Frequency Domain Decomposition
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The frequency domain decomposition (FDD) is an output-only
system identification The field of system identification uses statistical methods to build mathematical models of dynamical systems from measured data. System identification also includes the optimal design#System identification and stochastic approximation, optimal de ...
technique popular in
civil engineering Civil engineering is a regulation and licensure in engineering, professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads ...
, in particular in
structural health monitoring Structural health monitoring (SHM) involves the observation and analysis of a system over time using periodically sampled response measurements to monitor changes to the material and geometric properties of engineering structures such as bridges ...
. As an output-only algorithm, it is useful when the input data is unknown. FDD is a
modal analysis Modal analysis is the study of the dynamic properties of systems in the frequency domain. It consists of mechanically exciting a studied component in such a way to target the Normal mode, modeshapes of the structure, and recording the vibration ...
technique which generates a system realization using the frequency response given (multi-)output data.


Algorithm

# Estimate the
power spectral density In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be decomposed into ...
matrix \hat_(j\omega) at discrete frequencies \omega = \omega_i. # Do a
singular value decomposition In linear algebra, the singular value decomposition (SVD) is a Matrix decomposition, factorization of a real number, real or complex number, complex matrix (mathematics), matrix into a rotation, followed by a rescaling followed by another rota ...
of the power spectral density, i.e. \hat_(j \omega_i) = U_i S_i U_i^H where U_i = _,u_,...,u_/math> is a
unitary matrix In linear algebra, an invertible complex square matrix is unitary if its matrix inverse equals its conjugate transpose , that is, if U^* U = UU^* = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate ...
holding the singular vectors u_, S_i is the diagonal matrix holding the singular values s_. # For an n degree of freedom system, then pick the n dominating peaks in the power spectral density using whichever technique you wish (or manually). These peaks correspond to the mode shapes. ## Using the mode shapes, an input-output system realization can be written.


See also

*
Eigensystem realization algorithm The Eigensystem realization algorithm (ERA) is a system identification technique popular in civil engineering, in particular in structural health monitoring. ERA can be used as a modal analysis technique and generates a system realization using th ...
- an input/output identification technique


References

Systems theory {{engineering-stub