Research Papers

Nonlinear Mechanics of Interlocking Cantilevers

[+] Author and Article Information
Joseph J. Brown

Department of Mechanical Engineering,
University of Hawaii at Manoa,
2540 Dole St.—Holmes Hall 302,
Honolulu, HI 96822-2344
e-mail: jjbrown@hawaii.edu

Ryan C. Mettler

Department of Mechanical Engineering,
University of Colorado Boulder,
427 UCB,
Boulder, CO 80309-0427
e-mail: ryan.mettler@colorado.edu

Omkar D. Supekar

Department of Mechanical Engineering,
University of Colorado Boulder,
427 UCB,
Boulder, CO 80309-0427
e-mail: omkar.supekar@colorado.edu

Victor M. Bright

Fellow ASME
Department of Mechanical Engineering,
University of Colorado Boulder,
427 UCB,
Boulder, CO 80309-0427
e-mail: victor.bright@colorado.edu

1Corresponding author.

Manuscript received August 22, 2017; final manuscript received October 11, 2017; published online October 27, 2017. Assoc. Editor: Junlan Wang.

J. Appl. Mech 84(12), 121012 (Oct 27, 2017) (12 pages) Paper No: JAM-17-1458; doi: 10.1115/1.4038195 History: Received August 22, 2017; Revised October 11, 2017

The use of large-deflection springs, tabs, and other compliant systems to provide integral attachment, joining, and retention is well established and may be found throughout nature and the designed world. Such systems present a challenge for mechanical analysis due to the interaction of contact mechanics with large-deflection analysis. Interlocking structures experience a variable reaction force that depends on the cantilever angle at the contact point. This paper develops the mathematical analysis of interlocking cantilevers and provides verification with finite element analysis and physical measurements. Motivated by new opportunities for nanoscale compliant systems based on ultrathin films and two-dimensional (2D) materials, we created a nondimensional analysis of retention tab systems. This analysis uses iterative and elliptic integral solutions to the moment–curvature elastica of a suspended cantilever and can be scaled to large-deflection cantilevers of any size for which continuum mechanics applies. We find that when a compliant structure is bent backward during loading, overlap increases with load, until a force maximum is reached. In a force-limited scenario, surpassing this maximum would result in snap-through motion. By using angled cantilever restraint systems, the magnitude of insertion force relative to retention force can vary by 50× or more. The mathematical theory developed in this paper provides a basis for fast analysis and design of compliant retention systems, and expands the application of elliptic integrals for nonlinear problems.

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Grahic Jump Location
Fig. 1

Generalized problem and contact mechanics. (a) The cantilever moves past a rigid obstruction (the black circle in this diagram). Without stiction or friction forces, the obstruction applies a force F normal to the cantilever tip. Downward force P drives the motion and reaction force Q arises as a consequence of the constraint by the obstruction. As the interaction between the cantilever and obstruction progresses, it moves from scenario 1 to scenario 3. In scenario 1, the obstruction is above the cantilever and moving downward relative to the cantilever. In scenario 2, the obstruction is in contact with the cantilever, causing application of forces to the cantilever, until the cantilever slips past the obstruction. In scenario 3, after slipping past the obstruction, the cantilever is now above the obstruction and no load is applied to the cantilever as long as the obstruction continues to move downward. (b) The system in (a) may be extended to the case of two symmetric cantilevers moving past each other by rotating the original cantilever by 180 deg around the rigid obstruction. Only the portions of the cantilevers between the obstruction and the wall experience moments, forces, and bending. Any portion of the cantilever extending beyond the obstruction will be unloaded and straight. (c) If the radius of the cylindrical obstruction is allowed to approach 0, the cantilevers are in contact with each other at one point that does not vary in the x-direction, even as the cantilevers increase in deflection and move past each other.

Grahic Jump Location
Fig. 2

Coordinate systems and dimensions in analysis. (a) The contact mechanics problem. Assuming no friction or stiction, the cantilevers provide solely normal forces at one contact point. This is equivalent to providing a normal point force at the point of contact. (b) Typical model for solving cantilever bending due to applied end force. A cantilever of length L experiences downward force P and lateral force Q. Their resultant F is always perpendicular to the cantilever tip. (c) The cantilever bending problem with variable wall angle θ0. (d) Cantilever and force orientation for the general elliptic integral solution in coordinate system (x1,y1). The forces P1 and Q1 are projections of F into y1 and x1, respectively. (e) The elliptic integral solution in (x1,y1) space is transformed to (x2,y2) space to match the original cantilever bending problem. In (x2,y2), P2 and Q2 are the real loading and reaction forces, respectively, but they are also the projections of F into y2 and x2.

Grahic Jump Location
Fig. 3

Results of analytic and iterative modeling, and finite element simulation. (a) Cantilever shapes in iterative model. These plots show the cantilever shapes for different attachment angles θ0 as bending increases. (b) Finite element model results. Left: cantilever deflection for θ0 = 0 deg under the conditions identified in Sec. 5. Right: Downward tip displacement versus downward force during the simulation at three different wall angles. In this plot, the finite element model is plotted with the marker shapes and the solid lines are the predictions from the rescaled elliptic integral model. (c) Results of all investigations in nondimensional format. The elliptic (dotted line) and iterative (solid line) models are plotted along with rescaled data from the finite element model (markers), for three different values of θ0.

Grahic Jump Location
Fig. 4

Mechanical experiment. (a) Photo of experiment, at the start of loading for wall angle θ0=0deg. (b) Equivalent lumped element diagram of the experimental mechanical test system. In the experiment as pictured in (a) the motion and force are applied downward, but F and y in (b) are oriented upward in order to provide consistent notation with other derivations. (c)–(f) Further examples of interlocking and large deflection behavior in experiment. Arrows indicate direction of motion during experiment. (c) Cantilevers for θ0=0deg experiment just before the center cantilevers slip past the outer cantilevers. (d) Cantilevers just before slip for θ0=−30deg. (e) After slip in the θ0=−30deg experiment, (d), the angled cantilevers are locked together. Reversing motion allows data collection for the θ0=+30deg experiment. This subfigure shows the cantilevers just after they have come into contact in the θ0=+30deg experiment. (f) Cantilevers in the θ0=+30deg experiment just before they slip past each other. All experimental images in this figure were recorded as still frames from videos of experiments on acrylonitrile butadiene styrene (ABS) material. Larger versions of (c), (d), and (f) are provided in the Supplemental Figs. S3, S5, and S6 which are available under the “Supplemental Data” tab for this paper on the ASME Digital Collection, respectively. Video files are also provided in “Supplemental Data” tab for this paper on the ASME Digital Collection: (a) and (c) are still images from video SV3.mov, (d) is an image from SV1.mov, and (e) and (f) are images from SV4.mov. Part dimensions are defined in the Supplemental Fig. S2 which is available under the “Supplemental Data” tab for this paper on the ASME Digital Collection.

Grahic Jump Location
Fig. 5

Normalized experimental test results (solid or large dashed lines) in comparison with analytical models (dotted lines). The curves are normalized relative to the nominal 0 deg experimental and analytical curves.

Grahic Jump Location
Fig. 6

Analytically derived curves useful for design predictions, plotted for θ0 from −60 deg to +60 deg, with 5 deg steps in this range. (a) Dimensionless force versus dimensionless displacement. This set of curves presents the nonlinear cantilever stiffness. If an interlocking system is force-limited rather than displacement-limited, interlocking cantilevers will experience snap-through after reaching the peaks on this plot. (b) Dimensionless arc length versus dimensionless displacement for θ0 from −60 deg to +60 deg. These curves denote the arc length contained within the distance 0≤x≤L0 (between the wall and the tip constraint) during experiments. For negative θ0, the length increases until the total length of the cantilever is incorporated within the constraint space, at which point the cantilevers slip to create an interlocking configuration. For positive θ0, the arc length decreases until a minimum at which the cantilever begins to bend backward, after which the constrained arc length again increases until reaching the full cantilever length, at which point the interlocking slip occurs.

Grahic Jump Location
Fig. 7

Ratio of maximum retention force (for +angle) to maximum insertion force (for −angle) for various fixture angles and dimensionless cantilever overlaps (Δ0/Lc)



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