This relates to a stiffening nonlinear spring formed by a flexible cantilever member wrapping around a curved surface as it deflects.
Several technological processes such as energy harvesting from ambient vibrations, shock absorption from external loads, and passive control or suppression of mechanical instabilities involve targeted energy transfer from one component of a structure to another. In particular, energy harvesting is the process of using ambient energy sources to generate useful forms of energy such as electricity. The energy in these ambient sources is usually spread over a range of frequencies. Applications of energy harvesting may include MEMs sensors implanted in remote locations. Shock absorption is the process of protecting a primary structure from an ambient force or external pressure load. Applications include passive protection of buildings from earthquake excitations, offshore platforms from water waves impacts, or a delicate instrument from external loads. Passive control of mechanical instabilities in another important area that has recently emerged in the context of targeted energy transfer. Examples may include the suppression of aeroelastic instabilities on wings due to fluttering and the elimination of aeroelastic instabilities in suspension bridges.
In all of these cases, one aims to design elements that are capable of transferring the energy irreversibly and efficiently. In typical applications (especially energy harvesting), the ambient vibration can be described as a stochastic, multi-frequency signal that is often characterized by time-varying features. However, traditional single degree of freedom linear vibration harvesters are efficient only close to their design point; that is, when the excitation frequency matches the harvester's natural frequency. Therefore, linear harvesters respond inefficiently to ambient vibrations. In order to absorb ambient vibrations effectively, it is essential for an energy harvester to be characterized by adaptivity (i.e. the ability to adjust its resonance frequency/ies depending on the input spectrum) and robustness (i.e. the ability to maintain its energy harvesting performance even if the excitation varies significantly).
Methods for overcoming this mistiming problem include: designing systems that do not use a spring, control theory of linear spring systems, 2 degree-of-freedom linear systems, continuous linear systems, and nonlinear springs. Below, we give a critical overview of these techniques, focusing on their different advantages and disadvantages.
Mitcheson et al [1] describe a micro-scale coulomb-force parametric generator (CFPG) that absorbs ambient energy without using a spring. Instead of using a spring, the CFPG uses a charged capacitor plate that snaps away from a counter-electrode when excited by large accelerations. Since the CFPG does not have a spring, it does not have a resonant frequency and responds similarly to acceleration signals that have the same magnitude but different frequencies. The CFPG, however, only functions well when the excitation displacement greatly exceeds the allowable travel length of its sliding plate. Another shock absorption device that functions without a spring is the MEMS-fabricated hydraulic valve that fits inside a shoe, as described in [2]. A controller allows hydraulic fluid flowing in between two chambers to pulse on a piezoelectric element. Resulting strain in the piezoelectric element converts the mechanical energy into electric energy. Additionally, [3] discusses a device small enough to fit in a shoe that consists of a clamshell made from two piezoelectric elements. The device flattens with each heel-strike. [3] also reviews other energy harvesting devices that absorb ambient energy without vibrating.
The performance (i.e. peak power output, adaptivity, and robustness) of energy harvesters with linear springs can be improved by using control strategies to alter the oscillator's resonance frequency [4] or creating linear devices with two or more degrees of freedom so that they have multiple resonant frequencies [5]. [4] and [5] present devices with better performance than traditional single linear springs. However, the controlled devices consume some of the collected power, and the multiple degree of freedom systems are bulky and have limited robustness.
Another approach is to use a nonlinear spring. Essentially nonlinear springs—that is nonlinear springs without linear stillness components—do not have preferential linear frequencies. Therefore, they are more robust to variations in the external excitation and preserve their good performance level for a wide range of conditions [6], [7]. The simplest form of an essentially nonlinear spring is a cubic one. It may be implemented by linear springs supporting a proof mass at nonperpendicular angles. For example, MacFarland et al. [8] investigate the dynamics of a nonlinear oscillator realized by a thin elastic rod (piano wire) clamped at its ends without pretension that performs transverse vibrations at its center. To leading order approximation, the stretching wire produces a cubic stiffness nonlinearity. Despite its success in various applications, this design can suffer from significant frictional losses, especially in small scale applications, due to the guided motion of the moving mass [7]. In addition, there are limitations related to the spring breaking or yielding when the external forces become too large.
A different class of nonlinear springs includes those with negative linear stiffnesses, which are usually characterized by bi-stable configurations. Cottone et al. [9] describe a nonlinear spring implemented by an inverted pendulum with a tip magnet that faces an opposing static magnet. For a small enough gap between the magnets, the cantilever has two equilibria. For small base input accelerations, the tip magnet oscillates linearly about one of the equilibria. For Sufficiently large accelerations, the tip magnet cycles between the two equilibria. This resonance is insensitive to noise.
As described in [10] nonlinear springs may be physically implemented by helical springs with thickening coil wires or changing overall diameters. Another way to achieve nonlinear behavior is by employing multiple linear components that interact more strongly the further they deflect. For example, in the leaf springs of automobile suspensions, several layers of arc-shaped spring steel are clamped together. As the center of the upper are deflects, it contacts the arc below it, and both springs further deflect in contact. As more and more arcs deflect, the spring effectively stiffens. However, the many arcs of the leaf spring result in a lot of friction [10].
Mann and Sims [11] describe an oscillator that is implemented by a magnet sliding in a tube with two opposing magnets as the end caps. This configuration causes the stiffness to be the summation of a linear and cubic component. A disadvantage of this device is that the sliding magnet loses energy due to friction as it slides along the tube.
In [12], an ultra-wide bandwidth resonator is made out of a doubly-clamped piezoelectric beam. The double-clamps cause the cantilever to stretch as it bends, resulting in a nonlinear stiffness. However, the beam also has a linear stiffness. The linear stiffness is negligible compared to the nonlinear stiffness when the beam's residual stiffness is minimized. Consequently, efforts to minimize the linear stiffness component hinder the system optimization.
A nonlinear spring is also useful for the application of measuring forces as a load cell. Load cells are useful for applications ranging from material strength testing to prosthetic limb sensing (Sanders et al.) [13], monitoring infusion pumps delivering drugs (Mokhbery) [14], agricultural product sorting (Chang and Lin) [15], suction cup strength measuring (Messina) [16], and human-robot collision force sensing (Cordero et al) [17].
Load cells can measure forces via several different methods, including hydraulic or pneumatic pistons and deforming materials. For hydraulic or pneumatic load cells, the force is applied to a piston that covers an elastic diaphragm filled with oil or air respectively, and a sensor converts a pressure measurement to a force measurement. Use of hydraulic load cells is limited by high cost and complexity. Pneumatic load cells are limited by slow response times and a requirement for clean, dry air, Smith [18]. The most common load ceils are solid materials that deform when subject to an applied force.
Deforming load cells come in many different shapes, such as bending beams (a cantilever), S-beams (an “S”-shaped configuration of beams), single point load cells (a double-clamped beam, for which the force measurement is insensitive to the position of the load along the beam), shear beam load cells (an I-beam produces a uniform shear 15 across its cross-section that can be measured by strain gauges), and “pancake” load cells (round, at beams) [18]. All of these load cells deflect linearly.
Traditional linear load cells can be designed for almost any force capacity. Bending beam load cells are typically used for force ranges of 5.0×10−1 to 2.5×104 N and pancake load cells can be used for force ranges up to 2.5×106 N, Smith [18]. Many linear load cells are designed to withstand a limited amount of force overcapacity using oversteps that prevent over-deflection; typically up to 50-500% load capacity before breaking [19]. Because they deform linearly, these load cells also have constant resolution (that is, the smallest force increment that they can measure) for their entire force range.
There are several challenges to designing a load cell. One wants to reduce the load cell mass and volume in order to minimize its effect on the test sample. Additionally, the load cell should have minimal hysteresis for accurate measurements in both up-scale and down-scale, and low side-load sensitivity (response to parasitic loads) [18]. One of the most critical design challenges is the trade-off between force sensitivity and range: It is desirable to maximize strain or deflection in the load cell in order to increase force measurement resolution because strain and deflection sensors have limited resolution; typically 14-bits between 0 and their maximum rated measurement [20, 21, 22]. Simultaneously, one wants to maximize the load cell's functional force range and protect it from breaking due to forces that exceed that range, which requires limiting its strain.
Different studies have made various modifications to the traditional linear load cell to increase its force range and sensitivity, and minimize side-load sensitivity. Chang and Lin [15] studied a “capital G-shaped” load cell with two force ranges: for small forces, a top sensitive flexure deflected alone. For large forces, the sensitive flexure contacted a stiffer flexure, and the two flexures deflected together at the higher stiffness. In this way, the load cell was more sensitive to small forces, and did not yield for large forces. Other devices use multiple linear load cells of increasing stiffnesses in series, as described in several U.S. patents, (Storace and Sette [23], Suzuki et al. [24]). The multiple load cells of a device deflect together until overload stops prevent the weaker load cells from deflecting too far, after which the suffer load cells continue to deflect. A microcontroller determines which load cell measurement to display. Using this approach, [23] was able to measure weight over a range of 1 g to 30 Kg. One way to minimize sensitivity to side-loads such as undesired moments is to use multiple load cells (i.e. 3) and take the average force measurement [23]. Challenges with these designs are that the linear load cell components have limited resolution, and using multiple load cells in one device may be bulky or expensive.
Another approach for designing a load cell with high force resolution and capacity is to use a nonlinear mechanism rather than a linear one. A nonlinear load cell may have a low stiffness for low forces (and therefore high force sensitivity) and a high stiffness at large forces (and therefore protection from yielding due to over-deflection). The design may also be volume compact and inexpensive due to requiring only one nonlinear spring and sensor per device.
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