Investigation of methods of determining structural damping from random vibrations

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Investigation of methods of determining structural damping from

random vibrations

11

s e r

TH

I

Building

~esearch

Note

no.

57

C

t

INVESTIGATION

OF METHODS OF

DAMPING

FROM

H.S. W a r d and

L.

INVESTIGATION O F METHODS O F DETERMlNING STRUCTURAL

DAMPING FROM RANDOM VIBRATIONS

by

H.S. W a r d and L, I-Ialabisky

The analysis of random vibrations of buildings, caused by

winds, has been u ~ e d to esti'mate the damping characteristics of these

buildings. A s the true values of the damping w e r e not known, there was no check on the accuracy or validity of the methods u s e d . A n analogue computer simulation of the r a n d o m vibrations of a theoretical

five

-

assumed.

So

as sumptian,

Band limited white noise excitation ( 0 to 35 cps) w a s used because analysis of wind records shows its power spectrum is

essentially flat in the range of building frequencies. Autocor relation

and power spectrum a n a l y s i s had both been u s e d to determine t h e damping values of actual buildings; therefore, e a c h of them was investigated in the analogue simulation. Cherry and Brady (1) have

recently reported some work involving the use of autacorrelation a n a l y s i s to determine structuraL damping values from random v i b r a t i o n records.

EXPERIMENTAL PROCEDURE

A five - s t o r e y shear-type structure w a s simulated on the

analogue computer. It was assumed to b e one bay wide and supported

on a rigid b a s e . A w e i g h t ,

W ,

each floor level and all t h e columns and s t o r e y heights w e r e assumed

t o be equal, The equation of motion for the nth floor of an N-storey

building when e a c h is subjected to a f o r c e of the form

?!.

g f

In Eq. ( I ) , g is the acceleration due t o gravity,

p

( S E I / L ~ ~

,

o

-4

E i s Young" modulus, 1 is the second moment of area of the columns,

and L is the distance between f l o o r s . For the investigation the value

of W L ~ / ~ E I ~ w a s taken equal to 1.0 x

l o q 2

five natural frequencies of undamped vibration are 0.907 cps, 2.65 c p s ,

4.18 c p s , 5 . 3 8 c p s and 6.20 c g s .

The analogue structure w a s excited for 15 min, during which time the v e l o c i t y outputs or displacements w e r e recorded f o r analysis

purposes. Each 15 min of simulation generally represented a case when

the damping value on each floor w a s the same. The 15-min recording

period corresponds approximately t o the recording time used i n acixal

buildings,

V e l o c i t i e s f r o m the analogue simulation w e r e measured with

0, 1, 2 , 3 , 4 and 10 per cent c r i t i c a l damping. Displacements were

recorded with 0, 1, 2, 3 , 4 and 5 per c e n t c r i t i c a l damping. T h e s e

damping values for any particular t e s t w e r e equal on each floor except

in ofie instance. 1n this c a s e , d i f f e r e n t values of damping w e r e assigned to each f l o o r to investigate the effect on the measured values.This had

some practical importance as it is likely that the damping in multi-

storey structures will vary throughout the building.

A seven-channel f r e q u e n c y -modulated tape recorder w a s u s e d

t o r e c o r d the output voltages from the analogue computer s a that a

power -spectrum d e n s i t y a n a l y s i s and autocorrelation of these outputs

could be performed. The output voltages w e r e recorded on every floor

f o r both velocities a n d displacements. A11 recordings on the F M tape

recorder were at 1 718 i n . / s e c ,

In o r d e r t.o do a p o w e r - s p e c t r u l n density analysis of each

recording, it was played back at 1 5

in.

loop at 1 7/8 i n . / s e c . T h e loop w a s t h e n played back at 60 i n . / s e c into a spectrum analyser. T h e total speed-up factor was 2 5 6 ; i.e,, 256 cps

(analysis time) = 1 cps (real t i m e ) . T h e p o w e r - s p e c t r u m analysis w a s

performed with a bandwidth of 1 c p s and an averaging time of 1. 5 s e c .

The d a m p i n g w a s f o u n d f r o m the power spectrum by measuring

the bandwidth a t half - p o w e r p o i n t s . W i t h a icnowledge of this bandwidth

i t was p o s s i b l e to c a l c u l a t e the Q value 01 the system f r o m Q = fJ(half- p o w e r b a n d w i d t h ) , w h e r e f, is the r e s o n a n t f r e q u e n c y . The percentage

The autocorrelation analysis was c a r r i e d out on the S . D.S, 9 2 0 computer. Ten min of r e c o r d of each s e l e c t e d track were played

into the cortlputer at a speed of I 7/8 in, /sec. The sampling time w a s

s e t at 6 1 milliseconds, which gave approximately 10,000 sample

readings over which the analysis was done. The autocorrelation function

was calculated for 190 time-delay values. An X - Y recorder w a s used to

record the output with time measured on the x axis and the relative

amplitude on the y axis. The damping w a s found from a chart by

measuring the ratios of different peaks to the f i r s t , i . ~ .

,

Xn is the amplitude of the nth peak, with xo the amplitude of the initial

peak. From the ratio %/xo the percentage of critical damping can be

found from a chart similar t o that shown in _{Figure 1 .}

The autocorrelation analysis was done f a r the fundamental

frequency only. T h i s w a s isolated by means of a band-pass filter with the low frequency cut-off at 0 . 1 cps and the high cut-off at 1.7 c p s / s e c . THEORY O F DAMPING

IN

L n ~ u t -Olttnut Relations for S D ~ ctral Densitie s

Consider t h e case of a linear system that has a complex

frequency response, H(rn). If i t i s excited by a function, f ( k ) , then the

response, x(t), is defined by

w h e r e is the F o u r i e r transform of f ( t ) , If f ( t ) i s a stationary

random p r o c e s s It c a n be shown ( 2 ) that the _{power spectrum of t h e} response. S x ( a ) , is related t o the power spectrum of the input. S f ( w ) ,

by the relation,

If it i s now assumed that the input has a uniform spectral density,

FY

,

From this result w e can conclude that the response spectral

density is equal to a constant times the absolute value of the square of the complex frequency response of the system in the c a s e of white

n o i s e excitation, This proves that i t i s possible to measure directly

the Q value of a system from t h e power spectrum of the output produced by an input of white noise. A comparison of the half-power bandwidth

obtained from the analysis of displacement and velocity of a high

system is considered in Appendix A , with the results shown in Figure Awl,

It can be seen that there is no significant error in the u s e of velocity measurements t o obtain the Q value of such a system.

Autocar relation Analysis

The autocorrelaticn function, Rf(T), of a function, f ( t ) , is defined by

T

~

~

=

(

~

1

_-L

_J

+

1

T j o

2T

Also by definition the autocorrelatian function is related to the power

spectrum, Sf(lu). of f(t) by

Consider the free vibrations of a single -degree -of-f reedom

system that h a s an _{initial displacement, A, and no initial velocity condition.} For a lightly damped system the displacement as a function of time is

given by

f ( t )

=

-at

.. .

w h e r e

a

(6)

( = I ,

f given b y h 2 _-a(2t

+

-

-

. .

T h e i n t e g r a l is o n l y pcrformccl b c t w e e ~ l the limits 0 to T because for

The i n t e g r a l can b e put into the f o r m ,

A~ n a ( Z t + s )

TI'

J

+

-

1

The separate terms in the integral can be integrated by parts and when

T = -

R

W f

A~ CI

C -a7

{,

+

2 C O S W T R i d

=

Ie

A

.

-

1

The f i r s t t e r m in Eq. (10) is dominant since

a

can be seen that the autocorrelation function has an exponentially-damped amplitude directly proportional t o the damping value of the s y s t e m .

If

the velocity of vibration i s analysed, then again the autocorrelation

function will have a dominant damped t e r m directly proportional t o the

damping of the system when this damping is small. This c a n be demon-

strated by d i f f e r e n t i a t i n g Eq. ( 8 ) to obtain the velocity of the system, v ( t ) ,

The second t e r m _{in Eq, (11) c a n be ignored because} (11 i s small and the dominant term of the autocorrelation function, R of v(t3 will

v

be given by

The preceding e x p r e s s i o n s for the autocorrelation $unction r e f e r to the case when t h e r e i s n o forcing function, If the system is forced t o vibrate then the expression g i v e n i n

Eq.

autocorrelation function, T h i s h a s been done in Appendix B f o r t h e

power spectrum. S ( " ] ) , of a lightly damped single-degree-of -freedom

system given by

L

tU

2

w h e r e w, is the natural frequency of the system,

C

Ccritical

-

example of the autocorrelation function when a.

=

c

/

c

-

~

~

~

~

~

~

~

~

0.0 1 is plotted in Figure B-I f r o m the data of Table B-1, It can b e

seen from Appendix

B

where A >, A 1 2 and -1 2

@

.

@

When

g

-

5

R(T) e

-"

p1

+

A

I

This dernonstxates that f c r small damping

the

function of the random vibrations of a single-degree-of-freedom system has an exponentially - d a m p e d amplitude directly proportional to the

damping.

The r e s u l t s obtained in this section indicate that it should be

possible to d e t e r m i n e damping values f r o m the autocor relation analysis of r a n d o m vibrations by m e a n s of the logarithmic decrement: approach s u m m a r i z e d in Figure 1 ,

RESULTS

The results obtained from the power spectrum analyses of the

fifth floor displacements f o r f i v e different values of damping are shown

in Figure 2. This g i v e s an indication of the relation that can be expected between t h e measured damping value and the actual damping value. The

measured damping values tend t o b e too high but the scatter most

probably is due to the statistical r e l i a b i l i t y of the results(there is very

little information a v a i l a b l e on this a s p e c t of the problem and i t requires further attention). The fourth and fifth modes did not appear on the

i

- 7 -

P o w e r spectrum analyses w e r e performed t o see how the

measured values of damping varied f r o m floor to floor when the

actual value s e t up on the simulation was the same for each floor,

T h i s comparison was achieved by a n a l y s i n g the displacements of

e a c h f l o o r with damping values of 1 and

5

F o r 1 per c e n t damping the average measured value obtained from

the five floors was 2 . 0 3 per cent wiih a standard deviation of 0 . 1 9

per cent; in the case of 5 _{per cent damping, the average value w a s}

2.68 per cent and the standard deviation w a s 0. I I per cent. This

indicates that when the same value of damping exists at each floor,

then the measured value is independent of the floor analysed.

In the field ineasurements made of the wind-induced

vibrations of buildings, the velocity of the vibrations i s measured.

It has been shown that far small values of damping the analysis of

displacement or velocity recordings should f u r n i s h the same -galue for the damping coefficient. A s the analysis of velocity r e c o r d i n g s

had been used for the f i e l d measurements i t was a l s o used f o r the analogue measurements as a check of the method, F i g u r e 3 shows the

results of the power spectrum analysis of the velocity recordings of the

fifth floor when the same damping coefficient was u s e d at each floor. It can be seen that there is reasonable agreement between the m e a s u r e d

value and the actual value up to 5 per cent critical damping. At 10 per cent critical damping the measured value is approximately one -half the

actual value; this could b e due t o scatter but s i n c e the higher values

of damping a r e a l s o underestimated b y the analysis of the displacements

the error may be due to a more basic cause.

T h e r e is every possibility that in the case of actual buildings

the damping coefficients may vary significantly from f l o o r t o floor,

In

to each floor of the analogue simulation; the values used were 5 , 4 , 3 ,

2 and

Z

The results of power spectrum analysis of velocity and displacement

and autocoxrelation analysis of velocity are shown in Figure 4. It

appears that the measured damping values tend to be an average value of the t r u e damping values. This couldbeexplained by considering the e f f e c t of the feedback terms f r o m t h e more lightly damped floors t o the

m o r e heavily damped floors.

In F i g u r e 4 the displacement results tend to increase with

increase in actual damping values; it s e e m s , however, that the analysis

The s o l i d s y m b o l s in F i g u r e 5 represent the results obtained

directly f r o m :he autocorrelation function b y means of the logarithmic decrement approach. The results w e r e generally obtained by taking the

average value of damping values obtained from different portions of the autocorrelation function. It can be seen that there is some correlation between measured and actual damping values but t h e tendency i s to undere s timate the t r u e damping value.

A c l o s e r investigation of the autocorxelation functians obtained

for the different damping values showed that there w a s a tendency for the amplitude of the function to remain at a constant level after a certain interval of time. T h i s feature of the autocorrelation functions is dernon-

strated in F i g u r e 6 where, after 8 s e c , the amplitude seems to remain constant. T h i s indicates the presence of an error s i g n a l , which may be caused by the r e s p o n s e of the fil:er to the signal that i s being analysed, A logarithmic pLot of the uncorrected amplitudes is shown b y solid

symbols in Figure 7. This plot has t w o distinct linear portions; the

slope of such a plot is proportional to the damping value and therefore

t h e r e should be only one linear section. A f u r t h e r indication that the autocorrelation function i s in error is provided by a consideration of

the t e r m Ae

-ed

given in Figure

6 ,

=

cent.

If it is a s s u m e d that the measured autocorrelation function

is of the f o r m ,

w h e r e K c o s ~ n t is the error signal, then the damping value can be

determined f r o m the ratio of successive peak values obtained by

s u b t r a c t i n g K f r o m the peak values of RET]. The c o s r e ~ t i o n level is

shown

in

sec is of the o r d e r of 5 per cent of that at 7 r 0. Furthermore, a

logarithmic plot of the amplitudes for this case is shown by open

symbols in F i g u r e 7 , and this g i v e s a single linear section. If the

constant level shown in F i g u r e 6 w e r e t o o great, the logarithmic plot would again have two s e p a r a t e s e c t i o n s , in which case the s e c o n d s e c t i o n

would possess a g r e a t e r slope than the first. This would be opposite to the c a s e for the uncorrected values when i t is assumed t h a t K = 0 .

The procedure of m e a s u r i n g a m p l i t u d e s w i t h respect to the

of the damping values investigated. The r e s u l t s for the measured

damping values obtained by this method a r e shown as the open symbols in F i g u r e 5. The a g r e e m e n t between the measured and

theoretical values is _{quite good f o r damping values up t o 10 per}

cent of the critical value. Both the c o r r e c t e d and uncorrected r e s u l t s show that similar damping values are obtained from the

analysis of velocity or displacement; the damping values a l s o

appear independent of the f l o o r an which t h e analysis is made if

the same value of damping exists at each floor.

DISCUSSION O F RESULTS

The results obtained f rorn the power spectrum analysis of

the random vibrations of the theoretical five -storey building indicate

a reasonable correlation between the measured damping value and the true value, There is a certain amount of s c a t t e r in the results,

h o w e v e r , and this aspect of the problem I s worth further study. The

problem is probably b e s t approached by analysing a s e r i e s of 10-min r e c o r d i n g s f o r a fixed damping value. T h i s would then provide a

statistical distribution f o x the rneasur ed damping values. L e n g t h of

record also i n f l u e n c e s the reliability of the r e s u l t s , but in practice t h e r e is a limitation imposed on t h e _{record length by the analysis}

equipment, because i t is not feasible to have an averaging time greater

than 5 s e c . Therefore, the b e s t approach f o r the p o w e r spectrum

method at this s t a g e appears to be a knowledge of the statistical

d i s t r i b u t i o n of damping values for original record lengths of the order of 1 0 rnin.

One of the disadvantages of t h e power spectrum method is

that for lightly damped systems the half-power bandwidth is small and

it must b e c a r e f u l l y measured from the r e c o r d . In the autocorrelation

method, however, the procedure of plotting the logarithmic values of successive amplitude peaks provides an accurate method of determining

the slope of a line that is d i r e c t l y proportional to the damping value.

E

function and a constant amplitude occurs, t h e n the correction which is

s u m m a r i z e d in F i g u r e s 6 and 7 provides v e r y good a g r e e m e n t between

the measured and actual damping values. Unlike the power spectrum

method, the autocorrelation method gives good correlation between

measured and a r t u a l damping up t o 1 0 p e r cent of the critical value. The

power spectrum method seems to u n d e r e s t i m a t e the damping between 5 t o 10 p e r cent of t h e c r i t i c a l value but t h i s a l s o needs further investi- g a t i o n because i t may bc due t o the statistical variations of the results,

T h e o r y indicates that t h e r e should be n a error involved

when velocity rather than displacement is analysed to provide damping values. The results that have been obtained support this statement,

and justify the methods that w e r e used to obtain the damping charac- teristics of buildings from measurements recorded from velocity

transducers. The results also indicate that if the damping value is the same on each f l o o r , then there is very little variation in the measured damping value obtained from the analysis of the different f l o o r vibrations,

If

In

Actual damping values and measured values based on the

autocorrelation analysis a g r e e within about 1 5 per cent for values

up to at least 10 per cent of critical damping. The power spectrum

analysis of a similar record d o e s not give such good agreement and

the error is perhaps close to 20 p e r cent.

Both methods, however, will g i v e an estimate of the order

of damping from a single 10-min record. If vibrations are recorded for an hour or more, this record could be broken down into a series

of 10-min records to provide a more confident estimate of the damping, The results have shown that in the case of lightly damped systems ( l e s s than 10 per cent of the critical value), the damping may

be determined f r o m the velocity or displacement records.

R E F E R E N C E S

1 . C h e r r y , S . and A,G. Brady. Determination of structural

dynamic properties b y statistical analysis of random vibrations.

Proc, T h i r d W o r l d Conference on Earthquake E n g i n e e r i n g ; New

Zealand, 1965.

2. Crandall, S. H. and W, D. Mark. Random vibrations in mechantcal

A M P L I T U D E R A T I O

2

F I G U R E 1

C U R V E S TO P R O V I D E DAMPING V A L U E S FROM A U T O C O R R E L A T I O N

F I G U R E

2

D A M P

l N G R E S U L T S O B T A I N E D FROM

P O W E R

S P E C T R U M

A N A L Y S I S

OF

THE

F l

F T H

F L O O R

D I S P L A C E M E N T S

n

S p e c t r a l

D e n s i t y A n a l y s i s

of

V e l o c i t y

I

I

2

F

I

0

1

2

3

4

5

6:

A C T U A L

P E R C E N T A G E

O F

C R I T I

C A L

D A M P I N G

F I G U R E 4

DAMP

l

N G M E A S U R E M E N T S O B T A l

N E D

WHEN

D l FFERENT

V A L U E S

D A M P I N G

I N T R O D U C E D

A T

E A C H

A n a l y s i s o f 5 t h

F l o o r

Diisplacernent

D i r e c t V a l u e

f r o m

AuZocorrePation

A n a l y s i s

o f 1st

Floor

V e l o c i t y

D i r e c t

V a l u e f r o m A u t o c o r r e l a t i o n

A n a l y s i s

of

5 t h

F l o o r V e l o c i t y

C o r r e c t e d V a l u e , 5 t h F l o o r

0

1

2

3

4

5

6

7

8

9

1 0

A C T U A L

P E R C E N T A G E O F

C R l

T I

C A P DAMP

l

N G

F I G U R E

5

D A M P I N G R E S U L T S

O B T A I N E D F R O M A U T O C O R R E L A T ~ O N

A N A L Y S I

s

2

0 2.02 4.04

6.06

T I ME, S E C Q N

OS

EFFECT OF A P P L V l N G C O R R E C T l O N LEVEL TO

T H E

IN

A P P E N D I X A

POWER

The power spectrum SdCW ) , of a viscously damped single

-

degree-of-freedom system is d i r e c t l y proportional to the expression

given by

The expression in Eq. (13 represents the power spectrum of the displace-

ment as a function of the angular frequency, w . the natural angular

frequency, w , and the fraction of critical damping,

P.

0

The power spectrum of the velocity output, S,(w), of a single

-

given in E;q. (21,

2

The value of SJW) is divided through by w, in Eq. (2) because

i t p e r m i t s t h e d i r e c t comparison of Sd(w) and

S,(,,)

This comparison has been plotted in Figure A - l for the c a s e when U ] ~ '

8

F I G U R E A - l

C O M P A R l S O N

OF

1

4 4

/ $ I z

= I

-

z.)

n I

( A ,

-8,)

=

C.

+ J

-

( - .

=

I.

( p a

-

A,)

=

d.

r r r

+;

.

a r ~ ~ s )

- [ -

.W

.

=

(4,

-

a,)

( -

+;

.

-

(.

-i

F I G U R E

B - 1

A C T T O C O V A R l A N C E F U N C T I O N O F 1