Asymptotic analysis of kinematically excited dynamical systems near resonances

Starosta, R.; Sypniewska–Kamińska, Grażyna; Awrejcewicz, Jan

doi:10.1007/s11071-011-0229-6

Cited by 48 publications

(35 citation statements)

References 6 publications

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“…The nonlinear two-degree-of-freedom system has been examined in [19]. The analytical approximate solution up to the third order is obtained using the same previous method.…”

Section: Introductionmentioning

confidence: 99%

The Dynamical Behavior of a Rigid Body Relative Equilibrium Position

Amer

2017

Advances in Mathematical Physics

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In this paper, we will focus on the dynamical behavior of a rigid body suspended on an elastic spring as a pendulum model with three degrees of freedom. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity. The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange’s equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the fourth-order Runge-Kutta algorithms through Matlab packages. These solutions are represented graphically in order to describe and discuss the behavior of the body at any instant for different values of the physical parameters of the body. The obtained results have been discussed and compared with some previous published works. Some concluding remarks have been presented at the end of this work. The importance of this work is due to its numerous applications in life such as the vibrations that occur in buildings and structures.

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“…The nonlinear two-degree-of-freedom system has been examined in [19]. The analytical approximate solution up to the third order is obtained using the same previous method.…”

Section: Introductionmentioning

confidence: 99%

The Dynamical Behavior of a Rigid Body Relative Equilibrium Position

Amer

2017

Advances in Mathematical Physics

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“…After arranging the components of both equations according to the powers of the small parameter, this requirement is realized by equating to zero all coefficients standing at the succeeding powers of ε. Then, the obtained system of equations is solved recursively [22][23][24].…”

Section: Approximate Analytical Solution For Non-resonant Casementioning

confidence: 99%

“…4 are depicted the approximate solutions obtained using MSM and calculated numerically. The values of error (23) Initial value problem (50)-(54) arose in process of solving motion equations (10)-(12) using MSM is a good basis for studying the steady-state forced vibration of the micromechanical gyroscope. For this purpose, it is convenient to introduce modified phases Ψ 1 (τ ) and Ψ 2 (τ ) as follows…”

Section: Resonant Vibrationmentioning

confidence: 99%

Complexity of resonances exhibited by a nonlinear micromechanical gyroscope: an analytical study

2018

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Dynamics behavior of the micromechanical gyroscope designed for measuring one component of the angular velocity is studied in the paper. The Cardan suspension is applied to connect the sensing plate with the substrate whose angular velocity is measured. The gimbal and the plate with sensors are connected via torsional joints. Vibrating motion of the sensing plate is excited mainly by a torque resulting from the Coriolis effect. The mathematical model equations have been derived using the Lagrange equation of the second kind. Both nonlinear effects of the geometrical nature and the nonlinear characteristics of the torsional joints are taken into account. The governing equations are solved with help of the method of multiple scales in time domain that belongs to the broad class of asymptotic methods. The approximate solution of analytical form has been obtained for non-resonant vibration as well as for the case of the main and internal resonances that occur simultaneously. Analytical form of solution allows for extensive analysis of the behavior of the system. The desirable state of the gyroscope work is steady-state vibration in resonance that is discussed in detail.

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“…An analysis of the resonance solutions in multi-degree-offreedom systems is made in numerous publications due to exclusive importance of the problem for theory and applications. In particular, the multiple-scale method allows us to analytically solve the equations of motion and recognize resonances [28][29][30]. It is known that all analytical transformations are essentially complicated with an increase of the number of the degrees of freedom of the system under consideration.…”

Section: Iteration Procedures To Construct Forced Non-linear Normal VImentioning

confidence: 99%

“…Equations of the rotor motion are the following: x ; k 2 ð Þ y are similar coefficients for the right support; β is a coefficient of damping in supports; ρ 1 ; ρ 2 are coefficients of damping during the disk motion; m is the disk mass; ε is an eccentricity of the disk mass center. Note that the cubic non-linearity of Duffing type can be considered as an acceptable approximation for different types of the support restoring forces [18,22,23,28,30].…”

Section: Principal Model Of the Rotor Dynamicsmentioning

confidence: 99%

Non-linear normal forced vibration modes in systems with internal resonance

Perepelkin

Mikhlin

Pierre

2013

International Journal of Non-Linear Mechanics

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The new method of the forced resonance vibrations construction in mechanical systems with internal resonance is represented. According to this approach, the generalized theory of non-linear normal vibration modes by Shaw and Pierre, the modified Rauscher method and the harmonic balance method are combined with a new iterative computation procedure. The proposed approach is used in analysis of the single-disk rotor system with the isotropic-elastic shaft and the non-linear supports of Duffing type. Gyroscopic effects, asymmetrical disposition of the disk on the shaft and internal resonance are also taken into account. The NNM approach allows reducing the 8-DOF problem of the rotor dynamics to the 2-DOF non-linear system for each non-linear normal mode. Both the model of massless supports and the model of supports with inertial effects are considered. It is shown that in last case all resonance regimes are separated into two different kinds. First kind corresponds to cyclic symmetric trajectories in a system's configuration space; the second kind corresponds to centrally symmetric ones. Regimes of the first kind can be evaluated by the use of the simplified mathematical model proposed in this work. Simplified model consists only of four generalized coordinates instead of the eight initial ones.

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Asymptotic analysis of kinematically excited dynamical systems near resonances

Cited by 48 publications

References 6 publications

The Dynamical Behavior of a Rigid Body Relative Equilibrium Position

The Dynamical Behavior of a Rigid Body Relative Equilibrium Position

Complexity of resonances exhibited by a nonlinear micromechanical gyroscope: an analytical study

Non-linear normal forced vibration modes in systems with internal resonance

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