Chaotic frictional vibrations excited by a quasiperiodic load

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The vibrations of a single-disk rotor on nonlinear flexible supports are modeled taking into account the gyroscopic moments acting on the disk. The dynamics of this nonlinear system is analyzed using the multiple scales method Introduction. Elements of power equipment are rotors that sustain critical vibrations. Models describing the vibrations of rotor systems include various nonlinearities such as bearing clearance, forces exerted by the substrate oil film on pins, nonlinearity of bearing supports.Dimentberg [4] examined the effect of internal friction on the motion of rotors. Grobov [2] proposed to use asymptotic methods of nonlinear mechanics to analyze the dynamics of rotor systems. Filippov [6] studied the nonstationary vibrations of a single-disk rotor with one nonlinear support. Bolotin [1] proposed to consider the nonlinear lag in the single-disk rotor model. Tondl [5] modeled the vibrations of rotors and analyzed their motion for stability. The quasiperiodic vibrations of rotors were studied in [3]. The subharmonic vibrations of a rotor with two degrees of freedom were examined in [11] considering bearing clearances. Synchronization in vibrations of rotor systems was discussed in [10].A literature review indicates that the nonlinear vibrations of rotor systems were studied considering only two degrees of freedom and disregarding the gyroscopic moments.We will analyze the vibrations of a rotor on nonlinear supports as systems with four degrees of freedom and take into account the gyroscopic moments. It will be shown that if the rotor is in the middle of a shaft, it will undergo precession motion because of nonlinearities and internal resonances.

“…To analyze the dynamics of system (6), we will modify the multiple-scales method [8,9]. The motion of the system is described bỹ …”

Section: Development Of the Multiple-scales Methodmentioning

confidence: 99%

“…The approach proposed here is essentially different from the standard procedure of the multiple-scales method [8,9] due to the presence of gyroscopic terms in (6).…”

Section: Development Of the Multiple-scales Methodmentioning

confidence: 99%

“…Substituting Eqs. (8a) into (8c) and equating the secular terms to zero [8,9], we get two complex modulation equations for the variables ( , ) A A …”

Section: Development Of the Multiple-scales Methodmentioning

confidence: 99%

mentioning

confidence: 99%

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Vibrations of a single-disk rotor on nonlinear supports

Аврамов

2009

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“…The nonlinear modes [12][13][14][15][16] are used to study the sixteen degree-of-freedom system (4.5). The method of nonlinear modes is not presented here.…”

Section: Example Of Calculationsmentioning

confidence: 99%

Nonlinear equations of flexural-flexural-torsional oscillations of rotating beamswith arbitrary cross-section

Аврамов¹,

Pierre

Shyriaieva³

2008

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A system of three nonlinear partial differential equations describing the flexural-flexural-torsional vibrations of a rotating slender cantilever beam of arbitrary cross-section is derived using Hamilton's principle. It is assumed that the center of gravity and the shear center are at different points. The interaction between flexural and torsional vibrations is accounted for in the linear and nonlinear parts of model Keywords: flexural-flexural-torsional vibrations, rotating beam, cross-section 1. Introduction. Pretwisted rotating beams are elements of wind-driven equipment, helicopter rotor blades, and robot's arms. While operating, these structures often undergo intensive oscillations. Turbine and helicopter rotor blades are designed to avoid resonance vibrations. Experimental investigations testify that the deformations of helicopter rotor blades during vibrations are geometrically nonlinear [7,9]. We stress that most of such systems have internal resonances [9]. The non-symmetry of the blade cross-section is often taken into account. In this case, the center of gravity and the shear center are considered as different points [10]. Timoshenko [10] derived the equations of linear flexural-flexural-torsional vibrations of beams with nonsymmetric cross-sections considering that the center of gravity and the shear center are at different points. Filippov [6] obtained a linear model of flexural-flexural-torsional oscillations of beams with arbitrary cross-sections taking shear strains and rotary inertia into account. Dzhanelidze [5] suggested equations of flexural-torsional vibrations of thin-walled rods. Hodges and Dowell [8] derived a system of nonlinear partial differential equations for geometrically nonlinear flexural-flexural-torsional-extensional vibrations of rotating beams assuming that the center of gravity and the shear center are at the same point. Crespo da Silva and Glynn [1, 2] considered equations of flexural-flexural-torsional vibrations of inextensible beam assuming that the center of gravity and the shear center are at the same point.We will derive a system of three nonlinear partial differential equations of flexural-flexural-torsional vibrations of a slender rotating beam with arbitrary cross-section, assuming that the center of gravity and the shear center of the cross-section are at different points. Hamilton's principle is used to obtain the equations of motion. We stress that in this case the interaction between flexural and torsional vibrations takes place in both the linear and nonlinear parts of model. Main Geometrically Nonlinear Relations.It is suggested that a twisted cantilever beam is rotating with constant angular velocity. The Euler-Bernoulli beam model is considered, i.e., the cross-sections of the beams remain planar during nonlinear oscillations [3]. The coordinate system ( $ , $ , $ )x h V is attached to the beams' cross-section to predict their motions (Fig. 1).

Dynamic-model reduction for a multimass system with nonideal constraint in a seismic damper

Antonyuk

Timokhin

2008

A dynamic model for a multimass system with a seismic damper as a nonlinear inertial component is reduced via balancing. The seismic damper is subject to a nonideal constraint. As an example, the order of the system of differential equations is reduced by one for a dynamic three-mass system. A comparison of the processes in the original and reduced systems confirms that it is possible to use balancing to simplify such models Introduction. Most kinematic pairs are subjected, to a greater or lesser extent, to friction, which in certain cases has a strong effect on dynamic processes in various engineering devices, machines, and mechanisms [2,8,10,14,18]. The friction in seismic dampers and structural elements of buildings and other structures contributes to the damping of seismic vibrations and to the decrease of the extreme dynamic loads [9,11,12,15]. Theoretical studies of dynamic processes play an important role in designing seismic isolation systems. This is why there is an objective need for mathematical models that would adequately describe their major properties and meet the requirements of numerical integration (see, e.g., [17]) such as that the system of differential equations must not be stiff [5,6,16,19], which frequently leads to the need to reduce the system.Due to the increasing interest to the seismoprotection of buildings and other structures [9, 20], we will outline the model of an in-line system that includes a passive seismic damper as a nonlinear inertial via balancing [16,19] for not only linear and nonlinear systems [5,13], the latter including inertialess and inertial components, but also systems that have a nonlinear inertial component with a nonideal constraint. The balancing method is detailed in [5,6,16,19].It is natural that the approach outlined here can be applied to not only buildings subject to seismic effects, but also mining, metallurgy, and farm machinery, machining tools, etc. subjected to their typical loads. Mathematical Model of a System with a Seismic Damper that Includes a Nonideal Constraint.First, we will briefly outline a model for a system with seismic dampers that include ideal constraints. With such an assumption, the reactions R i at the contact points between the bodies I and O are normal to the surface, i.e., the angles of friction r i between the normal and total reactions R are zero (Fig. 1).The body O undergoes plane translation and has only one degree of freedom because two of the three possible degrees of freedom of a free body in plane motion are excluded by the two normal constraints to the surfaces at the contact points between the bodies I and O. Each additional damper of the same shape and size (third, fourth, …, nth) would introduce a repeated (passive) constraint [4] not affecting the law of motion of the system, but making it (singly, doubly, (n -2)-tuply) statically indeterminate. The reactions in each damper cannot be found through kinetostatics. For the translating system in Fig. 1, it is convenient to use two equations of motion of the body O and an ...