130859711

Mathematical model analysis of sample from...

b. The solution of the equations describing the model

As noted in the abstract the mathematical model of sample was shown in

papcr [1].
a)		pressure piąte	^b)
i
		~~foam	'—r-^a-H 1 1 %
		^\foam / coating	: 1 1 I*
		.base

Fig. 2. Sample tested: a) cross-section of the actual sample, b) created physical model Rys. 2. Badana próbka: a) przekrój próbki rzeczywistej, b) utworzony model fizyczny

Based on Figurę 2, it is possible to write the eąuations of motion [1]:

m₃x₃ +b₂ (x₃ - x,) + k₂ (x₃ - x,) = F (cd )

(w, + m₂)x, +(Z>, +6₂)(x, -x₄) + (fc, + &₂)(x, -x₄)=b₃ (x₃ -x,) + fc₃ (x₃ -x,)    (3)

w₄x₄ + b₄x₄ + k₄x₄ = (ó, + b₂) (x, - x₄) + (k_x + k₂) (x, - x₄)

where:

x₍ - displacement of the upper layer of the sample element, m, m_i - mass, kg,

b, - damping coefficient, Ns m'², k. - stiffiiess, N m'²,

F[cd) - driving force, according to DIN 45673-5, N, i = 1,2,3,4 the index that means respective: fundamental (porous) part of the sample, upper, side and lower skin layer.

It was assumed that the displacements are calculated from the static stable balance point. This assumption allow an adoption of zero initial conditions x₍ (0) = 0 and x. (0) = 0. After applying to the (3) Laplace transform and rearranging data formula was given by:

m_is²X_i +(6₃s^,+£₃)(X₃ -X,) = F(s)

(m, + m₂)s²X, +[(6, +b₂)s + (k_l + k₂)](X, -X₄) = (b,s + k,)(X₃ -X,)    (4)

[m₄s² +b₄s + k₄^X₄ =[(6, +b^)s + {k_x + &₂)](X, -X₄)

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