Automated Upgraded Generalized Full-Discretization Method …
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spatially reduced dynamics is still infinite dimensional on the temporal axis. The
solution of the reduced model according to the extended Floquet theory is generally
w(t) = ϒ(t)w(0) = P(t)e
B f t w(0),
(12)
where ϒ : R ≥0 → R
∞×∞ is the transition matrix, B f ∈ R
∞×∞ is the fundamental matrix, and P : R ≥0 → R
∞×∞ is T -periodic with the initial condition being an
identity matrix, that is, P(0) = I. Making use of similarity transformation and matrix
exponential function gives
ϒ(t) = P(t)e
(V A DV
−1
A )t
= P(t)V A e
Dt V
−1
A ,
(13)
where V A ∈ R
∞×∞ is the matrix of all the eigenvectors of B f and D ∈ R
∞×∞ is
a diagonal matrix with all the eigenvalues of B f , called characteristic exponents
λ i = σ i + jω i , as the diagonal elements. It is obvious from Eqs. (12) and (13) that
the condition for asymptotic stability is that all λ i have negative real parts, that is,
σ i < 0. From Eq. (13), the monodromy matrix becomes
ϒ(T ) = V A e
DT V
−1
A .
(14)
The eigenvectors of ϒ(T ) are identical with those of B f , and therefore the similarity
transformation gives
M T = e
DT
,
(15)
where M T ∈ R
∞×∞ is a diagonal matrix with eigenvalues of ϒ(T ) as the diagonal elements. The eigenvalues of ϒ(T ), called the characteristic multipliers μ i , are
seen from Eq. (15) to be given as μ i = e
λ i T . Therefore, the condition for asymptotic
stability is that all characteristic multipliers μ i have modulus less than one, that is,
|μ i | = e
σ i T
< 1 ∀ i . The basis for identifying the stability boundary is max|μ i | = 1.
Stability analysis of the reduced order model based on Eq. (12) is not feasible;
therefore, finite-dimensional approximations of the rather infinite-dimensional monodromy matrices are needed for computational purposes. It can then be said that
in addition to spatial reduction, temporal reduction is also necessary for stability
analysis of the system.
3 Generalized Discrete Map
The discrete time delay τ is also the period of the model for the studied case
of uniform pitch milling. The period of the system is divided into k equal discrete time intervals [t i , t i+1 ] where i = 0, 1, 2, . . . , (k − 1) and t i = i
τ
k
= it =
i(t i+1 − t i ). Equation (11) is solved in each discrete interval to give
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spatially reduced dynamics is still infinite dimensional on the temporal axis. The
solution of the reduced model according to the extended Floquet theory is generally
w(t) = ϒ(t)w(0) = P(t)e
B f t w(0),
(12)
where ϒ : R ≥0 → R
∞×∞ is the transition matrix, B f ∈ R
∞×∞ is the fundamental matrix, and P : R ≥0 → R
∞×∞ is T -periodic with the initial condition being an
identity matrix, that is, P(0) = I. Making use of similarity transformation and matrix
exponential function gives
ϒ(t) = P(t)e
(V A DV
−1
A )t
= P(t)V A e
Dt V
−1
A ,
(13)
where V A ∈ R
∞×∞ is the matrix of all the eigenvectors of B f and D ∈ R
∞×∞ is
a diagonal matrix with all the eigenvalues of B f , called characteristic exponents
λ i = σ i + jω i , as the diagonal elements. It is obvious from Eqs. (12) and (13) that
the condition for asymptotic stability is that all λ i have negative real parts, that is,
σ i < 0. From Eq. (13), the monodromy matrix becomes
ϒ(T ) = V A e
DT V
−1
A .
(14)
The eigenvectors of ϒ(T ) are identical with those of B f , and therefore the similarity
transformation gives
M T = e
DT
,
(15)
where M T ∈ R
∞×∞ is a diagonal matrix with eigenvalues of ϒ(T ) as the diagonal elements. The eigenvalues of ϒ(T ), called the characteristic multipliers μ i , are
seen from Eq. (15) to be given as μ i = e
λ i T . Therefore, the condition for asymptotic
stability is that all characteristic multipliers μ i have modulus less than one, that is,
|μ i | = e
σ i T
< 1 ∀ i . The basis for identifying the stability boundary is max|μ i | = 1.
Stability analysis of the reduced order model based on Eq. (12) is not feasible;
therefore, finite-dimensional approximations of the rather infinite-dimensional monodromy matrices are needed for computational purposes. It can then be said that
in addition to spatial reduction, temporal reduction is also necessary for stability
analysis of the system.
3 Generalized Discrete Map
The discrete time delay τ is also the period of the model for the studied case
of uniform pitch milling. The period of the system is divided into k equal discrete time intervals [t i , t i+1 ] where i = 0, 1, 2, . . . , (k − 1) and t i = i
τ
k
= it =
i(t i+1 − t i ). Equation (11) is solved in each discrete interval to give