Nonlinear vibration behavior and resonance of a Cartesian manipulator system carrying an intermediate end effector

This paper studied the nonlinear vibration and resonance of a Cartesian manipulator system carrying an intermediate end effector under mixed excitations. The multiple scales method is applied to get the approximate solutions of this system of the second-order differential equation. Furthermore, the analytical solution obtained the amplitudes and phases of the response from the first-order differential equation governing. We extracted all worst resonance cases and studied it numerically. The numerical solutions and response amplitude of this system are also studied and discussed. We analyzed the stability of the steady-state solution of a Cartesian manipulator system using frequency response equations and phase plane technique at the worst resonance cases. Comparison between analytical and numerical solutions is obtained. We determined both bifurcation diagrams and stability using Poincaré maps. Also, the numerical results are obtained using MAPLE and MATLAB algorithms.     

Modal analysis and dynamic responses of a rotating Cartesian manipulator with generic payload and asymmetric load

Abstract

Here, investigation to explore the effect of generic payload and externally applied asymmetric load on the calculation of modal parameters and dynamic performance of a rotating flexible manipulator under prismatic motion has been established. We thus have developed a dynamic model of a rotating Cartesian manipulator with a payload whose center of gravity doesn’t coincide with the point of attachment, to determine the modal parameters i.e., natural frequency and corresponding mode-shape. These modal parameters are then illustrated graphically upon varying parameters like offset parameters (i.e., offset mass, offset inertia, offset length), mass and stiffness of rotary actuator, and amplitude and frequency of asymmetric load. An investigation into the nonlinear dynamics of the system accounting of geometric nonlinearity has been executed while obtained results have been validated numerically within the permissible error at the assorted critical points in frequency characteristic curves. Current research further investigates the influences of offset parameters, mass and stiffness of the actuator, frequency and amplitude of axial force on the steady state responses for the primary and sub-harmonic resonance conditions to reveal the built-in saddle-node and pitchfork bifurcation due to which the system losses its structural stability. This work enables an insight into the modal characteristics and nonlinear behavior of a rotating-Cartesian manipulator with a generic payload under asymmetric axial force and prismatic motion.     

Non-linear dynamics of a soft magneto-elastic Cartesian manipulator

This paper studies the non-linear dynamics of a soft magneto-elastic Cartesian manipulator with large transverse deflection. The system has been subjected to a time varying magnetic field and a harmonic base excitation at the roller-supported end. Unlike elastic and viscoelastic manipulators, here the governing temporal equation of motion contains additional two frequency forced, and linear and non-linear parametric excitation terms. Method of multiple scales has been used to solve the temporal equation of motion. The influences of various system parameters such as amplitude and frequency of magnetic field strength, amplitude and frequency of support motion, and the payload on the frequency response curves have been investigated for three different resonance conditions. With the help of numerical results, it has been shown that by using suitable amplitude and frequency of magnetic field, the vibration of the manipulator can be significantly controlled. The developed results and expressions can find extensive applications in the feed-forward vibration control of the flexible Cartesian manipulator using magnetic field.     

Non-linear dynamics of a flexible single link Cartesian manipulator

The present work deals with the non-linear vibration of a harmonically excited single link roller-supported flexible Cartesian manipulator with a payload. The governing equation of motion of this system is developed using extended Hamilton's principle, which is reduced to the second-order temporal differential equation of motion, by using generalized Galerkin's method. This equation of motion contains both cubic non-linearities of geometric and inertial type in addition to linear forced and non-linear parametric excitation terms. Method of multiple scales is used to solve this non-linear equation and study the stability and bifurcations of the system. Influence of amplitude of the base excitation and mass ratio on the steady state response of the system is investigated for both simple and subharmonic resonance conditions. Critical bifurcation points are determined from the fixed-point responses and periodic, quasi-periodic responses are also found for different system parameters. The results obtained using the perturbation analysis are compared with the previously published experimental work and are found to be in good agreement. This work will be useful for the designer of a flexible manipulator.     

Non-linear vibration of a single link viscoelastic Cartesian manipulator

In this present work, the non-linear behavior of a single-link flexible visco-elastic Cartesian manipulator is studied. The temporal equation of motion with complex coefficients of the system is obtained by using D’Alembert's principle and generalized Galarkin method. The temporal equation of motion contains non-linear geometric and inertia terms with forced and non-linear parametric excitations. It may also be found that linear and non-linear damping terms originated from the geometry of the large deformation of the system exist in this equation of motion. Method of multiple scales is used to determine the approximate solution of the complex temporal equation of motion and to study the stability and bifurcation of the system. The response obtained using method of multiple scales are compared with those obtained by numerically solving the temporal equation of motion and are found to be in good agreement. The response curves obtained using viscoelastic beams are compared with those obtained from a linear Kelvin-Voigt model and also with an equivalent elastic beam. The effect of the material loss factor, amplitude of base excitation, and mass ratio on the steady state responses for both simple and subharmonic resonance conditions are investigated.     

Configuration-dependent modal analysis of a Cartesian parallel kinematics manipulator: numerical modeling and experimental validation

In the design optimization of a robot the configuration-dependent modal analysis can be a powerful tool to be exploited when high stiffness and high dynamic performances are concurrently required. In this paper the elastodynamics of a lower-mobility Parallel Kinematic Machine for pure translational motions is analyzed. The vibrational modes and the natural frequencies of the robot are evaluated as functions of the end effector position inside the workspace. A finite element model including kinematic joints is used to perform a series of modal analyses in a grid of points inside the workspace. A polynomial regression gives continuous volume maps of the natural frequencies distributions. The numerical model is validated by comparison with experiments: a modal analysis is conducted on a set of inertance Frequency Response Functions acquired on several points of the machine components as a result of an excitation given by an instrumented hammer. A Natural Frequency Difference analysis validates the model under certain conditions and highlights some critical issues to be focused on in future works.     

Nonlinear stability analysis of a clamped rod carrying electric current

This paper is devoted to the analysis of the nonlinear stability of a clamped rod carrying electric current in the magnetic field which is produced by the current flowing in a pair of inifinitely long parallel rigid wires. The natural state of the rod is in the plane of the wires and is equidistant from them. Firstly under the assumption of spatial deformation, the governing equations of the problem are derived, and the linearized problem and critical currents are discussed. Secondly, it is proved that the buckled states of the rod are always in planes. Finally, the global responses of the bifurcation problem of the rod are computed numerically and the distributions of the deflections, axial forces and bending moments are obtained. The results show that the buckled states of the rod may be either supercritical or subcritical, depending on the distancz between the rod and the wires. Furthermore, it is found that there exists a limit point on the branch solution of the supercritical buckled state. This is distinctively different from the buckled state of the elastic compressive rods.Project supported by the Foundation of the Natural Science of China and Gansu Province     

A novel approach for forward position analysis of a double-triangle spherical parallel manipulator

In this paper, we introduce a new approach for forward position analysis of a double-triangle (DT) spherical parallel manipulator. Utilizing spherical geometry of the manipulator, two coupled trigonometric equations are obtained through using special form of Rodrigues formula. Next, the two coupled equations are solved using Bezout's elimination method which leads to a polynomial of eight degree. Finally, we provide an example having eight real solutions, the polynomial thus being minimal.     

Nonlinear dynamic analysis on rigid-flexible coupling system of an elastic beam

Previous work examined the effect of the attached stiffness matrix terms on stability of an elastic beam undergoing prescribed large overall motion. The aim of the present work is to extend the nonlinear formulations to an elastic beam with free large overall motion. Based on initial stress method, the nonlinear coupling equations of elastic beams are obtained with free large overall motion and the attached stiffness matrix is derived by solving sub-static formulation. The angular velocity and the tip deformation of the elastic pendulum are calculated. The analytical results show that the simulation results of the present model are tabled and coincide with the one-order approximate model. It is shown that the simulation results accord with energy conservation principle.     

Nonlinear dynamic analysis of a 3D guyed mast

Nonlinear Dynamics - To study the collective behavior and the regulation mechanism of memristor, the Hindmarsh–Rose neuron cells are selected as the units to construct a coupled system by...     

Nonlinear dynamic analysis of a photonic crystal nanocavity resonator

A nonlinear dynamic model of a one-dimensional photonic crystal nanocavity resonator is presented. It considers the internal tensile stress and the geometric characteristics of a photonic crystal with rectangular(and circular) holes. The solution of the dynamic model shows that the internal tensile stress can suppress the hardening and softening behaviors of the resonator. However, the stress can reduce the amplitude, which is not conducive to an improvement of the sensitivity of the sensor. It is demonstrated that with an optimized beam length, the normalized frequency drift of the beam can be stabilized within 1% when the optical power increases from 2 mW to 6 mW. When the hole size of the resonator beam is close to the beam width, its increase can lead to a sharp rise of the resonant frequency and the promotion of hardening behavior. Moreover,the increase in the optical power initially leads to the softening behavior of the resonator followed by an intensification of the hardening behavior. These theoretical and numerical results are helpful in understanding the intrinsic mechanism of the nonlinear response of an optomechanical resonator, with the objective of avoiding the nonlinear phenomena by optimizing key parameters.     

A new approach for dynamic analysis of flexible manipulator systems

In this paper, a new approach for dynamic analysis of the flexible multibody manipulator systems is described. The organization of the computer implementations which are used to automatically construct and numerically solve the system of loosely coupled dynamic equations expressed in terms of the absolute, joint and elastic coordinates is discussed. The main processor source code consists of three main modules: constraint module, mass module and force module. The constraint module is used to numerically evaluate the relationship between the absolute and joint accelerations. The mass module is used to numerically evaluate the system mass matrix as well as the non-linear Coriolis and centrifugal forces associated with the absolute, joint and elastic coordinates. At the same time, the force module is used to numerically evaluate the generalized external and elastic forces associated with the absolute, joint and elastic coordinates. Computational efficiency is achieved by taking advantage of the structure of the resulting system of loosely coupled equations. The absolute, joint and elastic accelerations are integrated forward in time using direct numerical integration methods. The absolute positions and velocities can then be determined using the kinematic relationships. The flexible 2-DOF double-pendulum and spatial manipulator systems are used as illustrated examples to demonstrate and verify the application of the computational procedures discussed in this paper.     

Nonlinear vibrations and stability of an axially moving beam with an intermediate spring support: two-dimensional analysis

The nonlinear coupled longitudinal-transverse vibrations and stability of an axially moving beam, subjected to a distributed harmonic external force, which is supported by an intermediate spring, are investigated. A?case of three-to-one internal resonance as well as that of non-resonance is considered. The equations of motion are obtained via Hamilton??s principle and discretized into a set of coupled nonlinear ordinary differential equations using Galerkin??s method. The resulting equations are solved via two different techniques: the pseudo-arclength continuation method and direct time integration. The frequency-response curves of the system and the bifurcation diagrams of Poincaré maps are analyzed.     

Nonlinear dynamic behavior analysis of an elastically restrained double-beam connected through a mass-spring system that is nonlinear

Some complex engineering structures can be modeled as multiple beams connected through coupling elements. When the coupling element is elastic, it can be simplified as a mass-spring system. The existing studies mainly concentrated on the double-beam coupled through elastic connectors, where the connector is simplified as the equivalent linear stiffness element or linear mass-spring system. Furthermore, many researches ignore rotational boundary restraints in analyzing dynamic behavior of the double-beam connected through elastic connectors, limiting their engineering generality. Considering the above limitations, this study attempts to employ the cubic nonlinear stiffness in the coupling mass-spring system and study the potential application of the mass-spring system that is nonlinear on the vibration control of the double-beam system. Using the variational method and the generalized Hamiltonian method build the corresponding system’s governing functions. Applying the Galerkin truncation method (GTM) obtains the dynamic behavior of the double-beam connected through a mass-spring system that is nonlinear. According to this study, the change of the mass-spring system that is nonlinear significantly influences the dynamic behavior of the double-beam system, where the complex dynamic behavior occurs under certain parameters of the mass-spring system that is nonlinear. Suitable parameters of the mass-spring system that is nonlinear are good at the vibration suppression at the boundary of the vibration system. Furthermore, the mass-spring system that is nonlinear can change the characteristics of the double-beam system’s kinetic energy transfer. For the vibration model established in this work, a quasi-periodic vibration state can be regarded as a sign of the occurrence of the targeted energy transfer of the double-beam connected through a mass-spring system that is nonlinear.

     

自由端受磁力吸引悬臂梁的非线性振动

研究了含分式函数的非线性磁力系统的受迫振动。磁铁吸引力横向作用在悬臂梁自由端处。根据含有非线性边界的连续体模型,给出了非平凡静平衡位形。以稳定位形作坐标变换后,计算了相应的固有频率,并求得微幅受迫振动稳态响应的近似解析解。研究了不同磁铁间距与频响曲线的关系,结果表明磁力方向和大小对它们均有较大影响。数值结果与解析结果吻合得很好。     

Nonlinear dynamic analysis of a structure with a friction-based seismic base isolation system

Many dynamical systems are subject to some form of non-smooth or discontinuous nonlinearity. One eminent example of such a nonlinearity is friction. This is caused by the fact that friction always opposes the direction of movement, thus changing sign when the sliding velocity changes sign. In this paper, a structure with friction-based seismic base isolation is regarded. Seismic base isolation can be employed to decouple a superstructure from the potentially hazardous surrounding ground motion. As a result, the seismic resistance of the superstructure can be improved. In this case study, the base isolation system is composed of linear laminated rubber bearings and viscous dampers and nonlinear friction elements. The nonlinear dynamic modelling of the base-isolated structure with the aid of constraint equations, is elaborated. Furthermore, the influence of the dynamic characteristics of the superstructure and the nonlinear modelling of the isolation system, on the total system’s dynamic response, is examined. Hereto, the effects of various modelling approaches are considered. Furthermore, the dynamic performance of the system is studied in both nonlinear transient and steady-state analyses. It is shown that, next to (and in correlation with) transient analyses, steady-state analyses can provide valuable insight in the discontinuous dynamic behaviour of the system. This case study illustrates the importance and development of nonlinear modelling and nonlinear analysis tools for non-smooth dynamical systems.