PSI - Issue 81

Ján Dižo et al. / Procedia Structural Integrity 81 (2026) 11 – 17

14

The mathematical model of vibrations of a platform wagon with containers in a vertical plane is described by Domin and Chernyak (2003), and it has the form:

1 1 1,5 5 1 ( ), MqCqCqCq Ft  +  +  +  = 1,1 1 1,3 3

(1)

2 2 2,5 5 2 ( ), MqCqCqCqFt  +  +  +  = 2,2 2 2,3 3

(2)

3 3 3,3 3 3 ( ), MqBqCqCqCqFt + + + + = 3,3 3 3,1 1 3,2 2

(3)

4 4 4,4 4 4 ( ), M q B q C q F t  +  +  = 4,4 4

(4)

5 5 5,5 5 5 ( ), MqBqCqCqCqFt + + + + = 5,5 5 5,1 1 5,2 2

(5)

6 6 6,6 6 6 ( ), M q B q C q F t  +  +  = 6,6 6

(6)

(

)

( )

1 sign sign fr F  +

1 F t

=− 

2 

(7)

(

)

2 fr F t F l =   ( )

1  sign sign , + 2 

(8)

(

)

(

)

3 fr F t F ( )

sign

,

1 1 1 k       + + + + 2 1 1 2

= 

(9)

(

)

(

) 2 ,

F

a k      =−   − − −

(10)

4

1

1

2

1 1

(

)

(

)

5 fr F t F ( )

sign

,

2 1 3 k       + + + + 4 1 3 4

= 

(11)

(

)

(

)

( )

,

6 F t

a   

4 1 k a − −  

=−  

  −

(12)

1

3

3

4

where M 1 , M 3 , M 5 – the mass of the frame of the flat wagon with four containers, the first in the course of movement and the second bogies, respectively; M 2 , M 4 , M 6 – respectively, the moment of inertia of the frame of the flat wagon with four containers, the first in the course of movement and the second bogies; C ij – parameter characterizing the elasticity of the elements forming the calculation model; B i – energy dissipation function; a – bogie semi-base (a model 18 – 100); q i – generalized coordinates characterizing the translational and angular displacement, respectively, of the frame of the flat wagon with four containers, the first in the course of movement and the second bogies; k i – stiffness of the springs forming the spring suspension of the bogies; β i – damping coefficient; F fr – friction force acting in the spring suspension of the bogies; η i – track unevenness. The derivation in performed based on the energetic method, which is described by Caban et al. (2023). The solution of the mathematical model (eq. 1 to 6) was carried out in the MathCad software package presented by Bogach et al. (2020) and Syasev (2004). The initial translational displacement of the platform car frame is taken to be 0.004 m, of the bogies – 0.003 mm, as written by Domin and Chernyak (2003), Rahimov and Zafarov (2025) and Opasiak and Hełka (2025). Angular displacements are zero. The initial velocities are also set to zero. The calculation was performed under the condition of the platform car moving at a speed of 100 km/h. The track irregularities lead to the wagon excitation as it is described by Ézsiás et al. (2024), Fischer (2025), Fischer et al (2024), Frej et al. (2023), Semenov et al. (2022), Semenov et al. (2023a, 2023b) and Zuska et al. (2021), with an unevenness length of 3 m, amplitude of 0.01 m. The calculation results are shown in Fig. 6. Therefore, the acceleration acting on the platform car, and accordingly on the containers, was about 4 m/s 2 . Such acceleration occurs when passing the joint. After that, the acceleration decreases and is about 2.5 m/s 2 . This acceleration was taken into account when applying the vertical load in the calculation scheme shown in Fig. 4. When calculating the strength of the fastening of the unloading hatch cover, it was done behind the hinges. In Fig. 7, the fastening zones are marked in green. In this case, rigid connections were used. The finite element (FE) method was applied for the strength analysis of the hatch cover, because it is the most widely used method for such a type of analysis, as it can be found in the research by Messoude et al. (2025) and Sae-Long et al. (2025). The created FE model is formed by tetrahedra (Fig. 8). It has 22514 nodes and 68378 elements with a maximum size of 10 mm and a minimum of 2 mm. The material for the discharge hatch cover is 09G2S steel with a yield strength of 345 MPa and permissible stresses for the applied loading regime of 210 MPa prescribed in DSTU (2014).