Observations Regarding Numerical Results Obtained by the Floquet-Method

Stability studies of parametrically excited systems are frequently carried out by numerical methods. Especially for LTP-systems, several such methods are known and in practical use. This study investigates and compares two methods that are both based on Floquet’s theorem. As an introductary benchmark problem a 1-dof system is employed, which is basically a mechanical representation of the damped Mathieu-equation. The second problem to be studied in this contribution is a time-periodic 2-dof vibrational system. The system equations are transformed into a modal representation to facilitate the application and interpretation of the results obtained by different methods. Both numerical methods are similar in the sense that a monodromy matrix for the LTP-system is calculated numerically. However, one method uses the period of the parametric excitation as the interval for establishing that matrix. The other method is based on the period of the solution, which is not known exactly. Numerical results are computed by both methods and compared in order to work out how they can be applied efficiently.

Parametric Resonance of a Self-Excited System Under External Force and Time Delay Influence

Vibrations of a nonlinear self-excited system driven by parametric excitation are presented in the paper. The considered model with one DOF includes a self-excitation term represented by a nonlinear Rayleigh function and also a periodically varied stiffness coefficient which represents parametric excitation. The influence of the external force or/and time delay, treated as a control signal, is demonstrated. Nonlinear parametric resonance is determined numerically and analytically by the multiple time scale method. The influence of time delay on the resonance zones and the frequency locking phenomenon is analysed.

Energy-Momentum Conserving Time Integration of Modally Reduced Flexible Multibody Systems

Many conventional time integration schemes frequently adopted in flexible multibody dynamics fail to retain the fundamental conservation laws of energy and momentum of the continuous time domain. Lack of conservation, however, in particular of angular momentum, may give rise to unexpected, unphysical results. To avoid such problems, a scheme for the consistent integration of modally reduced multibody systems subjected to holonomic constraints is developed in the present paper. As opposed to the conventional approach, in which the floating frame of reference formulation is combined with component mode synthesis for approximating the flexible deformation, an alternative, recently proposed formulation based on absolute coordinates is adopted in the analysis. Owing to the linear relationship between the generalized coordinates and the absolute displacement, the inertia terms in the equations of motion attain a very simple structure. The mass matrix remains independent of the current state of deformation and the velocity dependent term known from the floating frame approach vanishes due to the absence of relative coordinates. These advantageous properties facilitate the construction of an energy and momentum consistent integration scheme. By the mid-point rule, algorithmic conservation of both linear and angular momentum is achieved. In order to consistently integrate the total energy of the system, the discrete derivative needs to be adopted when evaluating the strain energy gradient and the derivative of the algebraic constraint equations.

Vertical-Plane Dynamics of Power-Augmented-Ram Vehicle

Power-augmented-ram vehicles represent novel air-assisted marine craft that can be used for high-speed amphibious transportation of heavy cargo. These vehicles rely on combined hydrodynamic and aerodynamic support that is also augmented by front air-based propulsors. Dynamic models for these craft in the presence of wind gusts and surface waves are needed for confident design of these vehicles, including motion control systems. This study addresses 3-DOF vertical-plane dynamics. The models for unsteady forces are based on the aerodynamic extreme-ground-effect theory and hydrodynamic added-mass strip theory. Modeling of the vehicle motions are carried out for cases of head and following wind gusts and waves of low and high amplitudes. Simulation results can be used for determining amplitudes of the vehicle responses, peak accelerations, and forward speed degradation.

Parallel Algorithm for Modeling Constrained Multi-Flexible Body System Dynamics

This paper presents an algorithm for modeling the dynamics of multi-flexible body systems in closed kinematic loop configurations where the component bodies are modeled using the large displacement small deformation formulation. The algorithm uses a hierarchic assembly disassembly process in parallel implementation and a recursive assembly disassembly process in serial implementation to achieve highly efficient simulation turn-around times. The operational inertias arising from the rigid body modes of motion at the joint locations on a component body are modified to account for the nonlinear inertial effects and body forces arising from the body based deformations. Traditional issues, such as motion induced stiffness and temporal invariance of deformation field related inertia terms, are robustly addressed in this algorithm. The algorithm uses a mixed set of coordinates viz. (i) absolute coordinates for expressing the equations of motion of a body fixed reference frame, (ii) relative or internal coordinates to express the kinematic joint constraints and (iii) body fixed coordinates to account for the body’s deformation field. The kinematic joint constraints and the closed loop constraints are treated alike through the formalism of relative coordinates, joint motion spaces and their orthogonal complements. Verification of the algorithm is demonstrated using the planar fourbar mechanism problem that has been traditionally used in literature.

Intrinsic Localized Modes of Harmonic Oscillations in Pendulum-Arrays Subjected to Horizontal Excitation

The behavior of intrinsic localized modes (ILMs) is investigated for an array with N pendula which are connected with each other by weak, linear springs when the array is subjected to horizontal, sinusoidal excitation. In the theoretical analysis, van der Pol’s method is employed to determine the expressions for the frequency response curves for fundamental harmonic oscillations. In the numerical calculations, the frequency response curves are presented for N = 2 and 3 and compared with the results of the numerical simulations. Patterns of oscillations are classified according to the stable steady-state solutions of the response curves, and the patterns in which ILMs appear are discussed in detail. The influence of the connecting springs of the pendula on the appearance of ILMs is examined. Increasing the values of the connecting spring constants may affect the excitation frequency range of ILMs and cause Hopf bifurcation to occur, followed by amplitude modulated motions (AMMs) including chaotic vibrations. The influence of the imperfections of the pendula on the system response is also investigated. Bifurcation sets are calculated to examine the influence of the system parameters on the excitation frequency range of ILMs and determine the threshold value for the connecting spring constant after which ILMs do not appear. Experiments were conducted for N = 2, and the data were compared with the theoretical results in order to confirm the validity of the theoretical analysis.

Constrained Multibody Dynamics With Python: From Symbolic Equation Generation to Publication

Symbolic equations of motion (EOMs) for multibody systems are desirable for simulation, stability analyses, control system design, and parameter studies. Despite this, the majority of engineering software designed to analyze multibody systems are numeric in nature (or present a purely numeric user interface). To our knowledge, none of the existing software packages are 1) fully symbolic, 2) open source, and 3) implemented in a popular, general, purpose high level programming language. In response, we extended SymPy (an existing computer algebra system implemented in Python) with functionality for derivation of symbolic EOMs for constrained multibody systems with many degrees of freedom. We present the design and implementation of the software and cover the basic usage and workflow for solving and analyzing problems. The intended audience is the academic research community, graduate and advanced undergraduate students, and those in industry analyzing multibody systems. We demonstrate the software by deriving the EOMs of a N-link pendulum, show its capabilities for LATEX output, and how it integrates with other Python scientific libraries — allowing for numerical simulation, publication quality plotting, animation, and online notebooks designed for sharing results. This software fills a unique role in dynamics and is attractive to academics and industry because of its BSD open source license which permits open source or commercial use of the code.

Torsional Vibration of a Spur Gear System With Time-Varying and Square Nonlinearities

This paper investigates the torsional vibration of a spur gear system with time-varying and square nonlinearities, by both the analytical method and numerical simulation. First, the equations of motion of a rotating spur gear system are established. Then a single-dof equivalent system is induced to describe the relative motion or torsional vibration of the gears. The harmonic balance method is used to obtain the steady-state response. Influence of the input torque on the response is discussed and a phenomenon, one resonant peak split up into two peaks when the input torque is high enough is revealed. Last, numerical simulations are carried out and bifurcation diagrams and amplitude-frequency curve is given by taking the excitation frequency as control parameter. Selected typical motions are also presented in detail by time-histories, phase portraits, Poincaré map and frequency spectra.

The Application of Parametric Excitation in MEMS Gyroscopes

Parametric excitation, via electrostatic stiffness modulation, can be exploited in resonant MEMS gyroscopes. In the case of the Rate gyroscope, which is by far the most common type of MEMS gyro, parametric excitation may be used to amplify either the primary mode of the gyro or the response to the angular rate. Both approaches will be discussed. In the more complex mode of operation, known as “Rate Integrating” the output of the gyro is angle directly as opposed to angular velocity in the case of Rate gyro. In this rate integrating mode of operation parametric excitation does offer an effective energy control used to initiate, sustain the vibration and minimise damping perturbations. Parametric amplification of the primary mode of the rate gyroscope is presented and supported with experimental results. In this implementation parametric excitation is combined with external harmonic forcing of the primary mode in order to reduce electrical feedthrough of the driving signal to the sense electrodes. A practical parametric excitation scheme implemented using Digital Signal Processing has been developed to enable either amplification of the primary mode of the gyroscope or amplification of the response to the applied angular velocity. Parametric amplification of the primary mode of the gyroscope is achieved by frequency tracking and regulation of the amplitudes of the harmonic forcing and parametric excitation to maintain a desired parametric gain by closed loop PID control. Stable parametric amplification of the primary mode by a factor of 20 is demonstrated experimentally. This has important benefits regarding the minimisation of electrical feedthrough of the drive signal to the sense electrodes of the secondary mode. By taking advantage of the phase dependence of parametric amplification and the orthogonality of the Coriolis force and quadrature forcing, the response to the applied angular velocity may be parametrically amplified by applying excitation of a particular phase directly to the sensing mode. The major advantage of parametric amplification applied to MEMs gyroscopes is that it can mechanically amplify the Coriolis response before being picked off electrically. This is particularly advantageous for sensors where electronic noise is the major noise contributor. In this case parametric amplification can significantly improve the signal to noise ratio of the secondary mode by an amount approximately equal to the parametric amplification. Preliminary rate table tests performed in open loop demonstrate a magnification of the signal to noise ratio of the secondary mode by a factor of 9.5.

Switchability in Discontinuous, Discrete Dynamical Systems

Tin this paper, a theory for switchability and singularity of discontinuous, discrete dynamical systems. G-functions for the discrete dynamical systems are introduced through the boundary, and the necessary and sufficient conditions for the switchability of discrete mappings are presented.