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 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.     

Vibration control of a transversely excited cantilever beam with tip mass

The problem of controlling the vibration of a transversely excited cantilever beam with tip mass is analyzed within the framework of the Euler–Bernoulli beam theory. A sinusoidally varying transverse excitation is applied at the left end of the cantilever beam, while a payload is attached to the free end of the beam. An active control of the transverse vibration based on cubic velocity is studied. Here, cubic velocity feedback law is proposed as a devise to suppress the vibration of the system subjected to primary and subharmonic resonance conditions. Method of multiple scales as one of the perturbation technique is used to reduce the second-order temporal equation into a set of two first-order differential equations that govern the time variation of the amplitude and phase of the response. Then the stability and bifurcation of the system is investigated. Frequency–response curves are obtained numerically for primary and subharmonic resonance conditions for different values of controller gain. The numerical results portrayed that a significant amount of vibration reduction can be obtained actively by using a suitable value of controller gain. The response obtained using method of multiple scales is compared with those obtained by numerically solving the temporal equation of motion and are found to be in good agreement. Numerical simulation for amplitude is also obtained by integrating the equation of motion in the frequency range between 1 and 3. The developed results can be extensively used to suppress the vibration of a transversely excited cantilever beam with tip mass or similar systems actively.     

Nonlinear response of a flexible Cartesian manipulator with payload and pulsating axial force

In this present work, the nonlinear response of a single-link flexible Cartesian manipulator with payload subjected to a pulsating axial load is determined. The nonlinear temporal equation of motion is derived using D’Alembert’s principle and generalised Galerkin’s method. Due to large transverse deflection of the manipulator, the equation of motion contains cubic geometric and inertial types of nonlinearities along with linear and nonlinear parametric and forced excitation terms. Method of normal forms is used to determine the approximate solution and to study the dynamic stability and bifurcations of the system. These results are found to be in good agreement with those obtained by numerically solving the temporal equation of motion. Influences of amplitude of the base excitation, mass ratio, and amplitude of static and dynamic axial load on the steady state responses of the system are investigated for three different resonance conditions. For some specific conditions, the results obtained in this work are found to be in good agreement with the previously published experimental work. The results obtained in this work will find applications in the design of flexible Cartesian manipulators with payload.     

Nonlinear dynamic analysis of a Cartesian manipulator carrying an end effector placed at an intermediate position

Nonlinear dynamic analysis of a Cartesian manipulator carrying an end effector which is placed at different intermediate positions on the span is theoretically investigated with a single mode approach. The governing equation of motion of this system is formulated by using the D??Alembert principle in addition to profuse application of Dirac delta function to indicate the location of the intermediate end effector. Then the governing equation is further reduced to a second-order temporal differential equation of motion by using Galerkin??s method. The method of multiple scales as one of the perturbation techniques is being used to determine the approximate solutions and the stability and bifurcations of the obtained approximate solutions are studied. Numerical results are demonstrated to study the effect of intermediate positions of the end effector placed at various locations on the link with other relevant system parameters for both the primary and secondary resonance conditions. It is worthy of note that the catastrophic failure of the system may take place due to the presence of jump phenomenon. The results are found to be in good agreement with the results determined by the method of multiple scales after solving the temporal equation of motion numerically. In order to determine physically realized solution by the system, basins of attraction are also plotted. The obtained results are very useful in the application of robotic manipulators where the end effector is placed at any arbitrary position on the robot arm.     

Non-linear response of a magneto-elastic translating beam with prismatic joint for higher resonance conditions

The non-linear response of a magneto-elastic translating beam having prismatic joint for higher resonance conditions is studied. A periodically varying transverse magnetic field is applied to the system. Two frequencies of prismatic motion and oscillating transverse magnetic field are implemented to the system. The method of multiple scales as one of the perturbation techniques is used to derive two first order ordinary differential equations that govern the time variation of the amplitude and phase of the response. Then a stability analysis is conducted for subharmonic resonance and simultaneous resonance conditions. A parametric study is performed to investigate the effect of magnetic field strength, amplitude of prismatic motion, damping and payload mass on the frequency response curves for both the resonance conditions. The catastrophic failure of the system may occur due to the presence of saddle-node and pitchfork bifurcations. The results obtained by method of multiple scales are compared with those obtained by numerically integrating the reduced equations and are found to be in good agreement. The developed results can be applied to control the vibration of a beam with prismatic joint subjected to magnetic field for third order subharmonic resonance and simultaneous resonance conditions.     

Non-linear parametric vibration and stability of axially moving visco-elastic Rayleigh beams

An axially moving visco-elastic Rayleigh beam with cubic non-linearity is considered, and the governing partial-differential equation of motion for large amplitude vibration is derived through geometrical, constitutive, and dynamical relations. By directly applying the method of multiple scales to the governing equations of motion, and considering the solvability condition, the linear and non-linear frequencies and mode shapes of the system are analytically formulated. In the presence of damping terms, it can be seen that the amplitude is exponentially time-dependent, and as a result, the non-linear natural frequencies of the system will be time-dependent. For the resonance case, through considering the solvability condition and Routh–Hurwitz criterion, the stability conditions are developed analytically. Eventually, the effects of system parameters on the vibrational behavior, stability and bifurcation points of the system are investigated through parametric studies.     

Non-linear normal modes,invariance, and modal dynamics approximations of non-linear systems

Non-linear systems are here tackled in a manner directly inherited from linear ones, that is, by using proper normal modes of motion. These are defined in terms of invariant manifolds in the system's phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems — one based on a purely continuous approach, and one based on a discretized approach to which the theory developed for discrete systems can be applied-are simultaneously applied to the same study case-an Euler-Bernoulli beam constrained by a non-linear spring-and compared as regards accuracy and reliability. Numerical simulations of pure non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the non-linear normal modes are demonstrated, and it is also found that, for a pure non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differential equation.     

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation are studied analytically. The axial speed, as the source of parametric vibrations, is assumed to involve a mean speed, along with small harmonic variations. The method of multiple scales is applied to the governing non-linear equation of motion and then the natural frequencies and mode shape equations of the system are derived using the equation of order one, and satisfying the compatibility conditions. Using the equation of order epsilon, the solvability conditions are obtained for three distinct cases of axial acceleration frequency. For all cases, the stability areas of system are constructed analytically. Finally, some numerical simulations are presented to highlight the effects of system parameters on vibration, natural frequencies, frequency-response curves, stability, and bifurcation points of the system.     

Nonlinear response of an initially buckled beam with 1:1 internal resonance to sinusoidal excitation

The nonlinear response of an initially buckled beam in the neighborhood of 1:1 internal resonance is investigated analytically, numerically, and experimentally. The method of multiple time scales is applied to derive the equations in amplitudes and phase angles. Within a small range of the internal detuning parameter, the first mode; which is externally excited, is found to transfer energy to the second mode. Outside this region, the response is governed by a unimodal response of the first mode. Stability boundaries of the unimodal response are determined in terms of the excitation level, and internal and external detuning parameters. Boundaries separating unimodal from mixed mode responses are obtained in terms of the excitation and internal detuning parameters. Stationary and non-stationary solutions are found to coexist in the case of mixed mode response. For the case of non-stationary response, the modulation of the amplitude depends on the integration increment such that the motion can be periodically or chaotically modulated for a choice of different integration increments. The results obtained by multiple time scales are qualitatively compared with those obtained by numerical simulation of the original equations of motion and by experimental measurements. Both numerical integration and experimental results reveal the occurrence of multifurcation, escaping from one well to the other in an irregular manner. and chaotic motion.     

An analytical study of the non-linear vibrations of cylindrical shells

The primary resonance response of simply supported circular cylindrical shells is investigated using the perturbation method. Donnell's non-linear shallow-shell theory is used to derive the governing partial differential equations of motion. The Galerkin technique is then employed to transform the equations of motion into a set of temporal ordinary differential equations. Considering only the primary resonance case, the method of multiple scales is used to study the periodic solutions and their stability. The necessary and sufficient conditions for appearance of the so-called companion mode are also discussed. To this end, a range of the possible multi-mode solution is obtained for response and excitation amplitudes and also excitation frequency as a function of damping, geometry and material properties of the shell. Parametric studies are performed to illustrate the effect of different values of thickness, length and material composition on the possibility of the companion mode participation in primary resonance response.     

Large oscillations of beams and columns including self-weight

Large-amplitude, in-plane beam vibration is investigated using numerical simulations and a perturbation analysis applied to the dynamic elastica model. The governing non-linear boundary value problem is described in terms of the arclength, and the beam is treated as inextensible. The self-weight of the beam is included in the equations. First, a finite difference numerical method is introduced. The system is discretized along the arclength, and second-order-accurate finite difference formulas are used to generate time series of large-amplitude motion of an upright cantilever. Secondly, a perturbation method (the method of multiple scales) is applied to obtain approximate solutions. An analytical backbone curve is generated, and the results are compared with those in the literature for various boundary conditions where the self-weight of the beam is neglected. The method is also used to characterize large-amplitude first-mode vibration of a cantilever with non-zero self-weight. The perturbation and finite difference results are compared for these cases and are seen to agree for a large range of vibration amplitudes. Finally, large-amplitude motion of a postbuckled, clamped–clamped beam is simulated for varying degrees of buckling and self-weight using the finite difference method, and backbone curves are obtained.     

A novel scheme for nonlinear displacement-dependent dampers

In this paper, a novel scheme for nonlinear displacement-dependent (NDD) damper is introduced. The damper is attached to a simple mass-spring-damper vibration system. The vibration system equipped with a NDD damper is mathematically modeled and the nonlinear governing differential equation of the system is derived. To obtain the displacement of the system, the approximate analytical solution of the governing equation is elaborated using the multiple scales method. The advised approximate analytical algorithm is performed for several case studies and is also verified by the numerical fourth-order Runge?CKutta method. In addition, the performance of the NDD damper is analyzed and compared with the performance of the traditional linear damper. It is found that the proposed NDD damper scheme along with the multiple scales method is not only feasible for vibration reduction but also yields satisfactory response performance rather than the existing traditional linear damper.     

Nonlinear Response of a Parametrically Excited System Using Higher-Order Method of Multiple Scales

Two fundamentally different versions of the method of multiple scales (MMS) are currently in use in the study of nonlinear resonance phenomena. While the first version is the widely used reconstitution method, the second version is proposed by Rahman and Burton [1]. Both versions of the second-order MMS are applied to the differential equation obtained for a parametrically excited cantilever beam with a lumped mass at an arbitrary position. The bifurcation and stability of the obtained response show the difference between the two versions. While the Hopf bifurcation phenomena with no jump is found in the case of second-order MMS version I, both jump-up and jump-down phenomena are observed in second-order MMS version II, which closely agree with the experimental findings. The results are compared with those obtained by numerically integrating the original temporal equation.     

On non-linear dynamics of a coupled electro-mechanical system

Electro-mechanical devices are an example of coupled multi-disciplinary weakly non-linear systems. Dynamics of such systems is described in this paper by means of two mutually coupled differential equations. The first one, describing an electrical system, is of the first order and the second one, for mechanical system, is of the second order. The governing equations are coupled via linear and weakly non-linear terms. A classical perturbation method, a method of multiple scales, is used to find a steady-state response of the electro-mechanical system exposed to a harmonic close-resonance mechanical excitation. The results are verified using a numerical model created in MATLAB Simulink environment. Effect of non-linear terms on dynamical response of the coupled system is investigated; the backbone and envelope curves are analyzed. The two phenomena, which exist in the electro-mechanical system: (a)?detuning (i.e. a natural frequency variation) and (b)?damping (i.e. a decay in the amplitude of vibration), are analyzed further. An applicability range of the mathematical model is assessed.     

Non-linear dynamics of a slender beam carrying a lumped mass with principal parametric and internal resonances

The non-linear behaviour of a slender beam carrying a lumped mass subjected to principal parametric base excitation is investigated. The dimension of the beam–mass system and the position of the attached mass are so adjusted that the system exhibits 3 : 1 internal resonance. Multi-mode discretization of the governing equation which retains the cubic non-linearities of geometrical and inertial type is carried out using Galerkin’s method. The method of multiple scales is used to reduce the second-order temporal differential equation to a set of first-order differential equations which is then solved numerically to obtain the steady-state response and the stability of the system. The linear first-order perturbation results show new zones of instability due to the presence of internal resonance. For low amplitude of excitation and damping Hopf bifurcations are observed in the trivial steady-state response. The multi-branched non-trivial response curves show turning point, pitch-fork and Hopf bifurcations. Cascade of period and torus doubling, crises as well as the Shilnikov mechanism for chaos are observed. This is the first natural physical system exhibiting a countable infinity of horseshoes in a neighbourhood of the homoclinic orbit.     

Higher order linearization in non-linear random vibration

In this paper a higher order linearization method for analyzing non-linear random vibration problems is presented. The non-linear terms of the given equation are replaced by unknown linear terms. These are in turn described by extra non-linear differential equations. The combined system of equations is then linearized to arrive at a higher degree-of-freedom equation for the original system. The method is illustrated by considering the Duffing oscillator under white noise input. The equivalent two d.o.f linear system is derived by the present method. Numerical results on steady state variance and PSD functions are obtained. These are found to be better than the simple linearization results.     

圆板非线性振动有限元分析的一种迭代方法

同时考虑横向振动和板平面内的运动,用3节点有限元研究均匀圆板的轴对称大振幅非线性振动,构造了一个避免发散加速收敛的平均迭代法,并将计算结果与文献的已有结果做了比较。     

A unified Krylov-Bogoliubov method for solving second-order non-linear systems

A unified theory is presented for obtaining the transient response of second-order non-linear systems by the Krylov-Bogoliubov method. The method is a generalization of Bogoliubov's asymptotic method and covers all three cases when the roots of the corresponding linear equation are real, complex conjugate, or pure imaginary. It is shown that by suitable substitution for the roots in the general result, that the solution corresponding to each of the three cases can be obtained. The solution for the equation governing the motion of a simple pendulum with and without damping derived from the general solution reduces to that obtained by Popov's [4] method.