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Research Papers

Nonlinear Model Based Estimation of Rigid-Body Motion Via an Indirect Measurement of an Elastic Appendage

[+] Author and Article Information
M. Senesh, A. Wolf

Department of Mechanical Engineering, Technion–Israel Institute of Technology, Haifa 32000, Israel

O. Gottlieb1

Department of Mechanical Engineering, Technion–Israel Institute of Technology, Haifa 32000, Israeloded@technion.ac.il

1

Corresponding author.

J. Vib. Acoust 132(1), 011007 (Jan 11, 2010) (12 pages) doi:10.1115/1.4000465 History: Received July 14, 2008; Revised August 12, 2009; Published January 11, 2010

In this paper, we develop and implement a nonlinear model based procedure for the estimation of rigid-body motion via an indirect measurement of an elastic appendage. We demonstrate the procedure by motion analysis of a compound planar pendulum from indirect optoelectronic measurements of markers attached to an elastic appendage that is constrained to slide along the rigid-body axis. We implement a Lagrangian approach to derive a theoretical nonlinear model that consistently incorporates several generalized forces acting on the system. Identification of the governing linear and nonlinear system parameters is obtained by analysis of frequency and damping backbone curves from controlled experiments of the decoupled system elements. The accuracy of the proposed model based procedures is evaluated and its results are compared with those of a previously reported point cluster estimation procedure. Two cases are investigated to yield 1.7% and 3.4% errors between measured motion and its model based estimation for experimental configurations, with a slider mass to pendulum frequency ratios of 12.8 and 2.5, respectively. Motion analysis of system dynamics with the point cluster method reveals a noisy signal with a maximal error of 3.9%. Thus, the proposed model based estimation procedure enables accurate evaluation of linear and nonlinear system parameters that are not directly measured.

Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 2

Pendulum decay with fixed slider mass in the first setup: (a) angular free vibration, (b) instantaneous frequencies, (c) frequency backbone, and (d) damping backbone

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Figure 3

Pendulum decay with fixed slider mass in the second setup: (a) angular free vibration, (b) instantaneous frequencies, (c) frequency backbone, and (d) damping backbone

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Figure 4

Slider mass decay on a horizontal rigid body in the first setup: (a) free vibration, (b) instantaneous frequencies, (c) frequency backbone, and (d) damping backbone

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Figure 5

Slider mass decay on a vertical rigid body in the second setup: (a) free vibration, (b) instantaneous frequencies, (c) frequency backbone, and (d) damping backbone

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Figure 6

Coupled motion pendulum decay in the first setup: (a) angle (rad) and (b) frequency response backbone curve

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Figure 21

Accuracy of the coupled system in the second setup: (a) estimated maximal angular versus real maximal angular amplitudes (b) relative error

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Figure 1

Experiment setup: (a) elastically restrained sliding mass, (b) system setup, (c) pendulum hinge, and (d) slider mass and markers for the point cluster method

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Figure 7

Coupled motion pendulum decay in the second setup: (a) angle (rad) and (b) frequency response backbone curve

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Figure 8

Coupled motion slider mass decay in the first setup: (a) free vibration decay and (b) frequency response backbone curve

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Figure 9

Coupled motion slider mass decay in the second setup: (a) free vibration decay and (b) frequency response backbone curve

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Figure 10

Pendulum angle calculated using the PCT method

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Accuracy: (a) Estimated maximal angular versus directly measured maximal angular amplitudes and (b) relative error

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Figure 12

Maximal instantaneous frequency: (a) measured angle and estimated angle based on PCT method, (b) directly measured angle and model based estimated angle. Circle (◼) indicates the estimated angle and plus (+) indicates the directly measured angle.

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Figure 13

Sketch of the dynamical system

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Figure 14

Coupled system model in the first setup—(1) μ=0.0477,η=0.0205: (a) slider mass decay (mm) and (b) pendulum angle (rad)

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Figure 15

Coupled system model in the first setup—(2) μ=0.0477,η=0.0256: (a) slider mass decay (mm) and (b) pendulum angle (rad)

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Figure 16

Nonresonant equivalent model in the first setup: pendulum angle (rad)

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Figure 17

Coupled system model in the second setup: (a) pendulum angle (rad) and (b) slider mass decay

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Figure 18

Accuracy of the coupled system model in the first setup—(1): (a) estimated maximal angular versus real maximal angular amplitudes and (b) relative error

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Figure 19

Accuracy of the coupled system model in the first setup—(2): (a) estimated maximal angular versus real maximal angular amplitudes and (b) relative error

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Figure 20

Accuracy of the nonresonant model in the first setup: (a) estimated maximal angular versus real maximal angular amplitudes and (b) relative error

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