# Effects of Phase Shift Errors in Recurrence Plot for Rotating Machinery Fault Diagnosis
## 3 Results  
### Phase Shift Effects on Multi-Source Functions  
When a mechanical object is subjected to vibration or oscillation caused by a force or moment, it tries to return to its equilibrium. Forces and moments are usually functions of displacement. The representation of a dynamic system by a second-order differential equation is expressed as:  
\[[M][x]+[C][x]+[C][x]=F(t) \tag{7}\]  
where \(M\), \(C\), and \(K\) are the structural mass, the stiffness, and the damping matrix, respectively, and \(F(t)\) represents the external loads. A simple example of a mechanical system with some vibrations sources is presented in Figure 3. The rotor assembly gives the primary vibration source, while the two bearings and the friction disk represent three more vibration sources. The most common rotating machinery is represented by an arrangement of a motor or turbine, a gear box, and one piece of equipment that transforms the dynamic torque into work. In linear systems, the answer is proportional to the entrance function \(F(t)\), which is composed of different periodic, synchronous and asynchronous functions. The sine and cosine functions often appear as a solution to Equation (7). Both functions are related as \(\cos(wt)=\sin(wt+\pi/2)\). In this case, \(w\) is the frequency of oscillation of the function expressed in radians per unit time. The frequency in cycles per unit time is often denoted by \(f\). Both frequencies are related as \(w=2\pi f\). In the function \(A\)\(\sin(wt)\), A is the amplitude of oscillation. The period is the time between two adjacent peaks. Period and frequencies are related as \(P=1/f=2\pi/w\). In trigonometric functions, there are two types of shifts: phase shift and vertical shift. The phase shift is the movement of the function when it is displaced horizontally from the initial position. The vertical shift happens when the function moves vertically from its initial position. The solution for the simple harmonic motion of one particle with an initial phase angle \(\phi\) is expressed by the following sinusoidal form:  
\[x=A\,\sin(w\,t+\phi) \tag{8}\]  
Then, velocity and acceleration are obtained by differentiating Equation (8). The displacement, velocity and acceleration are shown in Figure 4, where \(A=1\), \(w=2\), \(f=1/\pi\), \(P=\pi\), and \(\phi=0\). Note that there is a phase offset of 90\({}^{\circ}\) between displacement-velocity and velocity-acceleration. However, \(\phi\) can take any value, and its effects are only moving the function backward or forward. Figure 3: Photo of one application of the proposed research. As commented before, the phase plane is used to describe multiple states of a system in which a single point describes a state of an arbitrary system. Assuming that a system behavior is characterized by the state \(x(t)\), where its rate of change is \(\dot{x}(t)=dx(t)/dt\), then the phase plane is given by plotting \((x(t),\dot{x}(t))\)[40]. Figure 5 shows the evolution and the phase plot for a single periodic function without \((\phi=0)\) and with \((\phi=\pi/8)\) phase shift. The functions are: (a) \(x_{1}=\sin(2t)\), \(\dot{x}_{1}=2\cos(2t)\) and (b) \(x_{2}=\sin(2t+\phi)\), \(\dot{x}_{2}=2\cos(2t+\phi)\), where \(\phi\) can have any value. The evolution in Figure 5a reveals a constant displacement at both the beginning and the end of the trajectory. The phase plane in Figure 5b shows the same geometry in both cases. The RP with a threshold distance \(\epsilon=0.13\) for \(x=\sin(2t+\phi)\), \(\phi=0\) and \(\phi=\pi/8\) is computed in Figure 6. It is a typical pattern for a simple harmonic function. The perpendicular lines moved along the main diagonal line and there are no variations in shape. Figure 4: Displacement, velocity and acceleration for the function: \(x=\sin(2t+\phi)\). Figure 5: Evolution of the state variables and phase plane for \(x=\sin(2t+\phi)\), \(\dot{x}=2\cos(2t+\phi)\) and \(\phi=[0,\pi/8]\).