The Behaviour of Damped Linear Systems in Steady Oscillation

1956 ◽  
Vol 7 (4) ◽  
pp. 353-354
Author(s):  
R. E. D. Bishop
Keyword(s):  
Linear Systems ◽  
Harmonic Excitation ◽  
Steady Oscillation ◽  
Image Position ◽  
Limiting Condition ◽  
Partial Fractions ◽  
Generalised Coordinates

In this paper (published in The Aeronautical Quarterly, Vol. VII, Part 2, p. 156, May 1956), a set of equations is derived giving the responses at the generalised coordinates q1, q2, … qn which are caused by a harmonic generalised force Φseiωt corresponding to qs. They are equations (26), namelyThis note is concerned with a sentence which follows these equations which says (of them) that “ … there is no longer any assurance that the ratios between the constants mBrs in any column of partial fractions are the same for all co-ordinates qs at which excitation is applied, since the previous limiting condition cannot now be applied. Thus the shapes of the ‘ modes ’ which are associated with the various columns depend upon the nature of the applied harmonic excitation.”

scholarly journals LINEAR SYSTEMS ON IRREGULAR VARIETIES

2019 ◽  
Vol 19 (6) ◽  
pp. 2087-2125 ◽  
Author(s):  
Miguel Ángel Barja ◽  
Rita Pardini ◽  
Lidia Stoppino
Keyword(s):  
Continuous Function ◽  
Line Bundle ◽  
Linear Systems ◽  
Projective Variety ◽  
Geometrical Properties ◽  
Complex Projective Variety ◽  
Wide Range ◽  
Image Position ◽  
Irregular Varieties ◽  
Complex Projective

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety of dimension $m$ (possibly $T=X$) and $a:X\rightarrow A$ a morphism to an abelian variety such that $\text{Pic}^{0}(A)$ injects into $\text{Pic}^{0}(T)$; let $L$ be a line bundle on $X$ and $\unicode[STIX]{x1D6FC}\in \text{Pic}^{0}(A)$ a general element.We introduce two new ingredients for the study of linear systems on $X$. First of all, we show the existence of a factorization of the map $a$, called the eventual map of $L$ on $T$, which controls the behavior of the linear systems $|L\otimes \unicode[STIX]{x1D6FC}|_{|T}$, asymptotically with respect to the pullbacks to the connected étale covers $X^{(d)}\rightarrow X$ induced by the $d$-th multiplication map of $A$.Second, we define the so-called continuous rank function$x\mapsto h_{a}^{0}(X_{|T},L+xM)$, where $M$ is the pullback of an ample divisor of $A$. This function extends to a continuous function of $x\in \mathbb{R}$, which is differentiable except possibly at countably many points; when $X=T$ we compute the left derivative explicitly.As an application, we give quick short proofs of a wide range of new Clifford–Severi inequalities, i.e., geographical bounds of the form $$\begin{eqnarray}\displaystyle \text{vol}_{X|T}(L)\geqslant C(m)h_{a}^{0}(X_{|T},L), & & \displaystyle \nonumber\end{eqnarray}$$ where $C(m)={\mathcal{O}}(m!)$ depends on several geometrical properties of $X$, $L$ or $a$.

scholarly journals Remarks on the Asymptotic Behaviour of Perturbed Linear Systems

1970 ◽  
Vol 11 (1) ◽  
pp. 84-84 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Asymptotic Behaviour ◽  
Linear Systems ◽  
Image Position ◽  
Perturbed Linear Systems

Remarks 1, 3 and 5 are incorrect as stated. They should be supplemented by the following observations:(i) In case the perturbing term is linear in y, i.e. f(t, y) = B(t)y, the conclusion of Theorem 1 will follow from Lemma 1 when applied to equation (15) if we assume, instead of (6),The hypothesis given in Trench's theorem is sufficient to imply (*) but not (6). A similar comment applies to Remark 5.

Dynamic responses of clearance-type non-linear systems subjected to base harmonic excitation and their experimental verification

2021 ◽  
pp. 107754632110128
Author(s):  
K Renji
Keyword(s):  
Linear Systems ◽  
Experimental Verification ◽  
Harmonic Excitation ◽  
Dynamic Responses ◽  
Maximum Response ◽  
Base Excitation ◽  
Mathematical Expressions ◽  
Non Linear Systems ◽  
Propellant Tank ◽  
Different Levels

Realistic joints in a spacecraft structure have clearances at the interfacing parts. Many such systems can be considered to be having bilinear stiffness. A typical example is the propellant tank assembled with the structure of a spacecraft. However, it is seen that the responses of such systems subjected to base excitation are rarely reported. In this work, mathematical expressions for theoretically estimating the amplitude of its response, the frequency at which the response is the maximum and the maximum response when it is subjected to base sine excitation are derived. Several experiments are conducted on a typical such system subjecting it to different levels of base sine excitation. The frequency at which the response is the maximum reduces with the magnitude of excitation. The expressions derived in this work can be used in estimating the amplitudes of responses and their characteristics reasonably well.

scholarly journals Infinite linear systems with homogeneous kernel of degree –1

1965 ◽  
Vol 5 (2) ◽  
pp. 129-168
Author(s):  
T. M. Cherry
Keyword(s):  
Continuous Function ◽  
Linear Systems ◽  
Algebraic Function ◽  
Additional Restriction ◽  
Main Concern ◽  
Homogeneous Kernel ◽  
Infinite Linear Systems ◽  
Image Position ◽  
Infinite Linear

The main concern of this paper is with the solution of infinite linear systems in which the kernel k is a continuous function of real positive variables m, n which is homogeneous with degree –1, so that If k is a rational algebraic function it is supposed further that the continuity extends up to the axes m = 0, n > 0 and n = 0, m > 0; the possibly additional restriction when k is not rational is discussed in § 1,2.

Global attraction for systems with almost C1 vector fields

2005 ◽  
Vol 135 (4) ◽  
pp. 815-832
Author(s):  
Zhanyuan Hou
Keyword(s):  
Linear Systems ◽  
Global Attractor ◽  
Differential System ◽  
Vector Fields ◽  
Single Point ◽  
Sufficient Conditions ◽  
Global Attraction ◽  
Almost Everywhere ◽  
Image Position ◽  
Special Case

Sufficient conditions are given for an autonomous differential system to have a single point global attractor (repeller) with f continuously differentiable almost everywhere. These results incorporate those of Hartman and Olech as a special case even when the condition f ∈ C1(D, ℝN) is fully met. Moreover, these criteria are simplified for a class of region-wise linear systems in ℝN.

Bounds for the characteristic exponents of linear systems

1965 ◽  
Vol 61 (4) ◽  
pp. 889-894 ◽  
Author(s):  
R. A. Smith
Keyword(s):  
Differential Equations ◽  
Linear Systems ◽  
Characteristic Exponent ◽  
Zero Solution ◽  
System Of Differential Equations ◽  
Characteristic Exponents ◽  
Integrable Functions ◽  
Image Position ◽  
Complex Valued

For an n-vector x = (xi) and n × n matrix A = (aij) with complex elements, let |x|2 = Σi|xi|2,|A|2 = ΣiΣj|aij|2. Also, M(A), m(A) denoteℜλ1,ℜλn, respectively, where λ1,…,λA are the eigenvalues of A arranged so that ℜλ1 ≥ … ≥ ℜλn. Throughout this paper A(t) denotes a matrix whose elements aij(t) are complex valued Lebesgue integrable functions of t in (0, T) for all T > 0. Then M(A(t)), m(A(t)) are also Lebesgue integrable in (0, T) for all T > 0. The characteristic exponent μ of a non-zero solution x(t) of the n × n system of differential equationscan be defined, following Perron ((12)), aswhere ℒ denotes lim sup as t → + ∞. When |A(t)| is bounded in (0,∞), μ is finite; in other cases it could be ± ∞.

Power Conserving Bond Graph Based Modal Representations and Model Reduction of Lumped Parameter Systems

2014 ◽  
Vol 136 (6) ◽  
Author(s):  
Loucas S. Louca
Keyword(s):  
Model Reduction ◽  
Linear Systems ◽  
Controller Design ◽  
Physical Space ◽  
Order Reduction ◽  
Bond Graph ◽  
Harmonic Excitation ◽  
Model Complexity ◽  
Modal Decomposition ◽  
Lumped Parameter

Dynamic analysis is extensively used to study the behavior of continuous and lumped parameter linear systems. In addition to the physical space, analyses can also be performed in the modal space where very useful frequency information of the system can be extracted. More specifically, modal analysis can be used for the analysis and controller design of dynamic systems, where reduction of model complexity without degrading its accuracy is often required. The reduction of modal models has been extensively studied and many reduction techniques are available. The majority of these techniques use frequency as the metric to determine the reduced model. This paper describes a new method for calculating modal decompositions of lumped parameter systems with the use of the bond graph formulation. The modal decomposition is developed through a power conserving coordinate transformation. The generated modal decomposition model is then used as the basis for reducing its size and complexity. The model reduction approach is based on the previously developed model order reduction algorithm (MORA), which uses the energy-based activity metric in order to generate a series of reduced models. The activity metric was originally developed for the generic case of nonlinear systems; however, in this work, the activity metric is adapted for the case of linear systems with single harmonic excitation. In this case closed form expressions are derived for the calculation of activity. An example is provided to demonstrate the power conserving transformation, calculation of the modal power and the elimination of unimportant modes or modal elements.

scholarly journals Linear systems of nonoscillatory differential equations with delayed arguments

1984 ◽  
Vol 30 (2) ◽  
pp. 307-314
Author(s):  
K. Gopalsamy
Keyword(s):  
Differential Equations ◽  
Linear Systems ◽  
Sufficient Conditions ◽  
Delayed Arguments ◽  
Image Position

Sufficient conditions are obtained for a not necessarily scalar system of the formto be nonoscillatory.

Jacobians of linear systems on an algebraic variety

1956 ◽  
Vol 52 (2) ◽  
pp. 198-201 ◽  
Author(s):  
D. Monk
Keyword(s):  
Linear Systems ◽  
Algebraic Variety ◽  
Image Position

Let | Si | (i = 1, …, k) be k linear systems of hypersurfaces on an algebraic variety Vd, and letThe purpose of this note is to prove that the Jacobian of these systems is given by the equivalencewhere Kh is Eger's operator, defined as follows:

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