Ordinary Differential Equations Of Motion Research Articles

Modeling and free vibration behavior of rotating composite thin-walled closed-section beams with SMA fibers

Smart structure with active materials embedded in a rotating composite thin-walled beam is a class of typical structure which is using in study of vibration control of helicopter blades and wind turbine blades. The dynamic behavior investigation of these structures has significance in theory and practice. However, so far dynamic study on the above-mentioned structures is limited only the rotating composite beams with piezoelectric actuation. The free vibration of the rotating composite thin-walled beams with shape memory alloy(SMA) fiber actuation is studied. SMA fiber actuators are embedded into the walls of the composite beam. The equations of motion are derived based on Hamilton's principle and the asymptotically correct constitutive relation of single-cell cross-section accounting for SMA fiber actuation. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the Galerkin's method. The formulation for free vibration analysis includes anisotropy, pitch and precone angle, centrifugal force and SMA actuation effect. Numerical results of natural frequency are obtained for two configuration composite beams. It is shown that natural frequencies of the composite thin-walled beam decrease as SMA fiber volume and initial strain increase and the decrease in natural frequency becomes more significant as SMA fiber volume increases. The actuation performance of SMA fibers is found to be closely related to the rotational speeds and ply-angle. In addition, the effect of the pitch angle appears to be more significant for the lower-bending mode ones. Finally, in all cases, the precone angle appears to have marginal effect on free vibration frequencies. The developed model can be capable of describing natural vibration behaviors of rotating composite thin-walled beam with active SMA fiber actuation. The present work extends the previous analysis done for modeling passive rotating composite thin-walled beam.

Resonances of an in-extensional asymmetrical spinning shaft with speed fluctuations

In this study, main and parametric resonances of an asymmetrical spinning shaft with in-extensional nonlinearity and large amplitude are simultaneously investigated. The main resonance is due to inhomogeneous part of the equations of motion, which is due to dynamic imbalances of shaft whereas the parametric resonances are due to parametric excitations due to speed fluctuations and a shaft asymmetry. The shaft is simply supported with unequal mass moments of inertia and flexural rigidities in the direction of principal axes. The equations of motion are derived by the extended Hamilton principle. The stability and bifurcations are obtained by multiple scales method, which is applied to both partial and ordinary differential equations of motion. The influences of asymmetry of shaft, speed fluctuations, inequality between two eccentricities corresponding to the principal axes and external damping on the stability and bifurcation are studied. To investigate the effect of speed fluctuations on the bifurcations and stability the loci of bifurcation points are plotted as function of damping coefficient. The numerical solutions are used to verify the results of multiple scales method. The results of multiple scales method show a good agreement with those of numerical solutions.

An extended high-dimensional Melnikov analysis for global and chaotic dynamics of a non-autonomous rectangular buckled thin plate

By using an extended Melnikov method on multi-degree-of-freedom Hamiltonian systems with perturbations, the global bifurcations and chaotic dynamics are investigated for a parametrically excited, simply supported rectangular buckled thin plate. The formulas of the rectangular buckled thin plate are derived by using the von Karman type equation. The two cases of the buckling for the rectangular thin plate are considered. With the aid of Galerkin’s approach, a two-degree-of-freedom non-autonomous nonlinear system is obtained for the non-autonomous rectangular buckled thin plate. The high-dimensional Melnikov method developed by Yagasaki is directly employed to the non-autonomous ordinary differential equation of motion to analyze the global bifurcations and chaotic dynamics of the rectangular buckled thin plate. Numerical method is used to find the chaotic responses of the non-autonomous rectangular buckled thin plate. The results obtained here indicate that the chaotic motions can occur in the parametrically excited, simply supported rectangular buckled thin plate.

Nonlinear free vibrations of thin-walled beams in torsion

Nonlinear torsional vibrations of thin-walled beams exhibiting primary and secondary warpings are investigated. The coupled nonlinear torsional–axial equations of motion are considered. Ignoring the axial inertia term leads to a differential equation of motion in terms of angle of twist. Two sets of torsional boundary conditions, that is, clamped–clamped and clamped-free boundary conditions are considered. The governing partial differential equation of motion is discretized and transformed into a set of ordinary differential equations of motion using Galerkin’s method. Then, the method of multiple scales is used to solve the time domain equations and derive the equations governing the modulation of the amplitudes and phases of the vibration modes. The obtained results are compared with the available results in the literature that are obtained from boundary element and finite element methods, which reveals an excellent agreement between different solution methodologies. Finally, the internal resonance and the stability of coupled and uncoupled nonlinear modes are investigated. This study can be a preliminary step in the understanding of complex dynamics of such systems in internal resonance excited by external resonant excitations.

A new energy harvester using a cross-ply cylindrical membrane shell integrated with PVDF layers

In this paper, we proposed a smart cylindrical membrane shell panel (SCMSP) model for vibration-based energy harvester. The SCMSP is made of an orthotropic elastic core covered by outer PVDF layers with transverse polarization vector. Electrodynamics governing equations of motion are derived by applying extended Hamilton’s principle. The governing equations are based on Donnell’s linear thin shell theory. The SCMSP displacement fields are expanded by means of double Fourier series satisfying immovable edges with free rotation boundary conditions and coupled system of linear partial differential equations are obtained. The discretized linear ordinary differential equations of motion are obtained using Galerkin method. The output power is taken as an indicating criterion for the generator. A parametric study for MEMS applications is conducted to predict the power generated due to radial harmonic ambient vibration. Optimal resistance value is also obtained for the particular electrode distribution that gives maximum output power. A low vibration amplitude (5 Pa), and a low-frequency (471.79 Hz) vibration source is targeted for the resonance operation, in which the output power of 0.4111 μW and peak-to-peak voltage of 0.2952 V are predicted.

Non-linear vibrations of variable stiffness composite laminated plates

It is the objective of this work to analyze vibrations of variable stiffness composite laminated plates (VSCL), and investigate the differences between the oscillations of these plates and traditional laminates. The analysis is based on numerical experiments and a new p-version finite element with hierarchic basis functions, which follows first order shear deformation theory and considers geometrical non-linearity, is derived. Considering first linear oscillations, the natural frequencies and mode shapes of different VSCL are computed and compared with the ones of constant stiffness laminates. The linear natural frequencies of the present model are also compared with the ones computed using a recently developed higher-order model for VSCL. After, numerical tests are carried out in the time domain and, for the first time in VSCL, taking geometric non-linearity into account, to investigate the response to external forces. The non-linear ordinary differential equations of motion are solved by Newmark’s method. It is verified that the variation of the fibre orientation can lead to significant differences in the amplitudes of the non-linear response.

Maneuver Control and Vibration Suppression of a Smart Flexible Satellite Using Robust Passivity Based Control

In this paper a satellite with two flexible appendages and the piezoelectric layers which are attached to them and a central hub is considered. The piezoelectric layers are used as actuators. The governing equations of motion are derived based on Lagrange method. Using Rayleigh-Ritz technique ordinary differential equations of motion are obtained. A robust passivity based control is applied to the system to not only control the three axes maneuver of the satellite but also suppress the vibrations of the flexible appendages. Finally, the system is simulated and simulation results show the good performance of this controller.

Driving simulator with double-wishbone suspension using efficient block-triangularized kinematic equations

When modeled with ideal joints, many vehicle suspensions contain closed kinematic chains, or kinematic loops, and are most conveniently modeled using a set of generalized coordinates of cardinality exceeding the degrees-of-freedom of the system. Dependent generalized coordinates add nonlinear algebraic constraint equations to the ordinary differential equations of motion, thereby producing a set of differential-algebraic equations that may be difficult to solve in an efficient yet precise manner. Several methods have been proposed for simulating such systems in real time, including index reduction, model simplification, and constraint stabilization techniques. In this work, the equations of motion for a double-wishbone suspension are formulated symbolically using linear graph theory. The embedding technique is applied to eliminate the Lagrange multipliers from the dynamic equations and obtain one ordinary differential equation for each independent acceleration. Symbolic computation is then used to triangularize a subset of the kinematic constraint equations, thereby producing a recursively solvable system for calculating a subset of the dependent generalized coordinates. Thus, the kinematic equations are reduced to a block-triangular form, which results in a more computationally efficient solution strategy than that obtained by iterating over the original constraint equations. The efficiency of this block-triangular kinematic solution is exploited in the real-time simulation of a vehicle with double-wishbone suspensions on both axles, which is implemented in a hardware- and operator-in-the-loop driving simulator.

Directional Control of a Driver-Heavy-Vehicle Closed-Loop System

A two degree of freedom (DOF) lateral dynamic model for a three-axe heavy vehicle is set up and the vehicle ordinary differential equations of motion are derived. The nonlinear lateral tire forces are obtained by Gim model with vertical loads, slip angles and cornering performances of front and rear tires being input parameters. A revised closed-loop single-point preview method is proposed to model the driver’s directional control performance. In this proposed method, the steering angle of front wheels is calculated in real time according to the track error between a certain point ahead of the vehicle and the required route. Then the steering angle is input into the vehicle model to gain the dynamic responses and position of the vehicle in next time step. Thus the driver-heavy-vehicle closed-loop system is built. The dynamic responses of the system are simulated on the condition of double lane change and the effects of system parameters on the path following behavior of the vehicle are researched. Then the advice on how to improve the vehicle directional control ability can be brought forward.

Nonlinear free and forced vibration analysis of a single-walled carbon nanotube using shell model

In this Paper, the nonlinear free and force vibration of a single-walled carbon nanotube (SWCNT) with simply supported ends is investigated based on von Karman’s geometric nonlinearity. The SWCNT described as an individual shell and the Donnell’s equations of cylindrical shells are used to obtain the governing equations. The Galerkin's procedure is used to discretized partial differential equations of the governing into the ordinary differential equations of motion. The method of averaging is applied to analyze the nonlinear vibration of (10, 0), (20, 0) and (30, 0) zigzag SWCNTs in the analytical calculations. The effects of the nonlinear parameters, different aspect ratios, different circumferential wave numbers and longitudinal half-wave numbers are investigated. Both free and forced motions (due to harmonic excitation) are considered. It is shown that (30, 0) zigzag SWCNT has less nonlinear behavior than the other CNTs for a constant aspect ratio. The type of nonlinearity is determined by the aspect ratio. It is seen from the results that for Small values of aspect ratios, the vibration behavior is softening type for the low amplitudes, and it is hardening type for the large amplitudes. And for large value of the aspect ratio, the vibration behavior is hardening type for all amplitudes.

Life-Threatening Arrhythmia Verification in ICU Patients Using the Joint Cardiovascular Dynamical Model and a Bayesian Filter

In this paper, a novel nonlinear joint dynamical model is presented, which is based on a set of coupled ordinary differential equations of motion and a Gaussian mixture model representation of pulsatile cardiovascular (CV) signals. In the proposed framework, the joint interdependences of CV signals are incorporated by assuming a unique angular frequency that controls the limit cycle of the heart rate. Moreover, the time consequence of CV signals is controlled by the same phase parameter that results in the space dimensionality reduction. These joint equations together with linear assignments to observation are further used in the Kalman filter structure for estimation and tracking. Moreover, we propose a measure of signal fidelity by monitoring the covariance matrix of the innovation signals throughout the filtering procedure. Five categories of life-threatening arrhythmias were verified by simultaneously tracking the signal fidelity and the polar representation of the CV signal estimations. We analyzed data from Physiobank multiparameter databases (MIMIC I and II). Performance evaluation results demonstrated that the sensitivity of the detection ranges over 93.50% and 100.00%. In particular, the addition of more CV signals improved the positive predictivity of the proposed method to 99.27% for the total arrhythmic types. The method was also used for false arrhythmia suppression issued by ICU monitors, with an overall false suppression rate reduced from 42.3% to 9.9%. In addition, false critical ECG arrhythmia alarm rates were found to be, on average, 42.3%, with individual rates varying between 16.7% and 86.5%. The results illustrate that the method can contribute to, and enhance the performance of clinical life-threatening arrhythmia detection.

Large displacement analysis of elastically constrained rotating disks with rigid body degrees of freedom

A central aspect of the linear vibration theory of rotating disks involves the concept of critical speeds. At such rotation speeds an axisymmetric disk can support a standing wave as recorded by a stationary observer. In such situations an applied space fixed constant force can give rise to a resonance in the disk. Such a response is of concern in industrial applications as diverse as circular saw blades and computer floppy disks. In such situations the magnitude of response may exceed the limits of linear theory. The present paper is concerned with the effects of large displacements upon the disk response in the neighborhood of such critical speeds. The effects of geometric nonlinearities and the influence of rigid body tilting and translation (caused by the boundary conditions) are considered. The equations of motion are based on Von Karman plate theory. The eigenfunctions of two self-adjoint eigenvalue problems, corresponding to the stress function and the transverse displacement, are determined and used as approximation functions in a numerically efficient Galerkin formulation. The coupled nonlinear ordinary differential equations of motion are solved using the Runge–Kutta method. Numerical results are presented for disks that are free to translate and rotate at their inner boundary and are constrained from lateral motion by space fixed linear springs. The effects of vibration magnitude on system response in the sub and super-critical speed regions are computed and the effects of large displacements on critical speed behavior and forced response are investigated. Experiments are conducted to verify the accuracy of the numerical results obtained in this paper.

A new mathematical construction of the general nonlinear ODEs of motion in rigid body dynamics (Euler’s equations)

It is shown that the three nonlinear dynamic Euler ordinary differential equations (ODEs), concerning the motion of a rigid body free to rotate about a fixed point, are reduced, by means of a subsidiary function which is to be determined, to three Abel equations of the second kind of the normal form. Based on a recently developed mathematical construction concerning exact analytic solutions of the Abel nonlinear ODEs of the second kind, we perform a new mathematical solution for the classical dynamic Euler nonlinear ODEs.

Nonlinear dynamic behavior and instability of slender frames with semi-rigid connections

The free and forced nonlinear vibrations of slender frames with semi-rigid connections are studied in this work. Special attention is given to the influence of static pre-load on the natural frequencies and mode shapes, nonlinear frequency–amplitude relations, and resonance curves. An efficient nonlinear finite element program for buckling and vibration analysis of slender elastic frames with semi-rigid connections is developed. The equilibrium paths are obtained by continuation techniques, in combination with the Newton-Raphson method. The ordinary differential equations of motion of the discretized frame are solved by the Newmark implicit numerical integration method using adaptive time-step strategies. Three structural systems with important practical applications are analyzed: an L-frame, a shallow arch, and a pitched-roof frame. The results highlight the importance of the static pre-load and the stiffness of the semi-rigid connections on the buckling and vibration characteristics of these structures.

Limit-cycle Oscillations of Functionally Graded Material Plates Subject to Aerodynamic and Thermal Loads

The nonlinear flutter and thermal buckling of a functionally graded material (FGM) plate panel subjected to combined thermal and aerodynamic loads are investigated using a finite element model based on the thin plate theory and von Karman strain-displacement relations to account for moderately large deflection. The thermal load is assumed to be steady-state constant temperature distribution, and the aerodynamic pressure is modeled using the quasi-steady first-order piston theory. The governing nonlinear equations of motion are obtained using the principle of virtual work adopting an approach based on the thermal strain being a cumulative physical quantity to account for temperature dependent material properties. The static nonlinear equations are solved by Newton-Raphson numerical technique to get the thermal post-buckling deflection. The dynamic nonlinear equations of motion are transformed to modal coordinates to reduce the computational efforts. The Newmark implicit integration scheme is employed to solve the second order ordinary differential equations of motion. Finally, the buckling temperature, post-buckling deflection and the nonlinear limit-cycle oscillations of an FGM panel are presented, illustrating the effect of volume fraction exponent, dynamic pressure, temperature rise, and boundary conditions on the panel response.

Deployment and retrieval of tethered satellite system under J2 perturbation and heating effect

Tethered satellite system is subject to a variety of perturbations from the geophysical environment and other planets, such as the J 2 perturbation and heating effect. Those perturbations are likely to yield new dynamics of tethered satellite system during deployment and retrieval. However, the J 2 perturbation and heating effect in studying the deployment and retrieval of tethered satellite have drawn little attention. This paper presents the effect of J 2 perturbation on the deployment and retrieval of tethered satellite and gives the new dynamics with the heating effect taken into account. The viscoelastic tether is modeled as a series of elements with lumped masses, which yields a set of ordinary differential equations of motion having finite degrees of freedom by using Newton’s laws. The forces on the system are on the basis of the J 2 perturbation and the thermal flux arising from the solar radiation, Earth’s infrared radiation as well as tether’s infrared radiation. The case studies show that the J 2 perturbation has a great influence on the deployment process, which depends mainly on the magnitude of sliding friction force between tether and deployment device. The heating effect, however, gives rise to quite different dynamics during retrieval of a tethered satellite compared with that of a tethered satellite without heating effect.

Free periodic vibrations of beams with large displacements and initial plastic strains

This paper intends to analyse free vibrations of beams in the geometrically non-linear regime and with plastic strains. The specific goal is to find how plastic strains combined with large displacements influence the non-linear modes of vibration, by analysing the influence of the former two factors in mode shapes and natural frequencies of vibration. The geometrical non-linearity is represented by the Von Kármán type strain–displacement relations. A stress–strain relation of the bilinear type, with isotropic strain hardening, is assumed, the Von Mises yield criterion is employed and the flow theory of plasticity applied. To obtain the time domain ordinary differential equations of motion the principle of virtual work is used and a Timoshenko p-version finite element model with hierarchical basis functions is adopted. The equations of motion are naturally different from the usual large displacement equations, due to the appearance of matrices and vectors related with plastic terms. In the cases studied, plastic strains are imposed on the beam by equally distributed static forces; the forces are then removed and a study on the free vibrations is carried out. It is assumed that, once defined, the plastic strain field does not change. The time domain equations are transformed to the frequency domain by the harmonic balance method and these frequency domain equations are solved by an arc-length continuation method. The variations of mode shapes of vibration and of natural frequencies with vibration amplitude are investigated. It is found that the plastic strain distribution defines if and how much softening spring effect occurs. Hardening spring effect always appears, but with some plastic strain fields hardening spring takes place only at large vibration amplitudes. Plastic deformations also have an important effect in the vibration shapes.

Nonlinear vibrations and imperfection sensitivity of a cylindrical shell containing axial fluid flow

The high imperfection sensitivity of cylindrical shells under static compressive axial loads is a well-known phenomenon in structural stability. On the other hand, less is known of the influence of imperfections on the nonlinear vibrations of these shells under harmonic axial loads. The aim of this work is to study the simultaneous influence of geometric imperfections and an axial fluid flow on the nonlinear vibrations and instabilities of simply supported circular cylindrical shells under axial load. The fluid is assumed to be non-viscous and incompressible and the flow to be isentropic and irrotational. The behavior of the thin-walled shell is modeled by Donnell's nonlinear shallow-shell equations. It is subjected to a static uniform compressive axial pre-load plus a harmonic axial load. A low-dimensional modal expansion, which satisfies the relevant boundary and continuity conditions, and takes into account all relevant nonlinear modal interactions observed in the past in the nonlinear vibrations of cylindrical shells with and without flow is used together with the Galerkin method to derive a set of eight coupled nonlinear ordinary differential equations of motion which are, in turn, solved by the Runge–Kutta method. The shell is considered to be initially at rest, in a position corresponding to a pre-buckling configuration. Then, a harmonic excitation is applied and conditions for parametric instability and dynamic snap-through are sought. The results clarify the marked influence of geometric imperfections and fluid flow on the dynamic stability boundaries, bifurcations and basins of attraction.

キャピラリージェット形状を利用した動的表面張力の測定法 : 第1報,噴流形状の理論的解析(流体工学,流体機械)

A theoretical and experimental consideration is conducted to propose a novel method to measure the dynamic surface tension of surfactant solution. The trajectory of laminar jet injected from a horizontal capillary tube is considered from an approximate theory. The momentum balance is discussed between surface tension, gravitational force, pressure and viscous force. Since the effect of gravitational force is relatively small compared with the surface tension, the cross section of jet is assumed as a circle and the pressure inside the jet is approximately obtained from the Laplace equation. The ordinary differential equation of motion is solved numerically by the simple Runge Kutta's method to obtain the profile of jet trajectory. The trajectory profiles are measured experimentally for water and two kinds of ethanol solutions with smaller surface tension than that of water. The theoretical profiles agree well with experimental ones for each condition and the trajectories are strongly dependent on the surface tension. It is demonstrated that the change of surface tension of surfactant solution could be measured by the method proposed in this study.

On the influence of membrane inertia and shear deformation on the geometrically non-linear vibrations of open, cylindrical, laminated clamped shells

It is the objective of this work to verify the validity of different approximations in the analysis of geometrically non-linear vibrations of open, cylindrical, laminated, fully clamped shells. Namely, it is intended to verify if membrane inertia and transverse shear deformation can be neglected and when. p-version finite elements with hierarchic basis functions are employed to define the models. A simple comparison of the different stiffness and mass terms is carried out first, to assess the relative importance of membrane inertia and shear deformation. Then a few numerical tests are carried out in non-linear vibrations by solving the ordinary differential equations of motion by Newmark’s method. It is found that what are usually considered to be thin panels actually require the consideration of shear deformation.