Degree Of Freedom System Research Articles

Limit cycle oscillations at resonances

This paper deals with limit cycles in one degree of freedom systems. The van der Pol equation is an example of an equation describing systems with clear limit cycles in the phase space (displacement-velocity 2 dimensional plane). In this paper, it is shown that a system with nonlinear loading, representing the drag load acting on structures in an oscillatory flow (the drag term of the Morison equation), will in fact exhibit limit cycles at resonance and at higher order resonances. These limit cycles are stable, and model self-excited oscillations. As the damping in the systems is linear and constant, the drag loading will to some degree work as negative damping. The consequences of the existence of these limit cycles are that systems starting at lesser amplitudes in the phase plane will exhibit increased amplitudes until the limit cycle is obtained.

Minimal time splines on the sphere

An interesting problem in geometric control theory arises from robotics and space science: find a smooth curve, controlled by a bounded acceleration, connecting in minimum time two prescribed tangent vectors of a Riemannian manifold Q. The state equation is $$ \nabla _{\dot{\gamma }} \dot{\gamma } = u \in TQ, |u| \le A$$ . Applying Pontryagin’s principle one gets a Hamiltonian system in $$T^*(TQ)$$ . We consider this problem in $$S^2(r)$$ . Seemingly, it has not been addressed before. Via the SO(3) symmetry, we reduce the four degrees of freedom system to the five variables $$(a,v, M_1,M_2,M_3),$$ where v is the scalar velocity, conjugated to a costate variable a and $$(M_1,M_2,M_3)$$ are costate variables that satisfy $$\{ M_i, M_j\} = \epsilon _{ijk} M_k$$ . We derive the reduced equations and find special analytical solutions, that are organizing centers for the dynamics. Reconstruction of the curve $$\gamma (t)$$ is achieved by a time dependent linear system of ODEs for the orthogonal matrix R whose first column is the unit tangent vector of the curve and whose last column is the unit normal vector to the sphere.

An influence of temperature on reconstructed middle ear with shape memory prosthesis

In the paper idea of reconstructed middle ear with a prosthesis made of shape memory alloy is proposed. The new design of shape memory prosthesis is used to enable adjusting its length to individual patient’s needs which is a novel contribution of the paper. In order to make a proper fit of prosthesis, a surgeon has to adjust its size and position by cutting step by step classical prosthesis. It takes time of a operation and enlarges a period of patient’s esthesia. Therefore, a shape memory prosthesis (SMP) is proposed to shorten operation time and improve fitting through precisely selected length. A reconstructed middle ear is modelled as a two degree of freedom system with nonlinear shape memory element. Finding advisable periodic and undesirable a periodic and irregular behaviour in various temperature is the main aim of the paper. Results of the study should give an answer whether SMP can be useful in medical practice and should also explain dynamics of the middle ear with SMP. The properties of the shape memory prosthesis, in the form of helical spring, are described here by a polynomial dependence. Firstly, free vibrations are investigated and equilibrium points, next forced vibrations are studied under different parameters of external excitations and temperature range. Bifurcation analysis and stability of periodic solutions are performed in order to reveal the system behaviour. Finally, interesting dynamical findings of chaotic vibrations pure regular and regular oscillations with fluctuation are presented. However, from practical point of view only harmonic response of the stapes is advisable. That can be achieved at the normal temperature of a human body only for small excitation amplitude.

The Influence of Overhead Cranes in the Seismic Performance of Industrial Buildings

The paper investigates the influence of overhead cranes with a hanging mass under earthquake type loading, considering the Emilia 2012 seismic sequence. The structural layout of precast concrete industrial buildings typical of the Italian territory is considered. The equations of motion describing the behaviour of the hoist load are derived and a sensitivity analysis is carried out on simplified three degrees of freedom systems by solving the governing differential equations. The influence of various parameters on the roof displacement and on the horizontal load transferred by the hanging mass is addressed. The considered parameters are the relative damping of the hanging mass, the length of the hoist ropes, the earthquake record, the hysteretic type of the plastic hinges at the column base, and the behaviour factor of the structural system. The results show that for a horizontal component of the considered seismic sequence the structural displacements are amplified in the case of a behaviour factor greater than 2.5. A simplified modelling strategy considering small displacements is also investigated. Such model is suitable for response spectrum analyses. Finally, a three-dimensional case study is analysed by means of non-linear time history analyses. The results show the influence of the overhead crane on the local performance of some structural and non-structural elements, such as columns and cladding panels, especially when the assumption of rigid roof diaphragm does not apply.

Analytical solutions of a nonlinear two degrees of freedom model of a human middle ear with SMA prosthesis

A nonlinear two degrees of freedom system with shape memory element (SMA) is presented in the paper. Thermomechanical properties, shape memory effect and pseudoelasticity of the SMA joint, are modelled by phenomenological approach with the help of a fifth order polynomial approximation. The analytical solution of the mathematical model is obtained by means of the multiple time scales method. Next, a numerical study of a biomechanical system of human middle ear is performed. It has been demonstrated that for a selected rod type prothesis the cubic nonlinearity has the most essential effect on the system dynamics in the first order approximation, which converges with direct numerical simulations.

Blind velocities mitigation for MIMO GMTI radar with Doppler division multiple access waveforms

Multiple-input multiple-output (MIMO) radar with multiple transmitters and multiple receivers can achieve a larger virtual antenna array and more system degrees of freedom; thus applying it to ground moving target indication (GMTI) radar can improve the performance of GMTI. Doppler division multiple access (DDMA) waveforms are approximately orthogonal providing good minimum detectable velocity (MDV) performance. However, in such DDMA systems, a sufficient pulse repetition frequency (PRF) design freedom is required. Furthermore, these waveforms suffer from blind velocities which are serious problems, especially in radar systems with high carrier frequency or low PRF. This paper analyses the blind velocities problem and show that blind velocities are relative to variation of the PRF and/or the carrier frequency. Variable PRF techniques are widely used in conventional GMTI radar including multiple PRFs and variable pulse repetition intervals (PRI). Combined with the characteristics of the DDMA MIMO GMTI radar, this paper proposed two methods to mitigate blind velocities: “multi-PRF DDMA” which employs multiple PRFs over successive coherent processing intervals, and “PRI-dithered DDMA” which employs nonuniform sampling by dithered PRI in slow time. Simulation results demonstrate that both the methods are effective ways to mitigate blind velocities in DDMA MIMO GMTI radar systems.

Parallel Transport Along Seifert Manifolds and Fractional Monodromy

The notion of fractional monodromy was introduced by Nekhoroshev, Sadovskií and Zhilinskií as a generalization of standard (‘integer’) monodromy in the sense of Duistermaat from torus bundles to singular torus fibrations. In the present paper we prove a general result that allows one to compute fractional monodromy in various integrable Hamiltonian systems. In particular, we show that the non-triviality of fractional monodromy in 2 degrees of freedom systems with a Hamiltonian circle action is related only to the fixed points of the circle action. Our approach is based on the study of a specific notion of parallel transport along Seifert manifolds.

State Feedback Switching Control of an Underactuated Planar Manipulator with Partial Feedback Linearization

In this paper, position control problem of a two-degree-of-freedom underactuated manipulator is considered and a state feedback control structure with energy-based switching is proposed. The mechanism has two revolute joints that move the two links on horizontal plane. Difficulties in system control arise with the fact that the manipulator has less control input signals than system degrees of freedom, and has complex nonholonomic structure. Furthermore, in horizontal operational conditions, position control of the system is a challenging work since free joint is not affected by the gravity. The system has only one actuator at the shoulder joint and the elbow joint is completely free. Therefore, the system cannot be stabilized around any equilibrium point by a linear state feedback control method. In this study, a control system that utilizes two stabilizing state feedback controllers and an energy-based supervisory switching mechanism is proposed. Stabilizing controllers are obtained utilizing the partial feedback linearization method. Proposed control is tested by computer simulations. First the open-loop plant dynamic model is obtained by Euler-Lagrange formulation and state-space modeling approach. Then, a simulation model of the system in closed loop with proposed control scheme is developed using the dynamic model and control law. Simulation tests are performed with respect to three different initial conditions. Performance of the control system is observed and revealed via simulations.

Feasibility of motion laws for planar one degree of freedom linkage mechanisms at dead point configurations

Abstract This paper proposes an analytical solution of the Inverse Kinematics (IK) problem at dead point configurations for any planar one degree of freedom linkage mechanism, with regard to the continuity C n of the motion law. The systems analyzed are those whose elements are linked with lower pairs and do not present redundancies. The study aims to provide the user with some rules to facilitate the design of feasible motion profiles to be reproduced by conventional electrical actuators at these configurations. During the last decades, several methods and techniques have been developed to study this specific configuration. However, these techniques are mainly focused on solving numerically the IK indeterminacy, rather than analyzing the motion laws that the mechanisms are able to perform at these particular configurations. The analysis presented in this paper has been carried out differentiating and applying l’Hopital’s rule to the system of constraint equations ϕ ( q ) of the mechanism. The study also considers the feasibility of the time-domain profiles to be reproduced with conventional electrical actuators (i.e. AC/DC motors, linear actuators, etc.). To show the usefulness and effectiveness of the method, the development includes the analytical application and numerical simulations for two common one degree of freedom systems: a slider-crank and a four linkage mechanisms. Finally, experimental results are presented on a four linkage mechanism test bed.

Partial hydrodynamic representation of quantum molecular dynamics.

A hybrid method is proposed to propagate system-bath quantum dynamics that use both basis functions and coupled quantum trajectories. In it, the bath is represented with an ensemble of Bohmian trajectories while the system degrees of freedom are accounted through reduced density matrices. By retaining the Hilbert space structure for the system, the method is able to capture interference processes that are challenging to describe in Bohmian dynamics due to singularities that these processes introduce in the quantum potential. By adopting quantum trajectories to represent the bath, the method beats the exponential scaling of the computational cost with the bath size. This combination makes the method suitable for large-scale ground and excited state fully quantum molecular dynamics simulations. Equations of motion for the quantum trajectories and reduced density matrices are derived from the Schrödinger equation and a computational algorithm to solve these equations is proposed. Through computations in two-dimensional model systems, the method is shown to offer an accurate description of subsystem observables and of quantum decoherence, which is difficult to obtain when the quantum nature of the bath is ignored. The scaling of the method is demonstrated using a model with 21 degrees of freedom. The limit of independent trajectories is recovered when the mass of bath degrees of freedom is much larger than the one of the system, in agreement with mixed quantum-classical descriptions.

A THEORETICAL-EXPERIMENTAL APPROACH FOR ELASTO-DAMPING PARAMETERS ESTIMATION OF CONE INERTIAL CRUSHER MOUNTING

The present paper deals with estimation of the elasto-damping parameters of a cone inertial crusher mounting. The numerical values of these parameters are crucial for accurate reproduction of the machine vibrational behavior and dynamical model adequacy. Due to the significant difficulties arising during the purely theoretical determination of the stiffness and damping parameters of the rubber vibroisolators it is well-suited to use a theoretical-experimental approach. The developed approach is based on the theoretical determination of the mounting stiffness parameters as a function of two experimentally measured natural frequencies of the mechanical system. The crusher is represented as a six degrees of freedom system with two planes of symmetry. By using the system characteristic polynomial, the theoretical derivation of mathematical relationships for the mechanical system natural frequencies as a function of stiffness, inertial and geometrical parameters is performed. A good agreement is shown when comparing the experimental and the theoretical results for the system kinematical characteristics.

Chatter and dynamic cutting force prediction in high-speed ball end milling

ABSTRACTMachine tool chatter is a serious problem which deteriorates surface quality of machined parts and increases tool wear, noise, and even causes tool failure. In the present paper, machine tool chatter has been studied and a stability lobe diagram (SLD) has been developed for a two degrees of freedom system to identify stable and unstable zones using zeroth order approximation method. A dynamic cutting force model has been modeled in tangential and radial directions using regenerative uncut chip thickness. Uncut chip thickness has been modeled using trochoidal path traced by the cutting edge of the tool. Dynamic cutting force coefficients have been determined based on the average force method. Several experiments have been performed at different feed rates and axial depths of cut to determine the dynamic cutting force coefficients and have been used for predicting SLD. Several other experiments have been performed to validate the feasibility and effectiveness of the developed SLD. It is found that the proposed method is quite efficient in predicting the SLD. The cutting forces in stable and unstable cutting zone are in well agreement with the experimental cutting forces.

Optimization of passive control performance for different hard disk drives subjected to shock excitation

Laptop personal computers (LPCs) and their components are vulnerable devices in harsh mechanical environments. One of the most sensitive components of LPCs is hard disk drive (HDD) which needs to be protected against damages attributable to shock and vibration in order to have better magnetic read/write performance. In the present work, a LPC and its HDD are modeled as two degrees of freedom system and the nonlinear optimization method is employed to perform a passive control through minimizing peak of HDD absolute acceleration caused by a base shock excitation. The presented shock excitation is considered as half-sine pulse of acceleration. In addition, eleven inequality constraints are defined based on geometrical limitations and allowable intervals of lumped modal parameters. The target of the optimization is to reach optimum modal parameters of rubber mounts and rubber feet as design variables and subsequently propose new characteristics of rubber mounts and rubber feet to be manufactured for the HDD protection against shock excitation. The genetic algorithm and the modified constrained steepest descent algorithm are employed in order to solve the nonlinear optimization problem for three widely-used commercial cases of HDD. Finally, the results of both optimization methods are compared to make sure about their accuracy.

Effect of Automatic Ball Balancer on Unbalanced Rotor Vibration

Rotor vibration due to unbalance causes a lot of problems during operation. One of the simplest ways to reduce rotor vibration is the passive balancing. Unbalanced vertical rotor is modeled by a two degrees of freedom system considering Jeffcott model assumption. A (2+n) degrees of freedom mathematical model is derived with respect to a Cartesian co-ordinate system for the unbalanced rotor with the automatic ball balancer. The model equations are expressed as state equations then solved numerically to calculate vibration amplitudes and angular positions of balancing balls. Model validation is achieved by comparing these results with a three-dimensional model. The effect of the automatic ball balancer on vibration amplitudes is explained at different speed ranges. A parametric study is done to explain the effect of the system damping coefficient on the ball balancer behaviors.

Dynamic analysis of a cantilever beam with a multiple degrees of freedom system

A beam attached to a multiple degrees of freedom system is a common and complex structure in a broad field of engineering and science. The paper presents a novel and efficient computer model for dynamic analysis of a cantilever beam structure attached to a multiple degrees of freedom system by means of the recurrence matrix method. In this model, the forced responses and natural characteristics of the coupled beam are achieved and numerical simulations of the established model agree well with the early results. Finally, the vibration reductions of a beam by a dynamic vibration absorber are numerically studied.

Application of linear and nonlinear vibration absorbers in micro-milling process in order to suppress regenerative chatter

In this paper, linear and nonlinear vibration absorbers are designed to suppress regenerative chatter in micro-milling process. Suppressing regenerative chatter leads to improve surface finish. Micro-milling process is considered as a two degree of freedom system, and the run-out effects of cutting tool are also taken into account. Linear and nonlinear absorbers in two directions are composed of mass, spring and dashpot elements. The optimum values of the absorber parameters are determined by means of an optimization algorithm in order to minimize the cutting tool vibration. The effectiveness of different types of the absorbers in micro-milling process is illustrated. The optimum parameters for each type of absorbers are presented.

Vibrations of Elastically Supported Masses Separated by a Textile Layer

In this paper the problem of the transmission of vibration through a textile layer is presented. A mathematical model of a two degree of freedom system containing a textile layer and excited to vibrate by an electromagnet is formulated. The numerical simulation showsthat the textile layer increases the resonance frequency.

Vibration of Elastically Connected Multiple Timoshenko Beams under a Moving Six Degrees of Freedom System

Vibration of Elastically Connected Multiple Timoshenko Beams under a Moving Six Degrees of Freedom System

Why We Need More Degrees of Freedom

Mechanical systems encountered in biology typically have many more degrees of freedom (DOF) than the 6 DOF required to manipulate a body in space. Even the relatively rigid arthropods and crustaceans have at least 5 DOF in each limb; tentacles and human hands have many more. Robotics engineers are routinely required to choose the number of DOF in a robot in the early design stages, potentially limiting the robot's future uses. We theoretically motivate the definition of “mechanical versatility” as the ability of a mechanical system to express distinct static configurations and switch among them rapidly. Requiring versatility and assuming that the systems are power and force limited, and must furthermore resist finite energy environmental disturbances to their state, we show that such multiuse 1 mechanical systems have a lower bound on the number of DOF they require. For bio-mechanics, this suggests which organs and organisms will be driven to become more complex mechanically by indicating domains where higher DOF systems would intrinsically out-compete lower DOF systems.

Influence of tire stiffness on automotive quarter car suspension system

Quarter car automotive suspension model is the model with two degrees of freedom system, which is used for the analysis of ride comfort and vertical movement of the vehicle. It represents the analysis of vehicle suspension system at any one wheel out of all four wheels; it means the vertical motion of the vehicle body at any one wheel of the vehicle is being analyzed to check the performance. Automotive suspension designs have been always compromise between road handling, road holding and passenger comfort. This paper describes the dynamic model of quarter car automotive suspension system for bump condition prepared in MATLAB using Simulink. The influence of varying values of tire stiffness on vertical acceleration of the vehicle has been discussed in this paper. The results obtained from the simulation of simulink model are used for performance investigation and analysis of the dynamic behavior of the vehicle.