## 2 Structural Characteristics and Dynamic Behaviors of Cylinder Liners
### Excitation Configurations
#### 2.2.2 Piston Side Thrust  
Driven by the combustion pressure \(P_{c}\) and inertial force \(P_{i}\), the piston assembly moves laterally and impact the liner surface ceaselessly while it moves up and down longitudinally. Figure 4 shows the dependence of the side-thrust force \(F_{y}\) upon combustion and inertial forces. In addition because time-varying nature of these forces, the side-thrust force pushes the piston assembly away from one side to the other, slapping the cylinder liners intermittently [9]. Figure 4: Dynamic model of cylinder assembly. Figure 3: Short-time Fourier transform (STFT) of measured in-cylinder pressure; (**a**) STFT result of the pressure measured at engine speed of 1800 rpm and load of 40 Nm; (**b**) Raw pressure signal and the pressure rise rate. For more accurate analysis, the piston side-thrust force can be calculated according to the kinematic connection, the specific expression for the longitudinal displacement of piston \(x_{p}\) in terms of the crank angle \(\omega t\) can be derived as [18]:  
\[x_{p}=r_{c}(1-\cos\omega t)+l(1-\sqrt{1-\lambda^{2}\sin^{2}\omega t)} \tag{1}\]  
where \(\lambda=r_{c}/l\), \(r_{c}\) is the crank radius and \(l\) is the length of the connecting rod. Furthermore, Equation (1) can be simplified into a first-term approximation form [9]:  
\[x_{p}\approx r_{c}(1-\cos\omega t)+\frac{1}{2}\lambda^{2}\sin^{2}\omega t \tag{2}\]  
The side-thrust force of the piston assembly toward the cylinder wall can be decomposed from the longitudinally acting combustion force \(P_{c}\) and the inertial force of the moving piston assembly \(P_{i}=m_{l}r_{c}\omega^{2}(\cos\omega t+\lambda\cos 2\omega t)\) (\(m_{1}\) is the equivalent mass of the piston assembly). Along with the periodically changing pendular angle of the connecting rod, the side-thrust force is also periodically changing, with a periodic term \(\sin\omega t/\sqrt{1+\left(\lambda\sin\omega t\right)^{2}}\), which pushes the piston assembly to skip from one side to the other. Finally, the piston side-thrust force \(F_{y}\) can be calculated by [9]:  
\[ \begin{array}{ll}F_{y}&=(P_{c}-P_{i})\lambda\sin\omega t/\sqrt{1+\left( \lambda\sin\omega t\right)^{2}}\\ &=P_{c}\times\lambda\sin\omega t/\sqrt{1+\left(\lambda\sin\omega t\right)^{2}} +P_{i}\times\lambda\sin\omega t/\sqrt{1+\left(\lambda\sin\omega t\right)^{2}} \\ &=F_{c}-F_{i}\end{array}  \tag{3}\]  
The crankshaft and connecting rod mechanism govern the longitudinal movement of piston assembly. The side-thrust force calculated in Equation (3) is the force driving lateral movement of the piston. Since the sliding friction between the piston and liner surfaces is less than 2% of the piston side-thrust force [9,19], the friction behaviors between the piston and liner surfaces are not considered in the model. The clearance between the matching surfaces was modelled with a frictionless contact pair to limit lateral movement of the piston. As shown in Figure 5a, the piston's translational degree of freedom is perpendicular to the moving plane and is set as free in order to avoid over-constraint. The calculation is simplified by setting rotational degree of freedom (DOF) of the piston tilting motion as free, while ignoring all other rotational freedoms. In this way, the key motions of the piston are modelled closer to the reality. Figure 5: Excitation configurations of cylinder model; (**a**) Piston slap; (**b**) Combustion shock. Combustion excitation is represented by the in-cylinder pressure data collected from the pressure sensor mounted on the test engine. To accelerate the simulation process, sampling points were resampled to 1800, from 3600 points of raw cylinder pressure signal by polyphase implementation, to show the compression and power strokes. As shown in Figure 5b, the application zone for the combustion force was restricted to the top zone of the liner's inner surface. The height of this zone is 15 mm, enough to simulate the areas of effective contact at the moment of combustion shock.