The decoupling of defective linear dynamical systems in free motion

Kawano, Daniel T.; Morzfeld, Matthias; Ma, Fai

doi:10.1016/j.jsv.2011.05.013

Cited by 18 publications

(26 citation statements)

References 51 publications

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“…This system possesses at least one real, defective eigenvalue. Decoupling of the system by phase synchronization confirms that one of the decoupled degrees of freedom is critically damped, which implies that one eigenvalue is real and defective [17]. The other degree of freedom is overdamped with viscous damping ratio 3.24.…”

Section: Examplementioning

confidence: 73%

“…It follows that system (1) generally cannot be decoupled into a set of mutually independent, real, scalar, second-order equations by a linear mapping q(t) → Lp(t), with the linear operator L independent of t. However, it was recently shown that any system can be decoupled if one utilizes time-dependent transformations [17,20,21,23]. Here, we follow [23] closely to review how system (1) is decoupled.…”

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confidence: 99%

“…Because a damped linear system selected at random is non-defective with probability one [20], we assume for the reminder of this paper that (1) is non-defective. However, this assumption can be relaxed [17]. Upon solving the quadratic eigenvalue problem (5), the solution of the differential equation (1) can be written in terms of the solution of the algebraic equation (5):…”

mentioning

confidence: 99%

“…To determine whether all or some decoupled coordinates p j (t) are oscillatory or not, we can use the viscous damping ratio (2). When system (1) is non-defective, the number of overdamped, underdamped, or critically damped coordinates is an essential and invariant characteristic of system (1) [17]. The decoupling of (1) by phase synchronization thus represents a complete solution to the problem of determining the response characteristics of the linear system (1).…”

mentioning

confidence: 99%

“…Moreover, it can be shown [17,20,21,23] that all decoupled coordinates of (1) are oscillatory if and only if all eigenvalues are complex and all decoupled coordinates are non-oscillatory if and only if all eigenvalues are real. Because the decoupled coordinates determine the response characteristics of (1), we have the following theorem.…”

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confidence: 99%

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Characterization of damped linear dynamical systems in free motion

Morzfeld¹,

Kawano²,

Ma³

2013

Numerical Algebra, Control &Amp; Optimization

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Abstract. It is well known that the free motion of a single-degree-of-freedom damped linear dynamical system can be characterized as overdamped, underdamped, or critically damped. Using the methodology of phase synchronization, which transforms any system of linear second-order differential equations into independent second-order equations, this characterization of free motion is generalized to multi-degree-of-freedom damped linear systems. A real scalar function, termed the viscous damping function, is introduced as an extension of the classical damping ratio. It is demonstrated that the free motion of a multidegree-of-freedom system is characterized by its viscous damping function, and sometimes the characterization may be conducted with ease by examining the extrema of the viscous damping function.1. Introduction. We consider the set of homogeneous linear second-order equationswith initial conditions q(0) = q 0 andq(0) =q 0 . All quantities in (1) are real and the superposed dots denote derivatives with respect to the independent variable t ≥ 0 (time). The coefficients M, C and K are symmetric positive definite (SPD) n × n matrices, and q(t) is an n-dimensional column vector. Equation (1) is a cornerstone in vibration theory and, for example, models the motion of particles around their equilibrium positions, or the currents and voltages in electrical networks [13,14,16,18,22,27]. Adopting vibration terminology, we refer to (1) as an n-degree-offreedom linear system, or simply a system, for short. The response of the system (1) can exhibit oscillations, i.e., the components of the solution q(t) can cross zero infinitely often before settling to zero as t → ∞. The decay of these vibrations

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Section: Examplementioning

confidence: 73%

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confidence: 99%

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confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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