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1

On High-Frequency Self-Excitation in Paper Calenders

Brommundt E.

Machine Dynamics Research

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2011

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Vol. 35, No. 4

5-10

EN

Oscillations at the nip of paper calenders spoil the quality of the paper, cross streaks – bars - occur, the machine will suffer damage. The paper outlines briefly the results of Brommundt (2009). The stability of the stationary motion of a machine with ideally circular elastic rolls is investigated. The paper compression in the nip has a hysteretic characteristic, for the small slip between rolls and paper holds a Coulomb type friction. Two thick-walled hollow cylinders serve as rolls. With respect to polar coordinates, in the unstable, the self-oscillating nonlinear system the third and second order Fourier terms of the noncircular deformation get the largest amplitudes. Thus, the frequency of the selfoscillation lies far above the basic critical frequencies of the rolls in their suspensions. The model can be applied to select stabilizing parameter variations.

2

A Simple Model for Friction Induced High-Frequency Self-Excitation of Paper Calenders

Brommundt E.

Machine Dynamics Problems

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2007

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Vol. 31, No. 2

25-45

EN

The model shows the possibility of friction induced high-frequency self-excitation at an ideally cylindrical roll of a paper calender (the ensuing corrugation by wear will amplify and finally govern the process). The basic investigation shows that the geometric and kinematical relations at the nip, together with friction and the material characteristics of the paper, here linearly visco-elastic, favour the excitation of higher order modes of the elastic ring which is taken as roll. The ring is attached to a flexibly mounted hub by a Winkler suspension; (all suspensions visco-elastic). The upper half only of a two roll calender is modelled, and oscillations symmetrical with respect to a horizontal middle plane are analyzed. The oscillations are restricted to the rigid body motions of the system and to a second order Fourier polynomial for the circumferential waves of the bending and the extensional displacements of the ring. Non-linear equations of motion of these 13 degrees of freedom system are established, simplified and, as initial value problem, numerically solved for estimated parameter values. The resulting limit cycle confirms the presumption.

3

Stabilization of a Rotor with Internal Friction by a Helical Damper Strip

Brommundt E.

Machine Dynamics Problems

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2005

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Vol. 29, No. 4

21-46

EN

The damper strip consists of a visco-elastic material which contains at its upper side, inlayed, several wires which are inextensible compared to the matrix. The strip is helically wound around and attached to the surface of the shaft. The influence of this device to the oscillations of a simple overhung rotor is investigated. Seen from a rotating frame, the equations of motion contain coefficients which are periodical with respect to the longitudinal coordinate of the shaft (but do not depend explicitly on the time). Numerical evaluations of the equations show that, depending on the parameters especially the pitch, there are regions of stable stationary operation at speeds well above the natural frequencies. Thus, the destabilizing effects of internal damping can be overcome selectively.

4

Some Dynamic Effects of Gear Couplings in Rotor Systems

Brommundt E.

Machine Dynamics Problems

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2004

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Vol. 28, No. 1

41-44

EN

This is a brief survey on dynamic effects due to the presence of a strongly non-linear gear coupling in an otherwise linear rotor system. Numerical results show strong influences of the gear coupling on the system behavior, and a high sensitivity to parameter variations. Proper misalignment can stabilize the rotor (the model of the gear coupling characteristic is the topic of a paper under preparation).

5

Moving Contact with Friction at an Elastic Strip

Brommundt E.

Machine Dynamics Problems

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2001

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Vol. 25, No. 3/4

21-34

EN

Careful investigations of sliding contact between flexible bodies require detailed formulations of the instantaneous local geometrical and equilibrium conditions at the touching surfaces. For a linearly elastic strip we discretize the dependent variables (displacements, strains, stresses) uniformly with respect to the transversal direction by cubic base-splines but retain the differential equations with respect to the longitudinal coordinate x. Thus, we arrive at a system of PDE's - second order in time, first order in x - which allows to display the essential boundary terms at the surfaces explicitly (that is convenient for the formulation of the contact conditions) and is well suited for numerical integration.

6

Stability of Motion of an Elastic Wheel Rolling on a Rigid Plane

Brommundt E.

Machine Dynamics Problems

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2000

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Vol. 24, No. 1

33-45

EN

In the paper by Brommundt (1998) the stationary motion of a rolling on an elastic strip was studied. Here we investigate the stability of such motion for the “inverted problem” : an elastic wheel rolls on a rigid plane. The goal of the paper is mostly methodical. Therefore, the deformations of the wheel are approximated by a very rough single layer linear elastodynamic model. The contact conditions are formulated by nonlinear equations. Local Coulomb friction is taken into account. The stability is studied by means of the pertinent linearized variational equations. They consist of two partial differential equations and several contact conditions. The corresponding eigenvalue problem is solved numerically. Many of the calculated eigenvalues of this model have (small) positive real parts, the stationary motion is always unstable (at least for our parameter set). In case that these eigenvalues lead to multifrequency selfexcited oscillations our result could explain the generation of rolling noise.

7

Self-Excited Oscillations of Disk Brakes

Römmich C., Brommundt E.

Machine Dynamics Problems

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1998

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Vol. 20

231-243

EN

The squealing of disk brakes is an effect of self-excited oscillations that is often explained by the presence of a coefficient of friction that decreases with increasing relative velocity, but also occurs in cases with an almost velocity-independent coefficient. We present a model for a disk brake that shows self-excited oscillations even for a constant coefficient of friction. The model consists of a rotating annular plate and two solid brake blocks. The system behaviour is described by a set of differential algebraic equations. We study the effect of several parameters on the stability of the stationary solution and calculate self-excited oscillations for unstable cases.

8

Rolling Contract with Friction of a Cylindrical Wheel on a Strip

Brommundt E.

Machine Dynamics Problems

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1998

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Vol. 20

49-72

EN

We establish a model for the rolling motion of an elastic wheel on an elastic strip. It includes carefully formulated (nonlinear) contact conditions for a regularized form of Coulomb's friction. The system resembles a railway aheel on a rail. The rim of the wheel and the strip (the head of the rail) are substituted by three-layer-models deduced from linear elastodynamics for plane deformations. For stationary rolling motion the set of governing (ordinary differential) equations is worked out in detail, nonlinear inside the contact region and linear outside. Numerical solutions to the problem are under investigation and will be published shortly. However, some facts can be read directly from the equations. Different from the common Hertz approach the pressure jumps at the borders of the contact region ; beside the contact length, the longitudinal position and the inclination of the contact surface depend on all loads, on the friction and on the overall dimensions of wheel and strip. A drawback of the present model is that it is based on "linear" elastodynamics.