Dynamic Response of Mechanical Systems by a Weak Hamiltonian Formulation

Borri, M.; Ghiringhelli, G.L.; Lanz, M.; Mantegazza, Paolo; Merlini, T.

doi:10.1016/b978-0-08-032789-1.50055-0

Cited by 18 publications

(28 citation statements)

References 10 publications

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“…Borri [14] notes that the trailing terms should be written as momenta (P ) and not explicitly as (∂L/∂q). A mixed formulation makes this possible.…”

Section: Hamilton's Principlementioning

confidence: 99%

Simulation of a Moving, Elastic Beam Using Hamilton's Weak Principle

Leigh

Kunz

2006

47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

1
0
1
0

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Hamilton's Law is derived in weak form for slender beams with closed cross sections. The result is discretized with mixed space-time finite elements to yield a system of nonlinear, algebraic equations. An algorithm is proposed for solving these equations using unconstrained optimization techniques, obtaining steady-state and time accurate solutions for problems of structural dynamics. This technique provides accurate solutions for nonlinear static and steady-state problems including the cantilevered elastica and flatwise rotation of beams. Modal analysis of beams and rods is investigated to accurately determine fundamental frequencies of vibration, and the simulation of simple maneuvers is demonstrated.iv

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“…Borri [14] notes that the trailing terms should be written as momenta (P ) and not explicitly as (∂L/∂q). A mixed formulation makes this possible.…”

Section: Hamilton's Principlementioning
confidence: 99%

Simulation of a Moving, Elastic Beam Using Hamilton's Weak Principle
Leigh
¹
,
Kunz
²

2006
47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
1
0
1
0
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“…Further, let us denow with L(q, q, t ) the Lagrangean of the system, Q the set of nonconservative generalized forces applied to the system, and p = aL/aq the set of generalized momenta. Then tlie following variational equation, 1;nown as HWP [5], describes the real motion of tlie s p e m between the two known tiines t o and t f :…”

Section: Hamilton's Weak Principle For Dynamicsmentioning
confidence: 99%

“…However, in order to derive an uiicondition,?! !~-stable algoritlini in [5]$reduced/selective integration had to be used. Further c o l~l i~u~~-tional advantages may be obtcned in so-called mixed formulations of H U T in 11-hich the generalized coordinates and momenta appear as independent unknowns [ 71.…”

mentioning
confidence: 99%

Weak Hamiltonian finite element method for optimal control problems
Hodges
¹
,
Bless
²

1991
Journal of Guidance, Control, and Dynamics
80
0
34
0
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A temporal finite element method based on a mised form of the Hamiltonian n-eal; principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the nionient a and displacements do not appear therein; instead, only the virtual momenta and virtual displaceinents are differentiated with respect to time. Based on the duality that is obser\-cd to esist between the mixed form of Hamilton's weak principle and variational principles governing classical optiinal control problems, a temporal finite element forinulation of the latter can be developed in a rather straightforward manner. Several n-ell-l;nowl problelxs in dynamics and optimal control are illustrated. The esample dynamics problem inr-olws a time-marching problem. As optimal control esaniples, elementary trajectory optimizat ion problems are treated.

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“…Another possible approach is the use of the finite element in time method (FET) based on Hamilton's principle [14][15][16][17], in which the solution for all the spatial degrees of freedom at all time steps within a given time interval of interest is sought via a set of algebraic equations. Some recent development has shown that this approach offers several potential advantages over numerical solution of ordinary differential equations in the time domain, including the flexibility in formulating the problem directly from system energy expressions, the greater accuracy at specific time points of interest, and the use of adaptive finite elements to improve computational efficiency and accuracy [14,[17][18][19].…”

Section: Introductionmentioning
confidence: 99%

Dynamics of unsymmetric piecewise-linear/non-linear systems using finite elements in time
Wang
¹

1995
Journal of Sound and Vibration
20
0
14
0
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The dynamic response and stability of a single-degree-of-freedom system with unsymmetric piecewise-linear/non-linear stiffness are analyzed using the finite element method in the time domain. Based on a Hamilton's weak principle, this method provides a simple and efficient approach for predicting all possible fundamental and sub-periodic responses. The stability of the steady state response is determined by using Floquet's theory without any special effort for calculating transition matrices. This method is applied to a number of examples, demonstrating its effectiveness even for a strongly non-linear problem involving both clearance and continuous stiffness non-linearities. Close agreement is found between available published findings and the predictions of the finite element in time approach, which appears to be an efficient and reliable alternative technique for non-linear dynamic response and stability analysis of periodic systems.

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Dynamic Response of Mechanical Systems by a Weak Hamiltonian Formulation

Cited by 18 publications

References 10 publications

Simulation of a Moving, Elastic Beam Using Hamilton's Weak Principle

Simulation of a Moving, Elastic Beam Using Hamilton's Weak Principle

Weak Hamiltonian finite element method for optimal control problems

Dynamics of unsymmetric piecewise-linear/non-linear systems using finite elements in time

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