Earthquake Text App b Duhamel

8/9/2019 Earthquake Text App b Duhamel

1/8

APPENDIX B. DUHAMEL INTEGRAL FOR AN

UNDAMPED AND DAMPED SYSTEM

The unit impulse response procedure for approximating the response of a structure to amay be used as the basis for developing a formula for evaluating response to a generaldynamic loading. Consider the arbitrary general loading p(t) shown in Fig. B-1,specifically the intensity of loading p(τ) acting at time t = τ . This loading acting duringthe short interval of time dτ produces a short duration impulse p(τ )dτ on the structure,

and euation B-! can be used to evaluate the response to this impulse. "t should be notedcarefully that although the procedure is only approximate for impulses of finite duration,it becomes exact as the duration of loading approaches #ero. Thus for the differential timeinterval dτ, the response produced by the loading p(τ) is exactly $for t % τ &

&$sin&$

&$ τ t ωωm

τ d τ pt dx −= $B-1&

"n this expression, the term dx(t) represents the differential response to the differentialimpulse over the entire response history for t % τ ' it is not the change of x during a time

interval dt.


2/8

)uation $B-!& is generally 0nown as the Duhamel integral for an undamped system. "tmay be used to evaluate the response of an undamped (2F system to any form of

dynamic loading p(t), although in the case of arbitrary loadings the evaluation will haveto be performed numerically.)uation $B-!& may also be expressed in the form

τ d τ t hτ pt xt

∫ −=+

&$&$&$ $B-3&

where the new symbol has the definition

&$sin1

&$ τ t ω

ωm

τ t h −=− $B-4&

)uation $B-3& is called the convolution integral ' computing the response of a structureto an arbitrary loading using this integral is 0nown as obtaining the response through thetime domain. The function h(t - τ) is generally referred to as the unit-impulse response$defined in this case for an undamped system&, because it expresses the response of thesystem to an impulse of unit magnitude applied at time t = τ ."n ). $B-!& it has been tacitly assumed that the loading was initiated at time t 5 + andthat the structure was at rest at that time. For any other specified initial conditions, x$+& 6

+ and x

( +& 6 +, an additional free-vibration response must be added to this solution'thus, in general,

τ d τ t ωτ pωm

t ω xt ωω

xt x

t

∫ −++=+

&$sin&$1

cos&+$sin&+$

&$

$B-7&

Numerical Evaluation Of The Duhamel Integral For An Undamped

System

"f the applied-loading function is integrable, the dynamic response of the structure can bel t d b th f l i t ti f ) $B !& $B 7& " ti l


3/8

numerical calculation, the function has been evaluated at eual time increments τ ,successive values of the function being identified by appropriate subscripts. The value of

the integral can then be obtained approximately by summing these ordinates multiplied by appropriate weighting factors. )xpressed mathematically, this is

∑∫ ∆

== A

!

t

t ! ωm

τ τ d τ y

ωmt A &$

1&$

1&$

+

$B-/&

in which y(τ) = p(τ) cos ωτ and ∑ A

! ! .1 represents the numerical summation process,

the specific form of which depends on the order of the integration approximation being

used. For three elementary approximation procedures, the summations are performed asfollows*

p$=&

p$=&cos >=

5y$=&=

=

=

?=

cos>=

?=?=?=?=?=

p+

p1

p!

p3

p4 p7 p8

y+

y1

y!

y3

y4

y7

y8


4/8

@sing any of the summation processes of ). $B-;& with ). $B-/& leads to anapproximation of the integral for the specific time t under consideration. Aenerally,

however, the entire history of response is reuired rather than merely the displacement atsome specific time' in other words, the response must be evaluated successively at aseuence of times t $ , t %, ., where the interval between these times is τ $or ! τ if impsons rule is used&. To provide this complete response history it is more convenientto express the summations of ). $B-;& in incremental form*imple summation $: 5 1&*

∑∑ ∆−∆−+∆−= A A

τ t ωτ t pτ t t

11

&$cos&$&$&$ $B-1+a&

Figure B-3 Dater tower subEected to blast load.

Trape#oidal rule $:5!&* A A

& '

5!,9++ 0ft

p(t)

5;8.8 0

v

p(t)

;8.8 0

t

+.+!7 s +.+!7 s

Hoading history


5/8

The evaluation of the term B (t) can be carried out in exactly the same way, thatis,

∑∆= B

! ! ωm

τ t B 1&$ $B-11&

in which *! B (t) can be evaluated by expressions identical to )s. $B-1+& but with sinefunctions replacing the cosine functions. ubstituting )s. $B-/& and $BI11& into ). $B-8& leads to the final response euation for an undamped system*

−

∆= ∑ ∑

A B

t t t t m

t xζ ζ

ω ω ζ ω

τ cos&$sin&$

1&$ $B-1!&

EXAMPLE B.1 The dynamic response of a water tower subEected to a blastloading has been calculated to illustrate the numerical evaluation of the (uhamelintegral. The ideali#ations of the structure and of the blast loading are shown inFig. )9I1. For this system, the vibration freuency and period are

'

ω

+ , 'rad

)

(g ω !+;.+

!3+

8.;8

&!.3!$9++,!=====

The time increment used in the numerical integration was ?= 5 +.++7 s, whichcorresponds to an angular increment in free vibrations of >?= 5 +.17 rad$probably a longer increment would have given eually satisfactory results&. "nthis undamped analysis, the impsons rule summation was used' hence the factor : 5 3 was used in )s. $B-1+& to $B-1!&.

G hand solution of the first 1+ steps of the undamped response is presentedin a convenient tabular format in Table 1, pg. /. The operations performed in each

column are generally apparent from the labels at the top. ?J and ? B representthe summing of column 9 $or column 1!& by groups of three terms, as indicated by the braces. Column 19 is the term in suare brac0ets of ). $B-1!&, and thedisplacements given in column 1/ were obtained by multiplying column 19 by


6/8

Response Of Damped Systems

The derivation of the (uhamel integral euation which expresses the response of adamped system to a general dynamic loading is entirely euivalent to the undampedanalysis except that the free-vibration response initiated by the differential load impulse p(τ)dτ is subEected to exponential decay. Thus setting x$+& 5 + and letting x $+& 5p(τ)dτ#m in euation B-8 leads to

−=

−−&$sin

&$&$

&$&1$ τ ω

ω

τ τ τ ξ t

m

d pet dx D

D

t

t % = $B-13&

in which the exponential decay begins as soon as the load is applied at time t = τ .umming these differential response terms over the entire loading interval then results in

τ τ ω τ ω

τ ξω d t e pm

t x Dt

t

D

&$sin&$1

&$ &$

+

−= −−∫

$B-14&Comparing ). $B-14& with the convolution integral of ). $B-3& shows that the

unit-impulse response for a damped system is given by&$sin

1&$ &$ τ t ωe

ωmτ t h D

τ t /ω

D

−=− −−

$B-17&For numerical evaluation of the damped-system response, ). $B-14& may be

written in a form similar to ). $B-8&*t t Bt t At x D D ω ω cos&$sin&$&$ −= $B-18&

Dhere, in this case,τ d τ ω

e

eτ p

ωmt A

Dt /ω

/ωτ t

D

cos&$1

&$+

∫ =

/ωτt1


7/8

Trape#oidal rule $:5!&*

t t p

t t pt t

D

D

A A

ω

τ ξω τ ω τ τ

cos&$

&$exp&$cos&$&$&$!!

+

∆−

∆−∆−+∆−= ∑∑

$B-1;b&

impson


8/8

Table 1. Example problem rom !lo"#$ a%& Pe%'(e%

ichard . ay //

Documents

Earthquake Text App b Duhamel