The Sommerfeld effect of second kind: passage through parametric instability in a rotor with non-circular shaft and anisotropic flexible supports

Bharti, Saurabh Kumar; Sinha, Anubhab; Samantaray, Arun Kumar; Bhattacharyya, Ranjan

doi:10.1007/s11071-020-05681-9

Cited by 21 publications

(9 citation statements)

References 64 publications

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“…Note that one finds the same behavior in rotor dynamics, where the Sommerfeld rotor leads to the same driving characteristic [2] when in Equation ( 12) the road factor z 0 Ω is replaced by the mass ratio of the unbalanced rotor, the translation velocity by the rotation speed ratio, and the driving force by the moment. More details on Sommerfeld effects in rotor dynamics are given in [16][17][18][19][20].…”

Section: Speed Driving Force Characteristic Of Traveling Vehiclesmentioning

confidence: 99%

Speed Oscillations of a Vehicle Rolling on a Wavy Road

Wedig

2021

Applied Sciences

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Every driver knows that his car is slowing down or accelerating when driving up or down, respectively. The same happens on uneven roads with plastic wave deformations, e.g., in front of traffic lights or on nonpaved desert roads. This paper investigates the resulting travel speed oscillations of a quarter car model rolling in contact on a sinusoidal and stochastic road surface. The nonlinear equations of motion of the vehicle road system leads to ill-conditioned differential-algebraic equations. They are solved introducing polar coordinates into the sinusoidal road model. Numerical simulations show the Sommerfeld effect, in which the vehicle becomes stuck before the resonance speed, exhibiting limit cycles of oscillating acceleration and speed, which bifurcate from one-periodic limit cycle to one that is double periodic. Analytical approximations are derived by means of nonlinear Fourier expansions. Extensions to more realistic road models by means of noise perturbation show limit flows as bundles of nonperiodic trajectories with periodic side limits. Vehicles with higher degrees of freedom become stuck before the first speed resonance, as well as in between further resonance speeds with strong vertical vibrations and longitudinal speed oscillations. They need more power supply in order to overcome the resonance peak. For small damping, the speeds after resonance are unstable. They migrate to lower or supercritical speeds of operation. Stability in mean is investigated.

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Section: Speed Driving Force Characteristic Of Traveling Vehiclesmentioning

confidence: 99%

Speed Oscillations of a Vehicle Rolling on a Wavy Road

Wedig

2021

Applied Sciences

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“…The process of passing the rotor system through critical speeds, considering its interaction with the energy source, was considered in many studies. 1–11 In particular, it was found that under conditions when the power reserve of the energy source is small, the course of the process significantly depends on the characteristics of the energy source. Changes in the rotor speed turn out to be closely related to the change in vibration amplitude.…”

Section: Introductionmentioning

confidence: 99%

“…The Sommerfeld effect of the first kind refers to the resonance at a synchronous rotor vortex (critical speeds), while the second kind effect refers to the instability of the rotor vortex. 4…”

Section: Introductionmentioning

confidence: 99%

“…With such excess power, the transition from a lower limiting rate of instability to a much higher stable rate occurs as a nonlinear surge phenomenon. [1][2][3][4] These symptoms, commonly referred to as the Sommerfeld effect, arise from the internal energy coupling between the drive and the driven system and are important design considerations in the design of various rotating machines with flexible supports (bearings).…”

Section: Introductionmentioning

confidence: 99%

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Dynamic modeling of a non-ideal gyroscopic rotor system with nonlinear damping and nonlinear rigidity of an elastic support

Iskakov¹,

Jamalov

Bissembayeb

et al. 2022

Advances in Mechanical Engineering

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The article considers the effect of joint linear and nonlinear cubic damping on dynamics of a gyroscopic rigid rotor interacting with an electric motor with a rectilinear characteristic, taking into account the nonlinear rigidity of the support material. The method of regulating the control parameter (voltage on the motor), the amplitude of vibration, and the angular velocity of the shaft in the frequency equation, depending on the value of the coefficient of nonlinear cubic damping of the support, offers the most effective options for controlling resonant oscillations of large amplitudes. It is shown that the greater the value of the coefficient of nonlinear cubic damping, the easier it is to control these oscillations. Moreover, it is proved that the Sommerfeld effect (of the first kind) can also be weakened and eliminate with the help of joint linear and nonlinear damping. To do this, in the case of a rigid characteristic of the nonlinear elasticity of the support material, in a rotor system with a nonideal energy source to eliminate the bistability region, that is, jumping effects, more nonlinear damping of support or energy from a nonideal energy source will be required than in the case of an ideal rotor system.

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“…Analytical expressions for the stability regions of Mathieu's equation were studied recently [20]. Parametric instabilities, bifurcations and chaotic motion are not only important in various vibrating mechanical or electromechanical systems, e.g., [21,22]. They also continue to be of research interest in various pendulum systems [23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

Parametric instability of a magnetic pendulum in the presence of a vibrating conducting plate

2020

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A pendulum with an attached permanent magnet swinging in the vicinity of a conductor is a typical experiment for the demonstration of electromagnetic braking and Lenz’ law of induction. When the conductor itself moves, it can transfer energy to the pendulum. An exact analytical model of such an electromagnetic interaction is possible for a flat conducting plate. The eddy currents induced in the plate by a moving magnetic dipole and the resulting force and torque are known analytically in the quasistatic limit, i.e., when the magnetic diffusivity is sufficiently high to ensure an equilibrium of magnetic field advection and diffusion. This allows us to study a simple pendulum with a magnetic dipole moment in the presence of a horizontal plate oscillating in vertical direction. Equilibrium of the pendulum in the vertical position can be realized in three cases considered, i.e., when the magnetic moment is parallel to the rotation axis, or otherwise, its projection onto the plane of motion is either horizontal or vertical. The stability problem is described by a differential equation of Mathieu type with a damping term. Instability is only possible when the vibration amplitude and the distance between plate and magnet satisfy certain constraints related to the simultaneous excitation and damping effects of the plate. The nonlinear motion is studied numerically for the case when the magnetic moment and rotation axis are parallel. Chaotic behavior is found when the eigenfrequency is sufficiently small compared to the excitation frequency. The plate oscillation typically has a stabilizing effect on the inverted pendulum.

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The Sommerfeld effect of second kind: passage through parametric instability in a rotor with non-circular shaft and anisotropic flexible supports

Cited by 21 publications

References 64 publications

Speed Oscillations of a Vehicle Rolling on a Wavy Road

Speed Oscillations of a Vehicle Rolling on a Wavy Road

Dynamic modeling of a non-ideal gyroscopic rotor system with nonlinear damping and nonlinear rigidity of an elastic support

Parametric instability of a magnetic pendulum in the presence of a vibrating conducting plate

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