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MATHEMATICAL MODELLING AND STABILITY OF NONLINEAR MECHANICAL 
SYSTEM IN HARMONIC EXCITATIONS 
Buranov Hudayor Mahmadiyorovich
1
, Khodjabekov Muradjon Usarovich
2 
Uzbekistan, Samarkand State University
1
, Uzbekistan, Samarkand State Architectural and Civil 
Engineering Institute
2
.  
Annotation. This work dedicated that investigation condition and border of stability of 
nonlinear mechanical system. 
Results and discussions. We will use the free body diagram method [1] to derive the 
differential equation governing the motion of the nonlinear mechanical system of fig.-1 using 
as the 
generalized coordinate. Summing moments about the pin support leads to 
∑
∑
(1) 
(
)
(
)
̇ (
)
̈
̈ (
)
(2) 
where
is characteristics of dissipation of hysteretic type spring [2]; 
and 
are coefficients linearization;
is stiffness; 
is length of the 
beam; 
is amplitude of harmonic excitations; 
is damping coefficient; 
is the mass of the beam. 
We will write following from (2): 
̈ ̇
(3) 
(3) is the differential equation of the given system in fig.-1. We able to analyze behavior of 
given dynamic system completely from received differential equation. 
In [3] single and multi-degree of hysteretic dynamic systems have been analyzed completely 
and method of analyzing has been shown. We will use this method to analyze the given dynamic 
system. Then we will investigate normal equation. For this we will choose (3) the differential 
equation‘s solution as following: 
( ) 
(4) 
where 
are amplitude and phase of 
and they are slowly changing functions, namely 
̈ ̈ ̇ ̇
. 
We will calculate 
̇ ̈
from (4) and put in (3). 
̇ ( ̇) ̇
( ̇)
(5) 
We will receive followings from (5): 
̇
(6) 
̇
Received (6) are normal equations of the given system. 
We will investigate the steady-state solution. For this 
̇ ̇
will be put in (6). 

---

146 
√
We use vertical tangents method [4] to investigate the condition and border of stability. 
From (8) we will write following equation: 
(
)
(
)
( 
) 
(9) equation gives us that border of stability. 
We will write following square form of (9). 
(
)
(
)
where 
The steady-state solution will be stable when the sign of (10) is positive. For this we will write 
following matrix from (10): 
1 
0 
0 
1 
0 
0 
where 
(
)
(
)
(10) Square form will be positive when the matrix form‘s main diagonal matrixes are positive, 
namely 
1 
1 
0 
0 
1 
1 
0 
-
0 
1 
1 
0 
0 
1 
0 
0 
(
)
The received 
is condition of stability of steady-state solution. 
Conclusions. We can say that from (9) and (11), border and condition of stability of steady-
state solution depend on stiffness, damping coefficient, mass and amplitude.