Quasi-static Deflection Research Articles

Quasistatic microscanner with linearized scanning for an adaptive three-dimensional laser camera

This paper reports on a gimbaled MEMS scanning mirror with quasistatic resonant actuation, specially developed for adaptive raster scanning in an innovative three-dimensional (3-D) time-of-flight (ToF) laser camera with real-time foveation. Large quasistatic deflections of ±10 deg are provided by vertical comb drives in the vertical direction in contrast to resonant horizontal scanning. This mirror is 2.6×3.6 mm and operates at 1600 Hz with an 80-deg optical scan range. For position feedback, piezo-resistive position sensors are integrated on chip for both axes. To guarantee the full reception aperture of effectively 5 mm, a synchronized driven MEMS scanner array—consisting of five hybrid assembled MEMS devices—is used in an innovative 3-D ToF laser scanner. This enables a distance measuring rate of 1 MVoxel/s with an uncertainty in distance measurement of 3 to 5 mm for a 7.5-m measuring range for a gray target. Flatness-based open loop control is used for driving control of quasistatic axis in order to compensate for the dynamics of the low damped MEMS system.

Energy absorption characteristics of glass/epoxy nano composite laminates by impact loading

Laminated composites with nano fillers are subjected to medium velocity impact. The laminates are prepared by hand lay-up technique followed by compression moulding, for different thicknesses. Laminates are made of glass woven roving mates of 610 gsm, epoxy resin and nano clay. Clay dispersion is varied from 1% to 5% by weight. A piston type gas gun set up is employed to impact the composite laminates. Spherical nose cylindrical projectile, of diameter 9.5 mm and mass 7.6 g, is used for the study. The impact tests are conducted to predict the ballistic limits of the laminates with and without clay. Quasi-static deflection tests and dynamic tests are carried out to find the energy absorbed by the laminates. Dispersion of clay in polymer matrix shows considerable improvement in energy absorption and ballistic limit of the composite laminates.

Deflection and frequency monitoring of the Forth Road Bridge, Scotland, by GPS

The use of carrier phase kinematic GPS (global positioning system) has evolved into a reliable technique to measure both the three-dimensional magnitudes and frequencies of movements of structures. Techniques have been developed that tackle errors caused by multipath, tropospheric delay and issues relating to satellite geometry. GPS-derived movements compare well with data from both design predictions and structural models. Results from field trials carried out on the Forth Road Bridge are presented. This paper brings together key results that outline the procedure as well as a series of new data that indicate other potential applications. GPS data were collected continuously over a period of 46 h at a minimum rate of 10 Hz. During the trials wind speed, wind direction, relative humidity and temperature were also recorded. Frequently there was very heavy traffic flow, and at one point a special load (a 100-t lorry) passed over simultaneously to the heavy daytime flow of traffic. Data from a planned load trial during a brief bridge closure are reported and compared with the limited results available from a finite element model. Measured vibration frequencies are also computed from GPS data and compared with those given in the literature. In addition, results indicating the change in structural characteristics are also presented – in particular changes of mass associated with changes in traffic loading are observed. The results show the performance of GPS as it has developed in recent years, and that it can now reliably be used as a significant part of structural health monitoring schemes, giving both the magnitude of quasi-static deflections in known time periods and hence the frequency of dynamic movements of structures.

Thermally Loaded Two-Layer Beams with Fuzzy Interlayer Slip

The presence of uncertainty in the mechanical properties of composite structures is considered as fuzzy randomness. When extending the work on two layer beam-, plate- and shell structures based on the definition of an equivalent effective homogeneous model, to include fuzzy interface slip, we can avoid numerical analyses schemes and work out the effects of thermal loads on these fuzzy structures. Fully analyzed within the scope of this article is a simply supported beam with fuzzy interface stiffness making use of the superposition of the fuzzy quasi-static deflection and the modally expanded and truncated complementary fuzzy dynamic response. The analysis of this illustrative example is based on the interval representation (i.e., on the set of a-cuts) with a triangular membership function of the slip modulus prescribed. Membership functions of the quasi-static deflection are worked out using fuzzy set theory, however, avoiding artificial uncertainties. Maximum of the complementary dynamic deflection is approximately evaluated by means of the SRSS superposition of the fuzzy modal response.

Optically interrogated MEMS pressure sensor array

A novel pressure measurement technique is presented for wireless recording of time-averaged surface pressure distributions in wind tunnel surveys. An array of silicon micro-plate resonators acts as pressure sensing element. The pressure is recorded by measuring the sensor diaphragms’ resonance frequency using optical interferometry. Dependent on the quasi-static deflection caused by a pressure load, the resonance frequency varies with an average pressure sensitivity of 3 Hz/Pa in a frequency range between 30 and 150 kHz. A smart-pixel CMOS camera, narrow-band acoustic noise excitation and a specific sensor surface structure allow for the interrogation of a large number of sensors in parallel without the need for alignment between sensor and detector. Experimental tests reveal increased sensing performance with acoustic excitation of the higher vibration modes.

Evaluation of length-scale effects for mechanical behaviour of micro- and nanocantilevers: I. Experimental determination of length-scale factors

When the thickness of micro- and nanocantilevers, which are the building blocks of micro- and nanoelectromechanical systems, are below 10 µm, the natural frequency and quasi-static deflection of the cantilevers cannot be estimated accurately using the conventional natural frequency and quasi-static-deflection models. The effect of the thickness is summed into what is known as the length-scale factor, which is incorporated into the models in order to improve the accuracy of the models. With this in mind, this paper, which is the first of two parts, reports on the experimental determination of length-scale factors for micro- and nano-sized silicon cantilevers, the micro- and nano-sized length-scale factors are estimated using experimental data collected from nanoindentation and microindentation experiments. The same experimental data are used to cross-validate the empirical length-scale factors in the literature. The experimentally estimated length-scale factors are then incorporated into the natural frequency and static-deflection models. The experimental results presented suggest that the length-scale factors have a great influence on the natural frequency and quasi-static deflection of micro- and nano-sized silicon cantilevers, compared with the natural frequency and deflection calculated using the conventional models not incorporating the length-scale factor.

Dynamic and quasi-static bending of saturated poroelastic Timoshenko cantilever beam

Based on the three-dimensional Gurtin-type variational principle of the incompressible saturated porous media, a one-dimensional mathematical model for dynamics of the saturated poroelastic Timoshenko cantilever beam is established with two assumptions, i.e., the deformation satisfies the classical single phase Timoshenko beam and the movement of the pore fluid is only in the axial direction of the saturated poroelastic beam. Under some special cases, this mathematical model can be degenerated into the Euler-Bernoulli model, the Rayleigh model, and the shear model of the saturated poroelastic beam, respectively. The dynamic and quasi-static behaviors of a saturated poroelastic Timoshenko cantilever beam with an impermeable fixed end and a permeable free end subjected to a step load at its free end are analyzed by the Laplace transform. The variations of the deflections at the beam free end against time are shown in figures. The influences of the interaction coefficient between the pore fluid and the solid skeleton as well as the slenderness ratio of the beam on the dynamic/quasi-static performances of the beam are examined. It is shown that the quasi-static deflections of the saturated poroelastic beam possess a creep behavior similar to that of viscoelastic beams. In dynamic responses, with the increase of the slenderness ratio, the vibration periods and amplitudes of the deflections at the free end increase, and the time needed for deflections approaching to their stationary values also increases. Moreover, with the increase of the interaction coefficient, the vibrations of the beam deflections decay more strongly, and, eventually, the deflections of the saturated poroelastic beam converge to the static deflections of the classic single phase Timoshenko beam.

Robust Control of a Two-Link Flexible Manipulator with Quasi-Static Deflection Compensation Using Neural Networks

A robust control method of a two-link flexible manipulator with neural networks based quasi-static distortion compensation is proposed and experimentally investigated. The dynamics equation of the flexible manipulator is divided into a slow subsystem and a fast subsystem based on the assumed mode method and singular perturbation theory. A decomposition based robust controller is proposed with respect to the slow subsystem, and H ? control is applied to the fast subsystem. The overall closed-loop control is determined by the composite algorithm that combines the two control laws. Furthermore, a neural network compensation scheme is also integrated into the control system to compensate for quasi-static deflection. The proposed control method has been implemented on a two-link flexible manipulator for precise end-tip tracking control. Experimental results are presented in this paper along with concluding remarks.

Modeling and Control of a Singly Curved Active Aperture Antenna Using Curved Piezoceramic Actuators

There is an increasing need for aperture antenna devices that incorporate multifunctional capability. One popular solution to this problem is the use of an “active aperture antenna” that is able to vary the direction and shape of its radiation pattern by using actuators to alter the shape of the reflector. This study focuses on the use of a pre-curved piezoceramic actuator, referred to as a THUNDER actuator, to change the shape of a prototype reflector. These actuators offer greater force, deflection and higher strength than traditional PZT actuators, while maintaining cost effectiveness. In this study the design of a prototype antenna structure that can accommodate deflection experiments and far-field radiation pattern tests is presented. The theoretical relationship between the dish deflection and the input voltage is established in two stages. In the first stage, the deflection at the end of the actuator is determined using a combination of Hamilton’s principle and laminated composite curved beam theory. The piezoelectric properties of the actuator are also considered during this stage. In the second stage, the deflection at the tip of the dish is calculated using geometric relationships, with the assumption that the remaining portion of the dish moves as a rigid body. The resulting deflection equations yield results that closely match the experimental quasi-static deflection results for a given input voltage, despite the presence of hysteresis. A dynamic model for a THUNDER actuator is developed based on experimental frequency response measurements. Positive Positioned Feedback (PPF) control is implemented on the system, in order to reduce position overshoot and oscillations in the transient response of the structure. The controller yields an improvement in the transient response in both simulation and experiments. Finally, the controlled system is utilized to transmit radiation in the far field.

Piezoelectric actuation of PZT thin-film diaphragms at static and resonant conditions

The piezoelectric response of silicon diaphragms covered with sputter-deposited PbZr 0.45Ti 0.55O 3 (PZT) films has been investigated in view of their application in ultrasonic micro-actuators. The behaviour of resonance frequencies and quasistatic deflections has been studied as a function of membrane thickness and d.c. bias. The total stress in the films and the piezoelectric constant, d 31, have been derived by means of two different methods. The results are consistent with direct strain measurements by optical interferometry and with bulk ceramic values of identical composition.

Characterization of PZT thin films for micromotors

Piezoelectric membranes consisting of sputter deposited PbZr xTi 1−xO 3 (PZT) films on silicon diaphragms have been investigated for their resonance and piezoelectrical properties, in view of their application as stator of a micromotor. The behavior of resonance frequencies was studied as a function of membrane thickness and dc-bias, in order to derive the total stress in the films and the piezoelectric coupling constant (d 31≈40 pm/V). The latter was also derived from the quasi-static deflections.

Electronic speckle pattern interferometry on a microscopic scale

An electronic speckle pattern interferometer that incorporates a commercial, long-working-distance microscope is described which provides new opportunities to perform nondestructive inspection for applications in fields such as microelectronics. The long-working-distance microscrope was attached directly to the CCD camera to form a compact, portable system with a field of view that was variable over several mm in width. Alignment was greatly simplified because the skewed Michelson interferometer configuration used a speckled reference beam imaged from a diffusely reflecting reference surface that was positioned adjacent to the test object. To demonstrate the performance of the micro-ESPI instrument, fringe patterns were recorded for the quasi-static cantilever deflection of a 1 mm-wide interconnect clip from an electronics socket.

The synthesis of flexible linkages by balancing the tracer point quasi-static deflections using microprocessor and advanced materials technologies

The contribution of the quasi-static deformation to the total deviation of the tracer point of a flexible path generator linkage from the desired path may be reduced to zero by replacing one of the normally fixed ground link pivots by a sliding joint whose motion is controlled by a microprocessor. If the resulting output characteristics are still unacceptable to the analyst, the remaining high frequency oscillatory component may be attenuated by redesigning the mechanism links in a modern composite material to nominally generate the output associated with a rigid linked system. Illustrative examples are presented for four bar linkages.

Dynamic analysis of elastic-plastic beams by means of thermoelastic solutions

The equation of bending motion of the inelastic beam is put into a linear form by interpreting the nonlinear part of the operator as a fictitious temperature-moment loading, thus consistently considering the inhomogeneous character of the dynamic boundary conditions. The total solution is separated into two parts: a quasistatic deflection due to external loadings and fictitious temperature moments and a dynamic part of the deflection due to inertial forces and homogeneous boundary conditions. The numerical procedure is of an iterative and limestep type and is equivalent to the Newlon-Raphson method in matrix formulation.

A note on quasi-static thermal deflection of a thin clamped circular plate due to ramp-type heating of a concentric circular region of the upper face

A note on quasi-static thermal deflection of a thin clamped circular plate due to ramp-type heating of a concentric circular region of the upper face

Circular viscoelastic plates subjected to in- plane loads

This paper is concerned with circular viscoelastic plates subject to in-plane forces. The plates are assumed to have a small arbitrary initial curvature and the increase in curvature as a function of time is determined. The plate material is assumed to follow a linear viscoelastic stress-strain relation and quasi-static small deflection theory is utilized. The general equation for the deflection of a circular viscoelastic plate is derived from the equation of an elastic plate using the well-known correspondence principle. Special cases are: 1) a circular plate with a fixed edge; 2) a circular plate with a simply supported edge; 3) a circular segment; and 4) an annular plate. Circular symmetry is assumed in cases 1, 2 and 4 but the solution developed in case 3 could readily be applied to the other cases if symmetry did not exist. Numerical examples are given for all cases. The applied load is assumed to be uniformly distributed around the boundary of the plate. For an annular plate the load at the inner boundary may differ from the load at the outer boundary. This leads to a variation of the in-plane forces throughout the plate. The load may vary with time in any arbitrarily prescribed manner. The load is assumed to be compressive but the analysis applies equally well for tensile loads.

Dynamic Thermoelastic Response of Cylindrical Shells

The dynamic, thermoelastic response of cylindrical shells to suddenly applied and rotating thermal inputs is investigated. Fully coupled, dynamic, thermoelastic cylindrical shell equations are derived using Galerkin’s method. Identical results were obtained independently using a variational theorem. Analytical solutions to these equations are formulated for finite-length and infinite-length cylinders. Numerical results for the response of infinite-length cylindrical shells to suddenly applied and rotating longitudinal lines of heat flux are presented. It is shown that for many thermoelastic problems involving moving thermal inputs that the maximum ratio of dynamic to quasi-static deflection can be much greater than two, that dynamic effects can be important for all thicknesses within the realm of thin shell theory, and that semicoupled theory gives incorrect results in some cases for which a fully coupled theory is required.

Quasi-static thermal deflection in a solid circular plate in the axisymmetric case

Quasi-static thermal deflection in a solid circular plate in the axisymmetric case

Second-order effects as related to critical coolant flow velocities and reactor parallel plate fuel assemblies

Second-order effects, removed by the linearization process in the analyses of Miller, Rosenberg and Youngdahl, and others, are retained in a stability analysis of the equation of motion of a flat fuel plate in a reactor parallel plate fuel assembly. The second-order terms generate an additional stability criterion is the form of an upper bound (critical deflection on the amplitude of quasi-static deflections for stable oscillations, or neutral stability, about the undeflected fuel plate position. The critical flow velocities derived in the earlier studies, and again as a consequence of this analysis, are upper bounds to insure local stability about the undeflected plate position. A constant of proportionality, relating the critical deflection to the initial channel depth, is designated as a design constant. When assigned a value, the design constant defines new critical coolant flow velocities which are lower than those first arrived at by Miller.

Behavior of Viscoelastic Plates in Bending

Solutions of fundamental equation by use of Laplace transformation; quasi-static deflection of laterally loaded plates is obtained in terms of associated elastic deflection of similar plate; dynamic response of simply supported rectangular plates under no load; inertia forces due to deflection are taken into account and free vibrations of Maxwell and Kelvin type plate are given; simplifications in case of incompressible plate material.