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In theory of [[oscillation|vibration]]s, '''Duhamel's integral''' is a way of calculating the response of [[linear system]]s and [[structures]] to arbitrary time-varying external [[excitation]]s.
 
==Introduction==
===Background===
The response of a linear, viscously damped [[single-degree of freedom]] (SDOF) system to a time-varying mechanical excitation ''p''(''t'') is given by the following second-order [[ordinary differential equation]]
:<math>m\frac{{d^2 x(t)}}{{dt^2 }} + c\frac{{dx(t)}}{{dt}} + kx(t) = p(t)</math>
where ''m'' is the (equivalent) mass, ''x'' stands for the amplitude of vibration, ''t'' for time, ''c'' for the viscous damping coefficient, and ''k'' for the [[stiffness]] of the system or structure.
 
If a system is initially rest at its [[Mechanical equilibrium|equilibrium]] position, from where it is acted upon by a unit-impulse at the instance ''t''=0, i.e., ''p''(''t'') in the equation above is a [[Dirac delta function]] ''δ''(''t''), <math>x(0) = \left. {\frac{{dx}}{{dt}}} \right|_{t = 0} = 0</math>, then by solving the differential equation one can get a [[fundamental solution]] (known as a '''unit-impulse response function''')
:<math>h(t)=\begin{cases} \frac{1}{{m\omega _d }}e^{ - \varsigma \omega _n t} \sin \omega _d t, & t > 0 \\ 0, & t < 0 \end{cases}</math>
where <math>\varsigma  = \frac{c}{2\sqrt{k m}}</math> is called the [[damping ratio]] of the system, <math>\omega _n=\sqrt{\frac{k}{m}}</math> is the natural [[angular frequency]] of the undamped system (when ''c''=0) and <math>\omega _d  = \omega _n \sqrt {1 - \varsigma ^2 } </math> is the [[circular frequency]] when damping effect is taken into account (when <math>c \ne 0</math>). If the impulse happens at ''t''=''τ'' instead of ''t''=0, i.e. <math>p(t)=\delta (t - \tau )</math>, the impulse response is
:<math>h(t - \tau ) = \frac{1}{{m\omega _d }}e^{ - \varsigma \omega _n (t - \tau )} \sin [\omega _d (t - \tau )]</math>,<math>t \ge \tau </math>
 
===Conclusion===
Regarding the arbitrarily varying excitation ''p''(''t'') as a [[Superposition principle|superposition]] of a series of impulses:
:<math>p(t) \approx \sum {p(\tau ) \cdot \Delta \tau  \cdot \delta } (t - \tau )</math>
then it is known from the linearity of system that the overall response can also be broken down into the superposition of a series of impulse-responses:
:<math>x(t) \approx \sum {p(\tau ) \cdot \Delta \tau  \cdot h} (t - \tau )</math>
Letting <math>\Delta \tau  \to 0</math>, and replacing the summation by [[Integral|integration]], the above equation is strictly valid
:<math>x(t) = \int_0^t {p(\tau )h(t - \tau )d\tau } </math>
Substituting the expression of ''h''(''t''-''τ'') into the above equation leads to the general expression of Duhamel's integral
:<math>x(t) = \frac{1}{{m\omega _d }}\int_0^t {p(\tau )e^{ - \varsigma \omega _n (t - \tau )} \sin [\omega _d (t - \tau )]d\tau }</math>
 
===Mathematical Proof===
The above SDOF dynamic equilibrium equation in the case ''p(t)=0'' is the [[homogeneous differential equation|homogeneous equation]]:
:<math>\frac{{d^2 x(t)}}{{dt^2 }} + \bar{c}\frac{{dx(t)}}{{dt}} + \bar{k}x(t) = 0</math>, where <math>\bar{c}=\frac{c}{m},\bar{k}=\frac{k}{m} </math>
The solution of this equation is:
:<math>x_h(t) = C_1.e^{ -\frac{1}{2}.(\bar{c}+\sqrt{\bar{c}^2-4.\bar{k}}).t}+C_2.e^{ \frac{1}{2}.(-\bar{c}+\sqrt{\bar{c}^2-4.\bar{k}}).t}</math>
The substitution: <math>A = \frac{1}{2}.(\bar{c}-\sqrt{\bar{c}^2-4.\bar{k}}), \; B=\frac{1}{2}.(\bar{c}+\sqrt{\bar{c}^2-4.\bar{k}}), \; P=\sqrt{\bar{c}^2-4.\bar{k}}, \; P=B-A</math> leads to:
:<math>x_h(t) = C_1.e^{ -B.t} \; + \; C_2.e^{ -A.t}</math>
One partial solution of the non-homogeneous equation: <math> \frac{{d^2 x(t)}}{{dt^2 }} + \bar{c}\frac{{dx(t)}}{{dt}} + \bar{k}x(t) = \bar{p(t)}</math>, where <math>\bar{p(t)}=\frac{p(t)}{m}</math>, could be obtained by the Lagrangian method for deriving partial solution of non-homogeneous [[ordinary differential equations]].
 
This solution has the form:
:<math>x_p(t) = \frac{\int{\bar{p(t)}.e^{At}dt}.e^{-At}-\int{\bar{p(t)}.e^{Bt}dt}.e^{-Bt}}{P}</math>
Now substituting:<math>\int{\bar{p(t)}.e^{At}dt}|_{t=z}=Q_z, \int{\bar{p(t)}.e^{Bt}dt}|_{t=z}=R_z </math>,where <math> \int{x(t)dt}|_{t=z} </math> is the [[antiderivative|primitive]] of ''x(t)'' computed at ''t=z'', in the case ''z=t'' this integral is the primitive itself, yields:
:<math>x_p(t) = \frac{Q_t.e^{-At}-R_t.e^{-Bt}}{P}</math>
Finally the general solution of the above non-homogeneous equation is represented as:
:<math>x(t)=x_h(t)+x_p(t)=C_1.e^{ -B.t}+C_2.e^{ -A.t} +\frac{Q_t.e^{-At}-R_t.e^{-Bt}}{P}</math>
with time derivative:
:<math> \frac{dx}{dt}=-A.e^{-At}.C_2-B.e^{-Bt}.C_1+\frac{1}{P}.[\dot{Q_t}.e^{-At}-A.Q_t.e^{-At}-\dot{R_t}.e^{-Bt}+B.R_t.e^{-Bt}]</math>, where <math>\dot{Q_t}=p(t).e^{At},\dot{R_t}=p(t).e^{Bt}</math>
In order to find the unknown constants <math>C_1, C_2</math>, zero initial conditions will be applied:
:<math>x(t)|_{t=0} = 0: C_1+C_2+\frac{Q_0.1-R_0.1}{P}=0</math> ⇒ <math>C_1+C_2=\frac{R_0-Q_0}{P}</math>
:<math>\left. {\frac{{dx}}{{dt}}} \right|_{t=0} = 0: -A.C_2-B.C_1+\frac{1}{P}.[-A.Q_0+B.R_0]=0</math> ⇒ <math>A.C_2+B.C_1=\frac{1}{P}.[B.R_0-A.Q_0]</math>
Now combining both initial conditions together, the next system of equations is observed:
:<math>\left.{\begin{alignat}{5}
C_1 &&\; + &&\; C_2 &&\; = &&\; \frac{R_0-Q_0}{P} & \\
B.C_1 &&\; + &&\; A.C_2 &&\; = &&\; \frac{1}{P}.[B.R_0-A.Q_0]\end{alignat}} \right|{\begin{alignat}{5}
C_1 &&\; = &&\; \frac{R_0}{P} & \\
C_2 &&\; = &&\; -\frac{Q_0}{P}\end{alignat}}</math>
The back substitution of the constants <math> C_1 </math> and <math> C_2 </math> into the above expression for ''x(t)'' yields:
:<math>x(t)=\frac{Q_t-Q_0}{P}.e^{ -A.t}-\frac{R_t-R_0}{P}.e^{ -B.t}</math>
Replacing <math>Q_t-Q_0</math> and <math>R_t-R_0</math> (the difference between the primitives at ''t=t'' and ''t=0'') with [[integral|definite integrals]] (by another variable ''τ'') will reveal the general solution with zero initial conditions, namely:
:<math>x(t)=\frac{1}{P}.[\int_0^t{\bar{p(\tau)}.e^{A\tau}d\tau}.e^{-At}-\int_0^t{\bar{p(\tau)}.e^{B\tau}d\tau}.e^{-Bt}]</math>
Finally substituting <math> c=2.\xi.\omega.m, \; k=\omega^2.m</math>, accordingly <math> \bar{c}=2.\xi.\omega, \bar{k}=\omega^2</math>, where <u>''ξ<1''</u> yields:
:<math>P=2.\omega_D.i, \; A=\xi.\omega-\omega_D.i, \; B=\xi.\omega+\omega_D.i</math>, where <math>\omega_D=\omega.\sqrt{1-\xi^2}</math> and '''''i''''' is the [[imaginary unit]].
Substituting this expressions into the above general solution with zero initial conditions and using the [[Euler's formula|Euler's exponential formula]] will lead to canceling out the imaginary terms and reveals the Duhamel's solution:
:<math>x(t)=\frac{1}{\omega_D}\int_0^t{\bar{p(\tau)}e^{-\xi\omega(t-\tau)}sin(\omega_D(t-\tau))d\tau}</math>
 
== See also ==
*[[Duhamel's principle]]
 
==References==
* Ni Zhenhua, ''Mechanics of Vibrations'', Xi'an Jiaotong University Press, Xi'an, 1990 (in Chinese)
* R. W. Clough, J. Penzien, ''Dynamics of Structures'', Mc-Graw Hill Inc., New York, 1975.
* Anil K. Chopra, ''Dynamics of Structures - Theory and applications to Earthquake Engineering'', Pearson Education Asia Limited and Tsinghua University Press, Beijing, 2001
* Leonard Meirovitch, ''Elements of Vibration Analysis'', Mc-Graw Hill Inc., Singapore, 1986
 
==External links==
*[http://tosio.math.toronto.edu/wiki/index.php/Duhamel's_formula Duhamel's formula] at "Dispersive Wiki".
 
[[Category:Mechanics]]
[[Category:Structural analysis]]
[[Category:Integrals]]

Revision as of 05:50, 17 January 2014

In theory of vibrations, Duhamel's integral is a way of calculating the response of linear systems and structures to arbitrary time-varying external excitations.

Introduction

Background

The response of a linear, viscously damped single-degree of freedom (SDOF) system to a time-varying mechanical excitation p(t) is given by the following second-order ordinary differential equation

md2x(t)dt2+cdx(t)dt+kx(t)=p(t)

where m is the (equivalent) mass, x stands for the amplitude of vibration, t for time, c for the viscous damping coefficient, and k for the stiffness of the system or structure.

If a system is initially rest at its equilibrium position, from where it is acted upon by a unit-impulse at the instance t=0, i.e., p(t) in the equation above is a Dirac delta function δ(t), x(0)=dxdt|t=0=0, then by solving the differential equation one can get a fundamental solution (known as a unit-impulse response function)

h(t)={1mωdeςωntsinωdt,t>00,t<0

where ς=c2km is called the damping ratio of the system, ωn=km is the natural angular frequency of the undamped system (when c=0) and ωd=ωn1ς2 is the circular frequency when damping effect is taken into account (when c0). If the impulse happens at t=τ instead of t=0, i.e. p(t)=δ(tτ), the impulse response is

h(tτ)=1mωdeςωn(tτ)sin[ωd(tτ)]tτ

Conclusion

Regarding the arbitrarily varying excitation p(t) as a superposition of a series of impulses:

p(t)p(τ)Δτδ(tτ)

then it is known from the linearity of system that the overall response can also be broken down into the superposition of a series of impulse-responses:

x(t)p(τ)Δτh(tτ)

Letting Δτ0, and replacing the summation by integration, the above equation is strictly valid

x(t)=0tp(τ)h(tτ)dτ

Substituting the expression of h(t-τ) into the above equation leads to the general expression of Duhamel's integral

x(t)=1mωd0tp(τ)eςωn(tτ)sin[ωd(tτ)]dτ

Mathematical Proof

The above SDOF dynamic equilibrium equation in the case p(t)=0 is the homogeneous equation:

d2x(t)dt2+c¯dx(t)dt+k¯x(t)=0, where c¯=cm,k¯=km

The solution of this equation is:

xh(t)=C1.e12.(c¯+c¯24.k¯).t+C2.e12.(c¯+c¯24.k¯).t

The substitution: A=12.(c¯c¯24.k¯),B=12.(c¯+c¯24.k¯),P=c¯24.k¯,P=BA leads to:

xh(t)=C1.eB.t+C2.eA.t

One partial solution of the non-homogeneous equation: d2x(t)dt2+c¯dx(t)dt+k¯x(t)=p(t)¯, where p(t)¯=p(t)m, could be obtained by the Lagrangian method for deriving partial solution of non-homogeneous ordinary differential equations.

This solution has the form:

xp(t)=p(t)¯.eAtdt.eAtp(t)¯.eBtdt.eBtP

Now substituting:p(t)¯.eAtdt|t=z=Qz,p(t)¯.eBtdt|t=z=Rz,where x(t)dt|t=z is the primitive of x(t) computed at t=z, in the case z=t this integral is the primitive itself, yields:

xp(t)=Qt.eAtRt.eBtP

Finally the general solution of the above non-homogeneous equation is represented as:

x(t)=xh(t)+xp(t)=C1.eB.t+C2.eA.t+Qt.eAtRt.eBtP

with time derivative:

dxdt=A.eAt.C2B.eBt.C1+1P.[Qt˙.eAtA.Qt.eAtRt˙.eBt+B.Rt.eBt], where Qt˙=p(t).eAt,Rt˙=p(t).eBt

In order to find the unknown constants C1,C2, zero initial conditions will be applied:

x(t)|t=0=0:C1+C2+Q0.1R0.1P=0C1+C2=R0Q0P
dxdt|t=0=0:A.C2B.C1+1P.[A.Q0+B.R0]=0A.C2+B.C1=1P.[B.R0A.Q0]

Now combining both initial conditions together, the next system of equations is observed:

C1+C2=R0Q0PB.C1+A.C2=1P.[B.R0A.Q0]|C1=R0PC2=Q0P

The back substitution of the constants C1 and C2 into the above expression for x(t) yields:

x(t)=QtQ0P.eA.tRtR0P.eB.t

Replacing QtQ0 and RtR0 (the difference between the primitives at t=t and t=0) with definite integrals (by another variable τ) will reveal the general solution with zero initial conditions, namely:

x(t)=1P.[0tp(τ)¯.eAτdτ.eAt0tp(τ)¯.eBτdτ.eBt]

Finally substituting c=2.ξ.ω.m,k=ω2.m, accordingly c¯=2.ξ.ω,k¯=ω2, where ξ<1 yields:

P=2.ωD.i,A=ξ.ωωD.i,B=ξ.ω+ωD.i, where ωD=ω.1ξ2 and i is the imaginary unit.

Substituting this expressions into the above general solution with zero initial conditions and using the Euler's exponential formula will lead to canceling out the imaginary terms and reveals the Duhamel's solution:

x(t)=1ωD0tp(τ)¯eξω(tτ)sin(ωD(tτ))dτ

See also

References

  • Ni Zhenhua, Mechanics of Vibrations, Xi'an Jiaotong University Press, Xi'an, 1990 (in Chinese)
  • R. W. Clough, J. Penzien, Dynamics of Structures, Mc-Graw Hill Inc., New York, 1975.
  • Anil K. Chopra, Dynamics of Structures - Theory and applications to Earthquake Engineering, Pearson Education Asia Limited and Tsinghua University Press, Beijing, 2001
  • Leonard Meirovitch, Elements of Vibration Analysis, Mc-Graw Hill Inc., Singapore, 1986

External links