Multi-degrees Of Freedom Systems Research Articles

Nonlinear dynamics investigation of contact force in cam–follower system using Lyapunov exponent parameter, power spectrum analysis, and poincare’ maps

ABSTRACT Non-linear-linear dynamics behaviour of the contact force is investigated for different cam angular speeds (N) and different internal dimensions of the follower guide (F.G.I.D.). The largest Lyapunov exponent parameter conception is used to detect whether the follower will detach from the cam or the follower will stay in permanent contact. Power spectrum analysis and Poincaré maps with a phase-plane diagram are added to detect the non-periodic motion of the contact force. The follower is moved with three degrees of freedom. Multi degrees of freedom (spring-damper-mass) systems are added at the end of the follower stem to suppress the non-linear dynamics behaviour of the contact force of the follower. SolidWorks software is used in the numerical simulation of the contact force. Contact force and follower displacement are determined at the different coefficient of restitution values. The friction and impact are considered between the cam and the follower and between the follower and its guide. The contact force is measured experimentally using OPTOTRACK/3020 by tracking the position of the follower through an infrared camera device. The value of the contact force has declined with the increase of cam angular speeds (N) and the internal dimension of the follower guide (F.G.I.D.). The peak of non-linear response of the follower displacement is reduced to 15%, 32%, 45%, and 62% after using multi degrees of freedom systems.

Nonlinear dynamics phenomena in globoidal cam with roller follower mechanism

• A multi degrees of freedom systems are used to suppress the spiral orbits of the nonlinear dynamics phenomena of the follower movement. • Linear and nonlinear spring-damper systems are used to suppress the nonlinear dynamics phenomena. • An OPTOTRAK/3020 with an infrared camera device is used to track the follower position experimentally. • A bifurcation diagram is used to detect the non-periodic motion of the follower movement. • Nonlinear dynamic response of the follower movement is calculated using the theory of mechanical vibration . This paper study the nonlinear dynamics phenomena in a globoidal cam and roller follower mechanism. The influence of internal dimension of the follower guide (F.G.I.D.) and cam angular speeds (N) on the nonlinear dynamics behavior are considered. Power spectrum analysis, phase-plane mapping and Largest Lyapunov exponent parameter are used to investigate the non-periodic motion of the follower. A multi degrees of freedom of (spring-damper-mass system) are used to suppress the spiral orbits of the nonlinear dynamics phenomena for the follower movement. Nonlinear dynamic response of the follower movement is calculated using the theory of mechanical vibration. Linear and nonlinear spring-damper systems are used to suppress the nonlinear dynamics phenomena. A bifurcation diagram is used to detect the non-periodic motion of the follower movement. A numerical simulation is done using SolidWorks software. An OPTOTRAK/3020 with an infrared camera device is used to track the follower position experimentally. The peak of nonlinear response of the roller follower is reduced by (20.89 % , 47.76 % , 58.2 % , 67.16 % ) after using multi degrees of freedom systems.

Non-periodic motion reduction in globoidal cam with roller follower mechanism

Non periodic motion has been investigated using power spectrum analysis of Fast Fourier Transform (FFT), Lyapunov exponent parameter using Wolf algorithm, and Phase-plane mapping. A multi degrees of freedom (spring-damper system) are added at the end of the follower stem to reduce the nonlinear dynamic behavior of the follower. In the experiment setup, data acquisition technique with signal processing approach is used through an infrared camera device to capture the nonlinear dynamic movement of the follower. A numerical simulation is done using SolidWorks software. An OPTOTRAK/3020 with an infrared camera device is used to track the follower position experimentally. The peak of nonlinear response of the roller follower is reduced by (20.89%, 47.76%, 58.2%, and 67.16%) after using multi degrees of freedom systems.

An Inerter-Based Dynamic Vibration Absorber With Concurrently Enhanced Energy Harvesting and Motion Control Performances Under Broadband Stochastic Excitation Via Inertance Amplification

Abstract This paper examines the performance of a regenerative dynamic vibration absorber, dubbed energy harvesting-enabled tuned mass-damper-inerter (EH-TMDI), for simultaneous vibration suppression and energy harvesting in white-noise-excited damped linear primary structures. Both single-degree-of-freedom (SDOF) structures under force and base excitations and multi-degrees-of-freedom (MDOF) structures under correlated random forces are studied. The EH-TMDI includes an electromagnetic motor (EM), assumed to behave as a shunt damper, sandwiched between a secondary mass and an inerter element connected in series. The latter element resists relative acceleration at its ends through a constant termed inertance known to be readily scalable in actual inerter device implementations. In this regard, attention is herein focused on gauging the available energy for harvesting at the EM and the displacement variance of the primary structure as the inertance increases through comprehensive parametric investigations. This is supported by adopting simplified inertance-dependent tuning formulae for the EH-TMDI stiffness and damping properties and deriving in closed-form the response of white-noise-excited EH-TMDI-equipped SDOF and MDOF systems through linear random vibration analyses. It is found that lightweight EH-TMDIs, having 1% the mass of the primary structure, achieve improved vibration suppression and energy harvesting performance as inertance amplifies. For SDOF structures with grounded inerter, the rate of improvement is higher as the inherent structural damping reduces and the EM shunt damping increases. For MDOF structures with nongrounded inerter, improvement rate is higher as the primary structure flexibility between the two EH-TMDI attachment points increases.

Frequency tuning of weakly and strongly coupled micromechanical beams

Mechanical coupling forms the basis for many exciting applications in tuning the frequencies of multi-degrees of freedom systems starting from coupling the motion of fire-flies to MEMS arrays. In this paper, the influence of the weak and strong coupling between two micromechanical cantilever beams is performed using the experimental and analytical studies. To do the study, the two types of coupled micromechanical beams are taken, namely, a weakly coupled beams and a strongly coupled beams. While the weakly coupled beams are coupled along their lengths through a coupling beam of constant length and width, two strongly coupled beams are coupled through beams of varying length and constant width with one end of its length fixed at fixed portion of cantilever beam. Here, we fabricated both types of coupled beams through microfabrication process. After measuring their in-phase and out-of-phase frequencies with varying coupling parameters, the influence of coupling parameters on these two modes is investigated. It is found that one of the modes consist of a transverse mode and another a kind of torsional mode which lead to out-of-phase mode. After finding this interesting observation, further analysis is done using the corresponding analytical models of transverse and torsional modes. Furthermore, the generalized analytical models are developed to capture the effects of coupling in weakly and strongly coupled beams as a function of coupling element dimensions.

Approximate Floquet Analysis of Parametrically Excited Multi-Degree-of-Freedom Systems With Application to Wind Turbines

General responses of multi-degrees-of-freedom (MDOF) systems with parametric stiffness are studied. A Floquet-type solution, which is a product between an exponential part and a periodic part, is assumed, and applying harmonic balance, an eigenvalue problem is found. Solving the eigenvalue problem, frequency content of the solution and response to arbitrary initial conditions are determined. Using the eigenvalues and the eigenvectors, the system response is written in terms of “Floquet modes,” which are nonsynchronous, contrary to linear modes. Studying the eigenvalues (i.e., characteristic exponents), stability of the solution is investigated. The approach is applied to MDOF systems, including an example of a three-blade wind turbine, where the equations of motion have parametric stiffness terms due to gravity. The analytical solutions are also compared to numerical simulations for verification.

Transient Response of MDOF Systems With Inerters to Nonstationary Stochastic Excitation

Explicit, closed-form, exact analytical expressions are derived for the covariance kernels of a multi degrees-of-freedom (MDOF) system with arbitrary amounts of viscous damping (not necessarily proportional-type), that is equipped with one or more auxiliary mass damper-inerters placed at arbitrary location(s) within the system. The “inerter” is a device that imparts additional inertia to the vibration damper, hence magnifying its effectiveness without a significant damper mass addition. The MDOF system is subjected to nonstationary stochastic excitation consisting of modulated white noise. Results of the analysis are used to determine the dependence of the time-varying mean-square response of the primary MDOF system on the key system parameters such as primary system damping, auxiliary damper mass ratio, location of the damper-inerter, inerter mass ratio, inerter node choices, tuning of the coupling between the damper-inerter and the primary system, and the excitation envelope function. Results of the analysis are used to determine the dependence of the peak transient mean-square response of the system on the damper/inerter tuning parameters, and the shape of the deterministic intensity function. It is shown that, under favorable dynamic environments, a properly designed auxiliary damper, encompassing an inerter with a sizable mass ratio, can significantly attenuate the response of the primary system to broad band excitations; however, the dimensionless “rise-time” of the nonstationary excitation substantially reduces the effectiveness of such a class of devices (even when optimally tuned) in attenuating the peak dynamic response of the primary system.

Modified Differential Transform Method for Solving Vibration Equations of MDOF Systems

Vibration equations of discrete multi-degrees-of-freedom (MDOF) structural systems is system of differential equations. In linear systems, the differential equations are also linear. Various analytical and numerical methods are available for solving the vibration equations in structural dynamics. In this paper modified differential transform method (MDTM) as a semi-analytical approach is generalized for the system of differential equations and is utilized for solving the vibration equations of MDOF systems. The MDTM is a recursive method which is a hybrid of Differential Transform Method (DTM), Pade' approximant and Laplace Transformation. A series of examples including forced and free vibration of MDOF systems with classical and non-classical damping are also solved by this method. Comparison of the results obtained by MDTM with exact solutions shows good accuracy of the proposed method; so that in some cases the solutions of the vibration equation that found by MDTM are the exact solutions. Also, MDTM is less expensive in computational cost and simpler with compare to the other available approaches.

A negative derivative feedback design algorithm

Vibration control logics based on the modal approach allow damping to be increased on a certain number of modes. The main limit associated with these strategies is represented by spillover on non-modelled modes. Negative derivative feedback (NDF) proves particularly robust against spillover since modal velocity is fed back through a band-pass filter so that undesired effects can be limited both at high and low frequencies. Unfortunately, the definition of the control gains for this logic is generally more difficult than other resonant controls owing to the lack of physical meaning. In this paper a design strategy for an NDF controller based on an optimal approach is proposed for single and multi-degrees of freedom systems and is tested on a cantilever beam finite element model.

A wavelet-based approach for vibration analysis of framed structures

In this paper, an explicit time integration scheme is proposed for structural vibration analysis by using wavelet functions. Initially, the differential equation of vibration governing SDOF (single-degree of freedom) systems has been solved by wavelet operators, and later the proposed approach has been generalized for MDOF (multi-degrees of freedom) systems. For this purpose, two different types of wavelet functions have been exemplified including, complex Chebyshev wavelet functions and simple Haar wavelet functions. In the proposed approach, a straightforward formulation has been derived from the numerical approximation of response through the wavelet definition. Emphasizing on frequency-domain approximation, a simple step-by-step algorithm has been implemented and improved to calculate the response of MDOF systems. Moreover, stability and accuracy of results have been evaluated. The effectiveness of the proposed approach is demonstrated using three examples compared with some of the existing numerical integration schemes such as family of Newmark-β, Wilson-θ and central difference method. In all the procedures, computation time involved has also been considered. Finally, it is concluded that the vibration analysis of structures is improved by lesser computation time and high accuracy of proposed approach, particularly, in large-scaled systems.

An explicit time integration method for structural dynamics using cubic [formula omitted]-spline polynomial functions

This paper presents a scheme where the cubic B-spline method is developed for Multi-Degrees-Of-Freedom (MDOF) systems. In the proposed approach, a straightforward formulation in a fluent manner was derived from the approximation of the response of the system with a B-spline basis. In this way, there is no need to use a special pre-starting procedure to commence solving the problem. A simple step-by-step algorithm is implemented and presented to calculate the dynamic response of MDOF systems. Stability and accuracy analyses have been done in this paper. The results of the accuracy investigation were compared with some other state of the art methods. Actually, this method lies in cases of conditionally stable methods. The validity and effectiveness of the proposed method is demonstrated with two examples.

A robust approach for seismic damage assessment

We present a robust approach based on a convex analysis to take into account uncertainties appearing in evaluations of damage to structures during an earthquake. Two kinds of uncertainties are introduced: on the excitation and on the structure. The results obtained by the convex analysis are compared with a Monte Carlo simulation to evaluate the conservatism of the method. The analysis is applied to a reinforced concrete shear wall which is modeled as a single degree of freedom system. The non-linearity of the structure is taken into account by an uncertainty on a linearized system. An extension to multi-degrees of freedom systems is presented to allow the treatment of more complex structures, with an application to portal frames. The results of this analysis consist in an evaluation of the maximum possible damage of the structure according to the uncertainties taken into account.

Design of parallel robots in microrobotics

During the past few years, there has been an increasing demand in the field of precision engineering for fine motion in multi-degrees of freedom systems. These applications motivated the development of a new robotics field called microrobotics. In this paper, we review both the design guidelines for microrobots and the advantages of using parallel robots in very high precision applications. Parallel micromanipulators using elastic joints as well as structures manufactured in single solid and metallic bellows are introduced.