Duhamel's integral

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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.

Contents

[edit] Introduction

[edit] Background

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

m\frac{{d^2 x(t)}}{{dt^2 }} + c\frac{{dx(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 delta function δ(t), x(0) = \left. {\frac{{dx}}{{dt}}} \right|_{t = 0} = 0, then by solving the differential equation one can get a fundamental solution (known as a unit-impulse response function)

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}

where \varsigma = \frac{c}{{2m\omega _n }} is called the damping ratio of the system, ωn is the natural circular frequency of the undamped system (when c=0) and \omega _d = \omega _n \sqrt {1 - \varsigma ^2 } is the circular frequency when damping effect is taken into account (when c \ne 0). If the impulse happens at t=τ instead of t=0, i.e. p(t) = δ(t − τ), the impulse response is

h(t - \tau ) = \frac{1}{{m\omega _d }}e^{ - \varsigma \omega _n (t - \tau )} \sin [\omega _d (t - \tau )]，t \ge \tau

[edit] Conclusion

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

p(t) \approx \sum {p(\tau ) \cdot \Delta \tau \cdot \delta } (t - \tau )

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) \approx \sum {p(\tau ) \cdot \Delta \tau \cdot h} (t - \tau )

Letting \Delta \tau \to 0, and replacing the summation by integration, the above equation is strictly valid

x(t) = \int_0^t {p(\tau )h(t - \tau )d\tau }

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

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 }

[edit] See also

[edit] 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
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