Nonlinear Structural Stiffness Research Articles

Analytical study on hybrid thrust foil bearing, structural characterization and measurement of static performance

This work presents a novel hybrid foil thrust bearing (HFTB) with an outer diameter of 154 mm, along with simulation and test results. The HFTB incorporates a high-pressure air/gas injection to the taper portion of the thrust foil bearing, boosting the hydrodynamic effect on the main bearing surface and hydrostatic load capacity at zero speed. The bearing has high load capacity, low power loss, and no friction/wear during startup and shutdown. Firstly, push-pull tests were conducted on a double-acting HFTB configuration to evaluate the nonlinear structural stiffness of the bump foil structure. The measured nonlinear stiffness model was adopted to predict the overall bearing performance under various speeds and loads. The simulation uses an advanced model which predicts the 2D plate top foil deflection with physical bump locations mapped from the actual hardware. The simulation confirms that the hybrid operation significantly increases the load capacity compared to hydrodynamic bearing due to the boost effect of hydrostatic pressure in the main film. Tests were conducted at 10krpm, 15krpm, and 20krpm to measure load capacity, power loss, and average film thickness. At 20krpm, a maximum axial load of 1711 N was achieved before bearing failure. Ultimately, the novel HFTB shows excellent static performance with potential for use in Organic Rankine Cycle (ORC) generators and other large oil-free turbomachines.

Flutter prediction of its occurrence, amplitude and nonlinear behaviour

Flutter prediction of its occurrence, limit cycle amplitude and chaotic behaviour is investigated using a general binary (two degree of freedom) coupled modal model to provide new closed form analytical predictions and insight into the aeroelastic phenomenon. The generalised unsteady flutter model, based on flutter derivatives (for any cross-section) and an approximate continuous bilinear lift curve, is used in a mass decoupled form to formulate and develop analytical criteria for the onset and amplitude of flutter under damping and offset angle of attack. This is based on modal coupling through stiffness and damping terms, to clearly identify and clarify two forms of flutter based on stiffness and viscous mode coupling. Closed form analytical criteria for flutter onset are developed including a new necessary dynamic divergence criterion (analogous to spragging in brake squeal) based on flutter derivatives for any structure. The model is first validated with the well-known classical numerical and experimental airfoil results of Theodorsen and analytical results of Pines, which are also shown to correspond to stiffness mode coupling. In addition, the new modelling enables the detection and corresponding conditions of viscous mode coupling flutter which does not require closely spaced modes. The occurrence and conditions of both stiffness and viscous mode coupling flutter is verified numerically with results highlighting important structural parameters i.e. damping factors which could cause flutter outside classical conditions. In particular, the conditions for viscous mode coupling are newly identified and found to occur at flow velocities substantially lower (up to approximately 10%) than classical stiffness mode coupling values. Dynamic divergence is shown to be the lower bound for the critical flutter speed under a range of conditons. The viscous mode coupling is shown to occur under high levels of torsional damping (20 times the nominal 1% of critical), for the airfoil case, compared to vertical damping, but does not require the modal frequencies to be close. The efficient analytical prediction for flutter amplitude is then numerically verified for Theodorsen's airfoil. The results show that flutter amplitude is primarily dependent upon flow velocity and stall conditions without structural nonlinearity. Finally, a further nonlinear investigation of flutter identifies useful analytical criteria for the occurrence of chaotic flutter which is confirmed numerically. These results, based on a bilinear lift coefficient model, highlight that the occurrence of chaotic flutter can be due to simplified aerodynamics alone i.e. without nonlinear structural stiffness. The closed form analyses provide critical insight into; the occurrence and underlying mechanisms of flutter based on stiffness and viscous mode coupling, dynamic divergence as well as the nonlinear amplitudes and behaviour under an offset angle of attack and stall conditions.

Mechanical characteristics of a new type of cable-supported photovoltaic module system

Cable-supported photovoltaic (PV) modules have been proposed to replace traditional beam-supported PV modules. The new system uses suspension cables to bear the loads of the PV modules and therefore has the characteristics of a long span, light weight, strong load capacity, and adaptability to complex terrains. It provides an excellent supplement to traditional structures in special sites such as sewage treatment plants, highways, farms, fishponds, and roofs with poor load-bearing conditions. However, most of the traditional cable-supported PV systems use only two cables to support the PV modules. The settlement of the support cables due to self-weight of PV modules always reduces their power generation efficiency. Therefore, it is necessary to make a reasonable design to flatten the structures. Recently, the authors (He et al., 2020) proposed a new cable-supported PV system using three cables and four triangle brackets to form an inverted arch to reduce the vertical displacement of the PV modules. In this study, the structural characteristics of the new PV system with a span of 30 m are numerically investigated in terms of mode shapes, modal frequency, and nonlinear structural stiffness through using the finite element method. The increase of torsion stiffness when the torsion displacement rises benefits the stability of the new PV system. The load bearing capacity of the PV system is discussed under self-weight, static wind load, snow load, and their combination. The influences of row spacing, tilt angle, initial cable force, and cable diameter on the structural characteristics are further studied. The results verify that the new system has a strong load capacity and potential for wide application.

Chatter stability prediction of milling considering nonlinearities

Nonlinearities have been evidenced during the chatter vibration of milling. Machinability of the thin-walled part is feed rate and position-dependent, and is subject to process damping at low cutting speed. Therefore, chatter stability prediction of milling considering nonlinear cutting force, nonlinear structural stiffness and process damping is investigated. The cutting force and stiffness are established based on the polynomial model and the process damping is investigated based on the dissipated energy. The dynamic cutting force and stability lobes are solved in the time domain with coefficients updated at each iteration. By formulating the displacement as an expanded form via the perturbation method, the time-consuming solution of delay differential equations is avoided. After formulating the identification of the nonlinear model via cutting tests and modal tests, numerical simulations considering nonlinearities are carried out and compared with the analytical method. The proposed method attains high accuracy of classic time-domain solution, but with an improved computational efficiency. Finally, cutting tests are conducted to verify the prediction of cutting force and stability lobes.

Structural Dynamic Improvement for Petal-Type Deployable Solid-Surface Reflector Based on Numerical Parameter Study

Petal-type Deployable Solid-surface Reflector (PDSR) is a kind of important structure widely applied in deployable reflector antennas in aerospace engineering. The dynamic properties of this reflector structure in deployed state are significant to the reflector accuracy for antennas. However, the study of the dynamic evaluation of deployable structure with revolute joints is difficult and seldom concerned by researchers. In order to study dynamic properties of the PDSR, the Cable Replacement Method (CRM) was utilized to equivalently simulate the nonlinear structural stiffness of the revolute joint for numerical analyses. The Finite Element Model (FEM) of this reflector structure was established by commercial software ANSYS (ANSYS Inc., Canonsburg, PA, USA) and verified by the theoretical analysis and dynamic test of actual prototype model. The natural frequencies and mode shapes of deployed reflector were computed to study the influence of drag spring design parameters as stiffness, pre-tensioned force, and distance of two adjacent linkage butts. Finally, the analysis results were concluded that the drag springs between two adjacent petals can essentially improve the dynamic performance of reflector structure in deployed state. It can be a useful technical system for future engineering applications of PDSR antennas.

Adaptive RBF observer-sliding mode controller design for a two dimensional aeroelastic system with unsteady aerodynamics

An adaptive RBF observer-sliding mode controller is designed for the vibration suppression of a two-dimensional aeroelastic system, using a single trailing-edge control surface. The prototypical aeroelastic model describes the plunge and pitch motion of a wing, including cubic nonlinear structural stiffness and unsteady aerodynamics. The unsteady aerodynamics are modeled with an approximation to Theodorsen's theory. It's assumed that only the pitch angle is measured and the remaining state variables needed for full state feedback are estimated by the designed observer. A neural-network is employed to approximate the nonlinear dynamics of the observer system, and then a sliding surface is put forward on the estimation space. The finite-time reachability of the predesigned sliding surface is guaranteed by the designed sliding mode control law. The effectiveness of the proposed strategy is finally demonstrated by simulation results.

Nonlinear spectral design analysis of a structure for hybrid self-centring device enabled structures

Seismic dissipation devices can play a crucial role in mitigating earthquake damages, loss of life and post-event repair and downtime costs. This research investigates the use of ring springs with high-force-to-volume (HF2V) dissipaters to create damage-free, recentring connections and structures. HF2V devices are passive rate-dependent extrusion-based devices with high energy absorption characteristics. Ring springs are passive energy dissipation devices with high self-centring capability to reduce the residual displacements. Dynamic behaviour of a system with nonlinear structural stiffness and supplemental hybrid damping via HF2V devices and ring spring dampers is used to investigate the design space and potential. HF2V devices are modelled with design forces equal to 5% and 10% of seismic weight and ring springs are modelled with loading stiffness values of 20% and 40% of initial structural stiffness and respective unloading stiffness of 7% and 14% of structural stiffness (equivalent to 35% of their loading stiffness). Using a suite of 20 design level earthquake ground motions, nonlinear response spectra for 8 different configurations are generated. Results show up to 50% reduction in peak displacements and greater than 80% reduction in residual displacements of augmented structure compared to the baseline structure. These gains come at a cost of a significant rise in the base shear values up to 200% mainly as a result of the force contributed by the supplemental devices.

Dynamic performance of foil bearings with a quadratic stiffness model

Hydrodynamic foil bearings found various applications in high speed small turbomachinery due to several advantages compared with traditional oil lubricated bearing and rolling element bearings such as high speed, pollution free, high temperature, long service life, etc. Corrugated bump foils made of thin sheet metal is the most widely adopted foil structure used in many commercial foil bearings. However, due to various factors during manufacturing process of the foil structure and mounting methods of the foil structure on to the bearing sleeve, measured stiffness curves of the foil structure show very strong non-linearity. The non-linearity of the foil structure is often neglected in most simulation studies in the past, and single value of bump stiffness combined with appropriate structural loss factor were used in these simulation studies. Based on measured non-linear structural stiffness of entire foil bearing, a quadratic stiffness model of individual bump foil was derived and utilized in transient rotordynamic simulations in time domain by solving Reynolds equation, rotor equations of motion, and bump deflections together. The simulation results using the quadratic bump stiffness model were compared with those using linear bump stiffness models.

Structural and Rotordynamic Force Coefficients of a Shimmed Bump Foil Bearing: An Assessment of a Simple Engineering Practice

High speed rotors supported on bump-type foil bearings (BFBs) often suffer from large subsynchronous whirl motions. Mechanically preloading BFBs through shimming is a common, low cost practice that shows improvements in rotordynamic stability. However, there is an absence of empirical information related to the force coefficients (structural and rotordynamic) of shimmed BFBs. This paper details a concerted study toward assessing the effect of shimming on a first generation BFB (L = 38.1 mm and D = 36.5 mm). Three metal shims, 120 deg apart, are glued to the inner surface of the bearing cartridge and facing the underside of the bump foil strip. The shim sets are of identical thickness, either 30 μm or 50 μm. In static load tests, a bearing with shims shows a (nonlinear) structural stiffness larger than for the bearing without shims. Torque measurements during shaft acceleration also demonstrate a shimmed BFB has a larger friction coefficient. For a static load of 14.3 kPa, dynamic loads with a frequency sweep from 250 Hz to 450 Hz are exerted on the BFB, without and with shims, to estimate its rotordynamic force coefficients while operating at ∼50 krpm (833 Hz). Similar measurements are conducted without shaft rotation. Results are presented for the original BFB (without shims) and the two shimmed BFB configurations. The direct stiffnesses of the BFB, shimmed or not, increase with excitation frequency, thus evidencing a mild hardening effect. The BFB stiffness and damping coefficients decrease slightly for operation with rotor speed as opposed to the coefficients when the shaft is stationary. For frequencies above 300 Hz, the direct damping coefficients of the BFB with 50 μm thick shims are ∼30% larger than the coefficients of the original bearing. The bearing structural loss factor, a measure of its ability to dissipate mechanical energy, is derived from the direct stiffness and damping coefficients. The BFB with 50 μm thick shims has a 25% larger loss factor—average from test data collected at 300 Hz to 400 Hz—than the original BFB. Further measurements of rotor motions while the shaft accelerates to ∼50 krpm demonstrate the shimmed BFB (thickest shim set) effectively removes subsynchronous whirl motions amplitudes that were conspicuous when operating with the original bearing.

Linearization and tunability improvement of MEMS capacitors using flexible electrodes and nonlinear structural stiffness

This paper proposes solutions for high nonlinearity and structural instability in electrostatically actuated MEMS capacitors. The proposed designs use the flexibility of the moving electrode and nonlinear structural stiffness to control the characteristic capacitance–voltage (C–V) response. The moving plate displacements are selectively constrained by mechanical stoppers to prevent sudden jumps in the capacitance and to eliminate the pull-in. A symmetric double-humped electrode shape is utilized which results in a fairly constant sensitivity in the C–V curve and therefore a linearized response. An analytical and a finite-element coupled-field model are developed to study the behavior of the proposed capacitors and to optimize their design for maximum linearity. The experimental results verify that the designs introduced in this paper improve the linearity of the C–V response and increase the maximum tunability by three times compared to conventional MEMS parallel-plate capacitors. At the same time, they also eliminate the pull-in hysteresis of the response.

Active control of supersonic/hypersonic aeroelastic flutter for a two-dimensional airfoil with flap

The flutter, post-flutter and active control of a two-dimensional airfoil with control surface operating in supersonic/hypersonic flight speed regions are investigated in this paper. A three-degree-of-freedom dynamic model is established, in which both the cubic nonlinear structural stiffness and the nonlinear aerodynamic load are accounted for. The third order Piston Theory is employed to derive the aerodynamic loads in the supersonic/hypersonic airflow. Nonlinear flutter happens with a phenomenon of limit cycle oscillations (LCOs) when the flight speed is less than or greater than linear critical speed. The LQR approach is employed to design a control law to increase both the linear and nonlinear critical speeds of aerodynamic flutter, and then a combined control law is proposed in order to reduce the amplitude of LCOs by adding a cubic nonlinear feedback control. The dynamic responses of the controlled system are given and used to compare with those of the uncontrolled system. Results of simulation show that the active flutter control method proposed here is effective.

Development of parallel-plate-based MEMS tunable capacitors with linearized capacitance–voltage response and extended tuning range

This paper presents a design technique that can be used to linearize the capacitance–voltage (C–V) response and extend the tuning range of parallel-plate-based MEMS tunable capacitors beyond that of conventional designs. The proposed technique exploits the curvature of the capacitor's moving electrode which could be induced by either manipulating the stress gradients in the plate's material or using bi-layer structures. The change in curvature generates a nonlinear structural stiffness as the moving electrode undergoes out-of-plane deformation due to the actuation voltage. If the moving plate curvature is tailored such that the capacitance increment is proportional to the voltage increment, then a linear C–V response is obtained. The larger structural resistive force at higher bias voltage also delays the pull-in and increases the maximum tunability of the capacitor. Moreover, for capacitors containing an insulation layer between the two electrodes, the proposed technique completely eliminates the pull-in effect. The experimental data obtained from different capacitors fabricated using PolyMUMPs demonstrate the advantages of this design approach where highly linear C–V responses and tunabilities as high as 1050% were recorded. The design methodology introduced in this paper could be easily extended to for example, capacitive pressure and temperature sensors or infrared detectors to enhance their response characteristics.

Identification of limit cycles for piecewise nonlinear aeroelastic systems

This paper describes two methods for the analysis of aeroelastic systems with complex piecewise nonlinear structural stiffness. These methods are tested and compared for low speed incompressible and transonic flows. The first technique employed in this paper uses a new application of the analytical solution of linear algebraic systems, the second technique utilises logarithmic and tanh functions to both represent discrete nonlinearities and to act as a switch between different nonlinear areas. The transonic aerodynamic models used are generated using an eigenvalue realisation algorithm (ERA) which produces reduced order models (ROMs) from the pulse responses of time linearised Euler simulations. It is shown that such aerodynamic models are well suited to use with continuation methods. Flutter boundaries and limit cycle oscillations can then be rapidly identified with good accuracy.

Dynamic behaviour of 3D-panel single-storey system using shaking table testing

In the present investigation, a full-scale dynamic test of a single storey building constructed on a shaking table by a prefabricated 3D sandwich panel method is carried out under several ground motions. Objectives of the study are to obtain seismic performances of the described system under dynamic loading, such as linear and nonlinear structural characteristics, deformability, stiffness degradation and failure mechanism. In analysis, the primary frequencies, vibration modes and comparison of the results obtained from the test and numerical analysis are presented. In addition, the lateral deformation is measured experimentally, calculated numerically and compared. Accelerometers were mounted to measure accelerations in both orthogonal horizontal directions. The results revealed considerable resistance of 3D-wall system to the high levels of earthquake vibrations. The main components of its resistance to powerful earthquakes appear to be (1) the high over strength (2) the minor energy dissipation by inelastic deformations (3) the small drift of the crest under applied base ground motion, all approximately within the allowable limit. Analysis of the results of the cyclic pushover loading indicated that the model structure has the overall displacement ductility ratio of 4.50 and the over strength coefficient of approximately 6.0.

Global Robust Control of an Aeroelastic System Using Output Feedback

A EROELASTIC systems exhibit a variety of phenomena including instability, limit cycle, and even chaotic vibration. Researchers in aerodynamics, structures, materials, and control have made interesting contributions to the analysis and control of aeroelastic systems [1]. Readers may refer to an excellent article by Mukhopadhyay [2] for the analysis and control of aeroelastic systems. A benchmark active control technology (BACT) windtunnel model has been designed at the NASA Langley Research Center and control algorithms for flutter suppression have been developed [3–5]. At Texas A&Ma 2-degrees-of-freedom aeroelastic model has been developed and tests have been performed in a wind tunnel to examine the effect of nonlinear structural stiffness, and control systems have been designed using linear control theory, feedback linearizing technique, and adaptive control strategies [6– 10]. Based on the Euler–Lagrange theory, control of an aeroelastic model has been considered [11]. The state variable and output feedback designs of [8–10] are based on adaptive control techniques. The synthesis of adaptive controllers is not simple because a large number of parameters must be updated in the dynamic feedback loop. It is also well known that unmodeled dynamics of the system can cause parameter divergence and instability in the closed-loop system. Therefore, it is interesting to design nonadaptive control systems for uncertain aeroelastic models which can be synthesized relatively easily. The contribution of this paper lies in the design of a robust control system for the global regulation of an aeroelastic system which describes the plunge and pitch motion of a wing. The model has polynomial type structural nonlinearity and only the pitch angle is measured for feedback. It is assumed that all the system parameters are unknown to the designer, but the bounds on uncertainties are known. For the purpose of design, afirst-order dynamic compensator is introduced and a “chained” structure of the aeroelastic model including the dynamic compensator is obtained by an appropriate coordinate transformation. Then using the Lyapunov stability theory, a control law for robust output regulation of the transformed system including the compensator is derived. In the closed-loop system, the controller accomplishes global robust stabilization of the aeroelastic system and system trajectories converge to the origin. Simulation results for various flow velocities and elastic axis locations are obtainedwhich show that the control system is effective in flutter suppression in spite of the large parameter uncertainties. An attractive feature of this control system lies in its simplicity from the point of view of implementation.

AEROELASTIC OSCILLATIONS OF A SEESAW-TYPE OSCILLATOR UNDER STRONG WIND CONDITIONS

This paper is concerned with one-degree-of-freedom aeroelastic oscillations of a seesaw- type structure in a steady wind flow. Here it is assumed that strong wind conditions induce nonlinear aeroelastic stiffness forces that are of the same order of magnitude as the structural stiffness forces. As a model equation for the aeroelastic behaviour of the seesaw-type structure, a strongly nonlinear self-excited oscillator is obtained. The bifurcation and the stability of limit cycles for this equation are studied using a special perturbation method. Both the case with linear structural stiffness and the case with nonlinear structural stiffness are studied. For both cases is assumed a general cubic approximation to describe the aerodynamic coefficient. Conditions for the existence, the stability, and the bifurcation of limit cycles are given.