Modelling of Web Dynamics in Paper Making Processusers.jyu.fi/~tyjosaks/seminar/Saksa_seminar2011b.pdf · · 2011-11-25Modelling of Web Dynamics in Paper Making Process ... Eigenvalue

MotivationResearch ProblemNumerical Results

Future Plans and Expected ResultsSummary

Modelling of Web Dynamics in Paper Making

Process

Tytti Saksa

Department of Mathematical Information Technology

University of Jyväskylä

Postgraduate Seminar in Information Technology,November 28, 2011

Tytti Saksa Paper Web Dynamics



Background and Current Status

Name: Tytti Saksa

Master's degree in Mathematics, December 2009

Postgraduate studies begun in January 2010

Status of the Ph.D. Thesis

Two years behind � two years ahead

Expected dissertation, December 2013

Format: collection of articles

Two articles published, two submitted

57 ECTS / 60 ECTS completed




Research Collaborators

Research group members

Lead by Prof. Pekka Neittaanmäki (supervisor)Prof. Nikolay Banichuk, RAS (2 one-month visits per year,supervisor)M.Sc. Juha JeronenM.Sc. Tero TuovinenM.Sc. Maria TirronenTech.Lic. Matti KurkiMe

Prof. Raino Mäkinen is also supervising my Ph.D. studies




Outline

1 Motivation

2 Research Problem

3 Numerical Results

4 Future Plans and Expected Results




MotivationAbout this presentation

Paper Making

Desire for high productivity and controlled machine running.

Press

nip

Dryer

section

Detail of a paper machine





Avoiding Web Breaks

Web breaks are expensive.

Non-productive machinerunning causes signi�canteconomic losses.





Original Research

This presentation is based on the following research:

Banichuk, Jeronen, Saksa and Tuovinen. Journal of StructuralMechanics (Rakenteiden mekaniikka). 44:172�185, 2011.




ModelObjectivesPhenomenon

Model

The travelling paper web ismodelled as a moving beam orpanel taking into account the(Earth's) gravity.

Panel: the webdisplacement does notvary in the cross directionto the movement.

V0 = web velocity, T =tension, g = the standardgravity, m = mass perunit area.





Research Objectives in Figures

How fast can we run the traveling plate?

What is the shape of the buckled plate?





Research Objectives in Words

Stability analysis of thin, elastic panels.

Critical conditions and critical velocity estimations.

E�ects of the gravity on the web behaviour and stability.

The e�ect of gravity has usually been neglected.





The Governing Equations

The dynamic equation for the transversedisplacement w of the panel:

mwtt +2mV0wxt +mV 20 wxx = T wxx +Tx wx −Dwxxxx

Tension varies due to the gravity:

T (x) = T0 +mgx

Pinned (hinged) boundary conditions:w(0) = wxx(0) = 0, w(`) = wxx(`) = 0.





Buckling Problem

Dimensionless static form of the buckling problem:

α wxxxx +(c20 −1−βx)wxx −β wx = 0 ,

w(0) = wxx(0) = 0 , w(1) = wxx(1) = 0 .

Energy:

α

∫ 1

0w2xx dx +(1− c20)

∫ 1

0w2x dx +β

∫ 1

0xw2

x dx = 0 .

Goal: to �nd the minimal (dimensionless) velocity c0 at whichthe energy equation (above) has a solution.





Optimization Problem

First, the buckling problem is solved minimizing the energy:

Solve

(c∗0)2 = min

w∈K

(1+α

∫ 10 (wxx)

2 dx∫ 10 (wx)2 dx

+β

∫ 10 x (wx)

2 dx∫ 10 (wx)2 dx

),

whereK = {w ∈ C ([0,1]) : w(0) = 0 , w(1) = 0} .





Estimations for the Critical Velocity

From the minimization problem on the previous slide, we may derivethe following analytical estimates for the critical buckling velocity

1+α π2 ≤ (c∗0)

2 ≤ 1+α π2 +

β

2.

The parameters (above) are de�ned as

c0 := V0/√T0/m ,

α :=D

`2T0, β :=

mg`

T0.





Eigenvalue Problem

Secondly, the buckling problem is solved directly as an eigenvalueproblem:

α wxxxx −β (xwxx +wx) = λ wxx ,

w(0) = wxx(0) = 0 , w(1) = wxx(1) = 0 ,

whereλ := 1− c20

is the eigenvalue.

Goal: �nd the largest eigenvalue λ (corresponding to the minimalcritical velocity c0) of the eigenvalue problem.




Observations madeIllustrations

Overall Discussion

Two di�erent solution methods were presented and realizednumerically.

There was a very good agreement in the results given by theenergy minimization and the eigenvalue problem.

The optimization approach was found to be trusted.

The gravity had a large e�ect on the buckling mode, while thee�ect on the critical velocity was minor.





Buckling Mode Dependence on the Band Angle to the

Gravity

! = 0! = " / 8

! = " / 4

! = 3 " / 8

! = " / 2





E�ect of the Problem Parameters on the Critical Velocity

α corresponds to bending rigidity, β corresponds mass(assuming that the problem geometry does not change)

!

"

Critical velocity as fraction of sqrt(T0/m)

0 0.002 0.004 0.006 0.008 0.010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−5

1

1.0001

1.0002

1.0003

1.0004

1.0005

1.0006

1.0007

1.0008

1.0009




Future Plans

To continue studies on 1D viscoelastic model

one submitted paper, continuation to the work I presented inthis seminar in March 2011.

Optimization of the paper productivity

taking into account limitations in the web speed and growingof cracks at the web edges.




Expected Results

Basic research.

Increase the understanding of the physical phenomenon.

Real time simulations.

Direct application in paper making process.

About 5 journal papers.




Summary

Stability of thin, elastic panels (beams) travelling between twosupports was studied taking into account the e�ect of thegravity.

Analytical estimations for the critical velocities were found.

Energy minimization was found to be a reliable method forsolving this type of problems, but it was seen to be slowcompared to the (direct) eigenvalue problem solver.

The e�ect of the gravity on the buckling shapes was seen tobe large.




Thank You for Your Attention!


Appendix For Further Reading

For Further Reading

N. Banichuk, J. Jeronen, M. Kurki, P. Neittaanmäki, T. Saksa and T.

Tuovinen.

On the limit velocity and buckling phenomena of axially moving

orthotropic membranes and plates.International Journal of Solids and Structures. 48:2015-2025, 2011.doi:10.1016/j.ijsolstr.2011.03.010.

T. Saksa.

Dynamic behaviour of an axially moving viscoelastic panel in contact with

supporting rollers.In Proceedings of ECCOMAS Thematic Conference on ComputationalAnalysis and Optimization. Eds. S. Repin, T. Tiihonen and T. Tuovinen.University of Jyväskylä, 2011.

N. Banichuk, J. Jeronen, T. Saksa and T. Tuovinen.

Static instability analysis of an elastic band travelling in the gravitational

�eld.

Journal of Structural Mechanics (Rakenteiden mekaniikka). 44:172�185,2011. Tytti Saksa Paper Web Dynamics

Documents

Modelling of Web Dynamics in Paper Making Processusers.jyu.fi/~tyjosaks/seminar/Saksa_seminar2011b.pdf · · 2011-11-25Modelling of Web Dynamics in Paper Making Process ... Eigenvalue