Nonlinear Modeling of Cracked Beams for Impedance Based Structural Health Monitoring

One of the main objectives of the structural health monitoring by piezoelectric wafer active sensor (PWAS) using electromechanical impedance method is continuously damage detection applications. In present work impedance method of beam structure is considered and the effect of early crack using breathing crack modeling is studied. In order to model the effect of a crack in beam, the beam is connected with a rotational spring in crack location. The Rayleigh–Ritz method is used to generate ordinary differential equation of cracked beam. Firstly, only open crack is considered that this is leads to linear system equation. In linear system, time domain system equations are converted to frequency domain, and then impedance of PWAS in frequency domain is calculated. Secondly, the breathing crack is modeled to be fully open or fully closed. This phenomenon leads to the nonlinear system equations. These nonlinear equations are solved using pseudo-arc length continuation scheme and collocation method for any harmonic voltage applied to actuator. Then impedance of PWAS is calculated. Two methods are used to detect early crack using breathing crack modeling on PWAS impedance. At the first, frequency response of breathing crack in the frequency range with its sub-harmonics is calculated. Second, only frequency response of one harmonic is computed with its super-harmonics. Finally, the detection method of linear is compared with nonlinear model.

Casimir Effect on Voltage Response of Superharmonic Resonance of NEMS Resonators

This paper investigates the voltage response of superharmonic resonance of the second order of electrostatically actuated nano-electro-mechanical system (NEMS) resonator sensor. The structure of the NEMS device is a resonator cantilever over a ground plate under Alternating Current (AC) voltage. Superharmonic resonance of second order occurs when the AC voltage is operating in a frequency near-quarter the natural frequency of the resonator. The forces acting on the system are electrostatic, damping and Casimir. To induce a bifurcation phenomenon in superharmonic resonance, the AC voltage is in the category of hard excitation. The gap distance between the cantilever resonator and base plate is in the range of 20 nm to 1 μm for Casimir forces to be present. 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) and Reduced Order Model (ROM) are 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. ROM, based on the Galerkin procedure, uses the undamped linear mode shapes of the undamped cantilever beam as the basis functions. The influences of parameters (i.e. Casimir, damping, fringe, and detuning parameter) were also investigated.

Vibration of Rectangular Single-Layered Black Phosphorus

Two-dimensional layered crystal material black phosphorus (BP) has attracted extensive attention due to its excellent property and practical applications. Single-layered BP has a characteristic puckered structure which leads to two anisotropic in-plane directions. The vibration properties of this puckered structure material would be very interesting. Thermal vibration of a rectangular single-layered BP is studied by using continuum orthotropic plate models together with molecular dynamics (MD) simulation. Five elastic constants including two bending moduli, two Poisson’s ratios, and one shear modulus of BP are calculated by using MD method. The natural frequencies of BP are obtained by orthotropic plate models and MD simulation via fast Fourier transformation (FFT). The result of MD simulation shows that continuum orthotropic plate models can predict the natural frequencies well.

Exploration of the Response of Electrically Actuated MEMS Arches Under the Effect of Mechanical Shock Loads

This work investigates the modeling and simulation of the dynamic response of MEMS shallow arches under the combined effects of mechanical shock waves and electrostatic actuating forces. The possible instabilities and/or failures that can be considered in any reliability study of such bi-stable structures are numerically examined. The results demonstrate that the simultaneous effects of shock loads and the actuating force can make the bi-stability and/or the instability thresholds of electrically actuated MEMS arches devices much lower than the predicted values when considering their effects independently. The outcomes of this investigation can be very useful to design smart MEMS bi-stable sensors/accelerometers activated at a pre-programmed level of shock and/or abrupt change in the acceleration.

Adaptive Backstepping Control of the Gripping Process of Heavy Manipulators

A valve-controlled asymmetrical cylinder model was established to study the gripping hydraulic drive system of the grip device of heavy manipulator. Due to the strong nonlinear characteristics and uncertain parameters of the model, the Lyapunov stability principle was used to design a multistage inversion adaptive controller based on backstepping method and by introducing the virtual control parameter. The simulation results reveal that the tracking control and adaptive of uncertain parameters are very effective, which confirm that the designed controller can guarantee the stability of the closed-loop clamping hydraulic drive system.

New Insights Into Slip-Stick Vibrations From Compact, Closed-Form Analysis

A mechanical system sliding on a moving surface with Coulomb friction is a rich area for study. Despite much past work, there is still something to be gleaned by closed-form expressions for the system behavior. Consider a spring-mass-damper system (K, M, C) with deflection x, base moving in the +x direction at velocity V, sliding friction F, and sticking friction Fs. An initial condition of x0 at rest can be considered general because all possible motions will follow. Two dimensionless schemes are used. For the abstract, we focus on the scheme normalized by x0 with variable z = x/x0, τ = (ωnt, ωn = [K/M]1/2, ζ = c/[2(KM)1/2], ν̄ = V / (ωnx0), f = F/(Kx0), and fs = Fs/(Kx0). Since the solution is piecewise linear, this allows closed-form results. For this abstract, we consider C = 0, Fs = F. (Other cases are in the paper.) There are three critical ground speeds. The first, ν̄d, is when sticking first occurs (at z = f). At the second speed, ν̄c, sticking has moved to z = −f. Thereafter, the sticking point again increases, reaching z = f at the third speed, ν̄b. For higher ν̄, there is no sticking. In this paper, closed form expressions are presented for the three critical speeds:(1)ν¯d=[(1+3f)(1−5f)]12,ν¯c=[(1+f)(1−3f)]12,ν¯b=1−f These formulas are verified by numerical simulation. The insight is that there is a limited range of f for which certain critical points can be reached. Thus, 0 < f < 1/5 has different dynamics than 1/5 < f < 1/3. Formulas are also derived for the second maximum of z, which gives an indication of decay or growth of the system. For example, with f = fs and C = 0, the second maximum z with f < 1/5 is:(2)zmax=f+((1−f)2−ν¯2−4f)2+ν¯2ν¯d<ν¯<ν¯czmax=ν¯+fν¯c<ν¯<ν¯bzmax=1ν¯>ν¯c Formulas will also be given for the times at which the maximum occurs and the times at which a transition occurs from static to sliding for all cases.

Modeling and Analysis of Nonlinear Wave Propagation in One-Dimensional Phononic Structures

Different from elastic waves in linear periodic structures, those in phononic crystals with nonlinear properties can exhibit more interesting phenomena. Linear dispersion relations cannot predict band-gap variations due to intensity of wave motion; creating nonlinear phononic crystals remains challenging and few examples have been studied. Recent studies in the literature mainly focus on discrete chain-like structures and consider weak nonlinear regimes; they cannot accurately obtain some relations between wave propagation characteristics and nonlinearities. Our models are based on exact Green-Lagrange strain relations for a structure using the B-spline wavelet on the interval (BSWI) finite element method. Numerical examples show that the proposed method performs well for band structure problems with general nonlinearities. This study can provide good support for engineering applications, such as sound and vibration control using tunable band gaps of nonlinear phononic crystals.

Development of a Nonlinear, C1-Beam Finite-Element Code for Actuator Design of Slender Flexible Robots

For actuator design and motion simulations of slender flexible robots, planar C1-beam elements are developed for Reissner’s large deformation, shear-deformable, curbed-beam model. Internal actuation is mechanically modeled by a rate-form of beam constitutive relation, where actuation curvature is prescribed at each time. Geometrically, a curbed beam is modeled as a frame bundle, whereby at each point on beam’s curve of centroids a moving orthonormal frame is attached to a cross section. After a finite element discretization, a curve of centroids is modeled as a C1-curve, employing cubic shape functions for both planar coordinates with an arc-parameter. The cubic shape functions have already been utilized in linear Euler-Bernoulli beams for the interpolation of transverse displacement. To define the rotation angle of each cross section or the attitude of the moving frame, quadratic shape functions are used introducing a middle node, resulting in three angular nodal displacements. As a result, each beam element has total eleven nodal coordinates. The implementation of a nonlinear finite element code is facilitated by the principle of virtual work, which yields Reissner’s large deformation curbed beam model as the Euler-Lagrange equations. For time integration, the Newmark method is utilized. Finally, as applications of the code, a few inchworm motions induced by different actuation curvature fields are presented.

Nonlinear Vibrations of a Beam-Spring-Large Mass System

This paper presents the nonlinear vibration of a simply supported Euler-Bernoulli beam with a mass-spring system subjected to a primary resonance excitation. The nonlinearity is due to the mid-plane stretching and cubic spring stiffness. The equations of motion and the boundary conditions are derived using Hamiltons principle. The nonlinear system of equations are solved using the method of multiple scales. Explicit expressions are obtained for the mode shapes, natural frequencies, nonlinear frequencies, and frequency response curves. The validity of the results is demonstrated via comparison with results in the literature. Exact natural frequencies are obtained for different locations, rotational inertias, and masses.

System Identification of Force of a Silver Coated Twisted and Coiled Polymer Muscle

The demand of using artificial muscle similar to the human muscle is significantly increased during past decades. Recently, silver-plated Twisted and Coiled Polymer (TCP) muscle was employed in many research projects. A first order differential equations (1st ODE) was used to predict the force of this muscle, assuming that the TCP muscle acts similar to a mechanical spring that has variable stiffness depending on the electrical power supplied. Thus, extensive study should be performed on different types of TCP muscles to reach a conclusion. In this paper, a black box system identification method is used to examine the behavior of TCP muscles under different input conditions. Different order differential equations are compared with experimental results. Prediction error method (PEM) is used for estimation of the force of silver-plated TCP muscle with several linear time invariant (LTI) discrete time state space system. In addition, we suggest a fast method (rule of thumb) to model a TCP muscle. Moreover, two key parameters have been introduced to compare the quality of the TCP muscle from force perspective.