FIBRES & TEXTILES in Eastern Europe 2014, Vol. 22, No. 2(104): 76-79.
Resonance Vibrations of an Elastic System
Supported on a Textile Layer
Jerzy Zajączkowski
Technical University of Lodz
116 Zeromskiego Street, 90-924 Lodz, Poland
http://www.bhp-k412.p.lodz.pl/k412/index.htm
Key words: nonlinear vibration, layer of fibres, electromagnet excitation
Strzeszczenie
Drgania Rezonansowe Układu Sprężystego Podpartego Warstwą Włóknistą
W pracy badane są drgania układu sprężystego spoczywającego na warstwie włókien.
Sformułowano model matematyczny układu wzbudzanego elektromagnesem zasilanym prądem
zmiennym. Znaleziono dwa obszary rezonansowe. W jednym z nich częstotliwość siły nacisku
jest dwukrotnie większa od częstotliwości prądu zasilania. W drugim obszarze częstotliwość
drgań jest równa częstotliwości prądu. W rezonansie system traci kontakt z warstwą w chwili
gdy siła reakcji staje się równa zeru.
Abstract
The vibrations of a flexible system resting on a layer of fibres are investigated in this
paper. Mathematical model of the system excited by an electromagnet supplied with an
alternating current is formulated. Two resonance regions are found. In one region the frequency
of the varying contact force is twice the supply electric current frequency. In the other region the
frequency of vibration is equal to the current frequency. At the resonances the system is losing
contact with the layer at the moment when the reaction force becomes equal to zero.
Introduction
Properties of textiles subject to the action of compressive forces were investigated in
works reported in papers [1,2,3]. Dynamic properties of compressed textiles were studied
numerically in paper [4] by subjecting them to electromagnet excited vibrations. The
mathematical model of a textile layer subject to compression was proposed in paper [5]. The
formulas were derived on the assumption that the properties of the layer were determined by the
bending elasticity of fibres and by the resistance to the fluid flow that is squeezed out of the
layer. The problem of vibration of a rigid object supported on a textile layer has been studied in
paper [6]. In this paper, the elasticity of the vibrating object is included. The paper is a
continuation of works [4,6].
Equations of motion
A schematic diagram of the considered system is shown in Figure 1. The system is used
for measuring the fatigue of textiles subject to oscillating compression forces. The similar system
was previously studied in paper [4]. The element of mass m2 is connected to the element of mass
m1 by the flat spring of stiffness k2. The elements are attracted to one another by the force of the
electromagnet having the coil of inductance L and resistance R. The coil is powered by voltage
u=u(t), being a function of time. The element m1 rests on textile layer k1.
Figure 1. A schematic diagram of an elastic system resting on a textile layer and subject to
electromagnetic oscillating force.
The relationship between the compressive force and the contraction of a textile layer was
assumed to be of the form (1). The parameters (k1, c1, L1, H1) are material constants of the textile
layer and are defined in paper [5].
2
 dx  dx 
c1 sgn  1  1 
k1 x1
 dt  dt  , y  H , y  L ,
Fk 
, Fc 
1
1
1
1
3
3


x1 
x1 
1  
1 

 L1 
 H1 
F  Fk  Fc for Fk  Fc  0 , F  0 for Fk  Fc  0.
(1)
Summing up all the forces acting on mass m1 and m2, one obtains the set of equations governing
their motion (2).
2
F1  m1
d 2 x1
k1 x1

3
2
dt

x1 
1  
 L1 
 dx  dx 
c1 sgn  1  1 
 dt  dt   c  dx2  dx1   k x  x   F  m g  0,

 2 2 1
2
E
1
3
dt 
 dt

x1 
1 

 H1 
d 2 x2
 dx dx 
F2  m2 2  c2  2  1   k2 x2  x1   FE  m2 g  0.
dt
dt 
 dt
(2)
In equations (2) (k2, c2) are stiffness and damping coefficients of the flat spring. The
electromagnet force FE is defined by relationships (3).
L
di d L

i  R i  u,
dt d x
1 dL
FE   i 2
,
2 dx
x    x2  x1 .
(3)
In equations (3), i is the intensity of the current, R is the circuit resistance, u the feed voltage,
coordinate x specifies the position of the movable core centre and has its origin in the centre of
the coil,  denotes the distance from the of centre of coil to the centre of core at rest. In order to
determine the inductance L the coil was supplied with the voltage u=Umsint for several
stationary positions x of core and the current I was measured. The values of inductance L=L(x)
was then calculated as a function of the mutual position x of the core and the coil (4). The
discrete function L(x) (4) was then approximated by continuous function (5).
L x  
1

2
U 
2
  R
 I 
(4)
L x  
Lmaz  Lmin 2 2 
lx
lx
r0  l

 2
2
2
2l
r02  l  x 
 r0  l  x 
dL Lmax  Lmin 2 2 
1

r0  l
 2
2
dx
2l
 r0  l  x 

 L ,
 min

2




l

x
1
1 

2
2
 r  l  x  
2
2
0

 r0  l  x 
2

 


l

x
 1 

2 
2



r

l

x
0


(5)
It was found that the shape of the function describing the intensity of the magnetic field of a
stationary solenoid suited for this purpose if the constant parameters were properly chosen. In
formula (5) Lmax is the measured maximum inductance of the coil when the centre of the core
coincides with the centre of the coil, Lmin is the measured minimum inductance of the coil when
core is in the end position, 2l denotes the computational length of the coil equal to the distance
between the maximum and minimum of its first derivative dL/dx, r0 denotes the computational
radius of the coil.
Numerical results and discussion
Computer program was created in C++ and set of differential equations (2,3) was
integrated by using Runge-Kutta method. The calculations were carried out for each frequency
till the system achieved the steady-state vibrations. The following values of system parameters
were taken: masses m1=0.2kg and m2=4kg, gravity acceleration g=9.81m/s2, stiffness and
damping coefficients of flat spring k2=8257.5N/m, c2=160Ns/m, textile layer material constants
k1=500N/m, c1=1Ns2/m2, L1=0.03m, H1=0.03m, electromagnet parameters Lmax=0.364319H,
Lmin=0.04H, l=0.028m, r=0.032m, R=40, distance from the of centre of coil to the centre of
core at rest=l, voltage u=Umsin(t), Um=220V, initial conditions x1(0)=0, x2(0)=0, x3(0)=0,
x4(0)=0, i(0)=0.
The maximum amplitudes of vibrating masses (x1/L1, x2/L1) and the textile layer reaction
force F/(m2g) versus frequency  of the supply voltage relative to the constant 2=(k2/m2)0.5 are
shown in Figure 2. Here, two regions of the resonance amplification of vibrations can be
observed. The time history of the contact reaction force F, the current i and the electromagnet
core displacement with respect to coil centre x at the resonance regions are shown in Figure 3. At
the resonance regions the system loses contact with the layer and as a result the reaction force
vanishes for a moment. Comparing the frequency of the force with that of the current, one can
see that the frequency of the force is two times higher than the frequency of current for
2=0.3, while for 2=0.7 they are equal at the steady state. For both cases, however, the
number of electromagnet core oscillations is doubled comparing to number of current
oscillations. For 2=0.7, one core oscillation takes place in contact with the layer and one
when the system is above the layer not touching it.
Figure 2. Maximum compression magnitude of textile layer x1, contact compressive force F and
mass displacement x2 versus voltage angular frequency  at the steady-state, c1=1Ns2/m2,
.
Figure 3. The contact reaction force F relative to the gravity force of the mass m2, the current i
and the electromagnet core displacement x relative to have half length l of the coil at the
resonance
.
Conclusions
1. The system composed of two masses connected by elastic flat spring and supported on a layer
of fibres exhibits two resonances: a peak and a local maximum.
2. In the resonance regions the elastic system is losing contact with the fibre layer and as a result
the reaction force of the layer drops to zero for a short while.
3. In the neighbourhood of the peak resonance the frequency of varying contact force is twice the
frequency of the supply voltage. This is because the excitation force depends on square power of
the electrical current.
4. In the neighbourhood of the resonance associated with local maximum, the frequency of the
varying contact force is equal to the frequency of the supply voltage at the steady state, while
number of oscillations of electromagnet core is twice the number of oscillations of the supply
voltage at steady state. This phenomenon is a result of losing contact of the vibrating object with
the textile layer and its free motion.
References
1. J.I. Dunlop: On the compression characteristics of fiber masses, J. Textile Institute, 1983, 74,
92-7.
2. J. Hu, A. Newton: Low-load lateral-compression behaviour of woven fabrics, Journal of
Textile Institute, 1997, 88, Part I, 3, 242-253.
3. M. Matsudaira: Q. Hong: Features and characteristic values of fabric compressional curves,
International Journal of Clothing Science and Technology 1994, 6, no. 2/3, 37-43.
4. Jerzy Zajączkowski: Modelling of dynamical properties of textiles under compression. Fibres
& Textiles in Eastern Europe 2000, 8, 1(28), 57-58.
5. Jerzy Zajączkowski: Impact of an object on a layer of fibres submerged in a fluid.
Fibres & Textiles in Eastern Europe 2010, Vol. 18, No. 6 (83).
6. Jerzy Zajączkowski: Chaotic Vibrations of an Object Supported on a Layer of Fibres. Fibres
& Textiles in Eastern Europe 2012, 20, 6A (95): 75-77.