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Research Papers: Mechanisms and Robotics

Dynamical Limit of Compliant Lever Mechanisms

[+] Author and Article Information
Michele Bonaldi

 Istituto di Fotonica e Nanotecnologie CNR-FBK and INFN Trento, I-38050 Povo (Trento), Italybonaldi@science.unitn.it

Mario Saraceni

 Enginsoft s.p.a., Via Giambellino 7, 35129 Padova, Italy

Enrico Serra

 FBK-irst Microsystem Division, 38050 Povo (Trento), Italy

MODEFRONTIER by ES.TEC.O. (www.esteco.com).

ANSYS by ANSYS Inc. (www.ansys.com).

J. Mech. Des 130(4), 042302 (Feb 28, 2008) (8 pages) doi:10.1115/1.2838328 History: Received November 16, 2006; Revised September 18, 2007; Published February 28, 2008

The application of the mechanical energy conservation principle sets a dynamical limit to the performances of compliant lever mechanisms endowed with a positive definite strain energy. The limit applies to every linear compliant lever and is given as an upper bound on the product between the static effective gain of the device and its bandwidth. The relevant parameters of this relation are determined only by the structures surrounding the device and not by its design. This result is obtained on the basis of a linear two-port model, with coefficients determined by the static elastic constants of the device. The model and the dynamical limit are validated by multiobjective optimization analysis interfaced with a finite element model of a practical mechanism.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

(a) Schematic of an ideal planar compliant lever made by two rigid links L of length LL, connected at joints by four identical flexure hinges J of length Lj and thickness tj. The rigid joint is positioned at a rest angle θ0. The motion of the input and output ports, respectively, A and B, is restrained along the directions shown and no rotation of the output payload Mb is allowed. The guides needed to constrain the motion are not shown. The value of Mb is determined by the application of the device or by the sensor used to detect the displacement. (b) Pivot-based device with the same dimensions.

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Figure 2

Conventions of force and displacement in a two port

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Figure 3

Continuous lines: displacement gain (gf) and quality (qa) of the bridge lever with flexure hinges; the curves are evaluated with LL∕Lj=10 and tj∕Lj=0.5. Dashed line: displacement gain gf′ of the pivots based lever. The quality of the pivot-based lever has the constant value qa′=1.

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Figure 4

Relative forward gain loss (gf′−gf)∕gf′ of the compliant device with respect to the pivot-based device. The loss is plotted as a function of the relative quality loss (qa′−qa)∕qa′. The quality of the pivot-based lever has the constant value qa′=1 but is explicitly written for clarity.

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Figure 5

General measurement scheme: An external unknown force Fe is applied to a linear elastic test body P. We assume that the elastic properties of P are known, and then we obtain Fe by the measurement of the displacement xr produced at Point R. This displacement can be directly measured or used to drive a compliant amplifier. In the first case, the measurement result is the displacement xr0=Fe∕Ker, evaluated when the amplifier is not connected, and in the latter the measurement result is the displacement xb at the output port of the compliant device.

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Figure 6

Continuous line: reduced effective gain Ge∕G0 as a function of the frequency; the gain is evaluated with Ka∕Kr=0.1, qa=0.9 and Kb∕Mb∕(2π)=5000Hz. The peak is at the cutoff frequency given by Eq. 22. Dashed line: reduced effective gain of a critically damped harmonic oscillator, with quality factor Q=0.5; this curve is not obtained on the basis of our model and is shown for comparison only.

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Figure 7

Solid line: maximum reduced gain bandwidth Bmax as a function of the device quality. Dashed line: stiffness ratio Rmax needed to reach this best gain-bandwidth value. When the compliant amplifier is applied to a test body of output stiffness Kr, the maximum gain-bandwidth product is obtained as BmaxKr∕Mb, while the input stiffness Ka required to reach the maximum is evaluated as Ka=KrRmax.

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Figure 8

FE model of the general measurement scheme discussed in Sec. 3. The coupled links restrain the motion of the input and output ports, respectively, xa and xb, along the directions shown and do not allow any rotation of the output payload Mb. The FE model fully accounts for the elastic properties and the mass of the links. The model is solved in plane strain assuming a unit depth. The mechanical impedance of the test body and the payload are set, respectively, as Kr=4×109N∕m and Mb=0.28kg. The compliant device material is Al 5056 with a quality factor of 105(25).

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Figure 9

Details of the FE model. The full model is obtained by mirroring across the symmetry line A′−A″. All joints have the same dimensions, as well as the links. The design space of the compliant device is determined by the allowed values of the geometrical parameters: The angle θ0 may assume any value in the range 1⩽θ0⩽12deg, while the lengths of flexure hinges Lj and links LL can be modified under the constraint of total constant length 2Lj+LL+4Rj=13mm, with Rj=0.2mm. The flexure hinge and link thicknesses may assume any value in the ranges 0.2⩽tj⩽2mm and 1⩽tL⩽3mm, respectively.

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Figure 10

Gain-bandwidth relation for the system shown in Fig. 8. Continuous line: dynamical limit given by Eq. 31. Data points: FEM results, where the accumulation of solutions close to the Pareto front can be clearly seen.

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Figure 11

Comparison of FE results and model predictions. For each configuration in the optimal set, the FE software evaluates the static gain and the stiffness constants (Ka, Kb, Kab). These stiffness constants are then used to calculate the model prediction for the gain through Eq. 21. A point lie on the dashed line when the FEM result and model prediction are the same: The dashed line separates the areas where FEM results are greater or smaller than model predictions.

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Figure 12

Comparison of FE results and model predictions. For each configuration in the optimal set, the FE software evaluates the bandwidth and the stiffness constants (Ka, Kb, Kab). These stiffness constants are then used to calculate the model prediction for the bandwidth through Eq. 22. A point lies on the dashed line when the FEM result and model prediction are the same: The dashed line separates the areas where FEM results are greater or smaller than model predictions.

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