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.

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.

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

$a_v=\frac{{\partial }^{2}v}{\partial t^2}-2\frac{\text{d}\alpha }{\text{d}t}\frac{\partial w}{\partial t}+\frac{{\text{d}}^2\alpha }{\text{d}{t}^2}\left(r-w\right)-{\left(\frac{\text{d}\alpha }{\text{d}t}\right)}^{2}v$

$a_w=\frac{\partial {}^{2}w}{\partial t^2}+2\frac{\text{d}\alpha }{\text{d}t}\frac{\partial v}{\partial t}+\frac{{\text{d}}^2\alpha }{\text{d}{t}^2}v+{\left(\frac{\text{d}\alpha }{\text{d}t}\right)}^2\left(r-w\right)$(1)

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

$q_x=-c_1\frac{\partial u}{\partial t}-m\frac{{\partial }^{2}u}{\partial t^2}$

$q_{\varphi }=-c_2\frac{\partial v}{\partial t}-m\left(\frac{{\partial }^{2}v}{\partial t^2}-2\frac{\text{d}\alpha }{\text{d}t}\frac{\partial w}{\partial t}+\frac{{\text{d}}^2\alpha }{\text{d}{t}^2}\left(r-w\right)-{\left(\frac{\text{d}\alpha }{\text{d}t}\right)}^{2}v\right)+mg\cos \left(\alpha +\varphi \right)-q_s\delta \left(-\alpha \right)$
$q_z=-c_3\frac{\partial w}{\partial t}-m\left(\frac{\partial {}^{2}w}{\partial t^2}+2\frac{\text{d}\alpha }{\text{d}t}\frac{\partial v}{\partial t}+\frac{{\text{d}}^2\alpha }{\text{d}{t}^2}v+{\left(\frac{\text{d}\alpha }{\text{d}t}\right)}^2\left(r-w\right)\right)-mg\sin \left(\alpha +\varphi \right)+q_n\delta \left(-\alpha \right)$(2)

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

$N_{\varphi }^{\text{'}}=m{\left(\frac{\text{d}\alpha }{\text{d}t}\right)}^2{r}^2$
$N_x^{\text{'}}=N_{x^1}+N_{x^2}\cos \varphi$

$N_{x\varphi }^{\text{'}}=0$(3)

The internal torque is

$M_s\left(x\right)=m{\displaystyle \underset{0}{\overset{x}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\left(\frac{{\partial }^{2}v}{\partial t^2}-2\frac{\text{d}\alpha }{\text{d}t}\frac{\partial w}{\partial t}+\frac{{\text{d}}^2\alpha }{\text{d}{t}^2}\left(r-w\right)-{\left(\frac{\text{d}\alpha }{\text{d}t}\right)}^{2}v\right)\left(r-w\right)rd\varphi dx}}$(4)

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

$\frac{{\text{d}}^2\alpha }{\text{d}{t}^2}=\frac{\frac{M_s}{mr}-{\displaystyle \underset{0}{\overset{L}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\left(\left(\frac{{\partial }^{2}v}{\partial t^2}-2\frac{\text{d}\alpha }{\text{d}t}\frac{\partial w}{\partial t}-{\left(\frac{\text{d}\alpha }{\text{d}t}\right)}^{2}v\right)\left(r-w\right)\right)d\varphi dx}}}{{\displaystyle \underset{0}{\overset{x}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\left(r-w\right)}^{2}d\varphi dx}}}$(5)

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

$u\left(x.\varphi ,t\right)=U{T}_u,v\left(x.\varphi ,t\right)=V{T}_v,w\left(x.\varphi ,t\right)=W{T}_w$(6)

Where

$T_u=\left[\begin{array}{c}{T}_{un}^{(0)}\left(t\right)\\ T_{un}^{(CN)}\left(t\right)\\ T_{un}^{(SN)}\left(t\right)\end{array}\right],T_v=\left[\begin{array}{c}{T}_{vn}^{(0)}\left(t\right)\\ T_{vn}^{(SN)}\left(t\right)\\ T_{vn}^{(CN)}\left(t\right)\end{array}\right],T_w=\left[\begin{array}{c}{T}_{wn}^{(0)}\left(t\right)\\ T_{wn}^{(CN)}\left(t\right)\\ T_{wn}^{(SN)}\left(t\right)\end{array}\right],$(7)

$\begin{array}{l}U=\left[U_n^{(0)}\left(x\right),U_n^{(CN)}\left(x\right)\cos N\varphi ,U_n^{(SN)}\left(x\right)\sin N\varphi \right],\\ V=\left[V_n^{(0)}\left(x\right),V_n^{(SN)}\left(x\right)\sin N\varphi ,V_n^{(CN)}\left(x\right)\cos N\varphi \right],\\ W=\left[W_n^{(0)}\left(x\right),W_n^{(CN)}\left(x\right)\cos N\varphi ,W_n^{(SN)}\left(x\right)\sin N\varphi \right],\end{array}$(8)

$\begin{array}{l}{U}_n\left(x\right)=\left[U_1\left(x\right),U_2\left(x\right),...,U_n\left(x\right)\right],\\ V_n\left(x\right)=\left[V_1\left(x\right),V_2\left(x\right),...,V_n\left(x\right)\right],\\ W_n\left(x\right)=\left[W_1\left(x\right),W_2\left(x\right),...,W_n\left(x\right)\right],\end{array}$ (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

$\frac{{\text{d}}^2\alpha }{\text{d}{t}^2}=\frac{\left(\begin{array}{c}\frac{M_s}{mr}-\overline{V}\frac{d^2{T}_v}{d{t}^2}r+2\frac{\text{d}\alpha }{\text{d}t}\overline{W}\frac{d{T}_w}{dt}r+{\left(\frac{\text{d}\alpha }{\text{d}t}\right)}^2\overline{V}{T}_{v}r\\ +\langle \overline{V^{T}W},\frac{d^2{T}_v}{d{t}^2}{T}_w^T\rangle -2\frac{\text{d}\alpha }{\text{d}t}\langle \overline{W^{T}W},\frac{d{T}_w}{dt}{T}_w^T\rangle -{\left(\frac{\text{d}\alpha }{\text{d}t}\right)}^2\langle \overline{V^{T}W},T_v{T}_w^T\rangle \end{array}\right)}{2\pi L{r}^2-2r\overline{W}{T}_w+\langle \overline{W^{T}W},T_w{T}_w^T\rangle }$,

${\displaystyle \underset{0}{\overset{L}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\left(\right)d\varphi dx=\overline{\left(\right)}}},\langle A,B\rangle ={\displaystyle \sum _r^{}{\displaystyle \sum _c^{}{A}_{rc}{B}_{rc}}}$.(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

${\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{l}{\int }}{U}^T\left(\frac{\partial N_x}{\partial x}+\frac{1}{a}\frac{\partial N_{\varphi x}}{\partial \varphi }-Q_x\frac{{\partial }^{2}w}{\partial x^2}-N_{x\varphi }\frac{{\partial }^{2}v}{\partial x^2}\right)dxd\varphi }}+$

${\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{l}{\int }}{U}^T\left(-\frac{Q_{\varphi }}{a}\left(\frac{\partial v}{\partial x}+\frac{{\partial }^{2}w}{\partial x\partial \varphi }\right)-\frac{N_{\varphi }}{a}\left(\frac{{\partial }^{2}v}{\partial x\partial \varphi }-\frac{\partial w}{\partial x}\right)+q_x\right)dxd\varphi }}=0$,

${\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{l}{\int }}{V}^T\left(\frac{1}{a}\frac{\partial N_{\varphi }}{\partial \varphi }+\frac{\partial N_{x\varphi }}{\partial x}+N_x\frac{{\partial }^{2}v}{\partial x^2}-\frac{Q_x}{a}\left(\frac{\partial v}{\partial x}+\frac{{\partial }^{2}w}{\partial x\partial \varphi }\right)\right)dxd\varphi }}+$

$+{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{l}{\int }}{V}^T\left(\frac{N_{\varphi x}}{a}\left(\frac{{\partial }^{2}v}{\partial x\partial \varphi }-\frac{\partial w}{\partial x}\right)-\frac{Q_{\varphi }}{a}\left(1+\frac{\partial v}{a\partial \varphi }+\frac{{\partial }^{2}w}{a\partial {\varphi }^2}\right)+q_{\varphi }\right)dxd\varphi }}=0$,

${\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{l}{\int }}{W}^T\left(\frac{\partial Q_x}{\partial x}+\frac{1}{a}\frac{\partial Q_{\varphi }}{\partial \varphi }+\frac{\partial }{\partial x}\left(N_{x\varphi }\left(\frac{v}{a}+\frac{\partial w}{a\partial \varphi }\right)\right)+\frac{\partial }{\partial x}\left(N_x\frac{\partial w}{\partial x}\right)\right)dxd\varphi }}+$

$+{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{l}{\int }}{W}^T\left(\frac{N_{\varphi }}{a}+\frac{\partial }{\partial \varphi }\left(\frac{N_{\varphi }}{a}\left(\frac{v}{a}+\frac{\partial w}{a\partial \varphi }\right)\right)+\frac{\partial }{\partial \varphi }\left(\frac{N_{\varphi x}}{a}\left(\frac{\partial w}{\partial x}\right)\right)+q_z\right)dxd\varphi }}=0$. (11)

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

$K_{11}{T}_u+K_{12}{T}_v+K_{13}{T}_w+K_{12}^{(1)}{\left(\frac{\text{d}\alpha }{\text{d}t}\right)}^2{T}_v+K_{13}^{(1)}{\left(\frac{\text{d}\alpha }{\text{d}t}\right)}^2{T}_w+C_{11}\frac{\text{d}{T}_u}{\text{d}t}+M_{11}\frac{{\text{d}}^2{T}_u}{\text{d}{t}^2}=F_1$,

$$\begin{array}{c}{K}_{21}{T}_u+K_{22}{T}_v+K_{23}{T}_w+K_{22}^{(1)}{\left(\frac{\text{d}\alpha }{\text{d}t}\right)}^2{T}_v+K_{23}^{(2)}\frac{{\text{d}}^{\text{2}}\alpha }{\text{d}{t}^2}{T}_w+C_{23}\frac{\text{d}\alpha }{\text{d}t}\frac{\text{d}{T}_w}{\text{d}t}+\\ +C_{22}\frac{\text{d}T{}_v}{\text{d}t}+M_{22}\frac{{\text{d}}^{\text{2}}T{}_v}{\text{d}{t}^2}+M_{24}\frac{{\text{d}}^{\text{2}}\alpha }{\text{d}{t}^2}+F_{2c}\cos \alpha +F_{2s}\sin \alpha +F_2\left(\alpha \right)=0,\end{array}$$

$\begin{array}{l}{K}_{31}{T}_u+K_{32}{T}_v+K_{33}{T}_w+K_{32}^{(1)}{\left(\frac{\text{d}\alpha }{\text{d}t}\right)}^2{T}_v+K_{33}^{(1)}{\left(\frac{\text{d}\alpha }{\text{d}t}\right)}^2{T}_w+K_{32}^{(2)}\frac{{\text{d}}^{\text{2}}\alpha }{\text{d}{t}^2}{T}_v+\\ +C_{32}\frac{\text{d}\alpha }{\text{d}t}\frac{\text{d}{T}_v}{\text{d}t}+C_{33}\frac{\text{d}{T}_w}{\text{d}t}+M_{33}\frac{{\text{d}}^{\text{2}}{T}_w}{\text{d}{t}^2}+F_{3c}\cos \alpha +F_{3s}\sin \alpha +F_3\left(\alpha \right)=0.\end{array}$(12)

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

$M\frac{d^{2}T}{d{t}^2}+C\frac{dT}{dt}+KT+{\left(\frac{d\alpha }{dt}\right)}^2{K}^{(1)}T+\frac{d^2\alpha }{d{t}^2}{K}^{(2)}T+C^{(1)}\frac{d\alpha }{dt}\frac{dT}{dt}+F_c\cos \alpha +F_s\sin \alpha +F\left(\alpha \right)=0$(13)

Where T=[Tu,Tv,Tw].

Adding now equation governing the driving torque of the motor

$\frac{d{M}_s}{dt}=\frac{1}{T_s}\left(C_s\left({\Omega }_s-\frac{d\alpha }{dt}\right)-M_s\right)$(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

None.

Author declares there is no conflict of interest.

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