## 3 Numerical Analysis  
To investigate the performance of the aforementioned methods, the dynamic response of a geometrically nonlinear cantilever beam under a periodic excitation is studied. As shown in Figure 3, a friction contact is considered at its tip. The beam motion is described in plane \((\mathbf{e}_{x},\mathbf{e}_{y})\) which are the axial and transverse coordinates, respectively. Axis \(\mathbf{e}_{x}\) is along beam neutral axis in undeformed configuration while \(\mathbf{e}_{y}\) is perpendicular to it. Notably, this model can be assumed as a simplified model of blade-casing rubbing interaction in gas turbines with an assumed rigid casing and disk.  
The beam length is L = 2.54 m and internal structural damping is introduced as mass-proportional viscous damping, where \(\alpha=5\). The beam material properties are [PERSON]'s modulus \(E=207\) GPa, density \(\rho=7801\) kg/m\({}^{3}\), and [PERSON]'s ratio of \(\
u=0.28\). The beam cross section is square with area \(A=0.6451\) m\({}^{2}\). A harmonic force with the amplitude of \(F_{\text{e}}=111.29\) N in a constant direction (y coordinate) is applied on the beam. A FE beam model based on a total Lagrangian nonlinear formulation and a Timoshenko kinematics [37] is considered. This model is consist of 20 elements while each node has 3 DoFs.  
Besides the dynamic load applied on the contact, the contact state is defined by the static normal load \(N_{0}\) and contact parameters, including contact stiffness and the friction coefficient. The effects of these parameters have been studied extensively in  
Figure 3: Model of a planar cantilever beam with friction contact at its end.  
Figure 2: Newmark time integration algorithm.  
the literature [38]. Here, the friction coefficient and contact stiffness are considered as \(\mu=0.5\) and \(k_{\mathrm{t}}=k_{\mathrm{n}}=10\) N/mm.  
To provide a better understanding of the dynamics of the structure under study, initially, the forced response analysis of its full order model is studied. The frequency response functions of the beam at four values of the static normal preload (\(N_{0}=0,40,60\), and 250 N) are shown in Figure 4. In the case of blade-casing rubbing interaction, the normal preload \(N_{0}\) is due to the centrifugal force acting on a rotating blade. The reference system is the so-called open contact (\(N_{0}=0\) N) corresponding to the cantilever beam with no friction contact.  
It can be seen in Figure 4 that by increasing the value of the normal preload, initially (\(N_{0}=40\) and 60 N) only the resonance amplitude decreases, while the resonance frequency remains constant. With the higher value of normal preload (\(N_{0}=250\) N), the resonance frequency also increases due to the stiffening effect of the contact.  
To better assess the performance of the enhanced reduction methods, two completely distinct system configurations are analyzed, corresponding to different contact states, which are representative of possible configurations of a vibrating beam potentially rubbing against a rigid casing. The name of the configuration corresponds to the contact states that dominate the nonlinear dynamics during vibration. In detail: - Slip-liftoff configuration with \(N_{0}=40\) N and excitation frequency \(Freq=3.2\) Hz; - Stick-slip configuration with \(N_{0}=250\) N and excitation frequency \(Freq=8.1\) Hz.