Causality and Mathematical Models in Vibration and Acoustics, A Realistic Perspective

8/3/2019 Causality and Mathematical Models in Vibration and Acoustics, A Realistic Perspective

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Proceedings of Meetings on Acoustics

Volume 5, 2008 http://asa.aip.org

156th Meeting

Acoustical Society of AmericaMiami, Florida

10 - 14 November 2008

Session 4aSA: Structural Acoustics and Vibration

4aSA1. Causality and mathematical models in vibration and acoustics, a realistic perspective

Allan D. Pierce*

*Corresponding authors address: Mechanical Engineering, Boston University, Boston, MA 02215, [email protected]

Causality principles are defined in the context of linear systems with time-independent properties. The primary concepts

are those of a reponse function and of a transfer function. The existence of the latter strictly requires that the system be

stable, and a causal response function can exist without the system being stable. The definition of the transfer function can

be extended to the negative real axis and subsequently to the upper half plane for specific mathematical models of vibrat-

ing systems. Stability and consequently the use of causality considerations requires that there be no poles in the upper halfplane. Examples are given for a one-degree-of-freedom vibrating system with various types of damping terms. Negative

damping allows the existence of a causal response function but leads to an instable sytem. Fractional derivative damping

is all right if the fractional power is less than 2. Consideration of arbitrary multi-degree-of-freedom systems shows that the

usual causality considerations apply if the mass, damping, and stiffness matrices are all positive-definite. An extended

class of Kramers-Kronig relations esists, and the general relations in this class are derived which relate the real and imagi-

nary parts of the transfer function.

Published by the Acoustical Society of America through the American Institute of Physics

Allan D. Pierce

2008 Acoustical Society of America [DOI: 10.1121/1.3073668]

Received 25 Nov 2008; published 31 Dec 2008

Proceedings of Meetings on Acoustics, Vol. 5, 065001 (2008) Page 1


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These are the slides that were shown in the talk given in Miami on November 13. It wasthe first in a session of invited talks, and the scope was largely an introduction to the principalconcepts.

The scope is limited to passive linear systems, where all the properties are independent oftime. For any such system, one would expect there to be a response function that connects anygiven (sole) cause to any given effect.

Allan D. Pierce



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The second fundamental premise presumes that the system is such that the response to asinusoidally oscillating cause will eventually result in a response that is oscillating with the samefrequency. This is not necessarily so.

In this simple example where there is a dashpot with negative damping, any transient ex-citation will always result in a term, perhaps initially very small, but which grows exponentiallywithout bound and which overwhelms any constant frequency steady-state response.

Allan D. Pierce



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Here are a list of mathematical observations that one can make about the example in termsof what might ordinarily expect from uncritical causality considerations.

The next portion of the paper examines the consequences of the fundamental principals, giventhat they hold for a given system. What follows below includes a reminder of what is stated in aprevious slide.

Allan D. Pierce



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One important consequence is that the frequency dependent transfer function can be equatedto a one-sided Fourier transform of the response function. The sign of the exponent in the eit

factor is a matter of choice. The choice here is that customarily used in acoustics, but the oppositeof what is customarily used in vibrations and circuit theory.

The most important consequence is that the inverse Fourier transform of the transfer functionmust vanish when the time t is negative.

The present author vaguely remembers having been told, at one time or another many yearago that, for the oscillation of bubbles, it was sometimes considered appropriate to take the damp-ing term in the harmonic oscillator as being proportional to the third derivative with respect totime. The author remembered that without much critical thinking and proceeded to use such asa pedagogical example in a course he regularly taught on vibrations, sometimes as a homeworkproblem, and other times as a quiz problem. In retrospect, the example has intrinsic problemsin regard to causality. The author discussed this topic with Phillip Marston, who kindly suppliedwhat appears to be the original reference for this type of damping. The reference is: A. Pros-peretti, The equation of bubble dynamics in a compressible fluid, Phys. Fluids 30, 3626-3628(1987). Subsequent literature has improved on Prosperettis model and the causality paradox isno longer present. However, the original model presents a good example of how a well-intentionedand apparently rigorously correct derivation can lead to a model that violates causality.

Allan D. Pierce



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What appears below is the nonlinear ordinary differential equation originally derived by Pros-peretti, followed by a linearized version.

The linearized equation has the mechanical analog of a spring-mass-dashpot system only witha dashpot term with a third derivative with respect to time.

Allan D. Pierce



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What follows is a simple naive analysis of the bubble with damping example.

The transfer function that results is the reciprocal of a third order polynomial. For the modelto be consistent with causality considerations, there must be no poles in the upper half plane. Apole in the upper half plane would imply that the transient response would grow exponentially intime.

Allan D. Pierce



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When all the singularities of the transfer function are simple poles, it is all right if some poleslie in the lower half plane. In the present example, there are always two good poles, and alwaysone bad pole.

At this point, it is appropriate to address the common terminology that mathematical modesare causal. Such terminology is not always appropriate. A causal response function may exist, butthe onset of a sinusoidal excitation may lead to a unstable solution. Perhaps, it is more appropriateto say that the model is instable.

Allan D. Pierce



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In surveying the applicable acoustics literature related to causality, the author came across thefollowing abstract of a talk which Murray Strasberg presented in May 2001 at an ASA meeting inChicago. The present author does not remember whether he heard the talk, and does not attempthere to explain the theoretical device alluded to in the second part of the abstract. It is shownhere primarily to emphasize that the case of hysteretic damping is long-recognized as presentingcausality problems.

Allan D. Pierce



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Perhaps, as an alternate to hysteric damping, one might want to model vibrating systemswhere the damping is accounted for by a fractional derivative term, so that ( d/dt)a (i)a.

Allan D. Pierce



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A brief analysis shows that any fractional exponent greater than 0 but less than 2 is all right.If a > 2, the system is instable.

In generalized linear vibration analysis, one often tacitly assumes that system has a largenumber of degrees of freedom, with the overall system described by a set of simultaneous ordinarydifferential equations, which can be written in matrix form.

Allan D. Pierce



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Given such a general model, the question naturally arises as to what specific constraints mustit conform to, in order that causality considerations can be applied in the analysis of data takenwith respect to such a model.

In the analysis for this general question, the author has found the book by Peter Lancaster,Lambda Matrices and Vibrating Systems, to more or less give a comprehensive answer.

Allan D. Pierce



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Thus, sufficient conditions for the usual causality considerations to hold are that the three matriceseach be positive definite. A related problem and one suggested for future discussion is that whenthe damping in the system is all or partly caused by relaxation processes.

Allan D. Pierce



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Frequently in the literature, especially in regard to the interrelations between phase velocitydispersion and attenuation, one finds mention of Kramers-Kronig relations. What is not widelyknown is that there is actually an extensive family of relations that might warrant such a designa-tion. An outline is given here of how they might be derived. One starts with a hypothesized transferfunction H() which is analytic in the upper half of the complex- plane and which therefore hasno poles in the upper half plane. One seeks relations between the real and imaginary parts of thisfunction along the positive real axis. The beginning of the derivation is to form one of either twointegrals of a generic form where the denominator has an even number of symmetrically placedpoles slightly above the real axis, with the integration contour passing above all such poles.

Allan D. Pierce



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The contour is extended to include the upper half of the complex plane and the overall integralis consequently zero. Then the portion of the contour originally just above the real axis is deformedso that it consists of (i) a sequence of segments along the real axis, each seqment stopping justshort of a pole, and (ii) a sequence of semicircular arcs which go above each of the poles.

The development below shows how the integration contribution from the segments along thereal axis leads to the principal value of an integral along the positive real axis. For the type Iintegral, the integrand involves only the real part of the transfer function. For the type II integral,it involves only the imaginary part.

Allan D. Pierce



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Here is a formal expression for any Kramers-Kronig relation of the first type. The right side,arising from contribution from the semi-circular arcs around the poles, involves the imaginary partof the transfer function at a limited number of points on the real axis. The left side involves theprincipal value of an integral whose integrand involves only the real part. For the second type ofKramers-Kronig relation, the converse is the case.

Allan D. Pierce



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Allan D. Pierce

Causality and Mathematical Models in Vibration and Acoustics, A Realistic Perspective

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