# Estimation of Resonance Frequency for Systems with Contact Using Linear Dynamics Methods
## 1 Introduction  
The analysis of vibrations including contact is a demanding requirement in the field of industrial applications. A few methods of vibrations analysis exist, including non-linearity in the form of contact, which can be separated into two main groups: frequency domain and time domain analysis.  
For frequency domain analysis, the following might be listed: the Harmonic Balance Method described in [1, 2, 3, 4] and its modifications based on force response [5], or the System DOF reduction described in [6, 7, 8]. Other methods used in the frequency domain are the energy conservation schemes described in [9, 10]. In specific problems such as turbomachinery blades, a non-linear modal analysis was proposed [11]. All of those methods are applied or are for simple geometrical cases, or are relatively difficult to apply in industrial usage, because of the truncated Fourier expansion used for the frequency domain for each DOF, which would produce extremely large models [12].  
As a specific type of solution for the situation of vibrations with contact is the linearization of a model, which is used for beams with cantilever-Hertzian contact vibrations, as described in [13, 14, 15, 16]. However, these simplifications, always taking into account closed contact, are applicable only for specific cases, as presented in those papers.  
The time domain methods that can be pointed out for solving the problem are based on the full solution of dynamic equations including interactions, such as contacts, and their simplifications. The full solution for dynamic equations has been shown in many cases to be very accurate. In addition, some of the research has been completed in that field for delaminated sandwich-composite panels to simulate the behavior of a delaminated area subjected to vibrations, including surface contact [17].  
In addition, for that group, there are modifications of the method that use both the time and frequency domain shown in [18].  
Finite Element (FE) codes, widely used in industry, have no implementation on frequency domain methods except for classical modal analysis, and only the time domainsolution is available for use in engineering to include contact in vibrations. The explicit solution of dynamics shown in [16] also has some limitations for large models. The Courant criterion enforces a very short time step which will lead to a huge number of steps [19], which increases the integration error or limits the simulation to short-time events.  
Three main approaches could be observed to obtain resonance frequencies of the system in practice. The first is by neglecting contact and performing simulation, in the way that contact was always opened [20; 21]. Second, where simplification for modal analysis is completed, in a way similar to contact being always closed [22; 23; 24; 25]. A third approach is to simulate a situation between the two mentioned springs and springs are added to contact to introduce additional stiffness, based on earlier experimental tests; however, the method requires earlier physical testing which is impractical in many solutions.  
To study how resonance frequency reacts to the introduction of contact to the system, a cantilever beam was carried out with usage of an implicit dynamics solution. The implicit algorithm was selected over the explicit algorithm due to no problems with tiny time-stepping or simulation time limitation.  
With the extracted characteristic of resonance frequency to contact length dependence, a simplified method of prediction was proposed, using modal analysis.