Stability analysis of rigid multibody mechanical systems with holonomic and nonholonomic constraints

Pappalardo, Carmine Maria; Lettieri, Antonio; Guida, Domenico

doi:10.1007/s00419-020-01706-2

Cited by 29 publications

(20 citation statements)

References 78 publications

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“…On the other hand, if J r (v 0 ) has six nonzero eigenvalues, we must have rank J r (v 0 ) = 6. According to (35), J c has full rank, and thus according to Theorem 1, v 0 must belong to a one-parameter solution family of dynamic equilibria. We have the following corollary for the curves of the solution families.…”

Section: Stability Of Dynamic Equilibriamentioning

confidence: 99%

“…This will lead to different difficulties when solving the relative equilibria and analyzing their stabilities. Pappalardo et al [35] proposed a method for analyzing the stability characteristics of multibody mechanical systems with holonomic and nonholonomic constraints based on the direct linearization of the index-three form of the DAEs. García-Agúndez et al [36] developed three novel general numerical procedures to obtain the linearized equations of any multibody system: the linearization of the index-one form of the DAEs, the reduction of the linearized equations from DAEs to ODEs form, and a further reduction of the linearized equations by eliminating the holonomic constraints.…”

Section: Introductionmentioning

confidence: 99%

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Symmetry and relative equilibria of a bicycle system moving on a revolution surface

Xiong¹,

Liu²

2021

Preprint

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Finding the relative equilibria and analyzing their stabilities are of great significance to revealing the intrinsic properties of mechanical systems and developing effective controller. In this paper, we study the symmetry and relative equilibria of a bicycle system moving on a revolution surface. We note that the symmetry group of the bicycle is a three-dimensional Abelian Lie group, and the rolling condition of the two wheels produces four time-invariant first-order linear constraints to the bicycle system. Therefore, we can classify the bicycle dynamics as a general Voronets system whose Lagrangian and constraint distribution are kept invariant under the action of the symmetry group. Applying the Voronets equations to the bicycle dynamics, we obtain a seven-dimensional reduced dynamic system on the reduced constraint space. This system takes time-reversal and lateral symmetries, and has two kinds of relative equilibria: the static equilibria and the dynamic equilibria. Further theoretical analysis shows that both kinds of relative equilibria form one-parameter solution families, and their Jacobian matrices take some specific properties. We then show that a static equilibrium cannot be stable unless all the eigenvalues of the Jacobian matrix are located at the imaginary axis of the complex plane. The stability of the dynamic equilibria is studied by limiting the reduced dynamic system to an invariant manifold, which is established based on the conservation of energy of the system. We prove in a strict mathematical sense that the dynamic equilibria may be Lyapunov stable, but cannot be asymptotically stable. Finally, we employ symbolic computation to carry out numerical simulations in conjunction with the benchmark parameters of a Whipple bicycle. How the revolution surface affects the relative equilibria and their stabilities is then investigated through our numerical simulations.

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Section: Stability Of Dynamic Equilibriamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Symmetry and relative equilibria of a bicycle system moving on a revolution surface

Xiong¹,

Liu²

2021

Preprint

View full text Add to dashboard Cite

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“…In the floating frame of reference formulation, the flexible body's deformation can be defined with respect to its reference [19]. The constraints equation of MBS with rigid and flexible bodies can be written as:…”

Section: System Of Equations Of Motionmentioning

confidence: 99%

Development of Dynamics for Design Procedure of Novel Grating Tiling Device with Experimental Validation

et al. 2021

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The article proposes a dynamic for design (DFD) procedure for a novel aperture grating tiling device using the multibody system (MBS) approach. The grating device is considered as a rigid-flexible MBS that is built primarily based totally at the load assumptions because of grating movement. This movement is utilized in many industrial applications, such as the compression of laser pulse, precision measuring instruments, and optical communication. A new design procedure of tiling grating device frame is introduced in order to optimize its design parameters and enhance the system stability. The dynamic loads are estimated based on the Lagrange multipliers that are obtained from the solution of the MBS model. This model is fully non-linear and moves in the three-dimensional space, and the relative movement of its bodies is restricted by the description of the constraints function in the motion manifold. The mechanism of the grating device is structurally analyzed in keeping with the dynamic conduct and therefore the generated forces. The symbolic manipulation as well as the computational work of solving the obtained differential-algebraic equations (DAEs) is carried out using MATLAB Symbolic Toolbox. Once the preliminary design has been attained, the stress behavior of the grating device is examined using the MATLAB FEATool Multiphysics toolkit, regarding system stability and design aspects. Moreover, the design was constructed in real life, and the movement has been verified experimentally, which confirms the effectiveness of the proposed procedure. In conclusion, the DFD procedure with trade-off optimization is utilized successfully to design the grating unit for maximum ranges of grating movements.

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“…As a consequence, many numerical methods were developed to solve higher index DAEs such as range-Kutta [13,17], projected Taylor series methods [18], hybrid block algorithms [19], stabilization methods [20][21][22], augmented Lagrangian method [23], sequential regularization methods [24], and the differential transform method (DTM) [25,26]. For the solution of DAEs, one can also nd in the literature the power series method combined with the Adomian polynomials [27,28] and other methods [29][30][31][32][33]. A very popular approach to treat higher index DAEs is rst to reduce the index by di erentiating the constraints one or more times with respect to time to obtain an ordinary di erential system or an index-1 DAE.…”

Section: Introductionmentioning

confidence: 99%

The Differential Transform Method as an Effective Tool to Solve Implicit Hessenberg Index-3 Differential-Algebraic Equations

Benhammouda

2023

Journal of Mathematics

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Differential-algebraic equations (DAEs) are important tools to model complex problems in various application fields easily. Those DAEs with an index-3, even the linear ones, are known to cause problems when solving them numerically. The present article proposes a new algorithm together with its multistage form to efficiently solve a class of nonlinear implicit Hessenberg index-3 DAEs. This algorithm is based on the idea of applying the differential transform method (DTM) directly to the DAE without applying the traditional index reduction methods, which can be complex and often result in violations of the DAE constraints. Also, to deal with the nonlinear terms in the DAE, we approximate them using the Adomian polynomials. This new idea has given us a simple and efficient algorithm, which involves the solution of linear algebraic systems except for the initial recursion terms. This algorithm is easy to implement in Maple or Mathematica. Furthermore, to enlarge the interval of convergence of the power series solution obtained from the DTM, an algorithm for the multistage DTM is also given. Both algorithms are applied to solve two examples of highly nonlinear implicit index-3 Hessenberg DAEs. Numerical results show that the DTM can determine the exact solution in convergent power series, while the multistage DTM can compute accurate numerical solutions over large intervals.

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Stability analysis of rigid multibody mechanical systems with holonomic and nonholonomic constraints

Cited by 29 publications

References 78 publications

Symmetry and relative equilibria of a bicycle system moving on a revolution surface

Symmetry and relative equilibria of a bicycle system moving on a revolution surface

Development of Dynamics for Design Procedure of Novel Grating Tiling Device with Experimental Validation

The Differential Transform Method as an Effective Tool to Solve Implicit Hessenberg Index-3 Differential-Algebraic Equations

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