Weakly damped second order system under non-stationary random excitation

COM NC EME NT

OF TEXT

WEAKLY DARDED SECOND ORDER SYSTEMUNDER MONETATIONARY RANDOM EXCITATION

F.J.PasveertAME

The applicability of the frequency response method for estimates of the outDut

variance

of a weakly damped second order system subject to nonstationary random excitation,

dictated by spectrally prescribed properties, is examined by time domain computer

simulations.

1. INTRODUCTION

The validity of variance calculations with the

frequency response method applied on a weakly

damped linear second order system subject to

nonstationary stochastic excitation is

examined by time domain simulations on a hybrid

computer. Spectral properties of the random

excitation are obtained by appropriate

filtering of the output of a noise generator

being part of the hybrid computer. This time

function becomes input to the second order

system.

This study originated from a dissertation on

the vibratory response of the shipp's hull to

rionstationary wave excitation E i]

The question arises to what extent the

non-stationarity in the wave spectrum, caused by

quick changea of the energy in the waves over

small periods of time (so called short time

non-stationarity) effects the vibratory response.

The influence of the short time nonstationarity

is investigated by comparing the exact response

in the mean square sense from time domain

simulations with the response obtained by the

frequency response method, which is only

allowed for stationary excitation. It appears

that for damping ratios below 3% of the

critical damping the variance of the exact

response differs considerably from the variance

derived by the frequency response method.

Por validation purposes both a hybrid and a

pure digital simulation have been carried out

of which the results agree.

P'- -ahopag fLter

hths,4cnLp.

Department of Hybrid Computation

University of Technolor

Deift, The Netherlands.

-

.i_.

'1-CEENT flF

---Corro tion PT1.Spcctron,

hping

f Lt en

digtaL

compute

Figure 1: Block diagram of modelling.

t

2. GENERAL

With the aid of a pseudo random binary noise

generator and suitable analog computer

programming a time domain simulation of

unidirectional ocean wave behaviour is obtained.

Setting the noise generator at a sequence length

of 14096 and clock interval of 0.5 sec

(corresponding with the usual sampling interval

of ship motions in open sea) roughly half an

hour real time load can be simulated. Inclusion

of a time scale factor in the design of the

analog modules permits for simulation runs

within a few seconds, mainly restricted by the

digital computer.

The ocean wave power density spectrum is modelled

according to the Pierson-Moskowitz wave

spectrum (PM spectrum). The presence of an

exponential factor in its equation makes it

unsuitable for implementation with physical

realisable filters. Therefore the spectrum is

approximated by a (squared) transfer function

built up with simple operations like

differentiation and integration. In series with

the shaping filter programming the differential

equation of the second order springing model is

implemented. Its relevant parameters, damping

ratio and resonance frequency, are translated

into coefficient settings by the digital

partner. The time domain variance is calculated

by the analog part. Simultaneously the digital

part computes the frequency domain variance by

the frequency response method. Their ratio is

plotted on an oscilloscope as function of the

frepescy du"a ,VcpOrÇe

Lab. vScheepbouwkune

Technische Hogeschool

Deift

re s panse seo pi dsplay

-p 3.U49 radon, go flint a r .ock

I3JUNi1979

ARCHIEF

damping ratio at fixed values of the resonance frequency, which lie in -the descending region of the PM spectrum where springing occurs. The entire model operates under control of a digital program and is synchronioed by the reset option of the noise generator. See figure 1. Repeated operation with the "same excitating data" permits examination of this proportion as a function of the model parameters only. The pre-shaping filtering in figure 1 is

explained in the following section.

3. PRE-SHAPING THE PSEUDO RANDOM BINARY SEQUENCE

Though the noise generator is provided with Gaussian noise output it can not he applied for our shaping purposes, because this terminal

delivers an insufficient broad frequency spectrum at a clock interval of 0.5 sec. For this reason the binary sequence itself is fed

into a fourth order low pass filter with 2 rad/sec cut off frequency. The low pass filtering operation is implemented by a feed back integrator circuit being the equivalent of

the low pass RC filter. Its coefficients a and B (figure 2) follow from the differential equation of the RC filter by well known methods of problem scaling.

Figure 2: RC filter with analog equivalent.

The pre-shaped pseudo random binary sequence, preprocessed by four of these low pass filtering operations becomes analog input for the actual shaping filter, according to the PM wave spectrum.

14. SHAPING FILTER

The PM wave spectrum has the gencral form

S(w) = A.w.exp(-E.w)

S in m2sec (14.) where

s is circuìair wave frequency (rad/sec)

A = 8.x1O3.g2

g gravitational constant

(rn/s cc2)

B = 3.11.h2 h significant wave height (a).

In this form of the FM wave spectrum only dependency in h exists. Figure 3 shows some spectra for various values of h.

From linear system theory it is known that

= H(s)!2. S(w)

is a relation in the frequency domain between the input power density spectrum S(w) of the stationary random excitation; the system's

2

,6

S(w)

= A.s

2 2k/( 2+a )r+12

C.y c.ê / ¿ 2.o

Figure 3 : PM wave spectra.

transfer function

_H(s)

and the output power density spectrum Sy(s).

If

_IH(ts)I

is made equal to Spm(w) the linear

system will produce a time function with prescribed spectral properties under condition of excitation with (white) stationary noise with zero mean and unity variance.

According to Stearns 1 2] Sam(s) is approximated

by a linear system with a transfer function of the shape

H(s) = A.sk/(s+a)r (14.2)

with s the Laplace transform operator. Then

(14.3)

becomes an approximation of the PM wave spectrum. Reference E 21 delivers two general properties

of this S(5)

(k

r-k+1 r-k+1 A2.kk _r-k+1 (r+1 )r+1 _a

5max frequency where maximum power occurs. To obtain numerical values for k and r of the shaping filter to be designed a separate digital program has been written. The procedure is straight forward and operates in an interactive way.

With input parameters k, r significant wave height h of the PM spectrum together with a choosen radian frequency increment SmaX and 5max are searched in generated values of S with equation (14.1). Next A and a are

calculated with equations (14.14). Finally, the approximated opectrum is generated with (14.3k. For visual inspection Spa(s) and its

approximation are displayed on a memory scope. WEAKLY DAED SECOND ORDER SYSTEM UNDER NONSTATIDNABY RAITDOM EXCITATION

1'EAKLY DAMPED SECOND ORDER SYSTEM UNDER NONSTATIONARY RANDOM EXCITATION

It appears that with k=5 and r10 adequate

shaping occurs in the descending region of the PM wave spectrum, which is the region of interest for springing studies. Figure 14 gives an

example for a significant wave height of 1m and

k=5,

r10.

b

Figure 14: Shaping filter and PM spectrum.

In shaping filler formula (14.2) we recognise two analog realisable sections

i s

and

5+a s+a

The first one is simply a low pass filtering oporation, the second is recognised as a 'bad"

differentjator. But it is just this denominator which makes the s-typo of operation

implementable by analog components in combination with the (s+a)1 which is yet required.

Analog implementation of the complete shaping filter is done by serial connection of five

s/(si-a) sections and six 1/(s+a) sections

by their underlaying differential equations.

5.

CALIBRATION OF SHAPING FILTER OUTPUT

In stead of taking care of the input variance it is more feasible to measure the output variance and compare it by the variance derived from h in the dictated PM spectrum. Its

deviation is then compensated by a scale factor at input side. See figure

_5.

The relation between h and the PM wave spectrum is 1MFT OF TI Xii P. M. hpin

hite.

to 4prngn model

h = 14.

S(w)dw

co r,c.c t io n

Fietor

variance Comp*ruon

4-

rs o u sure ,ssnt

varunce from PM wuve Spectrum

Figure

5:

Matching output variance of shaping filter.

By means of Parseval's theorem (relating energy in time- and frequency domain by fourier transform pairs) and the above equation it follows

h2/16

a/N.

(c)2

(5.1)

where the time domain integral in Parseval's theorem is estimated by equidistant samples with the digital partner. Setting initially a to unity value and measurement of the variance, the correction directly follows from (5.1).

6.

SPRINGING MODEL (dynamic scaling) The frequency domain and time domoin

expressions of a second order system are

H(s) w

/ (s2+2s+2

(6.1)

and

ax = y + 2siy +ay

2 .. . 2 (6.2)

with w and resonance frequency resp. damping ratio.

The amplification at resonance frequency, governed by

= (2Y1

(6.3)

with 2 and maximum input resp. output, is applied for scaling purposes.

Because of the stochastic excitation the range of the second order model components will not

fully be utilized in spite of proper scaling with (6.3). Therefore an initial run is done with the input coefficient of the model scaled according to(6.3). Depending upon the maximum output this coefficient is scaled upwards to utilize the whole linear range of the components.

WEAKLY DAÌED SECOND ORDER SYSTEM UNDER NONSTATIONABY RANDOM EXCITATION

Analog implementation of the springing model is obviouo and is shown in figure

6.

Figure

6:

Springing model.

The time domain variance is calculated by squared integration (not shown in figure

6).

In the mean time the frequency response method variance is calculated digitally from

fS(w),IH(a)j2 cts

At this point the digital computation speed becomes critical. In fact this computation speed dictates the real profit in computation time relative to problem time.

7. VALIDATION AND RESULTS

By digitally controlled frequency sweeping methods the (PM) shaping filter and springing model have been tested. Figure 7 gives measured results together with the theoretical transfer functions, equations (c.2) and (6.1).

Figure 7: Transfer functions of springing model and shaping filter, from 0.0 in steps of 0.02 rad/sec.

(amplitude spectra)

The vertical lines in figure 7 are measured amplitudes at various frequencies, the envelope is the theoretical amplitude transfer computed by equations (i.2) and (6.1). A pure digital time domain simulation has been made too for comparison of the results from the

hybrid method. In principle the same modules have been applied. Random data generation was done by subroutines available in the Fortran library. Shaping filtering and springing modelling were carried out in the time domain

with their difference equations,

which

vere derived by means of z transform theory from the Laplace functions. Ref

[31 .

Trends in the proportion of both methods of variance calculation in dependency of the damping correspond with those from the hybrid simulation. See figuro

8.

Finally figure 9

depicts the input power density spectrum for the hybrid simulation.

a

C;,. 90 ¿O digital / ¿ (/ 20

Figure

8:

variance ranos, digital and hybrid. a peak freq in input spectrum.

resonance freq 2.nd order system.

SERT SCOLE O.Iø07E OC

ICOR SERLE IC.ICE;4IC SO

Figure 9: Input power density spectrum for 2.nd order system, hybrid method.

8. DISCUSSION

In his theoretical analysis Darnoski,f1cJ estimates the variance under nonstationary random load in terms of noise power density

spectrum, system's transfer function and the envelope with which the white (or vide band) noise was multiplied. With a step envelope

function of finito resp. infinite duration he analysed that the output variance reaches

stationary mean square value after a growing number of response cycles" _{at lower values} of the damping. 2. lic

=1.80

l.19

hybrid 2 3 Li

Similar results are observed in our simulations, and are confirmed by [14J

By the frequency response method the mean square response is obtained by superposition of the steady state response to regular wave components, constituting the total wave spectrum.

In case of relative quickly variing wave energy with time, short time nonstationarity, the

frequency response method will fail, particularly in the region of very low dumping ratios.

CONCLUSION

The frequency response method can not merely be applied for mean square response calculations of second order systems subject to non-stationary random excitation, in cases of damping ratios below 3% of the critical damping.

REFERENCES

[11 Gunsteren, F.F. van., Springing of ships in waves. Doctoral thesis, 9 november

_1978,

Delft, University of Technology.

121 Stearns, S.D., Digital simulation of frequency limited random functions. Simulation Vol 12, number 1 jan 1969.

1 31 Various text books on z transform theory. Alan.V.Oppenheim/ Ronald.W.Schafer., Digital signal processing. Prentice Hall

1975.

B.C.Kuo., Analysis and synthesis of Sm sampled data and controa systems. Prentice Hall 1963 (electronical engineering).

Barnoski ,R.L. and Maurer,J.L., Mean-square response of simple mechanical systems to nonstationary random excitation. Journal of Applied Mechanics, jurie

_1969.

EMENI DF TE XI