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Research Papers

Probing the Force Field to Identify Potential Energy

[+] Author and Article Information
Yawen Xu

Department of Mechanical Engineering and Materials Science,
Duke University,
Durham, NC 27708
e-mail: yawen.xu@duke.edu

Lawrence N. Virgin

Department of Mechanical Engineering and Materials Science,
Duke University,
Durham, NC 27708
e-mail: l.virgin@duke.edu

Contributed by the Applied Mechanics Division of ASME for publication in the Journal of Applied Mechanics. Manuscript received May 17, 2019; final manuscript received July 19, 2019; published online August 5, 2019. Assoc. Editor: George Haller.

J. Appl. Mech 86(10), (Aug 05, 2019) (10 pages) Paper No: JAM-19-1248; doi: 10.1115/1.4044305 History: Received May 17, 2019; Accepted July 20, 2019

A small ball resting on a curve in a gravitational field offers a simple and compelling example of potential energy. The force required to move the ball, or to maintain it in a given position on a slope, is the negative of the vector gradient of the potential field: the steeper the curve, the greater the force required to push the ball up the hill (or keep it from rolling down). We thus observe the turning points (horizontal tangency) of the potential energy shape as positions of equilibrium (in which case the “restoring force” drops to zero). In this paper, we appeal directly to this type of system using both one- and two-dimensional shapes: curves and surfaces. The shapes are produced to a desired mathematical form generally using additive manufacturing, and we use a combination of load cells to measure the forces acting on a small steel ball-bearing subject to gravity. The measured forces, as a function of location, are then subject to integration to recover the potential energy function. The utility of this approach, in addition to pedagogical clarity, concerns extension and applications to more complex systems in which the potential energy would not be typically known a priori, for example, in nonlinear structural mechanics in which the potential energy changes under the influence of a control parameter, but there is the possibility of force probing the configuration space. A brief example of applying this approach to a simple elastic structure is presented.

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Figures

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Fig. 1

Free body diagram of a small ball on an inclined surface. G is the gravitational force, N is the supporting force from the surface, and Fb is the external force required to maintain the position of the ball on the surface at this location. Friction is neglected.

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Fig. 2

Two views of an initial 3D-printed surface, with five locations at which the surface is locally flat. The flat areas around the perimeter are the results of “cropping” the surface to avoid very high vertical values.

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Fig. 3

(a) The cage of load cells that contains the steel ball and (b) the complete experiment setup

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Fig. 4

Experimental setup to measure the magnitude of friction

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Fig. 5

Relation between the location of the center of the ball bearing with the contact point

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Fig. 6

A photographic image of a 3D-printed version of curve 1

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Fig. 7

Results relating to curve 1. (a) Comparison of the direct force measurement from the load cells, differentiated from the laser scan, and theoretical shape dz(x)/dx and (b) comparison of the numerical integration of the force from the load cell, laser scan, and theoretical shape of z(x).

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Fig. 8

The photographic image of a 3D-printed version of curve 2

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Fig. 9

The test result of the second 1 DOF surface (curve 2). (a) Comparison of the direct force measurement from the load cell, differentiated from the laser scan, and theoretical shape dz(x)/dx and (b) comparison of the numerical integration of the force from the load cell, laser scan, and theoretical shape of z(x).

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Fig. 10

(a) A bifurcation diagram incorporating a pair of saddle-nodes, with the changing form of the potential energy (V) indicated at discrete values of the control and (b) a photographic image of curve 3—representing a one-dimensional system as a function of a control parameter

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Fig. 11

Bifurcation diagram of the system curve 3, superimposed on the measured contours of potential energy. Again, the potential energy is determined by numerically integrating the measured force from load-cell scans; the solid circles are stable equilibria, while the hollow circles are unstable equilibria.

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Fig. 12

The zero values of the partial derivatives with respect to x and y from both theory and experiment and superimposed to reveal the locations of equilibrium points from their crossing. The dashed red (light) and blue lines (dark) are from theory while the solid red and blue lines are from experimental data. The solid black circle indicates the stable equilibria, the black crosses are the saddle points, while the hollow circle is the unstable equilibrium.

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Fig. 13

(a) Photographic image of surface 4 and (b) contour plot based on the shape equation (Table 1). The unit of the contours is mm, with the zero value (height) located at the origin. The corners of the surface are cropped to avoid excessively large values.

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Fig. 14

(a) Contour plot of ∂z/∂x based on the equation. (b) The measured force in the x-direction from the experiment. The mild grid distortion is related to Eq. (1). (c) Their (vertical) difference. A different contour scheme is used for part (c), although the scale is the same as (a) and (b). Very similar results were obtained for ∂z/∂y. The contour scales are dimensionless.

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Fig. 15

(a) The contour plot of the surface (potential energy), computed by numerically integrating the measured values from the load-cells; the solid circles are stable equilibria, the hollow circles are unstable equilibria, and the crosses are saddle points—the superimposed locations of the equilibria are calculated based on the laser scan result. (b) The contour plot of “error,” i.e., the difference between the theoretical and experimental shapes, where error is defined as error = zexpztrue where ztrue is the actual shape of the surface and zexp is the experimental result. The unit for the contours is mm, with again an expanded scaling for part (b).

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Fig. 16

(a) A photographic image of surface 5 used in the experiment, (b) the contour plot of surface 5 based on the equation in Table 1, (c) a contour plot (produced within matlab) of experimental data after numerical integration. Data were acquired on a grid within the cropped region. In the contour plots, the unit is mm. The black solid dots are the stable equilibria, the black crosses are the saddle points, and the hollow black circle is the hill-top.

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Fig. 17

(a) An experimental setup designed to identify potential energy in a continuous elastic system, (b) fundamental and “snapped-through” equilibria, (c) a super-critical pitchfork bifurcation resulting in multiple equilibria, (d) the measured force–deflection relation, and (e) potential energy achieved through numerical integration

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