## 3 Methodology of Wire Rope Research  
The possibility to find a broken wire is based on the difference of natural frequencies between the rope and its broken wire. Due to wearing of the rope, its diameter can decrease about 5%, but a single wire takes less than 1% of the actual cross-section. Influence of single broken wire to overall stiffness  
Figure 1: Steel rope 6 \(\times\) 19 IWRC; d is the actual diameter of the rope. Figure 2: Test rig vibration measuring diagram: 1, frame; 2, rope support; 3, rope support; 4, test rope; 5, holder of transducer Tr4; 6, linear transducer “Hottiger Tr102”; 7, linear transducer “Hottiger Tr4”; 8, power amplifier 2706; 9, amplifier “Hottiger KWS 503 D”; 10, signal generator 1027; 11, electrodynamic mini vibrator 4810; 12, holder of transducer Tr102; and 13, broken wire. of entire rope is insignificant, therefore analysis of longitudinal and rotational rope stiffness brings poor results [6]. It is possible to find the presence and position of such defect increases in the case of vibration of free end of wire with different parameters as vibrations of entire rope surface [1]. The length of broken wire varies in the range from one winding of a strand to one-half of a winding. In addition, it is necessary to note that broken wires will float to the external rope surface due to bending and tension forces. The main parts of industrial ropes are single-wound or cross-wound; cross-linked types are not used in the mentioned technical installations. To define defect on the rope surface using dynamic method, there is a need to create a model of such broken wire, considering the connection between rope and wire body. Successful implementation of theoretical model of Timoshenko beam for such structures brings prospective to use such model in the presented case [51; 52]. Nevertheless, the boundary conditions in such model, as shown in Figure 3, are obtained from experimental results [5]. Timoshenko beam model can be implemented, when few conditions are respected. First, beam stiffness should conform to the condition in Equation (1) [53], which defines limits of object dimensions and material properties for the beam:  
\[\frac{EI}{\kappa L^{2}AG}<<1, \tag{1}\]  
where \(L\) is the length of the beam, \(A\) is the cross-section area of the beam, \(E\) is the elastic modulus of material, \(G\) is the shear modulus, \(I\) is the second moment of area, and \(\kappa\) is called the Timoshenko shear coefficient, which depends on the geometry. Normally, \(\kappa=9/10\) for a circular section. In the case of broken wire, this condition is obviously respected. If the condition is violated and beam is too long, elasticity of material cannot withstand its own gravitational force. An important second condition defines frequency limit, as stated in Equation (2) [54]. In our case, critical frequency is far beyond the limit, especially when diagnostic method is focused in first natural frequency modes. Such approximation of broken wire has critical angular frequency \(\omega_{c}\) and is given as:  
\[\omega_{c}=2\pi f_{c}=\sqrt{\frac{\kappa GA}{\rho I}}, \tag{2}\]  
where \(\rho\) is density of the material of the beam. Implementation of Timoshenko beam model lets us define ranges of testing frequencies for the broken wire, where vibration amplitudes of the rope surface and broken wire are significantly different. It is important to note that the entire rope body is treated as tensed wire with special properties [15], the behavior of which is different from Timoshenko beam, where shear properties of the rope remain unclear. Experimental research have proven the initial assumptions on frequency range; damping in this case plays little role. Figure 3: Type of Timoshenko beam connection to the ground: (**a**) fixed; and (**b**) through elastic spring with rotational stiffness \(C_{rot}\).