On the partial simulation of a nonconservative system by a conservative system

Anderson, N.; Done, G.T.S.

doi:10.1016/0020-7683(71)90042-4

Cited by 10 publications

(7 citation statements)

References 6 publications

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“…where Ω = ω T is the dimensionless eigenvalue andM (2) ,C (2) ,K (2) andG (2) are the 2 × 2 upper-left corner partitions of matrices (55), (56), (57) and (58), respectively. As a consequence of the determinant property det B = det B (holding for every matrix B), the same eigenvalues leading to the vanishing of the determinant (59), and therefore the same critical loads, are obtained for both nonholonomic constraints of the 'skate' and 'violin bow' type.…”

Section: The Double Pendulum Subject To the 'Skate' And 'Violin Bow' mentioning

confidence: 99%

“…Note that the dimensionless massM X of the sliding block does not appear in the mass matrixM (2) , Eq. (55), so that it does not influence the value of the critical loads.…”

Section: The Double Pendulum Subject To the 'Skate' And 'Violin Bow' mentioning

confidence: 99%

“…The coefficients ρ 0 and ρ 4 are assumed to be always greater than zero, a condition expressing a non-vanishing determinant for the matricesM (2) andK (2) , respectively. Such assumption is automatically satisfied whenk 1 > 0 and at least one of the positive input parametersd,M L andĨ r,L is different from zero.…”

Section: The Double Pendulum Subject To the 'Skate' And 'Violin Bow' mentioning

confidence: 99%

“…All these features and the presence itself of flutter and divergence instabilities in structures are believed to be strictly related to the action of non-conservative loads, 1 which are considered of difficult realization. 2 Indeed, quoting Anderson and Done [2], a 'conservative system can not become dynamically unstable since, by definition, it has no energy source from which to supply the extra kinetic energy involved in the instability'.…”

Section: Introductionmentioning

confidence: 99%

“…if they are conservative forces), and in general is not applicable if they do not. '2 The attempt by Willems[33] of producing a tangential follower load was indicated as misleading by Huang et al[13]. Elishakoff[10] reports that 'Bolotin felt -if my memory serves me well!-that it should be impossible to produce Becks column experiment via a conservative system of forces.'…”

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confidence: 99%

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Non-holonomic constraints inducing flutter instability in structures under conservative loadings

Cazzolli

Corso

Bigoni

2020

Journal of the Mechanics and Physics of Solids

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Non-conservative loads of the follower type are usually believed to be the source of dynamic instabilities such as flutter and divergence. It is shown that these instabilities (including Hopf bifurcation, flutter, divergence, and destabilizing effects connected to dissipation phenomena) can be obtained in structural systems loaded by conservative forces, as a consequence of the application of non-holonomic constraints. These constraints may be realized through a 'perfect skate' (or a non-sliding wheel), or, more in general, through the slipless contact between two circular rigid cylinders, one of which is free of rotating about its axis. The motion of the structure produced by these dynamic instabilities may reach a limit cycle, a feature that can be exploited for soft robotics applications, especially for the realization of limbless locomotion. . 1 Bolotin [9] writes 'The Euler method is [only] applicable if the external forces have a potential (i.e. if they are conservative forces), and in general is not applicable if they do not.' 2 The attempt by Willems [33] of producing a tangential follower load was indicated as misleading by Huang et al. [13]. Elishakoff [10] reports that 'Bolotin felt -if my memory serves me well!-that it should be impossible to produce Becks column experiment via a conservative system of forces.' Anderson and Done [2] write 'Sometimes, the creation of a force like [a follower force] in the laboratory presents awkward practical problems, and the simulation of this force wherever possible by a conservative force would be very convenient. However, because of the differing nature of the fundamental properties of the conservative and non-conservative systems, the simulation could only work in a situation where the two systems behave in similar ways; that is when the conservative system is not operating in a regime of oscillatory instability. (The conservative system can not become dynamically instable since, by definition, it has no energy source from which to supply the extra kinetic energy involved in the instability).' Koiter [19] states that '[...] it appears impossible to achieve any non-conservative loading conditions in the laboratory by purely mechanical means', because 'non-conservative external loads always require an external energy source, much as a fluid flow or an interaction with electro-dynamic phenomenon'. Koiter was strongly convinced that follower forces were a sort of 'physical non-sense' (Koiter,[20], [21]), so that Singer et al. [31] write 'An example in the field of elastic stability of what Drucker referred to as playing useless games was presented by Koiter, in his 1985 Prandtl lecture, where he discussed the physical significance of instability due to non-conservative, purely configurationdependent, external loads.' Several years after these negative views, Bigoni and Noselli [5] and Bigoni and Misseroni [8] respectively showed how to realize a tangentially follower force (Ziegler, 1952 [34]) and a fixed-line force (Reut, 1939 [30]) with devices involving Coulomb fr...

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Section: The Double Pendulum Subject To the 'Skate' And 'Violin Bow' mentioning

confidence: 99%

“…Note that the dimensionless massM X of the sliding block does not appear in the mass matrixM (2) , Eq. (55), so that it does not influence the value of the critical loads.…”

Section: The Double Pendulum Subject To the 'Skate' And 'Violin Bow' mentioning

confidence: 99%