = -39.1 G MAX = 44.8 G KURTOSIS = 2.1

Figure 3. A time history was synthesized in Figure 3 to meet the specification in Figure 2. The time history was shortened to 30 seconds for brevity. A close-up view is given in Figure 4. The corresponding histogram is shown in Figure 5. The histogram has a somewhat symmetric, bimodal shape.

4

-50

0

50

10.00 10.05 10.10 10.15 10.20

TIME (SEC)

ACC

EL

(G)

SYNTHESIZED TIME HISTORY SINE-ON-RANDOM SPECIFICATIONOVERALL LEVEL = 12.2 GRMS MIN = -39.1 G MAX = 44.8 G KURTOSIS = 2.1

Figure 4.

0

4000

8000

12000

16000

20000

-50 -40 -30 -20 -10 0 10 20 30 40 50

ACCEL (G)

CO

UN

TS

HISTOGRAM SINE-ON-RANDOM SYNTHESIS

Figure 5.

5

0.001

0.01

0.1

1

10

100

100 100020 200 2000

PSD Synthesis 12.2 GRMSPSD Specification 6.1 GRMS

FREQUENCY (Hz)

ACC

EL (G

2 /Hz)

PSD SINE-ON-RANDOM SPECIFICATION

Figure 6. The PSD of the synthesis is compared to that of the specification in Figure 6. Note that the G^2/Hz amplitude at 200 Hz depends on the ∆f bandwidth. There is no exact way to convert pure sine G levels into G^2/Hz levels.

6

Time Domain Response, fn =100 Hz

-250

-200

-150

-100

-50

0

50

100

150

200

250

0 5 10 15 20 25 30

TIME (SEC)

ACC

EL

(G)

SDOF RESPONSE (fn=100 Hz, Q=10) TO SYNTHESIZED TIME HISTORYOVERALL LEVEL = 8.6 GRMS MIN = -33.7 G MAX = 33.0 G KURTOSIS = 2.8

Figure 7.

0

10000

20000

30000

-50 -40 -30 -20 -10 0 10 20 30 40 50

ACCEL (G)

CO

UN

TS

HISTOGRAM SDOF RESPONSE (fn = 100 Hz, Q = 10)

Figure 8.

7

Time Domain Response, fn =200 Hz

-1500

-1000

-500

0

500

1000

1500

0 5 10 15 20 25 30

TIME (SEC)

ACC

EL

(G)

SDOF RESPONSE (fn=200 Hz, Q=10) TO SYNTHESIZED TIME HISTORYOVERALL LEVEL = 107.4 GRMS MIN = -189 G MAX = 191 G KURTOSIS = 1.5

Figure 9.

0

5000

10000

15000

-200 -150 -100 -50 0 50 100 150 200

ACCEL (G)

CO

UN

TS

HISTOGRAM SDOF RESPONSE (fn = 200 Hz, Q = 10)

Figure 10.

8

Time Domain Response, fn =400 Hz

-500

0

500

0 5 10 15 20 25 30

TIME (SEC)

ACC

EL

(G)

SDOF RESPONSE (fn=400 Hz, Q=10) TO SYNTHESIZED TIME HISTORYOVERALL LEVEL = 20.6 GRMS MIN = -87 G MAX = 83 G KURTOSIS = 2.7

Figure 11.

0

5000

10000

15000

20000

25000

-100 -80 -60 -40 -20 0 20 40 60 80 100

ACCEL (G)

CO

UN

TS

HISTOGRAM SDOF RESPONSE (fn = 400 Hz, Q = 10)

Figure 12.

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Frequency Domain Response The frequency domain calculations are performed using the equations in Appendix A. Each Combined Response value is the “square root of the sum of the squares.” The Combined Response values agree very closely with the corresponding levels calculated in time domain.

Table 2. Response Levels from Frequency Domain Analysis

fn (Hz) Response to

Sine (GRMS)

Response to Random (GRMS)

Combined Response (GRMS)

100 3.6 7.82 8.6

200 106.8 11.18 107.3

400 14.1 14.9 20.5

References

1. T. Irvine, A Method for Power Spectral Density Synthesis, Rev B, Vibrationdata, 2000. 2. T. Irvine, An Introduction to the Vibration Response Spectrum Rev C, Vibrationdata,

2000.

3. T. Irvine, The Steady-state Response of Single-degree-of-freedom System to a Harmonic Base Excitation, Vibrationdata, 2004.

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APPENDIX A

Variables

ξ = fraction of critical damping

f n = natural frequency

f i = base excitation frequency

ρ i = non-dimensional frequency parameter

A = sine acceleration amplitude

PSDAY = PSD base input acceleration

GRMSx&& = overall acceleration GRMS response

Response to Sine Vibration The peak response is

( )

( )nf/ifi,

2i2

22i1

2i21Ax =ρ

ξρ+⎟⎠⎞⎜

⎝⎛ ρ−

ξρ+=&& (A-1)

The corresponding RMS value is calculated by multiply the peak response by 0.7071. Equation (A-1) is taken from Reference 3.

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Response to Random Vibration The overall acceleration response is

( )[ ] nf/ifi,

N

1iif)if(PSDAY

2]1[)2(1,nfGRMS 2

i22

i

2ix =ρ

=Δ

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

ξρ+ρ−

ξρ+=ξ ∑&& (A-2)

Equation (A-2) is taken from Reference 2.

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APPENDIX B

Kurtosis is a parameter that describes the shape of a random variable’s histogram or its equivalent probability density function.

The kurtosis for a time series iY is

Kurtosis =

[ ]

4

n

1i

4i

n

Y

σ

μ−∑=

where

μ = mean σ = standard deviation n = number of samples

The term in the numerator is the “fourth moment about the mean.”

A pure sine time history has a kurtosis of 1.5.

A time history with a normal distribution has a kurtosis of 3.

Some alternate definitions of kurtosis subtract a value of 3 so that a normal distribution will have a kurtosis of zero.

A kurtosis larger than 3 indicates that the distribution is more peaked and has heavier tails than a normal distribution with the same standard deviation.