Piston dynamics model

Figures 3.2 depicts the piston and pin’s free body diagrams. 𝑚𝑚_pis, 𝑚𝑚_pin and 𝐼𝐼_pis are the piston mass, pin mass and piston centroidal moment of inertia. The gas excitation force is distributed over the piston’s crown surface. The equivalent point force for the excitation is applied with an offset at distances 𝑟𝑟_𝑔𝑔 in the lateral direction with respect to the piston’s centre-line. The inertial forces are represented by 𝑚𝑚_pis𝑥𝑥̈ , 𝑚𝑚_pis𝑒𝑒̈_{𝐶𝐶𝐶𝐶𝑔𝑔} for the piston and 𝑚𝑚_pin𝑥𝑥̈, 𝑚𝑚_pin𝐻𝐻̈ for the pin, respectively. Other forces in Figure 3.2 are: the combustion force 𝐹𝐹_𝐺𝐺, the lubricant reaction 𝐹𝐹hyd = ∬ 𝑃𝑃d𝑥𝑥d𝐻𝐻, the oil film viscous friction force 𝐹𝐹𝑓𝑓, the connecting rod force 𝐹𝐹𝑐𝑐𝑟𝑟 and the pin reaction forces 𝐹𝐹_pin,𝑥𝑥 and 𝐹𝐹_pin,𝑒𝑒. The combustion force and piston inertia forces are known as the primary forces because they are aligned with the primary axes of motion.

Figure 3.2. Piston and pin free-body diagrams

The secondary piston motion is related to its primary dynamics. This can be established in any of the following approaches: (i) a purely kinematic analysis or (ii) including inertial dynamics.

In the former approach, the piston’s secondary motion and the lubricant film load carrying capacity are ignored when estimating the primary forces. In the latter approach (inertial dynamics) these effects are considered. Thus, in the kinematics approach, the lateral resultant force acting on the piston, 𝐹𝐹_𝑡𝑡, can be calculated using the forces in the primary axial direction.

The three equations of motion are shown as follows (Littlefair (2013), Zhu et al (1993), Lui et al (1998), Balakrishnan and Rahnejat (2005) and Gohar and Rahnejat (2008)

𝑚𝑚𝑝𝑝𝑑𝑑𝑑𝑑𝑥𝑥̈ = 𝐹𝐹𝑝𝑝𝑑𝑑𝑝𝑝,𝑥𝑥+ 𝐹𝐹𝐺𝐺 ∓ 𝐹𝐹𝑓𝑓 (3.18)

𝑚𝑚_{𝑝𝑝𝑑𝑑𝑑𝑑}𝑒𝑒̈_{𝐶𝐶𝐶𝐶𝑔𝑔} = 𝐹𝐹_{𝑝𝑝𝑑𝑑𝑝𝑝,𝑒𝑒}+ 𝐹𝐹_{ℎ𝑦𝑦𝑑𝑑} (3.19)

𝐼𝐼𝑝𝑝𝑑𝑑𝑑𝑑𝛽𝛽̈ + 𝑚𝑚𝑝𝑝𝑑𝑑𝑑𝑑𝑒𝑒̈𝐶𝐶𝐶𝐶𝑔𝑔(𝑎𝑎 − 𝑏𝑏) − 𝑚𝑚𝑝𝑝𝑑𝑑𝑑𝑑𝑥𝑥̈𝑟𝑟𝑝𝑝 = 𝐹𝐹𝐺𝐺𝑟𝑟𝑔𝑔+ 𝑀𝑀ℎ𝑦𝑦𝑑𝑑∓ 𝐹𝐹𝑓𝑓𝑅𝑅 (3.20)

The dynamics of the piston pin are described by the force components in x and z directions

𝑚𝑚_{𝑝𝑝𝑑𝑑𝑝𝑝}𝑥𝑥̈ = −𝐹𝐹_{𝑝𝑝𝑑𝑑𝑝𝑝,𝑥𝑥} − 𝐹𝐹_{𝑐𝑐𝑟𝑟}cos 𝜙𝜙 (3.21)

𝑚𝑚𝑝𝑝𝑑𝑑𝑝𝑝𝐻𝐻̈ = −𝐹𝐹𝑝𝑝𝑑𝑑𝑝𝑝,𝑒𝑒− 𝐹𝐹𝑐𝑐𝑟𝑟sin 𝜙𝜙 (3.22)

The combination of equations (3.18) and (3.21) and simple mathematical operations result in the evaluation of the connecting rod force (Equation (3.23))

𝐹𝐹𝑐𝑐𝑟𝑟 = 1

cos 𝜙𝜙 �−𝑚𝑚^{𝑝𝑝𝑑𝑑𝑝𝑝}𝑥𝑥̈ − 𝑚𝑚𝑝𝑝𝑑𝑑𝑑𝑑𝑥𝑥̈ + 𝐹𝐹𝐺𝐺 ∓ 𝐹𝐹𝑓𝑓� (3.23)

The lateral force acting on piston is the component of the connecting rod force in z-direction (Equation (3.24)). This lateral force, 𝐹𝐹_𝑡𝑡, is exploited in the aforementioned kinematic approach for the study of piston secondary dynamics. In this equation, only forces in the primary direction are presented and the change of direction in the lateral force imposes the conditions for occurrence of piston impacts, i.e.

𝐹𝐹_𝑡𝑡 = 𝐹𝐹_{𝑐𝑐𝑟𝑟}sin 𝜙𝜙 = �−𝑚𝑚_{𝑝𝑝𝑑𝑑𝑝𝑝}𝑥𝑥̈ − 𝑚𝑚_{𝑝𝑝𝑑𝑑𝑑𝑑}𝑥𝑥̈ + 𝐹𝐹_𝐺𝐺∓ 𝐹𝐹_𝑓𝑓� tan 𝜙𝜙 (3.24)

In the second approach (as described earlier), the dynamic equations are derived for the piston’s secondary motion using inertia dynamics and the oil film hydrodynamics. The combined form of equations (3.19) and (3.22) is shown in equation (3.25). Here, 𝐹𝐹_{𝑐𝑐𝑟𝑟}sin 𝜙𝜙 is the same as in equation (3.24), with the exception that piston’s secondary motion does not only depend on the connecting rod’s lateral force, but also on the oil film’s hydrodynamics and the piston/piston pin inertias, i.e.

𝑚𝑚_{𝑝𝑝𝑑𝑑𝑑𝑑}𝑒𝑒̈_{𝐶𝐶𝐶𝐶𝑔𝑔}+ 𝑚𝑚_{𝑝𝑝𝑑𝑑𝑝𝑝}𝐻𝐻̈ = −𝐹𝐹_{𝑐𝑐𝑟𝑟}sin 𝜙𝜙 + 𝐹𝐹_{ℎ𝑦𝑦𝑑𝑑} (3.25)

Equations (3.20) and (3.25) describe the piston’s secondary dynamics. These relations can be restated in terms of the eccentricity parameters using equations (3.16) and (3.17). The system dynamics can then be represented in a matrix form as:

�𝑚𝑚_pis�1 −𝑏𝑏

In order to include the effect of the piston’s secondary motion and the lubricant reaction using inertial dynamics, the set of equations of motion (3.26) is iteratively solved, coupled with Reynolds equation (Balakrishnan and Rahnejat (2005) and Littlefair et al (2014a)). Thus, the lateral force is obtained in a more realistic manner. The deviation between the results of methods (i) and (ii), when predicting the lateral force, is quite small at lower engine speeds. As the piston speed increases and inertial effects become significant, the aforementioned methods yield divergent results, as expected. The evaluated lateral force from equation (3.26) is used hereinafter to calculate the lateral impact force and approach velocity.

𝑀𝑀_hyd is the moment due to load capacity variation of the lubricant film 𝐹𝐹_hyd over the piston’s skirt area. 𝑀𝑀𝑓𝑓 is the moment due to the viscous friction of the lubricant film 𝐹𝐹𝑓𝑓. 𝑀𝑀𝑑𝑑 represents the tilting moment due to pin or crankshaft offset, which equates to: 𝐹𝐹𝐺𝐺𝑟𝑟𝑝𝑝+ 𝑚𝑚pis𝑥𝑥̈𝑟𝑟COG. It should be noted that the lubricant viscous friction and its generated moment are neglected in the current study as their influence on the impact force is quite small, constituting for less than 3% of the total impact force (Cho and Jang (2004)).

The flow chart for the iterative solution of the equations of motion is presented in Figure 3.3.

In the current research, the total simulation time corresponds to three engine cycles for all operational conditions (sufficient time to achieve steady state motion per engine cycle). The time step is invariable, equal to 5 μs. The convergence criterion is 1%, implemented on the eccentricity accelerations (𝑒𝑒̈𝑡𝑡 and 𝑒𝑒̈𝑏𝑏), which are the fastest system variables. The eccentricity velocity and displacement are evaluated using Newmark’s integration method (Newmark (1959)) for dynamic systems. This method is also known as the average acceleration method.

The implicit algorithm of Newmark integration is used in the predictor-corrector module. LAM I and LAM II are the predictor and corrector routines of the numerical solution. The estimated displacement and velocity at each time step are used in the dynamic equations of motion to obtain the new acceleration values for the continuation of the integrator. The aforementioned procedure is shown for a simple one degree of freedom system in equation (3.27) (Timoshenko (1974)). The index 𝑖𝑖 refers to the predicted values at each time step for 𝑗𝑗 = 1 (LAM I) and 𝑗𝑗 >

1 shows the number of corrective iterations upon the convergence within a specific time step (LAM II). 𝑓𝑓(𝑡𝑡, 𝑥𝑥, 𝑥𝑥̇) is the general form of the equation of motion for each degree of freedom, i.e. And if 𝑗𝑗 > 1, the corrector routine (LAM II) is defined as:

(𝑥𝑥̇_𝑑𝑑)_𝑗𝑗 = 𝑥𝑥̇_𝑑𝑑−1+1

2 �𝑥𝑥̈^𝑑𝑑−1+ (𝑥𝑥̈_𝑑𝑑)_𝑗𝑗−1�Δ𝑡𝑡_𝑑𝑑, (𝑥𝑥_𝑑𝑑)_𝑗𝑗 = 𝑥𝑥_𝑑𝑑−1+ (2𝑥𝑥̇_𝑑𝑑−1+ (𝑥𝑥̇_𝑑𝑑)_𝑗𝑗Δ𝑡𝑡_𝑑𝑑 3 +

1

6 𝑥𝑥̈^𝑑𝑑−1Δ𝑡𝑡_𝑑𝑑² The approximated error in eccentricity accelerations is 1% for the current problem (Figure 3.3).

This value ensures that the integrator converges very quickly, whilst the error in the residuals vector is constrained to very small values. The residual vector is defined as �𝑴𝑴(𝒆𝒆̈𝑑𝑑)_𝑗𝑗− 𝑭𝑭�_∞= 𝜖𝜖, using matrices of equation 3.26 and results of LAM II. 𝜖𝜖 is the residual vector error and it does not exceed 10⁻⁶ for approximation error of 1%.

Figure 3.3. Transient dynamic model flow chart