Numerical Modeling of the Motion of the Flat Textile Structures

Abstract

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.

1. Introduction

In most cases of dynamic analysis of textile structures, it is necessary to use the model of heave elastica, as a one-dimensional 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].

Figure 1 

The model of fabric approximate to elastica.

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.

2. 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, studyed in detail among other things in [2–5], are presented below:𝑑𝑑𝑡𝜕𝜕𝜑𝑇̇_𝑘−𝜕𝜕𝜑𝑇_𝑘+𝜕𝑈𝜕𝜑_𝑘=0,𝑘=1,2,…,𝑛,(1) where 𝑇-kinetic energy, 𝑈-potential energy, 𝜑_𝑘,𝜑̇_𝑘𝑘th 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:𝑥_𝑖=𝑙^𝑖_𝑗=1sin𝜑_𝑗,𝑥̇_𝑖=𝑙^𝑖_𝑗=1𝑦_𝑖cos𝜑_𝑗⋅𝜑̇_𝑗,=𝑙^𝑖_𝑗=1cos𝜑_𝑗,𝑦̇_𝑖=−𝑙^𝑖_𝑗=1sin𝜑_𝑗⋅𝜑̇_𝑗.(2) The kinetic energy of ith mass is 𝑇_𝑖=1/2𝑚(𝑥̇^2𝑖+𝑦̇^2𝑖).

Figure 2 

Discrete model in coordinate system.

Therefore, the total kinetic energy of the system of 𝑛 mass is𝑇=12𝑚𝑙^2𝑛_𝑖=1⎡⎢⎢⎣^𝑖_𝑗=1cos𝜑_𝑗⋅𝜑_𝑗²+^𝑖_𝑗=1sin𝜑_𝑗⋅𝜑_𝑗²⎤⎥⎥⎦.(3) The total potential energy of the system 𝑈 is the sum of two components𝑈=𝑉+𝑉_𝑏,(4) 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=∑^𝑛𝑖=1𝑚𝑔𝑦_𝑖=𝑚𝑔𝑙∑^𝑛𝑖=1∑^𝑖𝑗=1cos𝜑_𝑗 and the potential energy of the bending 𝑉_𝑏=1/2𝐶∫^𝐿0(𝑑𝜑/𝑑𝑠)²𝑑𝑠. After substitution the derivative by difference quotient, we have𝑉_𝑏=12𝐶^𝑛−1_𝑖=1𝜑^𝑖+1−𝜑_𝑖𝑙²𝑙=𝐶2𝑙^𝑛−1_𝑖=1𝜑_𝑖+1−𝜑_𝑖².(5) Thus, the total potential energy of the system 𝑈 is𝑈=𝑉+𝑉_𝑏=𝑚𝑔𝑙^𝑛_𝑖𝑖=1_𝑗=1cos𝜑_𝑗+𝐶2𝑙^𝑛−1_𝑖=1𝜑_𝑖+1−𝜑_𝑖².(6) The formula for derivative of kinetic energy 𝑇 in respect of 𝜑_𝑘 after transformations is as follows:𝜕𝑇𝜕𝜑_𝑘=⎧⎪⎨⎪⎩𝑚𝑙^2𝑛_𝑖𝑖=1_𝑗=10,sin𝜑_𝑗−𝜑_𝑘⋅𝜑̇_𝑗𝜑̇_𝑘,ffoorr𝑖𝑖≥<𝑘𝑘,.(7) After changing the limits of summation, we can write𝜕𝑇𝜕𝜑_𝑘=𝑚𝑙^2𝑛_{𝑖𝑖=𝑘}_𝑗=1sin𝜑_𝑗−𝜑_𝑘⋅𝜑̇_𝑗𝜑̇_𝑘.(8) Similarly, the other derivatives can be written as follows:𝑑𝑑𝑡=𝑚𝜕𝜕𝜑𝑇𝑙̇_𝑘_2𝑛_{𝑖𝑖=𝑘}_𝑗=1cos𝜑_𝑗−𝜑_𝑘⋅𝜑̈_𝑗−sin𝜑_𝑗−𝜑_𝑘⋅𝜑̇_𝑗−𝜑̇_𝑘𝜑̇_𝑗,(9)𝜕𝑉𝜕𝜑_𝑘=−𝑚𝑔𝑙^𝑛_𝑖=𝑘sin𝜑_𝑘=−(𝑛−𝑘+1)𝑚𝑔𝑙sin𝜑_𝑘,(10)𝜕𝑉_b𝜕𝜑_𝑘=𝐶𝑙^𝑛−1_𝑖=1𝜑^𝑖+1−𝜑_𝑖⋅𝜕𝜕𝜑_𝑘𝜑_𝑖+1−𝜑_𝑖.(11) The derivative 𝜕(𝜑_𝑖+1−𝜑_𝑗)/𝜕𝜑_𝑘 depending on the value of index 𝑖 is given by𝜕𝜕𝜑_𝑘𝜑_𝑖+1−𝜑_𝑖=⎧⎪⎪⎨⎪⎪⎩1−0,,1,fffooorrr𝑖𝑖𝑖==<𝑘𝑘𝑘,−−11,,𝑖>𝑘.(12) Using summation from 𝑖=1 to 𝑖=𝑛−1 the formula (11) can be written as follows:𝜕𝑉_𝑏𝜕𝜑_𝑘=⎧⎪⎪⎨⎪⎪⎩𝐶𝑙𝜑¹𝐶−𝜑₂,for𝑘=1,𝑙−𝜑^𝑘−1𝐶+2𝜑_𝑘−𝜑_𝑘+1,for𝑘=2,3,…,𝑛−1,𝑙𝜑^𝑛−𝜑_𝑛−1,for𝑘=𝑛.(13) Therefore, the derivative of total potential energy in respect of 𝜑_𝑘 is𝜕𝑈𝜕𝜑_𝑘=𝜕𝑉𝜕𝜑_𝑘+𝜕𝑉^𝑧𝑔𝜕𝜑_𝑘=−𝑚𝑔𝑙^𝑛_𝑖=𝑘sin𝜑_𝑘+𝐶𝑙^𝑛−1_𝑖=1𝜑^𝑖+1−𝜑_𝑖⋅𝜕𝜕𝜑_𝑘𝜑_𝑖+1−𝜑_𝑖.(14) Using (8), (9), and (14) in Lagrange's equations (1) and dividing by 𝑚𝑙², we have^𝑛_{𝑖𝑖=𝑘}_𝑗=1cos𝜑_𝑗−(𝑛−𝑘+1−)𝑔𝜑_𝑘𝜑̈_𝑗−sin𝜑_𝑗−𝜑_𝑘𝜑̇^2𝑗𝑙sin𝜑^𝑘+𝐶𝑚𝑙³×^𝑛−1_𝑖=1𝜑^𝑖+1−𝜑_𝑗⋅𝜕𝜕𝜑_𝑘𝜑_𝑖+1for𝑘=1,2,…−𝜑,𝑛_𝑗.=0(15) 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).

3. 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:𝚽=⎡⎢⎢⎢⎢⎣𝜑¹𝜑₂𝜑⋮_𝑛⎤⎥⎥⎥⎥⎦,𝑑𝚽𝑑𝑡=𝚽̇=⎡⎢⎢⎢⎢⎣𝜑̇¹𝜑̇₂𝜑⋮̇_𝑛⎤⎥⎥⎥⎥⎦,𝑑₂𝚽𝑑𝑡²=𝚽̈=⎡⎢⎢⎢⎢⎣𝜑̈¹𝜑̈₂𝜑⋮̈_𝑛⎤⎥⎥⎥⎥⎦.(16) In the matrix form the Lagrange's equations (15) are𝐀𝚽̈+𝐁+𝐂+𝐃=0,(17) where A is an appropriate matrix and 𝐁,𝐂,𝐃 are vectors as follows:𝐀=𝐀(𝚽)=⎡⎢⎢⎢⎢⎣𝐴¹¹𝐴₁₂⋯𝐴_1𝑛𝐴₂₁𝐴₂₂⋯𝐴_2𝑛𝐴⋯⋯⋯⋯_𝑛1𝐴_𝑛2⋯𝐴_𝑛𝑛𝐁=𝐁𝚽,𝚽̇=⎡⎢⎢⎢⎢⎣𝐵₁⎤⎥⎥⎥⎥⎦,𝐵₂𝐵⋮_𝑛𝐂=𝐂(𝚽)=⎡⎢⎢⎢⎢⎣𝐶₁⎤⎥⎥⎥⎥⎦,𝐶₂𝐶⋮_𝑛𝐃=𝐃(𝚽)=⎡⎢⎢⎢⎢⎣𝐷₁⎤⎥⎥⎥⎥⎦,𝐷₂𝐷⋮_𝑛⎤⎥⎥⎥⎥⎦.(18) On the basis of (15) we can write the matrix A and vectors 𝐁,𝐂 and 𝐃 as follows:𝐵_𝑘𝐴_𝑘𝑗=𝑆_𝑘𝑗cos𝜑_𝑗−𝜑_𝑘,=−^𝑛_𝑗=1𝑆_𝑘𝑗𝐶_𝑘sin𝜑_𝑗−𝜑_𝑘𝜑̇^2𝑗,=−(𝑛−𝑘+1)𝑔𝑙sin𝜑^𝑘𝐷_𝑘,=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝐶𝑚𝑙³𝐶𝜑₁−𝜑₂,for𝑘=1,𝑚𝑙³𝐶−𝜑_𝑘−1+2𝜑_𝑘−𝜑_𝑘+1,for𝑘=2,3,…,𝑛−1,𝑚𝑙³𝜑_𝑛−𝜑_𝑛−1,for𝑘=𝑛,(19) where the matrix 𝐒 of dimension 𝑛×𝑛 is defined by𝐒=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑛𝑛⋯−−321𝑛12𝑛𝑛𝑛⋯−−−321121𝑛𝑛𝑛⋯−−−321222⋯⋯⋯⋯⋯⋯⋯⋯321333⋯222122⋯111111⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(20) The element of 𝑘th row and 𝑗th column of matrix 𝐒 can be written as𝑆𝑆_𝑘𝑗=𝑛−𝑘+1for𝑗≤𝑘,_𝑘𝑗=𝑛−𝑗+1for𝑗>𝑘.(21) 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 form𝚽̈=−𝐀⁻¹(𝐁+𝐂+𝐃).(22)

4. Numerical Solution of the Problem

We can write the system of (22) in the form of system 2𝑛 1st-order ordinary differential equations, introducing additional vector of generalized velocities Φ̇=Ω=[_{𝜔1𝜔2⋯𝜔𝑛}]^𝑇. In this way, we obtain𝛀̇=−𝐀𝚽₋₁̇=𝛀,(𝐁+𝐂+𝐃).(23) 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 unknowns𝐘=⎡⎢⎢⎢⎢⎣𝑌¹𝑌₂𝑌⋮_2𝑛⎤⎥⎥⎥⎥⎦,(24) where𝑌𝑌_𝑖=𝜑_𝑖for𝑖=1,2,…,𝑛,_𝑖=𝜔_𝑖−𝑛for𝑖=𝑛+1,𝑛+2,…,2𝑛.(25) The right side of (22) has been denoted by vector 𝐅; that is,𝐅=−𝐀⁻¹(𝐁+𝐂+𝐃),𝐅=⎡⎢⎢⎢⎢⎣𝐹¹𝐹₂𝐹⋮_𝑛⎤⎥⎥⎥⎥⎦.(26) Finally, the following system of differential equations was solving:𝑌𝑌₁̇_𝑛̇=⋮𝑌_𝑛+1,𝑌̇_𝑛+1=𝑌_2𝑛,𝑌̇_2𝑛⋮=𝐹₁,=𝐹_𝑛,(27) with initial conditions for 𝑡=𝑡₀𝚽=𝚽⁰=⎡⎢⎢⎢⎣𝜑𝜑⋮⁰¹_0𝑛⎤⎥⎥⎥⎦,𝛀=𝛀₀=⎡⎢⎢⎢⎣𝜔𝜔⋮⁰¹_0𝑛⎤⎥⎥⎥⎦.(28)

5. 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:𝑦𝑥_𝑛=0,_𝑛−𝐻=0.(29)

Figure 3 

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

In accordance with (2) for 𝑥_𝑛 and 𝑦_𝑛 we can rewrite the following two conditions of constraints (29) obtaining𝑓₁≡𝑙^𝑛_𝑘=1𝑓sin𝜑_𝑘=0,₂≡𝑙^𝑛_𝑘=1cos𝜑_𝑘−𝐻=0.(30) Thus, we can write the Lagrange's equations in the following form:𝑑𝑑𝑡𝜕𝜕𝜑𝑇̇_𝑘−𝜕𝜕𝜑𝑇_𝑘+𝜕𝑈𝜕𝜑_𝑘+𝜆₁𝜕𝑓₁𝜕𝜑_𝑘+𝜆₂𝜕𝑓₂𝜕𝜑_𝑘𝑘=1,2,…,𝑛,=0,(31) where 𝜆₁,𝜆₂ are the unknown Lagrange's multipliers (their interpretations are the appropriate reaction forces in point B-horizontal and vertical).

Derivatives 𝜕𝑓₁/𝜕𝜑_𝑘 and 𝜕𝑓₂/𝜕𝜑_𝑘 are as below: 𝜕𝑓₁𝜕𝜑_𝑘𝜕𝑓₂=𝑙cos𝜑_𝑘,𝜕𝜑_𝑘=−𝑙sin𝜑_𝑘.(32) Let right side of the 𝑘th (15) be denoted by Ψ_𝑘(𝜑,𝜑̇). Therefore, using (31) and (32), we have the following system of equations: Ψ_𝑘(𝜑,𝜑̇)+𝜆¹𝑚𝑙cos𝜑^𝑘−𝜆²𝑚𝑙sin𝜑^𝑘=0,𝑘=1,2,…,𝑛.(33) Using matrix notation, we can write new matrix form of the system (33) 𝐀𝚽̈+𝐁+𝐂+𝐃+𝜆₁⋅𝐄+𝜆₂⋅𝐆=0,(34) where𝐄=𝐄(𝚽)=⎡⎢⎢⎢⎢⎣𝐸¹𝐸₂𝐸⋮_𝑛⎤⎥⎥⎥⎥⎦,𝐸_𝑘=cos𝜑^𝑘𝐆=𝐆(𝚽)=⎡⎢⎢⎢⎢⎣𝐺₁𝑚𝑙,𝑘=1,2,…,𝑛,𝐺₂𝐺⋮_𝑛⎤⎥⎥⎥⎥⎦,𝐺_𝑘=−sin𝜑^𝑘𝑚𝑙,𝑘=1,2,…,𝑛.(35) 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 𝜆₁,𝜆₂. To solve such problem, (30) have been differentiated two times in respect of time^𝑛_𝑘=1cos𝜑_𝑘⋅𝜑̈_𝑘−sin𝜑_𝑘⋅𝜑̇^2𝑘=0,_𝑛_𝑘=1−sin𝜑_𝑘⋅𝜑̈_𝑘−cos𝜑_𝑘⋅𝜑̇^2𝑘=0.(36) Next, from (34), the multipliers 𝜆₁ and 𝜆₂ 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 have𝐀𝚽̈+𝐁+𝐂+𝐃=0,(37) similarly as (17) in Section 3. The components of new matrix 𝐀 and vectors 𝐁,𝐂i𝐃 are as below:

For 𝑘=1,2,…,𝑛−2𝐴_𝑘𝑗=−𝐴𝐴_𝑘𝑗cos_𝑘+1𝑗𝜑_𝑘+1−𝐴_𝑘+1𝑗cos𝜑_𝑘sin𝜑_𝑘+1−𝜑_𝑘+2×sin𝜑_𝑘cos𝜑_𝑘+2−𝐴_𝑘+2𝑗cos𝜑_𝑘+1𝐵−𝜑_𝑘+1,for𝑗=1,2,…,𝑛,_𝑘=−𝐵𝐵_𝑘cos_𝑘+1𝜑_𝑘+1−𝐵_𝑘+1cos𝜑_𝑘sin𝜑_𝑘+1−𝜑_𝑘+2𝐶_𝑘cos𝜑_𝑘+2−𝐵_𝑘+2cos𝜑_𝑘+1sin𝜑_𝑘−𝜑_𝑘+1,=−𝐶_𝑘𝐶cos_𝑘+1𝜑_𝑘+1−𝐶_𝑘+1cos𝜑_𝑘sin𝜑_𝑘+1−𝜑_𝑘+2𝐷_𝑘cos𝜑_𝑘+2−𝐶_𝑘+2cos𝜑_𝑘+1sin𝜑_𝑘−𝜑_𝑘+1,=−𝐷𝐷_𝑘cos_𝑘+1𝜑_𝑘+1−𝐷_𝑘+1cos𝜑_𝑘sin𝜑_𝑘+1−𝜑_𝑘+2𝐴_𝑛−1𝑗cos𝜑_𝑘+2−𝐷_𝑘+2cos𝜑_𝑘+1sin𝜑_𝑘−𝜑_𝑘+1,𝐴_𝑛𝑗=cos𝜑_𝑗,𝐵=−sin𝜑_𝑗,for𝑗=1,2,…,𝑛,_𝑛−1=−^𝑛_𝑗=1sin𝜑_𝑗𝐵_𝑛⋅𝜑̇^2𝑗,=−^𝑛_𝑗=1cos𝜑_𝑗𝐶_𝑛−1⋅𝜑̇^2𝑗,𝐷=𝐶_𝑛=0,_𝑛−1=𝐷_𝑛=0.(38) The system of (37) has been transformed to the form similarly as in Section 5; that is,𝚽̈=−𝐀⁻¹𝐁+𝐂+𝐃.(39) Next, we can write𝛀̇=−𝐀𝚽₋₁̇=𝛀,𝐁+𝐂+𝐃.(40) 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 of𝑓₁≡𝑥_𝑛−𝑎=0.(41) The Lagrange's equations are simple in this case, because we have only one Lagrange's multipliers 𝜆₁.

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 constraints𝑓𝑓₁≡𝑥_𝑛−𝑎=0,₂𝑓≡𝑦_𝑛−𝑏=0,₃≡𝜑_𝑛−𝛼=0,(42) and three Lagrange's multipliers 𝜆₁, 𝜆₂, and 𝜆₃.

6. 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.

6.1. 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.

The properties of the elastica are as follows:(i)bending rigidity, 𝐶=0.0025 Nm²,(ii)linear mass density, 𝜌=0.01 kg/m,(iii)length, 𝐿= 1 m.

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:𝜑^0𝑘=𝜋𝜔2rad,_0𝑘=0rad/s,for𝑘=1,2,…,𝑛.(43) 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 𝑡₀,𝑡₁,𝑡₂,…. The shapes are presented in Figure 4. Figure 5 shows variations of generalized coordinates 𝜑₁,𝜑₂,…,𝜑₈ in the respect of time 𝑡. Figure 6 shows variations of generalized coordinates 𝜑₁,𝜑₂,…,𝜑₈ in the respect of time 𝑡 for elastica of bending rigidity 10 times less than previously (𝐶=0.00025 Nm²).

Figure 4 

The shape of elastica in the following time instant from 𝑡0 to 𝑡20.

Figure 5 

The variation of generalized coordinates 𝜑1,𝜑2,…,𝜑8 in the respect of time 𝑡.

Figure 6 

The variation of generalized coordinates 𝜑1,𝜑2,…,𝜑8 in the respect of time 𝑡 for bending rigidity of 𝐶=0.00025 Nm2.

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).

Figure 7 

The rigid rod as physical pendulum.

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 𝜑₈, its values marked by 𝜑^Δ𝑡1₈ with given time step Δ𝑡₁=0.005 s have been calculated. Next, the calculations have been repeated for the less time step Δ𝑡₂=1/2Δ𝑡₁. The difference Δ𝜑₈=𝜑^Δ𝑡1₈−𝜑^Δ𝑡2₈ has been determined for following 𝑡_𝑗=𝑡₀+𝑖⋅Δ𝑡₁.

The calculations have been made for 𝐶=0.0025 Nm². The graph of differences Δ𝜑₈ for following time instant is presented in Figure 8.

Figure 8 

Test of accuracy of generalized coordinate 𝜑8 in the following time instant.

The differences Δ𝜑₈ 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.

6.2. The Motion of the Elastica Fixed at Two Points

In this case, the elastica fixed at two points by joints as in Figure 3 will be considered. The motion is under the influence of the gravity and elastic forces.

The properties of the elastica are as follows:(i)bending rigidity, 𝐶=0.0015 Nm²,(ii)linear mass density, 𝜌=0.01 kg/m,(iii)length, 𝐿=1.2 m.

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:𝜑^0𝑘=𝜋𝜔21−2(𝑛𝑘−−11)rad,_0𝑘=0rad/s,for𝑘=1,2,…,𝑛.(44) Finally, the shape of the elastica in the following time instant 𝑡_𝑗 has been obtained. It is present in Figure 9.

Figure 9 

The shape of the elastica in the following time instant.

In order to check accuracy of calculations, we tested how the boundary conditions (29) change as time goes by. In this way, we can check how the algorithm secures fulfillment of the boundary conditions𝑓𝑓₁≡𝑥_𝑛=0,₂≡𝑦_𝑛−𝐻=0,(45) As a measure of deviation of boundary conditions, we can take Σ=|𝑓₁|+|𝑓₂|, which, if the boundary conditions are fulfilled, should be equal zero. The calculations have been made for two time steps: Δ𝑡₁=0.005 s i Δ𝑡₂=1/2Δ𝑡₁. The variation of Σ in respect of number of iterations are presented in Figure 10 for two time steps.

Figure 10 

The analysis of fulfillment of the boundary conditions ∑=|𝑓1|+|𝑓2|.

From Figure 10, we can conclude that for two times less time step deviation of boundary conditions was significant less. For Δ𝑡₁ after 300 iterations, deviation was about 2⋅10⁻⁴ m, and for Δ𝑡₂=1/2Δ𝑡₁, deviation was about 2⋅10⁻⁵ m (10 times less).

7. 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].

Figure 11 

The simulation of fabric folding using Lagrange's equations.

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.