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# 7-1 DUHAMEL INTEGRAL FOR AN UNDAMPED SYSTEM

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7-1 DUHAMEL INTEGRAL FOR AN UNDAMPED SYSTEM

APPENDIX B. DUHAMEL INTEGRAL FOR AN

UNDAMPED AND DAMPED SYSTEM

The unit impulse response procedure for approximating the response of a structure to a may be used as the basis for developing a formula for evaluating response to a general dynamic loading. Consider the arbitrary general loading p(t) shown in Fig. B-1,

the short interval of time dτ produces a short duration impulse p)dτ on the structure,

and equation B-2 can be used to evaluate the response to this impulse. It should be noted carefully that although the procedure is only approximate for impulses of finite duration, it becomes exact as the duration of loading approaches zero. Thus for the differential time interval dτ, the response produced by the loading p) is exactly (for t > τ)

p(τ)dτdx(t)sinω(tτ) (B-1)

mω

In this expression, the term dx(t) represents the differential response to the differential

impulse over the entire response history for t > τ; it is not the change of x during a time

interval dt.

t p(τ)

Figure B-1 Derivation of the Duhamel integral (undamped).

The entire loading history may be considered to consist of a succession of such short impulses, each producing its own differential response of the form of Eq. (B-1). For this linearly elastic system, then, the total response can be obtained by summing all the differential responses developed during the loading history, that is, by integrating Eq. (B-1) as follows:

t1x(t)p(τ)sinω(tτ)dτ (B-2) ?mω0

Richard P. Ray 1/8

Equation (B-2) is generally known as the Duhamel integral for an undamped system. It

may be used to evaluate the response of an undamped SDOF system to any form of dynamic loading p(t), although in the case of arbitrary loadings the evaluation will have to be performed numerically.

Equation (B-2) may also be expressed in the form

t

x(t)p(τ)h(tτ)dτ (B-3) ?0

where the new symbol has the definition

1 (B-4) h(tτ)sinω(tτ)mω

Equation (B-3) is called the convolution integral; computing the response of a structure

to an arbitrary loading using this integral is known as obtaining the response through the time 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 the system to an impulse of unit magnitude applied at time t = τ.

In Eq. (B-2) it has been tacitly assumed that the loading was initiated at time t = 0 and

that the structure was at rest at that time. For any other specified initial conditions, x(0) ?

;0 and (0) ? 0, an additional free-vibration response must be added to this solution; thus, x

in general,

t;x(0)1x(t)sinωtx(0)cosωtp(τ)sinω(tτ)dτ (B-5) ?ωmω0

Numerical Evaluation Of The Duhamel Integral For An Undamped System

If the applied-loading function is integrable, the dynamic response of the structure can be evaluated by the formal integration of Eq. (B-2) or (B-5). In many practical cases, however, the loading is known only from experimental data, and the response must be evaluated by numerical processes. For such analyses it is useful to note the trigonometric identity, sin (ωt- ωτ) = sin ωt cos ωτ - cos ωt sin ωτ, and to write Eq. (B-2) in the form

(zero initial conditions being assumed)

tt11x(t)sinωtp(τ)cosωτdτcosωtp(τ)sinωτdτ ??mωmω00

or

(B-6) v(t)A(t)sinωtB(t)cosωt

t1A(t)p(τ)cosωτdτwhere ?mω0

t1B(t)p(τ)sinωτdτ (B-7) ?mω0

Richard P. Ray 2/8

The numerical integration of the Duhamel integral thus requires the evaluation of the

(t) numerically. Consider, for example, the first of these; the function integrals Ā(t) and B

to be integrated is depicted graphically in Fig. B-2. For convenience of numerical calculation, the function has been evaluated at equal time increments Δτ, 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. Expressed mathematically, this is

tA1τ1 (B-8) A(t)y(τ)dτ(t);?mωmωζζ0

Ain which y(τ) = p(τ) cos ωτ and represents the numerical summation process, the 1/ζζ

specific form of which depends on the order of the integration approximation being used. For three elementary approximation procedures, the summations are performed as follows:

p(τ)

p3 ppp0 1 2

pτ pp4 5 6 cosωτ

τ Δτ Δτ Δτ Δτ Δτ Δτ

p(τ)cos ωτ

=y(τ)

y3 yyyτ 0 1 2

yyy4 5 6

Figure B-2 Formulation of numerical summation process for Duhamel integral.

Simple summation = 1):

A

(t)yyyy (B-9a) 012N11

Trapezoidal rule (ζ=2):

A

(t)y2y2y2yy (B-9b) 012N1N2

Simpson’s rule (ζ=3):

Richard P. Ray 3/8

A

(B-9c) (t)y4y2y4yy012N1N3

where N = t/Δτ must be an even number for Simpson’s rule.

Using any of the summation processes of Eq. (B-9) with Eq. (B-8) leads to an approximation of the integral for the specific time t under consideration. Generally,

however, the entire history of response is required rather than merely the displacement at

some specific time; in other words, the response must be evaluated successively at a sequence of times t, t, …., where the interval between these times is Δτ (or 2 Δτ if 12

Simpson's rule is used). To provide this complete response history it is more convenient to express the summations of Eq. (B-9) in incremental form: Simple summation (ζ = 1):

AA

(B-10a) (t)(tτ)p(tτ)cosω(tτ)11

v

W=96.6 k p(t)

96.6 k p(t)

t k=2,700 k/ft

0.025 s 0.025 s

Figure B-3 Water tower subjected to blast load.

Trapezoidal rule (ζ=2):

AA

(t)(tτ)！，p(tτ)cosω(tτ)p(t)cosωt(B-10b) 22

Simpson’s rule (ζ=3):

AAp(t2τ)cosω(t2τ)~?(t)(t2τ) (B-10c) ?(4p(tτ)cosω(tτ)p(t)cosωt33?)Ain which ? (t-Δτ) represents the value of the summation determined at the preceding ζ

time t - Δτ.

BThe evaluation of the term (t) can be carried out in exactly the same way, that is,

Bτ1 (B-11) B(t)mωζζBin which ? (t) can be evaluated by expressions identical to Eqs. (B-10) but with sine ζ

functions replacing the cosine functions. Substituting Eqs. (B-8) and (B?11) into Eq. (B-6)

leads to the final response equation for an undamped system:

Richard P. Ray 4/8

AB~?(1 (B-12) x(t)(t)sint(t)cost?(；；m：：?)

EXAMPLE B.1 The dynamic response of a water tower subjected to a blast loading has been calculated to illustrate the numerical evaluation of the Duhamel integral. The idealizations of the structure and of the blast loading are shown in Fig. E7?1. For this system, the vibration frequency and period are

The time increment used in the numerical integration was Δτ = 0.005 s, which

corresponds to an angular increment in free vibrations of ωΔτ = 0.15 rad

(probably a longer increment would have given equally satisfactory results). In this undamped analysis, the Simpson's rule summation was used; hence the factor ζ = 3 was used in Eqs. (B-10) to (B-12).

A hand solution of the first 10 steps of the undamped response is presented in a convenient tabular format in Table 1, pg. 8. The operations performed in each column are generally apparent from the labels at the top. ΔĀ

and Δ represent the summing of column 7 (or column 12) by groups of three B

terms, as indicated by the braces. Column 17 is the term in square brackets of Eq. (B-12), and the displacements given in column 18 were obtained by multiplying column 17 by G=Δτ/mωζ. The forces in the last column are given by f = kv(t). It s

should be noted that this is slide-rule work, so that the final results, which involve differences of large numbers, are rather rough.

Since the blast loading terminates at the end of these 10 time steps, the values of Ā and B remain constant after this time. If these constant values of the **integrals designated Ā and B, the free vibrations which follow the blast loading are given by [see Eq. (B-6)]

;; x(t)AsintBcost

1/2;;22and the amplitude of motion is ！，. v(A)(B)max

The Duhamel integral could easily have been evaluated by formal integration for

determined by experiment and cannot be expressed analytically.

Richard P. Ray 5/8

Response Of Damped Systems

The derivation of the Duhamel integral equation which expresses the response of a damped system to a general dynamic loading is entirely equivalent to the undamped analysis except that the free-vibration response initiated by the differential load impulse p)dτ is subjected to exponential decay. Thus setting x(0) = 0 and letting (0) = x;

[p(τ)dτ]/m in equation B-6 leads to

~?((p()dt(()(1) t > τ (B-13) dx(t)esin(t()D?(mD?)

in which the exponential decay begins as soon as the load is applied at time t = τ.

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

t1t，；(() (B-14) x(t)p(()esin(t()d(D?mD0

Comparing Eq. (B-14) with the convolution integral of Eq. (B-3) shows that the unit-impulse response for a damped system is given by

1ξωtτ() (B-15) h(tτ)esinω(tτ)DmωD

For numerical evaluation of the damped-system response, Eq. (B-14) may be written in a form similar to Eq. (B-6):

(B-16) x(t)A(t)sintB(t)costDD

Where, in this case,

tξωτ1e A(t)p(τ)cosωτdτDξωt?mωeD0

tξωτ1e (B-17) B(t)p(t)sinωτdτDξωt?mωeD0

These integrals can be evaluated by an incremental summation process equivalent to that used previously but taking account of the exponential decay in the process. The first integral is given by

Aτ1 (B-18) A(t)(t);mωζζD

in which the summations can be expressed for the different processes considered before as follows:

Simple summation (ζ = 1):

AA~? (B-19a) (t)(tτ)p(tτ)cosω(tτ)exp(ξωτ)D?(11?)

Trapezoidal rule (ζ=2):

Richard P. Ray 6/8

AA~?(((，；((t)(t)p(t)cos(t)exp()D?( (B-19b) ?)22

p(t)costD

Simpson’s rule (ζ=3):

AA~?(((，；((t)(t2)p(t2)cos(t2)exp(2)D?( (B-19c) ?)33

4p(t()cos(t()exp(，；()p(t)costDD

The B(t) term is given by similar expressions involving the sine functions.

Richard P. Ray 7/8

Table 1. Example problem from Clough and Penzien

Richard P. Ray 8/8

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