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Research Papers

A Pseudorigid-Body 3R Model for Determining Large Deflection of Cantilever Beams Subject to Tip Loads

[+] Author and Article Information
Hai-Jun Su

Department of Mechanical Engineering, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250haijun@umbc.edu

J. Mechanisms Robotics 1(2), 021008 (Jan 07, 2009) (9 pages) doi:10.1115/1.3046148 History: Received April 08, 2008; Revised November 01, 2008; Published January 07, 2009

In this paper, a pseudorigid-body (PRB) 3R model, which consists of four rigid links joined by three revolute joints and three torsion springs, is proposed for approximating the deflection of a cantilever beam subject to a general tip load. The large deflection beam equations are solved through numerical integration. A comprehensive atlas of the tip deflection for various load modes is obtained. A three-dimensional search routine has been developed to find the optimal set of characteristic radius factors and spring stiffness of the PRB 3R model. Detailed error analysis has been done by comparing with the precomputed tip deflection atlas. Our results show that the approximation error is much less than that of the conventional PBR 1R model. To demonstrate the use of the PRB 3R model, a compliant four-bar linkage is studied and verified by a numerical example. The result shows a maximum tip deflection error of 1.2% compared with the finite element analysis model. The benefits of the PRB 3R model include that (a) the model parameters are independent of external loads, (b) the approximation error is relatively small for even large deflection beams, and (c) the derived kinematic and static constraint equations are simpler to solve compared with the finite element model.

Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Large deflection of a cantilever beam subject to a combined end force and moment

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Figure 2

Trajectory of the deflected tip point arranged in terms of load ratio κ∊[0,50]. The maximum slope angle at the tip is determined by θ0 max=min[π,ϕ+cos−1(1−κ)].

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Figure 3

Limit of the tip slope θ0 max under pure moment loads determined by maximum stress

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Figure 4

The pseudorigid-body 1R model with parameter γ∊[0.735,0.85] depending on the load

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Figure 5

Actual and approximated characteristic radius factor γ versus load ratio κ for beams under large deflections. The actual values of γ are obtained by averaging over ϕ∊[0,π].

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Figure 6

A pseudorigid-body 3R model for a cantilever beam subject to a combined end force and moment

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Figure 7

Linear regression of spring stiffness kΘ1, kΘ2, and kΘ3 for end moment loads. The range of tip slope angle is θ0∊[0,3π/2].

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Figure 8

Comparison of the tip error of PRB 1R model with the PRB 3R model for beams under moment loads

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Figure 9

Comparison of the tip loci of the PRB 1R model and the PRB 3R model for beams under moment loads

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Figure 10

Comparison of tip locus of the PRB 1R model and the PRB 3R model for beams under force loads

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Figure 11

(a) Plot of stiffness kΘi versus the force directional angle ϕ∊[0,π] for κ=0 and (b) plot of stiffness kΘi versus the load ratio κ∊[0,25] for ϕ=π/2

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Figure 12

Approximation error of the PRB 3R model for various load indices κ with ϕ=π/2. The dots represent the deflection loci computed via numerical integration and the solid lines represent the approximated tip loci predicted the optimal PRB 3R model.

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Figure 13

Approximation error of the PRB 3R model for various force direction angles ϕ with fixed load ratio κ=0. The dots represent the actual deflection loci computed via numerical integration, and the solid lines represent the approximated tip loci predicted by the optimal PRB 3R model.

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Figure 14

A compliant four-bar linkage formed by a fixed-fixed flexible beam OQ and two rigid links BA and AQ connected by two pin joints A and B

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Figure 15

The internal force and moment at the connection point Q

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Figure 16

Plots of the deflection of a partially compliant four-bar mechanism computed by the optimized PRB 3R model and by the FEA model with the input crank ψ∊[0,2π]

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Figure 17

Tip error of the optimized 3R model compared with the FEA model for ψ∊[0,2π]

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Figure 18

Comparison of large deflection beams with a flection point (a) θ0=π/4, (b) θ0=π/6, (c) θ0=0, and (d) θ0=−π/8

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