Normal and Quasi-Normal Modes in Damped Linear Dynamic Systems

1966 ◽  
Vol 33 (2) ◽  
pp. 413-416 ◽  
Author(s):  
J. S. Maybee
Keyword(s):  
Linear Systems ◽  
Dynamic Systems ◽  
Normal Modes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Linear Dynamic Systems ◽  
Linear Dynamic ◽  
Necessary And Sufficient

A generalization of the concept of classical normal modes in damped linear systems is presented. It is then shown that a necessary and sufficient condition that such quasi-normal modes exist is that certain matrices associated with the system commute. Necessary and sufficient conditions of the same type are also obtained for the classical normal modes, but under more restrictive conditions.

Classical Normal Modes in Damped Linear Dynamic Systems

1965 ◽  
Vol 32 (3) ◽  
pp. 583-588 ◽  
Author(s):  
T. K. Caughey ◽  
M. E. J. O’Kelly
Keyword(s):  
Dynamic Systems ◽  
Normal Modes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Dynamic Systems ◽  
Linear Dynamic ◽  
Necessary And Sufficient

The purpose of this paper is to determine necessary and sufficient conditions under which both discrete and continuous damped linear dynamic systems possess classical normal modes.

Classical Normal Modes in Damped Linear Dynamic Systems

1960 ◽  
Vol 27 (2) ◽  
pp. 269-271 ◽  
Author(s):  
T. K. Caughey
Keyword(s):  
Dynamic Systems ◽  
Necessary Conditions ◽  
Normal Modes ◽  
Present Theory ◽  
Necessary And Sufficient Condition ◽  
Linear Dynamic Systems ◽  
Linear Dynamic ◽  
Damping Matrix ◽  
Necessary And Sufficient ◽  
Special Case

An analysis of the conditions under which a damped linear system possesses classical normal modes is presented. It is shown that a necessary and sufficient condition for the existence of classical normal modes is that the damping matrix be diagonalized by the same transformation that uncouples the undamped systems. Sufficient though not necessary conditions on the damping matrix are developed, and it is shown that Rayleigh’s solution is a special case of the present theory.

scholarly journals Equipartitioning and balancing points of polygons

Pythagoras ◽  
2010 ◽  
Vol 0 (71) ◽  
Author(s):  
Shunmugam Pillay ◽  
Poobhalan Pillay
Keyword(s):  
General Theorem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Sufficient Condition ◽  
Centre Of Mass ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

The centre of mass G of a triangle has the property that the rays to the vertices from G sweep out triangles having equal areas. We show that such points, termed equipartitioning points in this paper, need not exist in other polygons. A necessary and sufficient condition for a quadrilateral to have an equipartitioning point is that one of its diagonals bisects the other. The general theorem, namely, necessary and sufficient conditions for equipartitioning points for arbitrary polygons to exist, is also stated and proved. When this happens, they are in general, distinct from the centre of mass. In parallelograms, and only in them, do the two points coincide.

Necessary and sufficient conditions for asymptotic decay of oscillations in delayed functional equations

1981 ◽  
Vol 91 (1-2) ◽  
pp. 135-145
Author(s):  
Lu-San Chen ◽  
Cheh-Chih Yeh
Keyword(s):  
Differential Operator ◽  
Functional Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Oscillatory Solutions ◽  
Asymptotic Decay ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisThis paper studies the equationwhere the differential operator Ln is defined byand a necessary and sufficient condition that all oscillatory solutions of the above equation converge to zero asymptotically is presented. The results obtained extend and improve previous ones of Kusano and Onose, and Singh, even in the usual case wherewhere N is an integer with l≦N≦n–1.

Some results on the complement of the annihilating ideal graph of a commutative ring

2015 ◽  
Vol 14 (07) ◽  
pp. 1550099 ◽  
Author(s):  
S. Visweswaran ◽  
Hiren D. Patel
Keyword(s):  
Commutative Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Rings considered in this article are commutative with identity which admit at least one nonzero annihilating ideal. For such a ring R, we determine necessary and sufficient conditions in order that the complement of its annihilating ideal graph is connected and also find its diameter when it is connected. We discuss the girth of the complement of the annihilating ideal graph of R and prove that it is either equal to 3 or ∞. We also present a necessary and sufficient condition for the complement of the annihilating ideal graph to be complemented.

scholarly journals K-type slant helices on spacelike and timelike surfaces

2021 ◽  
Vol 25 (2) ◽  
pp. 201-220
Author(s):  
Santosh Kumar ◽  
Buddhadev Pal
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Timelike Surface ◽  
Spacelike Curve ◽  
Slant Helix ◽  
Darboux Frame ◽  
Necessary And Sufficient

We have derived a necessary and sufficient condition for a non-null normal spacelike curve lying in a spacelike or a timelike surface M ⊂ E13, so that the curve becomes a K-type spacelike slant helix with K ∈ {1,2,3}. We have used Darboux frame to define necessary and sufficient conditions. An example is given for a 1-type spacelike slant helix having a spacelike normal and a timelike binormal.

On a certain kind of reducible rational fractions

1980 ◽  
Vol 21 (3) ◽  
pp. 321-328
Author(s):  
Mordechai Lewin
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Rational Fraction ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Positive Coefficients ◽  
Positive Integers ◽  
Exact Positions

The rational fractiona, c, p, q positive integers, reduces to a polynomial under conditions specified in a result of Grosswald who also stated necessary and sufficient conditions for all the coefficients to tie nonnegative.This last result is given a different proof using lemmas interesting in themselves.The method of proof is used in order to give necessary and sufficient conditions for the positive coefficients to be equal to one. For a < 2pq, a = αp + βq, α, β nonnegative integers, c > 1, the exact positions of the nonzero coefficients are established. Also a necessary and sufficient condition for the number of vanishing coefficients to be minimal is given.

scholarly journals Generalising the Horodecki criterion to nonprojective qubit observables

2021 ◽  
Author(s):  
Michael J W Hall ◽  
Shuming Cheng
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Weak Measurements ◽  
Chsh Inequality ◽  
Bell Nonlocality ◽  
Necessary And Sufficient ◽  
Projective Measurements

Abstract The Horodecki criterion provides a necessary and sufficient condition for a two-qubit state to be able to manifest Bell nonlocality via violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. It requires, however, the assumption that suitable projective measurements can be made on each qubit, and is not sufficient for scenarios in which noisy or weak measurements are either desirable or unavoidable. By characterising two-valued qubit observables in terms of strength, bias, and directional parameters, we address such scenarios by providing necessary and sufficient conditions for arbitrary qubit measurements having fixed strengths and relative angles for each observer. In particular, we find the achievable maximal values of the CHSH parameter for unbiased measurements on arbitrary states, and, alternatively, for arbitrary measurements on states with maximally-mixed marginals, and determine the optimal angles in some cases. We also show that for certain ranges of measurement strengths it is only possible to violate the CHSH inequality via biased measurements. Finally, we use the CHSH inequality to obtain a simple necessary condition for the compatibility of two qubit observables.

Generalized concurrences do not provide necessary and sufficient conditions for entanglement detection

2009 ◽  
Vol 9 (1&2) ◽  
pp. 166-180
Author(s):  
L. Cattaneo ◽  
D. D'Alessandro
Keyword(s):  
Sufficient Conditions ◽  
Quantum Systems ◽  
Necessary And Sufficient Conditions ◽  
Entangled States ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bipartite Quantum Systems ◽  
Rank 2 ◽  
Entanglement Detection ◽  
Necessary And Sufficient

We study generalized concurrences as a tool to detect the entanglement of bipartite quantum systems. By considering the case of 2x4 states of rank 2, we prove that generalized concurrences do not, in general, give a necessary and sufficient condition of separability. We identify a set of entangled states which are undetected by this method.

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