Research Papers

Experimental Fitting of Rotor Models by Using a Special Three-Node Beam Element

[+] Author and Article Information
Mate Antali

Department of Applied Mechanics,
Budapest University of Technology
and Economics,
Budapest 1111, Hungary
e-mail: antali@mm.bme.hu

Denes Takacs

MTA-BME Research Group on Dynamics
of Machines of Vehicles,
Budapest 1111, Hungary
e-mail: takacs@mm.bme.hu

Gabor Stepan

Department of Applied Mechanics,
Budapest University of Technology
and Economics,
Budapest 1111, Hungary
e-mail: stepan@mm.bme.hu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 19, 2017; final manuscript received August 23, 2017; published online November 9, 2017. Assoc. Editor: Javier Cuadrado.

J. Comput. Nonlinear Dynam 13(2), 021009 (Nov 09, 2017) (9 pages) Paper No: CND-17-1270; doi: 10.1115/1.4038148 History: Received June 19, 2017; Revised August 23, 2017

In this paper, a special type of beam element is developed with three nodes and with only translational degrees-of-freedom (DOFs) at each node. This element can be used effectively to build low degree-of-freedom models of rotors. The initial model from the Bernoulli theory is fitted to experimental results by nonlinear optimization. This way, we can avoid the complex modeling of contact problems between the parts of squirrel cage rotors. The procedure is demonstrated on the modeling of a machine tool spindle.

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Fig. 1

The analyzed spindle with a squirrel cage mounted on the shaft. Top panel: drawing of the spindle of the tested milling machine. Bottom-left panel: computer tomography (CT) image of the spindle in a longitudinal section. Bottom-left panel: CT image of the spindle in a transversal section. The numbered circles denote the modeling challenges: 1—contact between the aluminum and steel parts of the cage, 2—contact between the cage and the shaft, 3—contact between the cage and the fixing ring, and 4—contact between the cage and the edge of the shaft.

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Fig. 2

Sketch of the three-node general beam element with translational DOFs

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Fig. 3

The nodal forces creating the elemental displacements. The equilibrium of these force systems has to be ensured in the stiffness matrix.

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Fig. 4

The effective model of the three-node beam element: three mass points connected by rigid massless bars and a torsional spring

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Fig. 5

Connection of the three-node beam elements

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Fig. 6

Effective model of several connected elements

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Fig. 7

The integration ranges of the three-node Bernoulli elements. The overlapping sections are divided at their midpoints.

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Fig. 8

Validation example 1: deflection of a double cantilever beam subjected to a constant distributed load

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Fig. 9

Validation example 2: free vibrations of a double cantilever beam

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Fig. 10

Model of a single cylindrical beam (see Sec. 5.1)

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Fig. 11

Suspension of the rotor by a rubber rope for the measurement of the natural frequencies. The frequency response functions between different points of the rotor (denoted bynumbers) were measured by accelerometers and a modal hammer.

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Fig. 13

Reduction of the cost function during the iteration starting from k0 = k*

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Fig. 14

Convergence of the stiffness values during the iteration starting from k0 = k*

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Fig. 15

Comparison of the first mode shape of the fitted models and the measured mode shape

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Fig. 16

Comparison of the second mode shape of the fitted models and the measured mode shape




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