NUMERICAL EVALUATION OF DUHAMEL’S INTEGRAL UNDAMPED SYSTEM

In many practical cases the applied loading function is known only from experimental data as in the case of seismic motion and the response must be evaluated by a numerical method. For this purpose we use the trigonometric identity wwtwwtt sincoscossin)(sin , in Duhamel’s integral.

Assuming zero initial conditions, we obtain Duhamel’s integral, eq. (4.4), in the form

dFm

tdFm

ttyt

t

t

t 00

sin)(1

coscos)(1

sin)( (4.14a)

or

ttBttAm

ty

cos)(sin)(1

)(

(4.14b)

where

dFtAt

0cos)()(

(4.15a)

t

dFtB0

sin)()(

(4.15b)

The calculation of Duhamel’s integral thus requires the evaluation of the integrals A(t) and B(t)

numerically. Several numerical integration techniques have been used for this evaluation. The most popular of

these methods are the TRAPEZOIDAL RULE and the SIMPSON’S RULE. Consider the integration of a

general function I()

dItAt

0)()(

For trapezoidal rule

nnO IIIIItA 121 2222

1)(

(4.16)

and for Simpson’s rule

nnO IIIIItA 121 4243

1)(

(4.17)

where

t

n must be an even number for Simpson’s rule.

An alternative approach to the evaluation of Duhamel’s integral is based on obtaining the exact

analytical solution of the integral for the loading function assumed to be given by a succession of linear

segments. This method does not introduce numerical approximations for the integration other than those inherent in the round off error, so in this sense it is an exact method.

In using this method, it is assumed that )(F , the forcing function may be approximated by a

segmentally linear function as shown. To provide a complete response history, it is more convenient to express the integrations in eq.

(4.15) in incremental form, namely

dFtAtAi

i

t

tii

1

cos)()()( 1 (4.18)

i

i

t

tii dFtBtB

1

sin)()()( 1 (4.19)

Where A(ti) and B(ti) represent the values of the integrals in eq. (4.15) at time ti. Assuming that the

forcing function F() is approximated by a piecewise linear functions as shown in Fig. 4.6, we may write

iii

i

ii ttt

t

FtFF

),()()( 111 (4.20)

)()( 1 iii tFtFF

1 iii ttt

The substitution of eq. (4.20) into eq. (4.18) and integration yield

dtt

FtFtAtA

i

i

t

ti

i

iiii

1

cos)()()( 111 (4.20a)

i

i

i

i

t

ti

i

it

tii dt

t

FdtFtA

11

coscos)()( 111 (4.20b)

i

i

i

i

i

i

t

ti

i

it

ti

it

tii dt

t

Fd

t

FdtFtA

111

coscoscos)()( 111 (4.20c)

Consider the 1st and 3rd integrals and let u and ddu

i

i

i

i

i

i

t

t

it

ti

t

ti

tFduutFdtF

111

sin)(

cos)(cos)( 111

(4.20d)

11 sinsin)(

ii

i tttF

(4.20e)

i

i

i

i

i

i

t

ti

i

it

ti

i

it

ti

i

i tt

Fduut

t

Fdt

t

F

111

sincoscos 111

(4.20f)

11 sinsin

iii

i

i tttt

F

(4.20g)

To evaluate the 2nd integral we use “integration by parts”.

Let u ddv cos

ddu

sinv

i

i

i

i

i

i

tt

ti

it

ti

i dt

Fd

t

F

11

1

sinsincos

(4.20h)

i

i

t

iiii

i

i duu

ttttt

F

1

sinsinsin 11

(4.20i)

i

i

t

t

iiii

i

i ttttt

F

1

cos1

sinsin 11

(4.20j)

1112coscossinsin

iiiiii

i

i ttttttt

F

(4.20k)

Therefore

1111

sinsin)()()( ii

i

iiiii

tt

t

FttFtAtA

1112sinsincoscos

iiiiii

i

i ttttttt

F

(4.21)

By the same token

ii

i

iiiii

tt

t

FttFtBtB

coscos)()()( 1

111

1112coscossinsin

iiiiii

i

i ttttttt

F

(4.22)

Equations (4.21) and (4.22) are recurrent formulas for the evaluation of the integrals in eq. (4.15) at any

time itt .

Determine the dynamic response of a tower subjected to a blast loading. The idealization of the

structure and the blast loading are shown. Neglect damping.

NUMERICAL EVALUATION OF DUHAMEL’S INTEGRAL UNDAMPED SYSTEM

The response of a damped system expressed by the Duhamel’s integral is obtained in a manner entirely

equivalent to the undamped analysis except that the impulse producing an initial velocity is substituted

into the corresponding damped free-vibration equation.

tm

dFetdy D

D

t sin)(

)( (4.23)

Summing these differential response terms over the entire loading interval results in

t

D

t

D

dteFm

ty0

sin)(1

)(

(4.24)

Which is the response for a damped system in terms of the Duhamel’s integral. For numerical

evaluation, we proceed as in the undamped case and obtain from eq. (4.24)

D

t

DDDDm

ettBttAty

cos)(sin)()( (4.25)

where

i

i

t

tD

t

iDiD tdeFtAtA1

cos)()()( 1 (4.26)

i

i

t

tD

t

iDiD tdeFtBtB1

sin)()()( 1 (4.27)