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Research Papers

A Method for Calculating and Continuing Static Solutions for Flexible Multibody Systems

[+] Author and Article Information
J. P. Meijaard

Precision Engineering,
University of Twente,
Enschede 7522 NB, The Netherlands
e-mail: J.P.Meijaard@utwente.nl

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 18, 2017; final manuscript received April 20, 2018; published online May 17, 2018. Assoc. Editor: Javier Cuadrado.

J. Comput. Nonlinear Dynam 13(7), 071002 (May 17, 2018) (8 pages) Paper No: CND-17-1379; doi: 10.1115/1.4040081 History: Received August 18, 2017; Revised April 20, 2018

A method to calculate static solutions for mechanical systems containing rigid and flexible bodies modeled by finite elements is described. The formulation of the equations makes use of generalized strains, which leads to an extended set of equations for both these generalized strains and nodal coordinates, together with constraint equations imposing the relations between these two groups of coordinates. The associated Lagrangian multipliers are the generalized stresses. The resulting iteration scheme appears to be quite robust in comparison with more traditional methods, especially if some displacements are prescribed. Once a static solution has been found, the linearized equations of motion about this solution can be obtained in terms of a set of minimal coordinates, that is, in the degrees-of-freedom (DOFs). In addition, a continuation method is described for tracing a branch of static solutions if some parameters are varied. The method is of the familiar predictor–corrector type with a linear or cubic predictor and a corrector with a step size constraint. Applications to a large-deflection problem of a curved cantilever beam, large deflections of a fluid-conveying tube and its resulting instability, and the buckling of an overconstrained parallel leaf-spring mechanism due to misalignment are given.

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References

Howell, L. L. , 2001, Compliant Mechanisms, Wiley, New York.
van der Werff, K. , 1977, “ Kinematic and Dynamic Analysis of Mechanisms, a Finite Element Approach,” Doctoral dissertation, Delft University Press, Delft, The Netherlands. https://repository.tudelft.nl/islandora/object/uuid:52eb5573-7993-4e05-a7bc-d951bc307c62
van der Werff, K. , and Jonker, J. B. , 1984, “ Dynamics of Flexible Mechanisms,” Computer Aided Analysis and Optimization of Mechanical System Dynamics, E. J. Haug ed., Springer-Verlag, Berlin, pp. 381–400. [CrossRef]
Jonker, J. B. , 1989, “ A Finite Element Dynamic Analysis of Spatial Mechanisms With Flexible Links,” Comput. Methods Appl. Mech. Eng., 76(1), pp. 17–40. [CrossRef]
Haselgrove, C. B. , 1961, “ The Solution of Non-Linear Equations and of Differential Equations With Two-Point Boundary Conditions,” Comput. J., 4(3), pp. 255–259. [CrossRef]
Riks, E. , 1970, “ On the Numerical Solution of Snapping Problems in the Theory of Elastic Stability,” Ph.D. thesis, Stanford University, Stanford, CA.
Riks, E. , 1972, “ The Application of Newton's Method to the Problem of Elastic Stability,” ASME J. Appl. Mech., 39(4), pp. 1060–1065. [CrossRef]
Wempner, G. A. , 1971, “ Discrete Approximations Related to Nonlinear Theories of Solids,” Int. J. Solids Struct., 7(11), pp. 1581–1599. [CrossRef]
Ramm, E. , 1981, “ Strategies for Tracing the Nonlinear Response Near Limit Points,” Nonlinear Finite Element Analysis in Structural Mechanics, W. Wunderlich , E. Stein , and K.-J. Bathe , eds., Springer-Verlag, Berlin, pp. 63–89. [CrossRef]
Crisfield, M. A. , 1981, “ A Fast Incremental/Iterative Solution Procedure That Handles ‘Snap-Through’,” Comput. Struct., 13(1–3), pp. 55–62. [CrossRef]
Fried, I. , 1984, “ Orthogonal Trajectory Accession to the Nonlinear Equilibrium Curve,” Comput. Methods Appl. Mech. Eng., 47(3), pp. 283–297. [CrossRef]
Cardona, A. , and Huespe, A. , 1998, “ Continuation Methods for Tracing the Equilibrium Path in Flexible Mechanism Analysis,” Eng. Comput., 15 (2), pp. 190–220. [CrossRef]
Masarati, P. , 2015, “ Robust Static Analysis Using General-Purpose Multibody Dynamics,” Proc. Inst. Mech. Eng., Part K: J. Multi-Body Dyn., 229(2), pp. 152–165.
Watson, L. T. , Billups, S. C. , and Morgan, A. P. , 1987, “ Algorithm 652, HOMPACK: A Suite of Codes for Globally Convergent Homotopy Algorithms,” ACM Trans. Math. Software, 13(3), pp. 281–310. [CrossRef]
Ritto-Corrêa, M. , and Camotim, D. , 2008, “ On the Arc-Length and Other Quadratic Control Methods: Established, Less Known and New Implementation Procedures,” Comput. Struct., 86(11–12), pp. 1353–1368. [CrossRef]
Meijaard, J. P. , 2014, “ Fluid-Conveying Flexible Pipes Modeled by Large-Deflection Finite Elements in Multibody Systems,” ASME J. Comput. Nonlinear Dyn., 9(1), p. 7.
Jonker, J. B. , 1991, “ Linearization of Dynamic Equations of Flexible Mechanisms—A Finite Element Approach,” Int. J. Numer. Methods Eng., 31(7), pp. 1375–1392. [CrossRef]
Fafard, M. , and Massicotte, B. , 1993, “ Geometrical Interpretation of the Arc-Length Method,” Comput. Struct., 46(4), pp. 603–615. [CrossRef]
Hellweg, H.-B. , and Crisfield, M. A. , 1998, “ A New Arc-Length Method for Handling Sharp Snap-Backs,” Comput. Struct., 66(5), pp. 705–709. [CrossRef]
Bathe, K. J. , and Bolourchi, S. , 1979, “ Large Displacement Analysis of Three-Dimensional Beam Structures,” Int. J. Numer. Methods Eng., 14(7), pp. 961–986. [CrossRef]
Jonker, J. B. , and Meijaard, J. P. , 2013, “ A Geometrically Non-Linear Formulation of a Three-Dimensional Beam Element for Solving Large Deflection Multibody System Problems,” Int. J. Non-Linear Mech., 53, pp. 63–74. [CrossRef]
Meijaard, J. P. , Brouwer, D. M. , and Jonker, J. B. , 2010, “ Analytical and Experimental Investigation of a Parallel Leaf Spring Guidance,” Multibody Syst. Dyn., 23(1), pp. 77–97. [CrossRef]

Figures

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Fig. 1

Curved spatial cantilever beam subjected to a tip force perpendicular to the plane of the curve

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Fig. 2

Semicircular tube conveying a fluid

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Fig. 3

Configurations of the centerline of the tube for flow rates of 0, 0.1, 0.2, 0.3, 0.4, 0.426, 0.5, and 0.6 kg/s

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Fig. 4

Eigenmode at the critical mass flow of 0.426 kg/s. The real and imaginary parts of the z-displacement are shown, with the amplitude at the tip normalized to one.

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Fig. 5

Parallel leaf-spring guidance with an adjustable misalignment

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Fig. 6

Applied moment T0 as a function of the absolute value of the prescribed rotation angle ϕ0 of the hinge. The fully drawn line represents a stable branch, whereas the dashed line represents an unstable branch of solutions. The maxima are indicated by the vertical lines.

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