Some Experiments on Process

Some Experiments on of the Weaving the Process Dynamics By E. Kuze and T. Sakai, Members, TMSJ Tokyo Institute of Technology, Tokyo Based on Journal of the Textile Machinery Society of Japan, Vol,18, No.4, 269-273(1965) Abstract It is assumed that a set of state variable is able to express the state of loom and, after one weaving cycle, the state of loom is determined only by the present state. Then, using the linearizing approximation, we have derived basic equations to express the dynamic property of the weaving process. Experimental results show the basic equations may be useful to discuss the dynamics of the weaving process and the design of loom. 1. General There are two methods of research into the weaving process. One[1i, widely tried, is to analyze directly various phenomena occurring during weaving. The other[21 is to analyze the function of the weaving process. This article concerns experiments based on the second method. A weaving loom is a system which converts yarn into a fabric and may be illustrated in Fig. 1. To describe this system, it seems empirically proper to take a cloth-fell position and warp tension as state variables. The variables are denoted, respectively, by 1(t) and T (t), where t is the number of weaving cycles. Assume that the variable with the suffix f or y refers to the fabric side or yarn side ; and that the variable with the suffix 0 refers to a tensionless state. Then, in the light of the weaving system shown in Fig. 1, the following equations are derivable : T=T f(ef)=t,(ey)...(1) lf±ly=l(const.) (2) where e= tl-...(3) o Generally, state variables change after one cycle of weaving operation, but their deviations are relatively slight. Assume T(t11) -T(t) =at...(4) l(t+1)-l(t)=al...(5) Fig. 1 Model of weaving system Then, in the light of Eqs. 1 and 3: 0T = dt1 d -Olf - dt 1 112 -Lllof ef lof def -of - dt dey y Lly-- loy dt dey y- l... oy and in the light of Eq. 2 : i.v 1± &,-O...(7) Combining Eqs. 5, 6 and 7 gives the following further equation : dt f 1 dt y 1 dt f l f def 'of -~ dry loy Alf def l2o Alof dty 1, -Aloy d5, l oy '.L11 f = (dt f 2f Alof-dT, 1, Oloy def l of dey l o, dtf 1 +dt, 1 de f l of dy loy / Vol. 12, No. 2 (1966) 63

N Factors which cause the deviation 0l o f are the take-up length Y, the newly produced fabric length X and the change in the length of the fabric part X'. Factors which cause the deviation & oy are the let-off length and the yarn length required to produce the new fabric part X. L1lof=-Yo+Xo+Xo'...(9) zloy=-xo+zo...(10) Of course, it may properly be assumed that those factors are functions of state variables. Assuming that T-1 is the inverse function of T, then : r=t-1(t) Therefore, X=X0T-'(T) Y =Y0T -'(T) etc. are obtained. Substituting for Xo, Yo, etc. into Eqs. 9 and 10 gives us : &01= (-Y+X +X')/T f-1(t) Using these relations transforms Eq. 8 into : elf = dt d f 1 (-Y+X+X') Ef lof _ dt y _1 (-x+z) dey loy (d I f 1 + _ y _1 drf 'of dry loy...(11) Substituting Eq. 6 into Dl f gives us : zt = (Y-X-X'+x-Z)/ (L. /dt f -,,. /dt,...(191 ~ agf - ~ dey,~-, Eqs. 11 and 12 are fundamental equations which express the dynamic characteristics of the weaving process. In practice, the change of state in the system is so small that, assuming dt =E (+ const.) de lo=const., Eqs. 11 and 12 can be rewritten thus : where l f(t+1)-l f(t)= --1- A -(-B{A(-Y+X-FX') +B(x-Z) }...(13) T (t+1) -T (t) =_A +B+B_{Y-X-X'±x-Z} (14) A =_L l (const.), B = Ey (const.) ot loy The next chapter applies the foregoing basic equations to a few cases. 2. Applied to Steady State No factors related to the weaving process change with cycles of weaving operation if the weaving process is steady. lf(t+1)=lf(t) T (t+1) =T (t)...(151 Eqs. 13 and 14 and condition 15, together, go to show that : A(-Y+X+X') +B(x-Z) = 0...i) -C- Y+X +X')-(x-Z) =0...ii) "'(16).. -Y+X +X'=x-Z=0 or X+X'=Y x=z...(17) X is defined as the fabric length produced during one cycle of weaving operation. In other words, x may be taken as the distance between the weft yarn inserted last and the penultimate weft yarn, or the last pick spacing. X' is defined as the change occurring in the length of the existing fabric part during one cycle of weaving operation. Assume that the existing fabric part is an accumulation of X, and that X, produced at t=k, transforms into X (t=k, t=t-1) at t=t-1. Then the following expression relating to l f is obtained : k=t-1 11(t)= X (t=k, t=t-1)...(18) By definition, the following expression relating to X' is obtained : X':= k=t-1 X {X (t=k, t=t) -X (t=k, =t-1)}...(19) Experience has shown that the smaller t-k is (the shorter the distance between a pick and the clothfell), the stronger the effect of beat-up force on the pick spacing. In other words, the effect of beatup force decreases rapidly as the distance from the cloth-fell increases. Here N is defined as the serial number of the last weft yarn counted from the clothfell and which shifts its position by the effect of beatup force. Accordingly : k=t-1 X+X'=X(t=t, t=t) - {X(t=k, t=t) -X (t=k, t=t-1) } =X (t=t, t=t) ±X(t=t-1, t=t) -X (t=t-1, t=t-1) +X (t=t-2, t=t) -X (t=t-2, t=t-1) etc. +X(t=N, t=t) -X (t=n, t=t-1).....(20) In steady state, X (t=k, t=t-1) -X (t=k+1, t=t) =0...(21) as a matter of course Eqs. 17, and 21, combined, transform into : X +X'=X (t=n, t=t) =Y...(22) This expression means that the finally determined pick spacing on the loom must be equal to the takeup length per pick. If a loom has a positive take-up 64 Journal of The Textile Machinery Society of Japan

mechanism, the final pick spacing equals the constant take-up length Y given by the gear ratio. 3. Applied to Transient State Consider a case where the system deviates from steady state, the deviations occurring in state variables being comparatively small. Assume that the deviations of tension and of the cloth-fell position at t cycle of weaving process are &1(t) and z T (t), respectively ; and that the values in steady state are T and l f, respectively. The deviations of functions X. Y, etc., in this case, are : X(11+&1, T+AT) -X (l f.t) Y(%f+&f. T+AT)-Y(lf,T) (...(23) etc.) Taking only the first-degree terms of Taylor's infinite product of functions, the above relations can be rewritten in simpler form as a first approximation : ax Ol + ax ~T ~f ;c- at ~, l f,~l f + ~ at -DT etc....(24) where `x a1, ax, etc. are the so-called influence 1 at coefficients determined by the relations of X versus if, x versus T, etc. Now consider the simplest weaving process in which the instantaneous pick spacing depends only on the instantaneous cloth-fell position, and in which the pick spacing in a woven fabric does not change its initial value. Let the displacement of cloth-fell from its correct position occurring in the system at t=0 be expressed as l(0). Let the displacement of clothfell from the correct position occurring at t=t be expressed as pl f (t). Then rearrange Eq. 13 on these assumptions. The result : ~lf(t+1)-&f(t)=-- A-l -+-B _ AaX elf -F B - l - A}z1(t)... (25) of The solution of Eq. 25 is : & AaX +B ax t 1(t+1)= 1+ - al A f pl(o) (26) +B If the interlacing structure of weft and warp yarns in the transient process differs only slightly from the similar structure in the correct state, then : ax = ax Substituting a for Eq. 25, we have : of f off Al f(t-i-1)=(1 -} a-xtal f(0)...(27) (,l f Fig. 2 Transient response of the simplest weaving system Fig. 2 shows the transient curve of J f (t) versus weaving cycle t given by Eq. 27. Figs. 2-i) ii) and iii), respectively, correspond tions : to the following condi- i) 0>l f >-1 ii) Qlf in) X <2 Olf (28) Vol. 12, No. 2 (1966) 65

f The following conditions for normal weaving are obtainable, according to experiments we have made : ax <0 and ax < 1...(29) alf alf Eq. 25, therefore, may be replaced approximately with the following differential equation : ddlf = l -- 1A X +B ax ~l f...)(30 dt A+B alf Similar analogies are presumably applicable to Eqs. 13 and 14. Now as to the influence coefficients of X and X'. X' expresses the amount of change in length occurring in an existing fabric because of beat-up force. The pick spacings from the last one to Nth in steady state are affected by beat-up force. Pick spacings influenced by beat-up force in transient state generally differ in number from N in steady state. It appears experimentally, however, pick spacings affected by beat-up force consolidate into N in number very rapidly. It may be assumed, then, that the number of pick spacings influenced by beat-up force is N for both states, steady and transient. The pick spacing X (t=k, t=k) is influenced by the displacement of cloth-fell position from (t=k+1) to (t=k+n), and makes final pick spacing X (t = k, t = k + N). Assume xx==x (t=t-i, t=t) -X (t=t-i. t=t-1) (2<t) in steady state. L\l f(t) is expressible in this form : a X alf i el f The change of vx; by the effect of where is an influence coefficient. Therefore, the effect of Oi f (t) in relation to X (t=k, t=k+n) is expressible in this form : N aox al i Dlf (t=k-i-i )... (31) f N being noticeably small and the transient change of LIl f being comparatively slow, Eq. 31 approximately equals the following : ~axi Ol f(t=k+n)...(32) i=1 alf Adding ax Ol f(t=k+n) a!1 gives us : = ax al ~l f + ~ N a~x; Dl f f i =1 at 1 ax N a0x i alf _-~ ~--alf Ol f to the above expression = a (x +X) Di f= f ~ (X+X')~l f...(33) This expression means the effect of zl f can be quantified by the influence coefficient ax f where X -FX'=X f the coefficient being determined from the slope of the cloth fell position l f versus the final pick spacing X f curve. Similar reasoning, it is believed, holds on the effect of tension deviation T. Eqs. 13 and 14 can be rewritten, by incorporating the various results cited, as follows : d&f = ax f _- A -- ay _- B- az 1 0. dt ;_1,T a; A+B a; A+B a; i f dint A B _ ay az d t f,t A+B a; a;...1)...(34) The take-up motion fitted to looms in common use is of a positive type, and the let-off motion, for the most part, operates irrespective of beating force. Hence : ay_az =O aj ai f Substituting ay and az into Eq. 34 gives us : a j a!1 dal f -ax1dl f-f ' axf- B a2- OT i dt alf ~, at A+B at...(35) dot _ A B az ~T... ii) dl -A+B at Eqs. 35-i and -ii can be put in matrix form thus : d dt CvJ = CgJ CvJ... (36) where rv1 _FDif L LT CgJ = ax a! f ax f _ B az 1, at A+B at _ A B az 0 A+B at Laplace transforms convert Eq. 36 into the following expression, similar to one relating to the variables of a four-terminal network [V] = CGJ [v(0) J... (37) where ~V JCL = L (Dl QT f) i w (0) = [~T 0l f (0) 0 _r s_axf axf_ B az -1 [G]=, at A + B at 0 S+A B A+B az at _ 66 Journal of The Textile Machinery Society of Japan

The validity of the tally inquired into the 4. Experiments basic equations is next chapter. experimen- axf being 0.55mm axf alf obtained -0.11 from Fig. 3 as a tangent at X1== Consider the simplest weaving process in which the instantaneous pick-spacing is fixed only by the instantaneous clothf ell position -a process which, we believe, is possible by using a loom built of iron gauge. Acting on this belief, we made experiments by using a loom made by the Nippon Iron Gauge Manufacturing Company. Fig. 4-i QXf-t curve in case forward displacement of clothf ell position given at t = 0. Fig 4-ii QX1-t curve in case backward of clothf ell given at t = 0 displacement Fig. 3 shows a curve of the pick spacing versus the clothfell position of the loom used in our experiments. The clothf ell position was measured as the distance between the clothfell and the steel wire line near the clothfell. The pick spacing or the clothfell position was read by photography or with a travelling microscope after enough gauge was woven for stable conditions. After weaving in steady state, the loom was stopped, then the cloth-fell was shifted slightly forward or backward by adjusting the let-off and take-up motions. Then the loom was started again and kept running until conditions returned to normal. After that, the pick spacing was measured. As an example of experimental results, we show in Figs. 4-i and ii a case in which the correct pick spacing is 0.55 mm and displacement given to the clothfell Qlf(0) is+1.0 mm or -1.5 mm. Theoretically, the relation between Fig. 3 X f-l f curve of wire-gauge loom Ql f and QXf is ax ft ax l Of f =f 4l f (o) e =X1 (t)...(38) Substituting these values into Eq. 38 makes it possible to calculate the transient values of pick spacing. The calculated and experimental values agree well with each other, as Fig. 4 shows. Similar experiments were made on an automatic loom for cotton in common use. Details of the loom and the yarn used in our experiments are : Loom : Automatic, for cotton, 50 inch in reed space, loose reed, underpick. Shedding : plain weave. Warp : 30/2s cotton Weft : 30s cotton The warp tension was measured with a tensiometer built of a three-line roller system pick-up, a strain gauge transducer and a pen-oscillograph. The clothfell position was measured by the method described in an earlier paragraph. The tension or clothfell position was read at the back point of the crank shaft. Fig. 5 Xf-lf curve if of automatic loom for cotton Vol. 12, No. 2 (1966) 67

To obtain the influence coefficients axf al and ax' f at the clothfell position versus pick spacing was measured for various warp tensions in stable weaving condition, with the results shown in Fig. 5. After weaving in steady state, the loom was stopped and the cloth-fell was slightly displaced by adjusting the take-up and let-off motions but 1 eeping the warp tension constant. Then the loom was re-started and the warp tension and the cloth-fell positions were measured every few picc s until weaving conditions returned to normal. The warp tension remained constant from beginning to end, thus bearing out the assumption used to derive Eq. 35. The transient response of the cloth-fell position for the normal pick spacing of 0.6 mm is shown in Fig. 6. down by adjusting the take-up and let-off motions but keeping the clothfell in a normal position. When the loom was re-started, the warp tension and the clothfell position returned to normal. Figs, 7-i and -ii give ecamples of the transient response of the warp tension and cloth-fell position for the normal pick spacing of 0.6 mm. These results call attention to the interesting fact that, if the initial tension deviation is plus or minus the clothfell temporarily shifts backward or forward, then returns to its normal position. Fig. 7-i Ql f-t & QT-t curve in changedup ward v ith kept constant at t - 0 case clothfell warp tension position Fig. 6 Ql f-t curve in case of backward with tension clothfell shifted kept constant at forward t =0 or Solution of Eq. 35 wth the initial condition =0 yields QT(t)=0,...i) Ql1(t) = l f (0) e p dx (t), ii) f which satisfactorily e plains e~ perimental results. QT(O) Assuming a`f= -0.080, then the theoretical relation Ql f(t) =Ql f (0) e:; p ( axf t agrees well with experimental results, and the value axf nearly equals the value determined from the slope ai f of the l f-x f curve shown in Fig. 5. After weaving in steady state, the loom was stopped and the warp tension was varied slightly up or of f Fig. 7-ii Ql f-t & QT-t curve in case with warp tension changed downward clothfell position kept cons taut at t = 0 Solution of Eq. QT (o) QTo, yields 35 with the l f(0) =0 initial condition 68 Journal of The Textile Machinery Society of Japan

OT=i To exp - A+B az_ at t )...li axf B az &f= at a--- f+- A+B - B_ a at -- OTo exp ax(ft alf.140) alf A+B at -exp _ A'B az t ii A+B at which satisfactorily explains experimental results. Eq. 40-ii shows that Dl f takes a maximum or minimum value at dal, ^0 d t A B az 1 I n A+B at or t= axf+ A _B az -axf (41) A+B at a11 and gradually declines to zero with time. The value of the index A+B A B- az at is obtainable from a comparison of the transient response of tension with Eq. 40-i. A A $ az =0.065 at ~To>0 +B at =0.040 z To<O In the light of the results of this experiment, we get axf = -0.080 a11 In the light of Fig. 5, we have axf-n at nni ~ mm /b he In the light of experimental conditions is obtainable : T0= +16 g or-27 Substituting these values into Eq. 40 ii and comparing them with the transient response of the clothfell position lead to: A+B ~Z-=0.021 at B mm/g at QTo>0 =0.014 mm/g QTo<0 Compare the value _A A+B B az at with the values A+B B az and we have : at, A+3 g/mm B=4 g/mm The value of A or B is obtainable also by the following experiment : After weaving in steady state, stop the loom, turn the cran'i handle several times without the lef-off or picking motion, and measure the variations in the warp tension and cloth-fell position. In this case, by using the initial condition which is expressible as follows : X =X'=x=Z=O we can rewrite basic Eqs. 13 and 14 in this form : deviation of tension _ A B take-up length A+B deviation of clothf ell position _ A j take-up length A+B ) The values on left side of the equations are obtainable by the experiment just described. A ~ B =1 A+B.7 /mm,.'.a=3 ~ /mm B+4 /mm - A A -i--b0 =.45 /mm These results agree well with the results obtained from the transient response curve of tension and the clothfell. B is defined as follows : B- Ey l o, The warp yarn used in our experiments had approximately a linear stress-strain curve, and the tensile elastic modulus of the yarn was calculated by the quotient of breaking strength and elongation. Tensilon showed the mean value of breaking strength and elongation to be 35Og and 7% respectively. Then, Ey _ 0350.07 The free length of the warp yarn on the loom l o, being about 13OOmm, giving : B = 0.07 x 350 13OOmm g -- - - +4g/mm This agrees well with the value of B obtained by the befogc-mentioned method. The evidence so far suggests that the basic equations on the weaving process are valid for cotton looms in common use ; and that, with the phenomenal coefficients given, the dynamic property of the weaving process of the loom can be e k plained. The validity of the basic equations on the weaving process has been investigated more closely and extensively by our coworker.[3] 5. Increasing the Number of State Variables Tension T and cloth-fell position l f were selected in the foregoing chapters as the state variables of the weaving process, but it is usually all right to select n factors as state variables. In that case, the following equation is obtainable on the analogy of Eq. 36: d dt x l i C11C12""' "' CI x1 ~ X2 C21 x2. j a'n vnl... Cnn xn Vol. 12, No. 2 (1966) 69

, where x1, X are the state variables of the weaving system C11, Cnn are the influence coefficients As an example, let us consider the diameter D of filling, besides l f and T, as one of the factors and study a case where, after weaving in steady state with a weft yarn D in diameter, weaving is continued, replaced with a weft yarn D+QD in diameter, the let-off and take-up motions being kept unchanged. The weaving process equation in this case is : used is : Ql f C11 C12 C13 ` r Ql f _d d _ t QT = C21 C22 C23 QT...(44) QD C31 C32 C33 l QD ' The weaving process equation on a loom commonly CaXf Ca 11- alf' 12=A+B axt f B-- az-caxf at, 13= ad C 21= 0, %-22 = A A B az C23 =0 +B at ', C31= 0, C32= 0, C33=O Assuming that the roots of the following equation are a1 and a2i the solution of Eq. 44 is written in the following form : a2- (C11 +C22)a+ (Cu C22-C12 C21) _0 Ql f=ae i'+be 21+ C C12.C13 - QD 11 C22-C12 C21...(45) QvltllC22-C12`C21 T=a'e + b/e 2t-r ~I~~_C ' C21. C13 QD whore YYllt.lV n. W. V h. W n' UJJ.i and b' uav are allvvblul integral constants...llvv E~ perience shows that al and a2 usually take real negatives. It follows, then, that, with the change of weft yarns, the system enters a new steady state in which the pick spacing is Xf(t=~)=X (lf~ T, D) f «Xf Qlf(t= ~) _I(f)T(t=) ax Qoo+ ax(f)d=x(71 Q at ad,,, D)...(46) which is e plained thus : az ax f QD az axf _ az axf at a11 a11 ` at az axf QD DT (t = oo) = - ai f - ad_ - az axf _ az axf at 1 a11 al; at This means that a change in the weft diameter does not change the normal pick spacing. In the case of a wire-gauge loom which permits us to ignore the influence of warp tension, we may omit the term relating to tension and re-arrange Eq. 46. The result will be : ax jql a11 f+ ax QD=O ad..ql f(t=o) =-axfqd/a ad f...(47) off This result means that a change in the filling diameter changes the position of the cloth-fell to the extent shown in Eq. 47. Let us make another e:: periment. Begin by weaving a filling yarn of a certain diameter in steady state. Change the diameter abruptly but carry on the process long enough for a new steady state to be reached. Then stop the loom and measure the change of the cloth-fell position and pick spacing. Fig. 8 QXf-t curve in case filling diameter changed at t=0 Fig, 8 V i or _ results of the measurement. Fig. 8 V a l ii shows the v surement. As soon as the diameter of the filling yarn was changed, the pick spacing changed, but the instantaneous value of the pick spacing returned gradually to the normal value. The difference between the cloth-fell positions in the old and new steady states was about 0.45mm. Put Ql f (t=oo) = + 0.45mm axf= a1-0.11 1 and substitute it into Eq. 47. Then we have axfqd~ ad +0.05mm...(48) which shows that, after the changing of the diameter of the filling yarn, the instantaneous pick spacing deviates about 0,05 mm from the normal value. In practice, the mean value of 15 instantaneous pick spacings before and after changing the diameter of the filling yarn (which value is shown by dotted lines in Fig. 8-i and 8-ii) is obtainable as follows : Count of wire used Mean value of 15 for filling pick spacings X35--~#33 0.511----~0.563mm X33--> 35 0.563 30.599mm 70 Journal of The Textile Machinery Society of Japan

These data amply support Eq. 48. By thus fixing the value air ad we can calculate the effect of variation in, the diameter of the filling yarn. 6. Conclusion We have derived basic equations to express the dynamic property of the weaving process and e perimentally investigated the validity of the equations in various cases. Results obtained have shown that theory and experiment agree well under some limitations ; and that the basic equations may be useful to discussions of the dynamics of the weaving process. References [1] F. Stein ; Melliand Textilber., 8, 994 (1927) Kuze, Sakai and Mihara ; J. Soc. Fiber Sci. & Tech,, 11, 192, 459 (1955), 14, 304 (1958) S. Matsuda ; ditto, 11, 567, 646 (1955) H. Tsuboi : Thesis for doctorate [2] K. C reenwood and W. T. Cowhig ; J. Text. Inst., 47, T241, T255, T266 (1956) Kubota and Matsumoto ; Bull. Text. Res. Inst., 46, 41 (1958) [3] Rai Tsuu Pin ; Thesis for master's degree Vol. 12, No. 2 (1966) 71