## 2 Atmospheric Corrosion and Impact on Natural Frequencies
Exfoliation involves intergranular weakening of the metal structure, affecting vibration modes at different frequencies compared to pitting. Wavelet analysis can highlight these differences through higher temporal and frequency resolution. Natural frequencies are essential characteristics of any structure, and they reflect its dynamic behaviour in the absence of external disturbances. They are influenced by the geometric properties of the structural elements, the material from which these elements are made, and the loading conditions. Figure 4: Exfoliation corrosion, \(\gamma\)-Fe\({}_{3}\)O\({}_{4}\) (brown rust): **(a)** SRs 1300-26/3.5 excavator bucket wheel arm; **(b)** exfoliation corrosion details. Figure 3: Pitting corrosion: **(a)** ERC 1400-30/7 excavator counter-arm caisson beam; **(b)** pitting corrosion details. Figure 2: General corrosion—\(\gamma\)-Fe\({}_{3}\)O\({}_{4}\) (brown rust): **(a)** wheel arm with type ERC1400-30/7 excavator buckets; **(b)** oscillating arm excavator tower ERC 1400-30/7. Corrosion can significantly affect the structural integrity of mining equipment [21; 23; 26]. The geometric elements of these machines, exposed to corrosive environments, can undergo significant changes over time due to the effects of corrosion. Thus, it is essential to understand how corrosion influences the natural frequencies and dynamic behaviour of mining machinery. The study of natural frequency analysis and the structural changes caused by corrosion gives us a deep insight into the performance and reliability of mining equipment. By identifying and evaluating these issues, we can develop effective maintenance and repair strategies, minimising the risk of structural failures and maximising the durability and operational efficiency of mining machinery in corrosive environments. We start from the equation of the motion of a linear harmonic oscillator, which is [27; 28; 29]:  
\[m\times\frac{d^{2}x}{dt^{2}}+c\times\frac{dx}{dt^{2}}+kx=F(t), \tag{1}\]  
\[m\times\frac{d^{2}x}{dt^{2}}+kx=0, \tag{2}\]  
The general solution of this equation has the form:  
\[x(t)=A\sin(\omega_{n}t+\phi), \tag{3}\]  
where \(A\)--initial amplitude; \(\omega_{n}\)--damped natural frequency; \(\Phi\)--initial phase; \(m\)--oscillator mass; \(c\)--damping coefficient; \(k\)--elasticity constant; \(F(t)\)--external force as a function of time. By substituting the general solution of (3) into (2), we get the equation:  
\[m\Big{(}-\omega_{n}^{2}A\sin(\omega_{n}t+\phi)\Big{)}+k(\sin(\omega_{n}t+\phi ))=0, \tag{4}\]  
After simplifying and isolating \(\omega\) we obtain:  
\[k=m\omega^{2}\ \Rightarrow\omega_{n}=\sqrt{\frac{k}{m}}, \tag{5}\]  
This formula shows that the natural frequency of oscillations is directly proportional to the square root of the stiffness constant and inversely proportional to the square root of the oscillator mass. Thus, greater stiffness or less mass will lead to a higher natural frequency of oscillation. For a simple harmonic oscillator, the frequency f (expressed in hertz) is given by the formula:  
\[f=\frac{1}{2\pi}\sqrt{\frac{k}{m}}, \tag{6}\]  
This formula expresses the relationship between the oscillation frequency, the stiffness constant, and the mass. The higher the stiffness constant and the lower the mass of the oscillator, the higher the frequency of oscillations will be. For a bar, embedded at one end and free at the other, acted upon by a force, the stiffness constant (k) is related to the material characteristics of the bar, such as the modulus of elasticity (E) and the moment of inertia (I), which is a function of the geometric characteristics of the bar determined according to the following relation:  
\[k=\frac{3 EI}{L^{3}}, \tag{7}\]  
For a plate, the stiffness constant depends on the geometry of the plate and the material from which it is made. In general, for a plate under uniform loading, the stiffness constant can be expressed as a function of the modulus of elasticity of the material, the thickness of the plate (t), and the moment of inertia of the cross section of the plate.