## 1 Introduction
[PERSON] and [PERSON] (2018) introduced a Hybrid Frequency-Time (HFT) method for analyzing the steady-state response of the large-scale dry-friction damped structural systems. The correlation of the static/dynamic coupling of the under-platform dampers was investigated by [PERSON], [PERSON] (2018). Mathematical relationships of dry friction force versus relative velocity in friction contact of two bodies were studied by [PERSON]\(\ddot{u}\)st et al. (2018). [PERSON] et al. (2018) focused their research on contact interface parameters in a nonlinear dynamic analysis of assembled structures. [PERSON] et al. (2018) studied the design and the calibration of a test rig specially developed to measure the in-plane forces transferred between the blade platforms through the under-platform damper and  
Figure 1: Dummy bladed disk for vibration analysis. their relative displacement. A nonlinear analysis based on an updated explicit damper model having different levels of details was performed [23], and the results were evaluated against a newly-developed under-platform damper test rig. A standard Scanning Laser Doppler Vibrometer (SLDV) technology to measure the mistuned bladed disk vibration was put forward by [PERSON] [24]. Then [PERSON] et al. [25] used a rotating laser Doppler Vibrometer to measure the performance of under-platform dampers for turbine blades. [PERSON] et al. [26] developed an experimental technique for damping efficiency estimation of gas-turbine blades. The experimental investigation into the dynamic response of the blades of a gas turbine for power generation carrying asymmetric under-platform dampers was presented [27]. [PERSON] et al. [28] proposed a novel approach which could directly measure the forces transmitted between the two platforms through the dampers. [PERSON] et al. [29] utilized the Bond graph model of dry friction dampers in structural analysis of turbine blades. An FE modal analysis was described and a simplified method to evaluate the under-platform damper effects was also presented by [PERSON] and [PERSON] [30]. Based on a set of newly introduced non-dimensional parameters that ensured a similar dynamic behavior of the test rig to a real turbine blade-damper system, a new experimental damper rig was developed [31]. Friction dampers are widely used in machines and structures, including under-platform dampers studied as a particular example in this paper, shrouds in steam engines [32], those used in buildings [33], washing machines [34] and train suspensions [35]. They share the same features of stick-slip vibration and are worth studying. Stick-slip vibration is a kind of nonsmooth nonlinearity and requires special mathematical treatment. [PERSON] et al. [36] made a comparison and classification of different contact models with friction that had been commonly used. In [37], the sticking and non-sticking orbits of a two-degree-of-freedom oscillator subjected to dry friction and a harmonic load were obtained in a closed form. [PERSON] et al. [38] developed a model of an elastic disc in sliding frictional contact with a rotating oscillator. Separation of the moving slider form the disc and its subsequent reattachment to the disc were considered, and various dynamic behaviors were discovered. One aim of this study is to propose a more accurate model to study the motion of a friction damper. Another aim is to study the general dynamic behaviour of friction dampers in various applications (using aero engine under-platform dampers as a particular example). A third aim is to establish an effective algorithm for dealing with stick-slip vibration. In this paper, the normal contact force and the resultant friction force at the contact interface is modeled as moving loads, which is what really happens in many friction dampers but has not been modeled as such in the past.