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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.
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| ==Introduction==
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| ===Background===
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| 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]]
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| :<math>m\frac{{d^2 x(t)}}{{dt^2 }} + c\frac{{dx(t)}}{{dt}} + kx(t) = p(t)</math>
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| 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.
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| 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''')
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| :<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>
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| 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
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| :<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>
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| ===Conclusion===
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| Regarding the arbitrarily varying excitation ''p''(''t'') as a [[Superposition principle|superposition]] of a series of impulses:
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| :<math>p(t) \approx \sum {p(\tau ) \cdot \Delta \tau \cdot \delta } (t - \tau )</math>
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| 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:
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| :<math>x(t) \approx \sum {p(\tau ) \cdot \Delta \tau \cdot h} (t - \tau )</math>
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| Letting <math>\Delta \tau \to 0</math>, and replacing the summation by [[Integral|integration]], the above equation is strictly valid
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| :<math>x(t) = \int_0^t {p(\tau )h(t - \tau )d\tau } </math>
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| Substituting the expression of ''h''(''t''-''τ'') into the above equation leads to the general expression of Duhamel's integral
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| :<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>
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| ===Mathematical Proof===
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| The above SDOF dynamic equilibrium equation in the case ''p(t)=0'' is the [[homogeneous differential equation|homogeneous equation]]:
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| :<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>
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| The solution of this equation is:
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| :<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>
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| 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:
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| :<math>x_h(t) = C_1.e^{ -B.t} \; + \; C_2.e^{ -A.t}</math>
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| 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]].
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| This solution has the form:
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| :<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:
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| :<math>x_p(t) = \frac{Q_t.e^{-At}-R_t.e^{-Bt}}{P}</math>
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| Finally the general solution of the above non-homogeneous equation is represented as:
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| :<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>
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| with time derivative:
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| :<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>
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| In order to find the unknown constants <math>C_1, C_2</math>, zero initial conditions will be applied:
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| :<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>
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| :<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>
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| Now combining both initial conditions together, the next system of equations is observed:
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| :<math>\left.{\begin{alignat}{5}
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| C_1 &&\; + &&\; C_2 &&\; = &&\; \frac{R_0-Q_0}{P} & \\
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| B.C_1 &&\; + &&\; A.C_2 &&\; = &&\; \frac{1}{P}.[B.R_0-A.Q_0]\end{alignat}} \right|{\begin{alignat}{5}
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| C_1 &&\; = &&\; \frac{R_0}{P} & \\
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| C_2 &&\; = &&\; -\frac{Q_0}{P}\end{alignat}}</math>
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| The back substitution of the constants <math> C_1 </math> and <math> C_2 </math> into the above expression for ''x(t)'' yields:
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| :<math>x(t)=\frac{Q_t-Q_0}{P}.e^{ -A.t}-\frac{R_t-R_0}{P}.e^{ -B.t}</math>
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| 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:
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| :<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>
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| 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:
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| :<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]].
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| 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:
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| :<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>
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| == See also ==
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| *[[Duhamel's principle]]
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| ==References==
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| * Ni Zhenhua, ''Mechanics of Vibrations'', Xi'an Jiaotong University Press, Xi'an, 1990 (in Chinese)
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| * R. W. Clough, J. Penzien, ''Dynamics of Structures'', Mc-Graw Hill Inc., New York, 1975.
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| * Anil K. Chopra, ''Dynamics of Structures - Theory and applications to Earthquake Engineering'', Pearson Education Asia Limited and Tsinghua University Press, Beijing, 2001
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| * Leonard Meirovitch, ''Elements of Vibration Analysis'', Mc-Graw Hill Inc., Singapore, 1986
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| ==External links==
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| *[http://tosio.math.toronto.edu/wiki/index.php/Duhamel's_formula Duhamel's formula] at "Dispersive Wiki".
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| [[Category:Mechanics]]
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| [[Category:Structural analysis]]
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| [[Category:Integrals]]
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Nice to meet you, my name is Refugia. Hiring is my profession. Years ago he moved to North Dakota and his family enjoys it. What I love performing is playing baseball but I haven't produced a dime with it.
My page; btcsoc.com