scholarly journals Dynamic and Stability of Harmonic Driving Flexible Cartesian Robotic Arm with Bolted Joints Based on the Sensitivity and Multiple Scales Method

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Yufei Liu ◽  
Wei Li ◽  
Xuefeng Yang ◽  
Yuqiao Wang
Keyword(s):  
Multiple Scales ◽  
Virtual Work ◽  
Multiple Scales Method ◽  
Bolted Joints ◽  
Parametric Stability ◽  
Robotic Arms ◽  
Average Motion ◽  
Duhamel Integral ◽  
Vibration Responses ◽  
The Stability

Flexible Cartesian robotic arms (CRAs) are typical multicoupling systems. Considering the elastic effects of bolted joints and the motion disturbances, this paper investigates the dynamic and stability of the flexible CRA. With the kinetic energy and potential energy of the comprising components, Hamilton’s variational principle and Duhamel integral are utilized to derive the dynamic equation and vibration differential equation. Based on the proposed elastic restraint model of the bolted joints, boundary conditions and mode equations of the flexible CRA are determined with using the principle of virtual work. According to the mode frequencies and sensitivities analysis, it reveals that the connecting stiffness of the bolted joints has significant influences, and the mode frequencies are more sensitive to the tensional stiffness. Moreover, describing the motion displacement of the driving base as combination of an average motion displacement and a harmonic disturbance, the vibration responses of the system are studied. The result indicates that the motion disturbance has obvious influence on the vibration responses, and the influence enhances under larger accelerating operations. The multiple scales method is introduced to analyze the parametric stability of the system, as well as the influences of the tensional stiffness and the end-effector on the stability.

Parametric Stability of Axially Accelerating Viscoelastic Beams With the Recognition of Longitudinally Varying Tensions

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
Li-Qun Chen ◽  
You-Qi Tang
Keyword(s):  
Governing Equation ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Finite Support ◽  
Governing Equations ◽  
Parametric Stability ◽  
Stability Boundaries ◽  
Axial Acceleration ◽  
The Stability ◽  
Longitudinally Varying Tension

In this paper, the parametric stability of axially accelerating viscoelastic beams is revisited. The effects of the longitudinally varying tension due to the axial acceleration are highlighted, while the tension was approximately assumed to be longitudinally uniform in previous studies. The dependence of the tension on the finite support rigidity is also considered. The generalized Hamilton principle and the Kelvin viscoelastic constitutive relation are applied to establish the governing equations and the associated boundary conditions for coupled planar motion of the beam. The governing equations are linearized into the governing equation in the transverse direction and the expression of the longitudinally varying tension. The method of multiple scales is employed to analyze the parametric stability of transverse motion. The stability boundaries are derived from the solvability conditions and the Routh-Hurwitz criterion for principal and sum resonances. In terms of stability boundaries, the governing equations with or without the longitudinal variance of tension are compared and the effects of the finite support rigidity are also examined. Some numerical examples are presented to demonstrate the effects of the stiffness, the viscosity, and the mean axial speed on the stability boundaries. The differential quadrature scheme is developed to numerically solve the governing equation, and the computational results confirm the outcomes of the method of multiple scales.

Analysis of Stability and Bifurcation of an Asymmetrical Rotor

2015 ◽  
Author(s):  
Majid Shahgholi ◽  
S. E. Khadem ◽  
Mahsa Asgarisabet
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Multiple Scales Method ◽  
Moments Of Inertia ◽  
Rotor Systems ◽  
Unequal Mass ◽  
Principal Axes ◽  
Analysis Of Stability ◽  
Stability And Bifurcations ◽  
The Stability

The effect of shaft and disk asymmetry on the harmonic resonances of a rotor system with the in-extensional nonlinearity and large amplitude are investigated. Two rotor systems, one of which has been comprised of a symmetrical shaft and an asymmetrical disk (SA), and the other one has been comprised of an asymmetrical shaft and an asymmetrical disk (AA) are investigated. The shaft in the AA rotor has unequal mass moments of inertia and flexural rigidities in the direction of principal axes. Also, in the AA system the rigid disk is asymmetric with unequal mass moments of inertia. The equations of motion are derived by the Hamiltonian principle. The stability and bifurcations are obtained using the multiple scales method. The influences of asymmetry of shaft, asymmetry of disk, inequality between two eccentricities corresponding to the principal axes, disk position and external damping on the stability and bifurcations of SA and AA rotors are investigated. The results achieved from multiple scales method show a good agreement with those of numerical simulations.

An hourglass-type electromagnetic isolator with negative resistance electromagnetic shunt circuit

2020 ◽  
Vol 64 (1-4) ◽  
pp. 549-556
Author(s):  
Yajun Luo ◽  
Linwei Ji ◽  
Yahong Zhang ◽  
Minglong Xu ◽  
Xinong Zhang
Keyword(s):  
Vibration Isolation ◽  
Electromechanical Coupling ◽  
Coupling Effect ◽  
Negative Resistance ◽  
Isolation System ◽  
Amplitude Vibration ◽  
Vibration Responses ◽  
The Stability ◽  
Isolation Performance ◽  
Shunt Circuit

The present work proposed an hourglass-type electromagnetic isolator with negative resistance (NR) shunt circuit to achieve the effective suppression of the micro-amplitude vibration response in various advanced instruments and equipment. By innovatively design of combining the displacement amplifier and the NR electromagnetic shunt circuit, the current new type of vibration isolator not only can effectively solve the problem of micro-amplitude vibration control, but also has significant electromechanical coupling effect, to obtain excellent vibration isolation performance. The design of the isolator and motion relationship is presented firstly. The electromechanical coupling dynamic model of the isolator is also given. Moreover, the optimal design of the NR electromagnetic shunt circuit and the stability analysis of the vibration isolation system are carried out. Finally, the simulation results about the transfer function and vibration responses demonstrated that the isolator has a significant isolation performance.

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

scholarly journals Perturbations of nonlinear autonomous oscillators

1994 ◽  
Vol 35 (4) ◽  
pp. 445-468 ◽  
Author(s):  
P. B. Chapman
Keyword(s):  
General Theory ◽  
Fourier Coefficients ◽  
Multiple Scales ◽  
Second Order ◽  
Multiple Scales Method ◽  
Suitable Parameter ◽  
Non Linear

AbstractA general theory is given for autonomous perturbations of non-linear autonomous second order oscillators. It is found using a multiple scales method. A central part of it requires computation of Fourier coefficients for representation of the underlying oscillations, and these coefficients are found as convergent expansions in a suitable parameter.

scholarly journals Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Bamadev Sahoo ◽  
L. N. Panda ◽  
G. Pohit
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Time History ◽  
Mean Value ◽  
Phase Portraits ◽  
Axially Moving Beam ◽  
Internal Resonances ◽  
Moving Beam ◽  
Axially Moving ◽  
The Stability

The nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. The beam velocity is assumed to be comprised of a constant mean value along with a harmonically varying component. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of second mode is approximately three times that of first mode; a three-to-one internal resonance is possible. The method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. The nonlinear steady state response along with the stability and bifurcation of the beam is investigated. The system exhibits pitchfork, Hopf, and saddle node bifurcations under different control parameters. The dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the influence of internal resonance.

The Multiple Scales Method

2011 ◽  
pp. 359-360
Author(s):  
Michael Paidoussis ◽  
Stuart Price ◽  
Emmanuel de Langre
Keyword(s):  
Multiple Scales ◽  
Multiple Scales Method

scholarly journals A new conceptual model of coral biomineralisation: hypoxia as the physiological driver of skeletal extension

2013 ◽  
Vol 10 (5) ◽  
pp. 2867-2884 ◽  
Author(s):  
S. Wooldridge
Keyword(s):  
Conceptual Model ◽  
Multiple Scales ◽  
High Growth ◽  
High Growth Rate ◽  
Coral Host ◽  
Structural Determinants ◽  
Night Time ◽  
Long Run ◽  
Animal Phyla ◽  
The Stability

Abstract. That corals skeletons are built of aragonite crystals with taxonomy-linked ultrastructure has been well understood since the 19th century. Yet, the way by which corals control this crystallization process remains an unsolved question. Here, I outline a new conceptual model of coral biomineralisation that endeavours to relate known skeletal features with homeostatic functions beyond traditional growth (structural) determinants. In particular, I propose that the dominant physiological driver of skeletal extension is night-time hypoxia, which is exacerbated by the respiratory oxygen demands of the coral's algal symbionts (= zooxanthellae). The model thus provides a new narrative to explain the high growth rate of symbiotic corals, by equating skeletal deposition with the "work-rate" of the coral host needed to maintain a stable and beneficial symbiosis. In this way, coral skeletons are interpreted as a continuous (long-run) recording unit of the stability and functioning of the coral–algae endosymbiosis. After providing supportive evidence for the model across multiple scales of observation, I use coral core data from the Great Barrier Reef (Australia) to highlight the disturbed nature of the symbiosis in recent decades, but suggest that its onset is consistent with a trajectory that has been followed since at least the start of the 1900s. In concluding, I outline how the proposed capacity of cnidarians (which includes modern reef corals) to overcome the metabolic limitation of hypoxia via skeletogenesis also provides a new hypothesis to explain the sudden appearance in the fossil record of calcified skeletons at the Precambrian–Cambrian transition – and the ensuing rapid appearance of most major animal phyla.

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