Numerical Modeling of the Motion of the Flat Textile Structures

In many problems from the field of textile engineering (e.g., fabric folding, motion of the sewing thread) it is necessary to investigate the motion of the objects in dynamic conditions, taking into consideration the influence of the forces of inertia and changing in the time boundary conditions. This paper deals with the model analysis of the motion of the flat textile structure using Lagrange’s equations in two variants: without constraints and with constraints. The motion of the objects is under the influence of the gravity force. Lagrange’s equations have been used for discrete model of the structure.


Introduction
In most cases of dynamic analysis of textile structures, it is necessary to use the model of heave elastica, as a onedimensional body of linear weight q and bending rigidity C in the state of large deflections.In this paper, it is assumed that during the run of bending effect the flat textile structure (e.g., fabric) will be represented as its longitudinal section.The mathematical model will be described as a flat deflection curve; this will be treated as a heavy elastica, as shown in Figure 1 and in [1].
The word "heavy" underlines the decisive influence of the gravity forces during the motion.It is assumed that the particular longitudinal sections do not act on each other by internal forces.Furthermore, the constancy of properties along the whole width of the bending strip is assumed.It will be assumed also that the elastica is inextensible.The analysis was made using Lagrange's equations, describing the motion of the system of n particles in conservative force field.Two variants of Lagrange's equations were considered: equations without constraints and equations with constraints.

Discrete Model of the Object and the Lagrange's Equations
The Lagrange's equations, describing the motion of the system of n particles in conservative force field, studied in detail among other things in [2][3][4][5], are presented below: where T-kinetic energy, U-potential energy, ϕ k , φk kth generalized coordinate and velocity.If we have n generalized coordinates, then we can write n Lagrange's equations (1).
The problem reduces to appropriate choosing of generalized coordinates ϕ k , deriving the formula for kinetic and potential energy, and next deriving and solving the Lagrange's equations.
During the analysis, we replace often continuous systems by discrete systems using partition into n elements.In considered problem partition was made as follows.The elastica of length L, mass M and bending rigidity C fixed at point A was replaced by system of n masses of identical value m = M/n, connected by n straight, inextensible, and weightless segments of length l = L/n, as shown in Figure 2 and work [6].The other end B of elastica is free.If ρ is linear mass of elastica, then M = ρL.The position of ith mass is described by generalized coordinate as an inclination angle ϕ i (i = 1, 2, . . ., n) of ith segment to the vertical direction.Coordinates of ith mass and its derivatives are as below:

ISRN Mechanical Engineering
( The kinetic energy of ith mass is Therefore, the total kinetic energy of the system of n mass is ( The total potential energy of the system U is the sum of two components where V is the potential energy of gravity forces and V b is the potential energy of the bending.The potential energy of gravity forces is V = n i=1 mg y i = mgl n i=1 i j=1 cos ϕ j and the potential energy of the bending V b = 1/2C L 0 (dϕ/ds) 2 ds.After substitution the derivative by difference quotient, we have Thus, the total potential energy of the system U is The formula for derivative of kinetic energy T in respect of ϕ k after transformations is as follows: After changing the limits of summation, we can write Similarly, the other derivatives can be written as follows: The derivative ∂(ϕ i+1 − ϕ j )/∂ϕ k depending on the value of index i is given by Using summation from i = 1 to i = n − 1 the formula (11) can be written as follows: Therefore, the derivative of total potential energy in respect of ϕ k is Using (8), (9), and ( 14) in Lagrange's equations ( 1) and dividing by ml 2 , we have Finally, it was obtained system of n 2nd-order ordinary differential equations with unknown generalized coordinates ϕ i (i = 1, 2, . . ., n).Formula (15) represents kth Lagrange's equation, which describes the motion of elastica as in Figure 2. Interpretation of boundary conditions results from formula for total potential energy of the system (6).In the presented problem, the end B of elastica is free; therefore, it is not loaded by any forces.The end A is fixed by joint; therefore, the bending moment M = 0, and displacements in point A are equal zero (it has not influence on potential energy).

The Lagrange's Equations in the Matrix Form
For further analysis, the Lagrange's equations (15) were written in the matrix form.The following denotations were introduced: In the matrix form the Lagrange's equations (15) are where A is an appropriate matrix and B, C, D are vectors as follows: On the basis of (15) we can write the matrix A and vectors B, C and D as follows: where the matrix S of dimension n × n is defined by The element of kth row and jth column of matrix S can be written as Its task is to realize the double summation from i = k to i = n and from j = 1 to j = i.Finally, it was obtained system of n 2nd-order ordinary differential equations with unknown generalized coordinates Φ in matrix form

Numerical Solution of the Problem
We can write the system of ( 22) in the form of system 2n 1storder ordinary differential equations, introducing additional vector of generalized velocities In this way, we obtain The system of (23) has been solved using a standard Runge-Kutta-Fehlberg 4th-order integration method.To solve this problem numerically, the following denotations were introduced.
The vector of unknowns where The right side of (22) has been denoted by vector F; that is, Finally, the following system of differential equations was solving: . . .
with initial conditions for t = t 0

The Lagrange's Equations with Constraints
In a previous section the elastica fixed only at point A was considered.The other end B of elastica was free (Figure 2).In this case, generalized coordinates ϕ k were completely independent in themselves.In many problems connected with the motion of elastic, there are some constraints concerning angles ϕ k .Then, the Lagrange's equations undergo modification.
In this section the motion of elastica with two ends fixed at two points A and B using joints as in Figure 3 will be considered.This case may be used for example to simulate fabric folding and so on.Then, the boundary conditions for the end B (which previously was free) are as follows: x n = 0, In accordance with (2) for x n and y n we can rewrite the following two conditions of constraints (29) obtaining (30) Thus, we can write the Lagrange's equations in the following form: where λ 1 , λ 2 are the unknown Lagrange's multipliers (their interpretations are the appropriate reaction forces in point B-horizontal and vertical).Derivatives ∂ f 1 /∂ϕ k and ∂ f 2 /∂ϕ k are as below: Let right side of the kth (15) be denoted by Ψ k (ϕ, φ).Therefore, using (31) and (32), we have the following system of equations: Using matrix notation, we can write new matrix form of the system (33) where The system of (34) with the (30) presents the system of n + 2 differential-algebraic equations with n + 2 unknowns, that is, n generalized coordinates ϕ i (i = 1, 2, . . ., n) and two Lagrange's multipliers λ 1 , λ 2 .To solve such problem, (30) have been differentiated two times in respect of time Next, from (34), the multipliers λ 1 and λ 2 have been eliminated successively, using twice Gaussian elimination.After this, (34) together with (36) present the system of n 2nd-order ordinary differential equations in respect of ϕ i (i = 1, 2, ..., n).Using matrix notation, we have similarly as ( 17) in Section 3. The components of new matrix A and vectors B, C i D are as below: The system of (37) has been transformed to the form similarly as in Section 5; that is, Next, we can write Finally, the system of (40) has been solved using a standard Runge-Kutta-Fehlberg 4th-order integration method.The considerations presented above concerned with the elastica for which the end B was fixed by joint.Of course, the other conditions of constraints may be introduced for this one.
If we want to allow vertical motion of the end B, then we have only one constraint for coordinate x of the end B : x must be always equal to value a = const., and we have only one algebraic equation (conditions of constraints) in the form of The Lagrange's equations are simple in this case, because we have only one Lagrange's multipliers λ 1 .
If we want to have the end B fixed at point B at an angle of ϕ = α, where x = a, y = b, then there are three conditions of constraints and three Lagrange's multipliers λ 1 , λ 2 , and λ 3 .

The Results of Calculations
To present the results of calculations and accuracy of numerical method two numerical cases were considered.
The first one concerned the motion of heavy elastica under the influence of the gravity force in conservative force field.The elastica was fixed byjoint at one end only as in Figure 2.
In the second one, the elastica was fixed by joint at two ends as in Figure 3.

The Motion of the Elastica Fixed at One Point.
In this case, the elastica fixed at one point by joint as in Figure 2 will be considered.The second end is free.The motion is under the influence of the gravity and elastic forces.
Too many points n of division can result in long time of calculations.In the example below, it has been assumed that n = 8.
The initial conditions (28) are as follows: In the initial instant, the elastica has rectilinear, horizontal shape.The velocity of all points is zero.After solving the (23) with the time step Δt = 0.005 s, we have obtained the shape of elastica in the following time instant t 0 , t 1 , t 2 , . ... The shapes are presented in Figure 4. Figure 5 shows variations of generalized coordinates ϕ 1 , ϕ 2 , . . ., ϕ 8 in the respect of time t. Figure 6 shows variations of generalized coordinates ϕ 1 , ϕ 2 , . . ., ϕ 8 in the respect of time t for elastica of bending rigidity 10 times less than previously (C = 0.00025 Nm 2 ).
In Figures 5 and 6, the heavy dashed line represents graph of angular displacement of rigid rod (physical pendulum) fixed in the same point as elastica (Figure 7).
From the graphs, it may be inferred that in the initial state of the motion the points of elastica oscillate around the dynamic equilibrium (as for physical pendulum).The rigid rod (Figure 7) can be treated as the elastica of bending rigidity C = ∞ for which all points have permanently the same angular displacements ϕ k = ϕ dla k = 1, 2, . . ., n.As the time goes, deviations from dynamic equilibrium become bigger and bigger.
In order to check accuracy of calculations, we follow as below.
For chosen coordinate ϕ 8 , its values marked by ϕ Δt1 8 with given time step Δt 1 = 0.005 s have been calculated.Next, the calculations have been repeated for the less time step has been determined for following t j = t 0 + i • Δt 1 .
The calculations have been made for C = 0.0025 Nm 2 .The graph of differences Δϕ 8 for following time instant is presented in Figure 8.
The differences Δϕ 8 are not big at all (between −0.00019 and 0.00022 rad), even after large number of iterations (i = 400).The accuracy turned out to be satisfactory.Number of points of division is n = 16.The system of (40) has been solved using Runge-Kutta-Fehlberg 4th-order integration method with time step Δt = 0, 005 s.The initial conditions (28) are as follows: Finally, the shape of the elastica in the following time instant t j has been obtained.It is present in Figure 9.
In order to check accuracy of calculations, we tested how the boundary conditions (29) change as time goes by.In this From Figure 10, we can conclude that for two times less time step deviation of boundary conditions was significant less.For Δt 1 after 300 iterations, deviation was about 2 • 10 −4 m, and for Δt 2 = 1/2Δt 1 , deviation was about 2• 10 −5 m (10 times less).

Conclusion
After carrying out series of numerical tests, it can be concluded that Lagrange's equations are practically useful for the analysis of the motion of heavy elastica.It can be used for simulation of many problems from the field of textile mechanics, for example, fabric folding (Figure 11), motion of the sewing thread and so on.Similar investigations, but not using Lagrange's equations, have been described by Lloyd in [7].The results of calculations show clearly wave nature of phenomenon.The bending rigidity has not influence on the convergence of the results.The analyzed motion of elastica is stable owing to initial conditions.

Figure 1 :
Figure 1: The model of fabric approximate to elastica.

Figure 2 :
Figure 2: Discrete model in coordinate system.

Figure 3 :
Figure 3: The elastica with two ends fixed at points A and B.

Figure 9 :Figure 10 :
Figure 9: The shape of the elastica in the following time instant.

Figure 11 :
Figure 11: The simulation of fabric folding using Lagrange's equations.
Figure 4: The shape of elastica in the following time instant from t 0 to t 20 .