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Advances in the Theory of Linear Dynamical Systems Through Coordinate Decoupling
 Author(s): Goncalves Salsa Junior, Rubens
 Advisor(s): Ma, Fai
 et al.
Abstract
Coordinate coupling in linear dynamical systems is a known barrier to analysis and design.
Using recent developments in the theory of decoupling, three problems on the theory of
linear systems are tackled. These independent problems have the common characteristic
that partial solutions documented in the open literature require explicit, or implicit, coordinate
decoupling. The rst problem studied is that of converting the equations of motion
of multidegreeoffreedom (MDOF) systems into a form devoid of the velocity term. In
this connection, it is shown that MDOF systems can always be converted by an invertible
transformation into a canonical form specied by two diagonal coecient matrices associated
with the generalized acceleration and displacement. As an important byproduct, a damped
linear system that possesses three symmetric and positive denite coecients can always
be recast as an undamped and decoupled system. Secondly, the characterization of the free
motion of MDOF damped systems is undertaken. Using the methodology of phase synchronization,
it is shown that the free response of a MDOF passive system can be completely
characterized by its spectrum. Furthermore, damping ratio for MDOF damped systems can
be constructed as a direct extension of the damping ratio for SDOF systems and it can be
used to predict oscillatory behavior. Lastly, a comprehensive study is reported on the inverse
problem of linear Lagrangian dynamics, which is concerned with nding a scalar function,
termed Lagrangian, such that the associated EulerLagrange equations are equivalent to the
assigned equations of motion. Contrary to popular beliefs, it is shown that many coupled
linear systems do not admit Lagrangian functions. In addition, Lagrangian functions generally
cannot be determined by system decoupling, but a scalar function that plays the role
of a Lagrangian function can be determined for any linear system by decoupling. This generalized
Lagrangian function produces the equations of motion and it contains information
on system properties, yet it satises a modied version of the EulerLagrange equations. A
necessary and sucient condition for generalized Lagrangian functions to be equivalent to
Lagrangian functions is also derived.
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