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Advances in the Theory of Linear Dynamical Systems Through Coordinate Decoupling


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 multi-degree-of-freedom (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 by-product, 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 Euler-Lagrange 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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