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Textile Engineering & Fashion Technology

Mini Review Volume 4 Issue 6

Mathematical modelling of direct drive of card machine drum

Jerzy Zajaczkowski

Lodz University of Technology, Poland

Correspondence: Jerzy Zajaczkowski, Lodz University of Technology, 116 Zeromskiego Street, 90-924 Lodz, Poland

Received: August 10, 2017 | Published: November 19, 2018

Citation: Zajaczkowski J. Mathematical modelling of direct drive of card machine drum. J Textile Eng Fashion Technol. 2018;4(6):364-366. DOI: 10.15406/jteft.2018.04.00166

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Abstract

The paper is concerned with the mathematical modelling of a directly driven card machine drum. The equations of the motion are derived taking account of an interaction of a drum and the electric motor.

Introduction

The problem of vibration of cylindrical shells is technically important. The equations of the motion can be found in works.1–6 The natural vibrations have been studied in papers.7–9 The vibrations of rotating shells have been investigated in papers.10,11 The phenomena of bifurcation and chaos in externally excited circular cylindrical shells have been reported in work.12 The stress and transverse deflection of a card machine drum, in a form of a cylindrical shell with wound wire, have been calculated in.13 The purpose of this work is to formulate a mathematical model of a direct drive of a card machine drum.

Equations of the motion

The rotating shell of mean radius r is shown in Figure 1. Here, m denotes unit mass, g - gravity acceleration and t - time. The rotation angle of shell is denoted as α, the longitudinal and the angular co-ordinates of shell elements are denoted as x and φ. The longitudinal, circumferential and radial components of the shell surface deflection are denoted as u, v and w respectively.

Figure 1 The rotating shell.

The components of the acceleration of the shell element have the form

av=2vt22dαdtwt+d2αdt2(rw)(dαdt)2vav=2vt22dαdtwt+d2αdt2(rw)(dαdt)2v

aw=2wt2+2dαdtvt+d2αdt2v+(dαdt)2(rw)aw=2wt2+2dαdtvt+d2αdt2v+(dαdt)2(rw)(1)

Taking into account the inertia, damping and gravity forces, the loading of an element can be written as

qx=c1utm2ut2qx=c1utm2ut2

qϕ=c2vtm(2vt22dαdtwt+d2αdt2(rw)(dαdt)2v)+mgcos(α+ϕ)qsδ(α)qϕ=c2vtm(2vt22dαdtwt+d2αdt2(rw)(dαdt)2v)+mgcos(α+ϕ)qsδ(α)
qz=c3wtm(2wt2+2dαdtvt+d2αdt2v+(dαdt)2(rw))mgsin(α+ϕ)+qnδ(α)qz=c3wtm(2wt2+2dαdtvt+d2αdt2v+(dαdt)2(rw))mgsin(α+ϕ)+qnδ(α)(2)

Here, (c1, c1, c1) are coefficients of damping. The internal forces are assumed to be of the form

N'ϕ=m(dαdt)2r2
N'x=Nx1+Nx2cosϕ

N'xϕ=0(3)

The internal torque is

Ms(x)=mx02π0(2vt22dαdtwt+d2αdt2(rw)(dαdt)2v)(rw)rdϕdx(4)

The integration over entire length L of the shell gives the driving torque Ms. The angular acceleration is found as

d2αdt2=MsmrL02π0((2vt22dαdtwt(dαdt)2v)(rw))dϕdxx02π0(rw)2dϕdx(5)

The components of the shell surface deflection can be expressed in the form

u(x.ϕ,t)=UTu,v(x.ϕ,t)=VTv,w(x.ϕ,t)=WTw(6)

Where

Tu=[T(0)un(t)T(CN)un(t)T(SN)un(t)],Tv=[T(0)vn(t)T(SN)vn(t)T(CN)vn(t)],Tw=[T(0)wn(t)T(CN)wn(t)T(SN)wn(t)],(7)

U=[U(0)n(x),U(CN)n(x)cosNϕ,U(SN)n(x)sinNϕ],V=[V(0)n(x),V(SN)n(x)sinNϕ,V(CN)n(x)cosNϕ],W=[W(0)n(x),W(CN)n(x)cosNϕ,W(SN)n(x)sinNϕ],(8)

Un(x)=[U1(x),U2(x),...,Un(x)],Vn(x)=[V1(x),V2(x),...,Vn(x)],Wn(x)=[W1(x),W2(x),...,Wn(x)], (9)

For n=1,2,3,...nmax, N=1,2,3,...Nmax.

Substituting (6-9) into equation (5) and carrying out the integration (5), one obtains

d2αdt2=(MsmrˉVd2Tvdt2r+2dαdtˉWdTwdtr+(dαdt)2ˉVTvr+¯VTW,d2Tvdt2TTw2dαdt¯WTW,dTwdtTTw(dαdt)2¯VTW,TvTTw)2πLr22rˉWTw+¯WTW,TwTTw,

L02π0()dϕdx=¯(),A,B=rcArcBrc.(10)

The partial differential equations governing the motion of shell element will now be replaced by ordinary differential equations using Ritz-Galerkin. Let’s pre-multiply those equations4,5 by the transposition of functions (8) and then integrate of the resultant expressions by parts with the use of the forces-displacement relationships and boundary conditions

2π0l0UT(Nxx+1aNϕxϕQx2wx2Nxϕ2vx2)dxdϕ+

2π0l0UT(Qϕa(vx+2wxϕ)Nϕa(2vxϕwx)+qx)dxdϕ=0,

2π0l0VT(1aNϕϕ+Nxϕx+Nx2vx2Qxa(vx+2wxϕ))dxdϕ+

+2π0l0VT(Nϕxa(2vxϕwx)Qϕa(1+vaϕ+2waϕ2)+qϕ)dxdϕ=0,

2π0l0WT(Qxx+1aQϕϕ+x(Nxϕ(va+waϕ))+x(Nxwx))dxdϕ+

+2π0l0WT(Nϕa+ϕ(Nϕa(va+waϕ))+ϕ(Nϕxa(wx))+qz)dxdϕ=0. (11)

As a result of the integration, we obtain a set of three ordinary differential equations of second order

K11Tu+K12Tv+K13Tw+K(1)12(dαdt)2Tv+K(1)13(dαdt)2Tw+C11dTudt+M11d2Tudt2=F1,

K21Tu+K22Tv+K23Tw+K(1)22(dαdt)2Tv+K(2)23d2αdt2Tw+C23dαdtdTwdt++C22dTvdt+M22d2Tvdt2+M24d2αdt2+F2ccosα+F2ssinα+F2(α)=0,

K31Tu+K32Tv+K33Tw+K(1)32(dαdt)2Tv+K(1)33(dαdt)2Tw+K(2)32d2αdt2Tv++C32dαdtdTvdt+C33dTwdt+M33d2Twdt2+F3ccosα+F3ssinα+F3(α)=0.(12)

Introducing matrices of three times higher order, than the matrices in equations (12), these set can be rewritten as one equation

Md2Tdt2+CdTdt+KT+(dαdt)2K(1)T+d2αdt2K(2)T+C(1)dαdtdTdt+Fccosα+Fssinα+F(α)=0(13)

Where T=[Tu,Tv,Tw].

Adding now equation governing the driving torque of the motor

dMsdt=1Ts(Cs(Ωsdαdt)Ms)(14)

We obtained the set of three equations (10,13,14) from which angle of drum rotation α, components of elastic deflection T (7-9)and driving torque Ms can be found through numerical integration. Similar calculations were performed by the author for a spindle directly driven by three-phase electric motor.14

Acknowledgements

None.

Conflict of interest

Author declares there is no conflict of interest.

References

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  14. Zajaczkowski J. Stability of a direct drive of a spindle in a spinning machine. Fibres & Textiles Eastern Europe. 2000;83(30):63–65.
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