When multi-DOF systems with arbitrary damping are modeled using the state-space method, then Laplace-transform of the state equations results into an eigen problem. In general the eigenvalues and vectors are complex. The IP (eigenvalue) represents the damped natural frequency.
In case of undamped systems the eigenvalues are complex with RP=0. But mode shapes are real and represents relation between various motion variables, when system vibrates with corresponding natural frequency.
I am interested in understanding the physical interpretation of complex mode shapes (complex eigenvectors).
You have to come back to the solution assumption. With a damped solution, your system trajectory is not periodic, given by
x(t) = Real( sum_j psi_j e^(l_j t) )
with psi_j and l_j the complex mode shape and complex eigenvalue of your j-th complex mode. To really visualize your mode shape you thus have to animate the trajectory.
Unlike real modes, you cannot properly define modal energies, an idea is then to express your complex modes as linear complex combination of your real modes to get some deeper insight.
You have to come back to the solution assumption. With a damped solution, your system trajectory is not periodic, given by
x(t) = Real( sum_j psi_j e^(l_j t) )
with psi_j and l_j the complex mode shape and complex eigenvalue of your j-th complex mode. To really visualize your mode shape you thus have to animate the trajectory.
Unlike real modes, you cannot properly define modal energies, an idea is then to express your complex modes as linear complex combination of your real modes to get some deeper insight.
I suppose to imagine that real mode shape represents a "standing wave", thus in every point at the same time the mode reaches the maxima with total phase or antiphase. A complex mode could be imagined as a "traveling wave", where the mode reach the maximum in every location in the different time instance. The complex mode can occur in case of non - proportional damping.
The real part of the eigen value gives the natural frequency and the ratio, imaginary part to the real part gives the systems loss factor.
1) http://books.google.co.in/books?id=ElwhqUtJUj8C&pg=PA171&lpg=PA171&dq=modal+loss+factor+from+complex+natural+frequencies&source=bl&ots=VeYyTt9aGE&sig=KvUOChNelyRBWh7movC5S6kZM6I&hl=en&sa=X&ei=VaA_VMzeHcyRuAS624GIDg&redir_esc=y#v=onepage&q=modal%20loss%20factor%20from%20complex%20natural%20frequencies&f=false
2) Longitudinal vibration of a damped rod. Part I:
Complex natural frequencies and mode shapes
"The real part of the eigen value gives the natural frequency and the ratio, imaginary part to the real part gives the systems loss factor"
the other way around
Thanks Xiaojun, for keeping the discussion alive.
Starting with the state model of single DOF system, it has two eigen values which are complex conjugates. It can be easily seen that the imaginary part IP = damped natural frequency.
The complex modes analysis is used for analysing not only damped structures, but also rotating structures, e.g. rotors, crankshafts, etc. Here, the spin effects come into play when considering the solution in coordinate system rotating with the body, which can considerably affect the motion of the system, without energy dissipation. These effects (centrifugal weakening, gyroscopic effect, etc.) are taken into account by adding respective matrices (centrifugal, gyroscopic) to the system of vibration equatons.
The resulting complex eigenvalues of rotor modes describe not only its flexible motion (vibration), but also its overall rotation (forward or backward whirl).
I've been reading all your comments, and by the way they've been very helpful. But in the case of a shear building model for example if I want to draw the complex mode shape of the structure, should I just take the proyection on the real axe of the components of my complex eigen vector and just draw the proyections?
i tend to draw: sign(real())*abs() and draw the phase separately.
also refer to "realisation of complex mode shape" by M. Imregun and D.J. Ewins
Hi Xiaojun Li,
This [sign(real(#))*abs(#), where # is the complex mode shape usually comes with conjugate pairs for underdampered mode] is exactly what I did when I was applying stochastic subspace identification (SSI) for modal identification. However, it seems that I havn't found related references on using this expression to equate the normal mode shape.
Actually, the complex mode shape normally comes with non-classical damping. One way I am also thinking to depict the complex mode shape is just to draw the real part and imaginary part separately.
Hi,
complex vibration modes arise whenever non-proportional damping occurs. Such is the case of mechanical systems, supported by assemblies of springs and dashpots, and also of civil engineering structures, contrary to what is commonly thought.
Such complex modes are responsible for the change of position of the "nodes" (modal configuration points with zero amplitude) that can be seen in many animated mode shapes.
I have prepared two notebooks (links below) to show the influence of these complex modes both in dynamic equilibrium and in state-space formulation that help you to visualize this phenomena.
https://github.com/pxcandeias/py-notebooks/blob/master/complex_vibration_modes_DE.ipynb
https://github.com/pxcandeias/py-notebooks/blob/master/complex_vibration_modes_SS.ipynb
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