Abstract This paper deals with electrostatically actuated Double-Walled Carbon Nanotubes (DWCNT) and Single-Walled Carbon Nanotubes (SWCNT) cantilever resonators. Frequency response of parametric resonance is investigated. Euler-Bernoulli cantilever beam model is used for both DWCNT and SWCNT. Electrostatic and viscous damping forces are applied on both types of resonators, DWCNT and SWCNT. In this investigation, soft AC voltage excitation is assumed. For the DWCNT, an intertube van der Waals force is present between the two concentric carbon nanotubes (CNTs), coupling their motion and acting as a nonlinear spring. The nonlinearities in the vibration are provided by the electrostatic (both SWCNT and DWCNT) and intertube van der Waals forces (DWCNT). The Method of Multiple Scales (MMS) is a perturbation method that provides uniformly valid approximations for weakly nonlinear systems. A Reduced-Order-Model (ROM) is developed and numerically solved using AUTO-07P (bifurcation and continuation software). Since large tip deflections are investigated in this paper, only coaxial vibration of the DWCNT is considered. Parametric resonance is investigated, as well as the influences of damping and voltage. Lastly, the effect of intertube van der Waals force on the bifurcation and stability of the DWCNT is reported.

Abstract This work deals with amplitude frequency response of MEMS cantilever resonators undergoing superharmonic resonance of third order. The cantilever resonator is parallel to a ground plate and under alternating current (AC) voltage that excites the cantilever into vibrations. The driving frequency of the AC voltage is near one sixth of the first natural frequency of the cantilever beam resulting into superharmonic resonance of third order. The cantilever beam is modeled using Euler-Bernoulli beam theory. The electrostatic force is modeled using Palmer’s formula to include the fringe effect. In order to investigate the amplitude frequency behavior of the system reduced order models (ROMs) are developed. Three methods are used to solve these ROMs they are 1) the method of multiple scales (MMS) for ROM with one mode of vibration, 2) homotopy analysis method (HAM) for ROM with one mode of vibration, and 3) direct numerical integration for 2 modes of vibration Reduced Order Model (2T ROM) producing time responses of the tip of the cantilever resonator. In this work the limitations of MMS and HAM are highlighted when considering large voltage values i.e hard excitations. For large voltage values MMS and HAM cannot accurately predict the amplitude frequency response; the results from 2T ROM time responses disagree significantly with the MMS and HAM solutions. The effect of voltage on the frequency response is investigated. As the voltage values in the system increase the responses shift to lower frequencies and larger amplitudes.

Abstract Inspired by the agility and maneuverability of running kangaroos, a prototype robot was developed using a reduced order model to constrain the system. Both passive and active models were used to understand the relationship between system parameters and gait performance. A frequency response experiment was performed on the prototype to quantify the relationship between design parameters and system responses. Additionally, preliminary tail controllers were tested. Based on the results of the initial platform, a new robot was designed and built as a platform for the study of three dimensional hopping.

A data driven control design approach in the frequency domain is used to design track following feedback controllers for dual-stage hard disk drives using multiple data measurements. The advantage of the data driven approach over model based approach is that, in the former approach the controllers are directly designed from frequency responses of the plant, hence avoiding any model mismatch. The feedback controller is considered to have a Sensitivity Decoupling Structure. The data driven approach utilizes H ∞ and H 2 norms as the control objectives. The H ∞ norm is used to shape the closed loop transfer functions and ensure closed loop stability. The H 2 norm is used to constrain and/or minimize the variance of the relevant signals in time domain. The control objectives are posed as a locally convex optimization problem. Two design strategies for the dual-stage hard disk drive are presented.

Due to the unique structure of TISO feedback systems, several closed loop properties can be characterized using the concepts of plant and controller “directions” and “alignment”. Poor plant/controller alignment indicates significant limitations in terms of closed loop performance. In general, it is desirable to design a controller that is well aligned with the plant in order to minimize the size of the closed loop sensitivity functions and closed loop interactions. Although the concept of alignment can be a useful analysis tool for a given plant/controller pair, it is not obvious how a controller should be designed to achieve good alignment. We present a new controller design approach, based on the well-known “PQ method”, which explicitly incorporates knowledge of alignment into the design process. This is accomplished by providing graphical information about the alignment angle on the Bode plot of the PQ frequency response. We show the utility of this approach through a design example.

This paper investigates the frequency response of superharmonic resonance of the second order of electrostatically actuated nano-electro-mechanical system (NEMS) resonator sensor. The structure of the MEMS device is a resonator cantilever over a ground plate under Alternating Current (AC) voltage. Superharmonic resonance insinuates that the AC voltage is operating in a frequency near one-fourth the natural frequency of the resonator. The forces acting on the system are electrostatic, damping and Casimir force. For the electrostatic force, the AC voltage is in the category of hard excitation in order to induce a bifurcation phenomenon. For Casimir forces to affect the system, the gap distance between the cantilever resonator and base plate is in the range of 20 nm to 1 μm. The differential equation of motion is converted to dimensionless by choosing the gap as reference length for deflections, the length of the resonator for the axial coordinate, and reference time based on the characteristics of the structure. The Method of Multiple Scales (MMS) is used to model the characteristic of the system. MMS transforms the nonlinear partial differential equation of motion into two simpler problems, namely zero-order and first-order. The influences of parameters (i.e. Casimir, damping, second voltage and fringe) were also investigated.

The brain is a highly complex network and analyzing brain connectivity is a nontrivial task. Consequently, the neuroscience community created a large-scale, customizable, mathematical model which simulates brain activity called The Virtual Brain (TVB). Using TVB, we seek to control electroencephalography (EEG) measured brain states using auditory inputs, through TVB. A safe non-invasive brain stimulation method is binaural beats (BB) which arise from the brain’s interpretation of two pure tones, with a small frequency mismatch, delivered independently to each ear. A third phantom BB, whose frequency is equal to the difference of the two presented tones, is produced. This paper details the development and proof-of-concept testing of a simulation environment for an EEG-based closed-loop control of TVB using BB. Results suggest that the connectivity networks, constructed from simulated EEG, may change with certain BB stimulation frequency. In this work, we demonstrate that a linear controller can successfully modulate TVB connectivity.

As composite materials are becoming increasingly applied in actively controlled flexible structures, the need for practical uncertainty bounding to capture the effect of normal manufacturing variations on their dynamic behavior is also increasing. Currently, there is a lack of quantification of manufacturing variation of composite materials cast in a robust control framework. This work presents a simple experimental study on a particular case of composite member. The modal parameters of a set of 12 unidirectional carbon fiber reinforce polymer beams are identified. A nominal finite element model is numerically fit to the average experimental natural frequencies and antiresonances. The model is augmented with real parametric uncertainties placed on the modal parameters. The bound on the uncertainties is found both deterministically, to capture all experimentally observed data, and stochastically using a predetermined confidence interval. The two uncertainty bounding approaches are compared through the resulting bound on the beam model frequency response. Also, simulations are conducted to compare possible time responses using the two uncertainty bounds. It is found that the utilized structure of parametric uncertainties is effective at capturing the experimentally observed behavior.

This paper deals with two different methods to analyze the amplitude frequency response of an electrostatically actuated micro resonator. The methods used in this paper are the method of multiple scales, which is an analytical method with one mode of vibration. The other method is based on system of odes which is derived using the partial differential equation of motion, as well as the boundary conditions. This system is then solved using a built in matlab function known as BVP4C. Results are then shown comparing the two methods, under a variety of parameters, including the influence of damping, voltage, and fringe.

Automated methods for deriving dynamic models from frequency response data of high-order dynamics systems are the default choice of most engineers. However, these methods themselves often require manual tuning of weighting parameters, a priori selection of system order, and even by hand removal of extraneous dynamics. On the other hand, manually matching complicated features in the Bode plot of the frequency response of high-order system is difficult with conventional first and second order numerators and denominators. In this papers we present a manual technique for systematically creating a dynamic model from Bode plots of frequency response data with complicated features. We apply the method to identifying dynamics of a piezoelectric stage holding the sample of an atomic force microscope (AFM). We show the manual method works better than the tfest command of Matlab ™ for this example system.

This paper investigates the frequency response of microplates under electrostatic actuation. The microplate is parallel to a fixed ground plate. The electrostatic force that actuates the system is given by both Alternate Current (AC) and Direct Current (DC) voltages. The AC frequency is set to be near half natural frequency of the structure. Damping influence is also investigated in this paper. The method of investigation is Reduced Order Model. The effects of various parameters on the response of the structure are reported.

This paper presents a model to explain complex non-minimum phase (CNMP) zeros seen in the non-collocated frequency response of a large displacement XY flexure mechanism, which employs multiple double parallelogram flexure modules (DPFM) as building-blocks. Geometric non-linearities associated with large displacement along with the kinematic under-constraint in the DPFM, lead to a coupling between the X and Y direction displacements. Via a lumped-parameter model that captures the most relevant geometric non-linearity, it is shown that specific combinations of the operating point (i.e. flexure displacement) and mass asymmetry (due to manufacturing tolerances) give rise to CNMP zeros. This model demonstrates the merit of an intentionally asymmetric design over an intuitively symmetric design in avoiding CNMP zeros. Furthermore, a study of how the eigenvalues and eigenvectors of the flexure mechanism vary with the operating point and mass asymmetry indicates the presence of curve veering when the system transitions from minimum phase to CNMP. Based on this, the hypothesis of an inherent correlation between CNMP zeros and curve veering is proposed.

A 2D electrohydraulic proportional directional valve is proposed, which integrates both direct and pilot operation of the valve. In this valve, the output magnetic force of the proportional solenoid is converted to rotate the spool through a thrust-torsion coupling and thus the pressure in the valve sensitive chamber is varied. The varied pressure exerted on the areas of the spool end produces a hydrostatic force to move the spool linearly, which will rotate the spool reversely. Theoretical analysis is carried to the proposed valve and the effects of the key geometric parameters on the dynamic characteristics of the 2D valve and stability are investigated. Experiments are also designed to access to the characteristics of the valve working under direct and pilot operation. The 2D electrohydraulic valve can work properly for both direct operation and pilot operation. The hysteresis and frequency response are measured and the results are within the acceptable range in practical engineering application required of the directional proportional valve.

The dynamic characteristics of the clamping system, which is composed of workpiece and chuck, have important effects on the stability of turning process. However, in current researches on cutting stability, the influence of the characteristics of clamping contact surface on the clamping system was rarely considered. In order to improve the prediction accuracy of stable cutting limits in turning process, the role of clamping contact surface in deciding the dynamics of the clamping system was analyzed in this paper. Then, the dynamics model of clamping system was established considering the characteristics of clamping contact surface between workpiece and chuck. The dynamics parameters of the clamping system were obtained with receptance coupling substructure analysis method. The frequency response function of clamping system at different cutting position was presented, which is a foundation for analyzing the cutting stability.

Polymeric smart materials exhibit viscoelastic behavior and their dynamic characteristics are dependent on both frequency and temperature. This allows the tuning of material properties (stiffness and loss factor) to manipulate the vibration behavior for a wide range of engineering applications. In this research, the effects of viscoelastic supports on the vibration of continuous structures such as axially vibrating rods and transversely vibrating beams are investigated. The governing equations of motion for harmonically excited rods with end supports, and the free vibration of beams with intermediate viscoelastic support are developed. The analytical response equation for a harmonically excited rod with viscoelastic ends is obtained. The resulting frequency response equations are then used to design the modification of the stiffness and loss factor of the viscoelastic materials in order to achieve the desired vibration response of the rod. By solving the resulting transcendental eigenvalue problems, the natural frequencies and damping ratios as a function of viscoelastic support parameters are computed for beams. The performance of structures with viscoelastic support is demonstrated with various numerical examples. The formulation and results can be utilized for estimating the optimal material tuning parameters as well as support locations for controlling and manipulating the vibration response of the structures.

Mechanical characterization of thin samples is now routine due to the prominence of the Atomic Force Microscope. Advances in amplitude modulation techniques have allowed for accurate measurement of a sample’s elastic properties by interpreting the changes in the vibration of a cantilevered beam in intermittent contact. However, the nonlinearities associated with contact complicate attempts to find an accurate time-history for the beam. Furthermore, the inclusion of viscous effects, common to soft samples, puts an explicit solution even farther from reach. A numerical method is proposed that analyzes the time-history and frequency response of a microcantilever beam with a viscoelastic end-condition. The mathematics can be simplified by incorporating the viscoelastic end-condition into the equation of motion directly by modeling it as a distributed load. A forcing function can then be derived from the Standard Linear Solid model of viscoelasticity and implemented in the non-conservative work term of Hamilton’s principle. The Galerkin method can separate the resulting nonlinear equation of motion into time and space components. Performing a numerical analysis of the time factor equation provide the beam’s response over time. The results demonstrate the distinctive effects of viscoelasticity and periodic contact on the beam’s motion and provide the framework for the determination of viscous properties using dynamic techniques.

A novel electromagnetic pendulum energy harvester with motion regulator is proposed and investigated in this paper. The motion regulator is a mechanical design which can mechanically rectify the irregular bidirectional swing motion of the pendulum into unidirectional rotational motion of the motor, so that no electrical rectifier is needed and voltage drops from diodes can be avoided. Therefore, the energy efficiency can be significantly improved. The working principle of this pendulum mechanical motion regulator is that when the input velocity of the motion regulator is smaller than the output velocity of the motion regulator, the output transmission shaft will disengage from the motion regulator. This disengage mechanism of the pendulum energy harvester will lead to a broad bandwidth frequency response because of the inclination of maximizing the velocity output from the system. The dynamic modeling of the switched linear system (either engaged or disengaged) is established. The simulation results illustrate the power generation performance of the pendulum energy harvester, and the comparison with the traditional energy harvester without motion regulator shows the broadband advantage of the proposed energy harvester. Experiment verification has also been carried out. The experiment results verified that at some frequencies over the natural frequency, the efficiency is increased.

In structural health monitoring, it is desirable to select sensor locations in order to minimize the number of sensors required for and the cost associated with an on-board monitoring system. When using a frequency response-based structural health monitoring technique, data measured at sensor locations which exhibit the greatest change in frequency response function (FRF) due to damage are expected to maximize the effectiveness of the chosen technique. In this work, an embedded sensitivity function is presented which identifies the sensor locations at which the maximum differences in FRFs due to damage at a known location will be observed. The formulation of the embedded sensitivity function is based on FRFs measured on a healthy structure in the frequency range in which the damage detection technique will be applied. The effectiveness of the embedded sensitivity functions in predicting the most effective sensor locations is demonstrated by applying a vibro-acoustic modulation (VAM) damage detection method to a residential-scale wind turbine blade. First, data from the healthy blade is acquired and the embedded sensitivity functions are calculated. Then, the blade is damaged and the VAM method is applied using several sensor locations. The data acquired using sensor locations identified by the embedded sensitivity functions as being most effective are shown to most clearly identify the damage on the blade.

This paper proposes a new method to the design of a controller to be embedded at a prescribed location in an otherwise unknown complex and multiple-loop Linear Time-Invariant (LTI) control system. The questions that arise naturally are: Will a controller proposed for this location stabilizes the overall system and, if so, what stability margins and response can be obtained? We address these questions by constructing an equivalent single-loop frequency domain representation of the original unknown complex system. This equivalent plant construction can be accomplished by a small set of frequency response measurements.

This paper presents an alternative approach to design PID controllers based on frequency response measurements. The proposed method does not require any mathematical model of the system and can handle the design process directly from a small set of frequency domain data. It is shown that for the class of Linear Time-Invariant (LTI) control systems, there exists a rational multilinear function for the frequency response between any two arbitrary breaking points in terms of the design controller. This function can be determined by conducting a small set of frequency response measurements and then will be used to synthesize a controller that guarantee a set of desired frequency-domain specifications. In this paper, we use this result to design a PID controller for a servomechanism control system. In particular, we show that such desirable PID controller can be calculated by solving an optimization problem.