A novel, experimental method is presented for measuring the coefficient of restitution during impact events. These measurements are used to indirectly validate a new model of elastic-plastic contact. The experimental setup consists of a stainless steel sphere that is attached at the bottom of a 2.2 m long pendulum. The test materials are of the form of 1 inch diameter pucks that the sphere strikes over a range of velocities. Digital image correlation is used to measure the displacement and velocity of the ball. From this data the coefficient of restitution is calculated as a function of velocity. This report details the experimental setup, experimental process, the results acquired, as well as the future work.
Measurements are presented from a two-beam structure with several bolted interfaces in order to characterize the nonlinear damping introduced by the joints. The measurements (all at force levels below macroslip) reveal that each underlying mode of the structure is well approximated by a single degree-of-freedom (SDOF) system with a nonlinear mechanical joint. At low enough force levels, the measurements show dissipation that scales as the second power of the applied force, agreeing with theory for a linear viscously damped system. This is attributed to linear viscous behavior of the material and/or damping provided by the support structure. At larger force levels, the damping is observed to behave nonlinearly, suggesting that damping from the mechanical joints is dominant. A model is presented that captures these effects, consisting of a spring and viscous damping element in parallel with a four-parameter Iwan model. The parameters of this model are identified for each mode of the structure and comparisons suggest that the model captures the stiffness and damping accurately over a range of forcing levels.
This memo documents the results and methodology of the high-frequency modal test performed on a solid metal cylinder, provided by Vibrant Corporation, in September 2014 at Sandia National Laboratories. The purpose of this test was to measure mode shapes of the unit (torsion, axial, and bending) as high in frequency as achievable with a Polytec PSV-400 scanning laser Doppler vibrometer.
A cantilever beam is released from an initial condition. The velocity at the tip is recorded using a laser Doppler vibrometer. The ring-down time history is analyzed using Hilbert transform, which gives the natural frequency and damping. An important issue with the Hilbert transform is vulnerability to noise. The proposed method uses curve fitting to replace some time-differentiation and suppress noise. Linear curve fitting gives very good results for linear beams with low damping. For nonlinear beams with higher damping, polynomial curve fitting captures the time variations. The method was used for estimating quality factors of a few shim metals and PZT bimorphs.
To study the rebound of a sphere colliding against a flat wall, a test setup was developed where the sphere is suspended with strings as a pendulum, elevated, and gravity-released to impact the wall. The motion of the sphere was recorded with a highspeed camera and traced with an image-processing program. From the speed of the sphere before and after each collision, the coefficient of restitution was computed, and shown to be a function of impact speed as predicted analytically.
To study the rebound of a sphere colliding against a flat wall, a test setup was developed where the sphere is suspended with strings as a pendulum, elevated, and gravity-released to impact the wall. The motion of the sphere was recorded with a highspeed camera and traced with an image-processing program. From the speed of the sphere before and after each collision, the coefficient of restitution was computed, and shown to be a function of impact speed as predicted analytically.
Quantitative studies of material properties and interfaces using the atomic force microscope (AFM) have important applications in engineering, biotechnology and chemistry. Emerging studies require an estimate of the stiffness of the probe so that the forces exerted on a sample can be determined from the measured displacements. Numerous methods for determining the spring constant of AFM cantilevers have been proposed, yet none accounts for the effect of the mass of the probe tip on the calibration procedure. This work demonstrates that the probe tip does have a significant effect on the dynamic response of an AFM cantilever by experimentally measuring the first few modes of a commercial AFM probe and comparing them with those of a theoretical model for a cantilever probe that does not have a tip. The mass and inertia of an AFM probe tip are estimated from scanning electron microscope images and a simple model for the probe is derived and tuned to match the first few modes of the actual probe. Analysis suggests that both the method of Sader and the thermal tune method of Hutter and Bechhoefer give erroneous predictions of the area density or the effective mass of the probe. However, both methods do accurately predict the static stiffness of the AFM probe due to the fact that the mass terms cancel so long as the mode shape of the AFM probe does not deviate from the theoretical model. The calibration errors that would be induced due to differences between mode shapes measured in this study and the theoretical ones are estimated.
Damping in a micro-cantilever beam was measured for a very broad range of air pressures from atmosphere (10 5 Pa) down to 0.2 Pa. The beam was in open space free from squeeze films. The damping ratio, due mainly to air drag, varied by a factor of 10 4 within this pressure range. The damping due to air drag was separated from other sources of energy dissipation so that air damping could be measured at 10 -6 of critical damping factor. The linearity of the damping was confirmed over a wide range of beam vibration levels. Lastly, the measured damping was compared with several existing theories for air-drag damping for both rarified and viscous flow gas theories. The measured data indicate that, in the rarefied regime the air damping is proportional to pressure, independent of viscosity, and in the viscous regime the damping is determined by viscosity.
One powerful method for measuring the motion of microelectromechanical systems (MEMS) relies on a Laser Doppler Vibrometer (LDV) focused through an optical microscope. Recent data taken under a very simple and common condition demonstrate that the velocity signal produced by the LDV with an optical microscope may be different from the velocity signal produced by the LDV without a microscope. This is especially important if one wishes to estimate acceleration by differentiating velocity. In this study, the time derivatives of LDV signals are compared against the signal from an accelerometer when the LDV is focused through an optical microscope and without the microscope system. The signal from the LDV without the microscope is almost identical to the accelerometer signal. In contrast, the signal from the LDV with the microscope exhibits a nonlinear relationship with the accelerometer signal. Both the LDV and the accelerometer were measuring a sinusoidal velocity generated by an electromechanical shaker. The Fourier transform of the acceleration from the LDV with the microscope shows a multitude of high harmonics of the excitation frequency, which have much higher amplitudes than the harmonics present in the accelerometer signal. Without the microscope, the LDV gives a much less distorted sinusoidal signal, even after time differentiation. The distortion of the signal from the LDV is periodic, with the same period as the sinusoidal drive signal. The largest distortion occurs near points of maximum negative acceleration, corresponding to the positive displacement peak of the sinusoidal oscillation. Because the measured oscillation is out of plane, pseudo-vibrations caused by speckle noise do not explain the distortion. Instead, the distortion appears to be caused by the optics of the microscope.
The shape control of thin, flexible structures has been studied primarily for edge-supported thin-plates. For applications such as electromagnetic wave reflectors, corner-supported configurations may prove more applicable since they allow for greater flexibility and larger achievable deflections when compared to edge-supported geometries under similar actuation conditions. Models of such structures provide insight for effective, realizable designs, enable design optimization, and provide a means of active shape control. Models for small deformations of corner-supported, thin laminates actuated by integrated piezoelectric actuators have been developed. However, membrane deflections expected for nominal actuation exceed those stipulated by linear, small deflection theories. In addition, large deflection models have been developed for membranes; however these models are not formulated for shape control. This paper extends a previously-developed linear model for a corner-supported thin, rectangular laminate to a more general large deflection model for a clamped-corner laminate composed of moment actuators and an array of actuating electrodes. First, a nonlinear model determining the deflected shape of a laminate given a distribution of actuation voltages is derived. Second, a technique is employed to formulate the model as a map between input voltage and deflection alone, making it suitable for shape control. Finally, comparisons of simulated deflections with measured deflections of a fabricated active laminate are investigated.
The shape control of thin, flexible structures has been studied primarily for edge-supported thin-plates. For applications such as electromagnetic wave reflectors, corner-supported configurations may prove more applicable since they allow for greater flexibility and larger achievable deflections when compared to edge-supported geometries under similar actuation conditions. Models of such structures provide insight for effective, realizable designs, enable design optimization, and provide a means of active shape control. Models for small deformations of corner-supported, thin laminates actuated by integrated piezoelectric actuators have been developed. However, membrane deflections expected for nominal actuation exceed those stipulated by linear, small deflection theories. In addition, large deflection models have been developed for membranes; however these models are not formulated for shape control. This paper extends a previously-developed linear model for a corner-supported thin, rectangular laminate to a more general large deflection model for a clamped-corner laminate composed of moment actuators and an array of actuating electrodes. First, a nonlinear model determining the deflected shape of a laminate given a distribution of actuation voltages is derived. Second, a technique is employed to formulate the model as a map between input voltage and deflection alone, making it suitable for shape control. Finally, comparisons of simulated deflections with measured deflections of a fabricated active laminate are investigated.
Current published models for predicting squeeze film damping (SFD), which are based on different assumptions, give widely different results in the free-molecule regime. The work presented here provides experimental data for validating SFD models in that regime. The test device was an almost rectangular micro plate supported by beam springs. The structure was base-excited. The rigid plate oscillated vertically while staying parallel to the substrate. The velocities of the plate and of the substrate were measured with a laser Doppler vibrometer and a microscope. The damping ratio was calculated by performing modal analysis of the frequency response functions. The test structures were contained in a vacuum chamber with air pressures controlled to provide a five-order-of-magnitude range of Knudsen numbers. The damping coefficients from the measurements were compared with predictions from various published models. The results show that the continuum-base Reynolds equation predicts squeeze-film damping accurately if used with correct boundary conditions. The accuracy of molecular-based models depends heavily on the assumptions used in developing the models.
Forces generated by a static magnetic field interacting with eddy currents can provide a novel method of vibration damping. This paper discusses an experiment performed to validate modeling [3] for a case where a static magnetic field penetrates a thin sheet of conducting, non-magnetic material. When the thin sheet experiences motion, the penetrating magnetic field generates eddy currents within the sheet. These eddy currents then interact with the static field, creating magnetic forces that act on the sheet, providing damping to the sheet motion. In the presented experiment, the sheet was supported by cantilever springs attached to a frame, then excited with a vibratory shaker. The recorded motions of the sheet and the frame were used to characterize the effect of the eddy current damping.
When a micro cantilever beam is excited by base shaking, electrostatic force makes the tip displacement response nonlinear with respect to the base acceleration input. This paper derives a single-degree-of-freedom model for the deflection in a micro cantilever due to electrostatic voltage for this excitation. The tip deflection due to electrostatic force is derived first as part of the total tip deflection, and then in terms of an equivalent base excitation. The relationship between electrostatic deflection and equivalent base excitation is determined numerically, but can be represented accurately by a simple curve-fit function.
This report summarizes a survey of several new methods for obtaining mechanical and rheological properties of single biological cells, in particular: (1) The use of laser Doppler vibrometry (LDV) to measure the natural vibrations of certain cells. (2) The development of a novel micro-electro-mechanical system (MEMS) for obtaining high-resolution force-displacement curves. (3) The use of the atomic force microscope (AFM) for cell imaging. (4) The adaptation of a novel squeezing-flow technique to micro-scale measurement. The LDV technique was used to investigate the recent finding reported by others that the membranes of certain biological cells vibrate naturally, and that the vibration can be detected clearly with recent instrumentation. The LDV has been reported to detect motions of certain biological cells indirectly through the motion of a probe. In this project, trials on Saccharomyces cerevisiae tested and rejected the hypothesis that the LDV could measure vibrations of the cell membranes directly. The MEMS investigated in the second technique is a polysilicon surface-micromachined force sensor that is able to measure forces to a few pN in both air and water. The simple device consists of compliant springs with force constants as low as 0.3 milliN/m and Moire patterns for nanometer-scale optical displacement measurement. Fields from an electromagnet created forces on magnetic micro beads glued to the force sensors. These forces were measured and agreed well with finite element prediction. It was demonstrated that the force sensor was fully functional when immersed in aqueous buffer. These results show the force sensors can be useful for calibrating magnetic forces on magnetic beads and also for direct measurement of biophysical forces on-chip. The use of atomic force microscopy (AFM) for profiling the geometry of red blood cells was the third technique investigated here. An important finding was that the method commonly used for attaching the cells to a substrate actually modified the mechanical properties of the cell membrane. Thus, the use of the method for measuring the mechanical properties of the cell may not be completely appropriate without significant modifications. The latest of the studies discussed in this report is intended to overcome the drawback of the AFM as a means of measuring mechanical and rheological properties. The squeezing-flow AFM technique utilizes two parallel plates, one stationary and the other attached to an AFM probe. Instead of using static force-displacement curves, the technique takes advantage of frequency response functions from force to velocity. The technique appears to be quite promising for obtaining dynamic properties. More research is required to develop this technique.
Experimental modal analysis (EMA) was carried out on a micro-machined acceleration switch to characterize the motions of the device as fabricated and to compare this with analytical results for the nominal design. Finite element analysis (FEA) of the nominal design was used for this comparison. The acceleration switch was a single-crystal silicon disc supported by four fork-shaped springs. We shook the base of the die with step sine type excitation. A Laser Doppler Velocimeter (LDV) in conjunction with a microscope was used to measure the velocities of the die at several points. The desired first three modes of the structure were identified. The fundamental natural frequency that we measured in this experiment gives an estimate of the actuation g-level for the specified stroke. The fundamental resonance and actuation g-level results from the EMA and the FEA showed large variations. The discrepancy prompted thorough dimensional measurement of the acceleration switch, which revealed discrepancies between the nominal design and tested component.
The shape control of thin, flexible structures has been studied primarily for edge-supported thin plates. For applications involving reconfigurable apertures such as membrane optics and active RF surfaces, corner-supported configurations may prove more applicable. Corner-supported adaptive structures allow for parabolic geometries, greater flexibility, and larger achievable deflections when compared to edge-supported geometries under similar actuation conditions. Preliminary models have been developed for corner-supported thin plates actuated by isotropic piezoelectric actuators. However, typical piezoelectric materials are known to be orthotropic. This paper extends a previously-developed isotropic model for a corner-supported, thin, rectangular bimorph to a more general orthotropic model for a bimorph actuated by a two-dimensional array of segmented PVDF laminates. First, a model determining the deflected shape of an orthotropic laminate for a given distribution of voltages over the actuator array is derived. Second, symmetric actuation of a bimorph consisting of orthotropic material is simulated using orthogonally-oriented laminae. Finally, the results of the model are shown to agree well with layered-shell finite element simulations for simple and complex voltage distributions.
Mechanical dynamics can be a determining factor for the switching speed of radio-frequency microelectromechanical systems (RF MEMS) switches. This paper presents the simulation of the mechanical motion of a microswitch under actuation. The switch has a plate suspended by springs. When an electrostatic actuation is applied, the plate moves toward the substrate and closes the switch. Simulations are calculated via a high-fidelity finite element model that couples solid dynamics with electrostatic actuation. It incorporates non-linear coupled dynamics and accommodates fabrication variations. Experimental modal analysis gives results in the frequency domain that verifies the natural frequencies and mode shapes predicted by the model. An effective 1D model is created and used to calculate an actuation voltage waveform that minimizes switch velocity at closure. In the experiment, the switch is actuated with this actuation voltage, and the displacements of the switch at various points are measured using a laser Doppler velocimeter through a microscope. The experiments are repeated on several switches from different batches. The experimental results verify the model.
The primary goals of the present study are to: (1) determine how and why MEMS-scale friction differs from friction on the macro-scale, and (2) to begin to develop a capability to perform finite element simulations of MEMS materials and components that accurately predicts response in the presence of adhesion and friction. Regarding the first goal, a newly developed nanotractor actuator was used to measure friction between molecular monolayer-coated, polysilicon surfaces. Amontons law does indeed apply over a wide range of forces. However, at low loads, which are of relevance to MEMS, there is an important adhesive contribution to the normal load that cannot be neglected. More importantly, we found that at short sliding distances, the concept of a coefficient of friction is not relevant; rather, one must invoke the notion of 'pre-sliding tangential deflections' (PSTD). Results of a simple 2-D model suggests that PSTD is a cascade of small-scale slips with a roughly constant number of contacts equilibrating the applied normal load. Regarding the second goal, an Adhesion Model and a Junction Model have been implemented in PRESTO, Sandia's transient dynamics, finite element code to enable asperity-level simulations. The Junction Model includes a tangential shear traction that opposes the relative tangential motion of contacting surfaces. An atomic force microscope (AFM)-based method was used to measure nano-scale, single asperity friction forces as a function of normal force. This data is used to determine Junction Model parameters. An illustrative simulation demonstrates the use of the Junction Model in conjunction with a mesh generated directly from an atomic force microscope (AFM) image to directly predict frictional response of a sliding asperity. Also with regards to the second goal, grid-level, homogenized models were studied. One would like to perform a finite element analysis of a MEMS component assuming nominally flat surfaces and to include the effect of roughness in such an analysis by using a homogenized contact and friction models. AFM measurements were made to determine statistical information on polysilicon surfaces with different roughnesses, and this data was used as input to a homogenized, multi-asperity contact model (the classical Greenwood and Williamson model). Extensions of the Greenwood and Williamson model are also discussed: one incorporates the effect of adhesion while the other modifies the theory so that it applies to the case of relatively few contacting asperities.
This report documents the results obtained during a one-year Laboratory Directed Research and Development (LDRD) initiative aimed at investigating coupled structural acoustic interactions by means of algorithm development and experiment. Finite element acoustic formulations have been developed based on fluid velocity potential and fluid displacement. Domain decomposition and diagonal scaling preconditioners were investigated for parallel implementation. A formulation that includes fluid viscosity and that can simulate both pressure and shear waves in fluid was developed. An acoustic wave tube was built, tested, and shown to be an effective means of testing acoustic loading on simple test structures. The tube is capable of creating a semi-infinite acoustic field due to nonreflecting acoustic termination at one end. In addition, a micro-torsional disk was created and tested for the purposes of investigating acoustic shear wave damping in microstructures, and the slip boundary conditions that occur along the wet interface when the Knudsen number becomes sufficiently large.
The constitutive behavior of mechanical joints is largely responsible for the energy dissipation and vibration damping in weapons systems. For reasons arising from the dramatically different length scales associated with those dissipative mechanisms and the length scales characteristic of the overall structure, this physics cannot be captured adequately through direct simulation of the contact mechanics within a structural dynamics analysis. The only practical method for accommodating the nonlinear nature of joint mechanisms within structural dynamic analysis is through constitutive models employing degrees of freedom natural to the scale of structural dynamics. This document discusses a road-map for developing such constitutive models.