Numerical study of piston group and crosshead guide system dynamics for a two‐stroke marine engine

Two stroke slow speed diesel engines are mainly used in deep sea going vessels such as container ships, bulk carriers, and tankers. In this article, a numerical model is developed and solved to relate impact occurrence and magnitude on engine structure to rigid body tribodynamics of the crosshead guide system for a two‐stroke marine engine. To achieve this, the reciprocating piston group is coupled to the crosshead guide shoes through the crosshead guide pin. The resulting second order nonlinear differential equations are solved for displacements, velocities, and friction characteristics of the system. The rigid body motions are important in identifying occurrences and the intensities of transient impacts between contacting surfaces. These impacts influence noise and vibration characteristics of the engine. A study of the coupled system at different engine running speeds as well as varying crosshead guide system clearances is performed. It is observed that secondary displacements, velocities, and friction characteristics increase with increase in speed and clearance on the crosshead guide system. Consequently, the impact on the engine structure increase with increase in these tribodynamic characteristics of the system. This impact is transferred to the engine surface as vibration and noise responses.

nonlinear models that are solved to reveal the tribodynamic behavior of the engine. In this study, tribodynamic analysis involves modeling the dynamics of the piston group (the piston-piston rod-crosshead pin system), and the crosshead guide system and coupling the resulting rigid body dynamic model with thin film lubrication to create a tribodynamic model. Thin film lubrication has been widely used to study different aspects of lubricated surfaces in internal combustion engines (ICE). It is important to mention a few contributions in this research area, because they form the foundation for this work. One of the early contributions on thin film lubrication for ICE piston-cylinder liner system was presented by Li et al. 3 In their work, an analytical model was developed to study the dynamics of a piston and the resulting piston-skirt frictional power loss. Zhu et al. 4 developed a mathematical model for mixed lubrication of piston skirt. The research focused on the effects of surface waviness, piston skirt surface profile and surface roughness on piston motion, lubrication, and friction. Based on this initial work, an improved model was developed to study the influence of bulk deformation of piston skirt on its secondary dynamics, lubrication and friction. 5 The effects of fully flooded, partially flooded, and starved regimes of lubrication on tribodynamics have been widely reported. 6,7 Most researches presented only considered the connecting rod force acting through the piston pin, ignoring the effects of connecting rod inertia. Meng et al. 8 investigated the effects of connecting rod parameters on piston skirt tribodynamic characteristics, namely, the side force, oil film thickness, and friction force. It is reported that taking the connecting rod parameters into consideration influences the tribodynamic characteristics of the piston skirt. Further, the influence of connecting rod inertia on piston skirt lubrication and dynamics is more pronounced when the engine runs at relatively high speeds. 9 Bo et al. 10 proposed an advanced model to simulate piston dynamics and lubrication behavior. The model couples a lubrication model for a piston skirt-liner system with a dynamic multibody system consisting of crank, connecting rod and piston. The influence of skirt profile on the piston secondary dynamics has also been investigated as reported references [11,12]. Piston slap impact on lubricated and unlubricated skirt-cylinder liner interfaces is captured as vibration and noise responses on the engine block surface. 2,13 This is useful in nonintrusive approach to condition monitoring of lubricated surfaces in internal combustion engines. The impacts between lubricated surfaces greatly influence the complex transient noise and vibration responses in relation to impact excitation within the engine. The correlation between the impacts and induced vibration can provide an effective approach to simulation of engine dynamic behavior and condition monitoring. Such a correlation would involve an in-depth investigation into impacts and the corresponding vibration responses starting with theoretical modeling to experimental verification. [14][15][16] The foregoing literature analysis on piston skirt tribodynamics reveals a trend in which focus is on the four stroke high speed ICES, which are different in construction and operation to the two-stroke slow speed crosshead engines considered in this article. However, the analyzes provide the basis for the current research. Abanteriba conducted rigorous analysis on crosshead guide system for two stroke slow speed engines. [17][18][19] In the publications, an algorithm to predict the oil film thickness of a single acting guide system for a slow speed engine was developed. In addition, an alternative simplified algorithm for accurate prediction of oil film thickness for the guide shoe surface was proposed. Further, the author dealt with friction and its minimization on a guide shoe. It should be noted that a quasi-static analysis was employed neglecting the influence of secondary dynamics of the crosshead guide shoe on tribological characteristics of lubricated surfaces. This shortcoming notwithstanding, the studies provide invaluable insights into the minimum oil film thickness (MOFT) and friction characteristics of crosshead guide system.
Recently, research interests on tribodynamic analysis of the crosshead guide system have increased. This is attributed to the need to develop energy efficient marine engines that comply with the stringent International Maritime Organization (IMO) requirements on engine emissions. Li et al. 20 developed a coupled model for crosshead guide and piston skirt-liner. The influence of crosshead guide-guide shoe clearance as well as guide shoe profile on the secondary motions, friction, frictional power, MOFT, and side thrust force were investigated. This work neglected the effects of connecting rod inertia. Li et al. 21 improved the tribodynamic model by considering connecting rod inertia. They presented a comparison for secondary dynamics behavior and lubrication characteristics between two models; one neglecting connecting rod inertia, while the other considered the effects of connecting rod inertia. Based on the coupled tribodynamic model, it is reported that the tribodynamic behavior is sensitive to connecting rod inertia. Another recent contribution investigated the tribodynamic behavior of the crosshead guide shoes during engine start-up. 22 In the current research, nonlinear second order tribodynamic models are developed and solved for secondary tribodynamic characteristics; displacements, velocities and friction characteristics, and related to occurrence and magnitude of structural impacts on the engine. The impacts considered are as a result of secondary motions of the coupled system consisting of piston group and crosshead guide system. Generally, transient impacts are severe during the start of combustion at top dead center (TDC). This article therefore, mainly focuses on this region of the engine cycle. To gain more insight on the influence of speed on tribodynamics and impact characteristics, the model is solved at four different engine running speeds. In addition, effects of tight and loose clearances of the guide system on tribodynamic characteristics are investigated by solving the model with three different clearances under constant speed and load. The impact occurrence on the engine structure due to the movement of guide shoes is predicted by maximum energy transfer approach, which depends on the lubricant squeeze action and the hydrodynamic forces generated on the lubricated surfaces. When the lubricant is squeezed, energy is transferred to the engine through the guide rails. As the squeeze action continues, hydrodynamic force generated increases and the work done on the surface increases. The increase in hydrodynamic pressure results to gradual decrease in squeeze action. The point of maximum work is assumed to be the point of impact occurrence. The intensity of impacts is determined by the magnitude of impact energy which considers both linear and angular dynamic characteristics of the guide shoes. Figure 1 shows the theoretical model of the piston group and crosshead guide systems used to develop the mathematical models for the dynamic system. The piston is rigidly connected to the piston rod using studs and nuts. It is then coupled to the crosshead guide system by a crosshead pin, rigidly connected on the lower end of the piston rod. The crosshead guide system consists of the crosshead guides, guide shoes, crosshead guide pin, and bearings. The piston group and the guide shoes form a coupled reciprocating system connected to the connecting rod through the pin bearing. Of great interest to the coupled system are the frictional pairs; the piston skirt-liner and the crosshead guide-guide shoes. Similar to the motions experienced by the piston within the liner, the guide shoes primarily reciprocate along the guide rails. It should be noted that, guide shoes also experience secondary motions. If the clearance between the crosshead pin and the guide shoes is ignored, the lateral displacement of the guide shoes, e l and that of the piston group taken at the center of the crosshead pin are equal. This is used to couple the piston group and the crosshead guide system. On the other hand, F I G U R E 1 Coupled system of piston group, crosshead guide, connecting rod, and crank swing angle of the guide shoes, γ c and that of the piston assembly, p are different. In reference to Figure 1, e t and e b are the lateral displacements of the guide shoe at the top and bottom edges, respectively, while e tp and e bp represent the secondary displacements at the top and bottom edges of the piston skirt, respectively. The origin of lateral displacement is taken as a vertical line through the crosshead guide shoe and the horizontal reference of displacement is taken as a line through the center of gravity (COG) of the guide shoes when the piston is at the top dead center.

Crosshead guide shoe
The forces and moments acting on a single guide shoe are show in Figure 2A. All moments are taken about the center of the crosshead pin. Applying the principle of force and moment equilibrium, the following equations are obtained: F c is guide shoe side force, M c is the moment due to F c . F cf and M cf are the friction force on a single guide shoe and its moment, respectively. F x and F y are forces from the crosshead pin acting on the guide shoe in x and y directions, respectively. G c represents the force due to gravity, while F icy and F icx are the reciprocating and traverse forces of inertia of the guide shoe with a moment of inertia M ic . These forces and moment are calculated as shown in Equations (4)- (6): The reciprocating velocity and acceleration are given by Equations (7) and (8), respectively; u = R c ω 2 cos θ + (R c ωR s cos θ) 2 (

Piston group
Piston rings are not considered in this analysis, since their tribodynamic influence is small compared to the piston skirt. 23 To analyze the dynamics of the rigid piston group system, the parts of the assembly are detached at the joints replacing the rigid joints with joint forces and moments. 20

Piston
The forces and moments on the piston are shown on Figure 2B. Applying the equilibrium of forces and moments, the following equations are obtained: The reciprocating force F ipy , lateral inertia force F ipx , and the moment of inertia M ip are calculated as: 2.2.2 Piston rod Figure 3 shows the forces and moments acting on the piston rod. Applying equilibrium of forces and moments the following equations are obtained.
The reciprocating force F ipry , lateral inertia force F iprx , and the inertia moment M ipr are obtained as below: F I G U R E 3 Forces and dimensions of piston rod 2.2.3 Crosshead pin Figure 2C shows the forces and moments acting on the crosshead pin. Applying the equilibrium of the forces and moments acting on the crosshead pin, the following models are obtained: The reciprocating force F icpy , inertia force F icpx , and the inertia moment M icp are obtained as: Eliminating joint forces F x1 and F x2 through rearrangement of Equations (9)-(11), (15)- (17) and (21)-(23), a set of three compound equations, Equations (27)-(29) are obtained: With Equations (1)-(3), F x , F y , and F l are eliminated from Equations (27) and (28). Writing the resultant equations into a matrix yields a system of equations: ) .
The coefficients M ij are given in the Appendix A. Equation (33) can be written in the form Më = b, which are nonlinear second order system of equations in e t , e b , e tp , and e bp . The nonlinearity of this equation comes from the presence of the thin film lubrication forces and moments in B F , B M , and B P .

HYDRODYNAMIC LUBRICATION MODEL
Two sets of frictional pairs are considered in this work; the crosshead guides-guide shoe and the piston skirt-liner. To calculate the forces F c , F cf , F p , F pf and moments M c , M cf , M p , and M pf , the lubrication pressure distribution over the computational domains is required. This is obtained by solving the average Reynold's equation for thin oil films. Neglecting the inertial forces of lubrication oil, the pressure over the computational domain is obtained as shown in Equation (34) 24 : In the above equation,u is the velocity of reciprocating components in the entraining direction, μ is the dynamic viscosity of the lubrication oil, ρ is the density of the lubrication oil, and σ is the root-mean square of surface roughness of a friction pair. The velocity of side leakage is considered insignificant compared to entrainment velocity and therefore, it is ignored. 25 Owing to the huge difference between the magnitudes of the variables in the Reynolds equation, difficulty in numerical convergence may be encountered during the solution process. This is resolved by converting the Reynolds equation into a nondimensional form using the following relations 23 : Assuming a constant density of the lubricant oil, the quantities in Equation (35) transform the Reynolds equation into a nondimensional form: The quantities x 1 and y 1 are the local coordinates in the traverse and axial directions on the computational domains for the guide shoes. y and x are pressure flow factors, c is the contact factor and s is the shear flow factor. 26-28

The crosshead
The nominal clearance between the crosshead guide and guide shoe is shown in Figure 4A while the lateral linear displacements at the top and bottom edges shown in Figure 4B. The instantaneous oil film thickness to calculate the pressure distribution over the guide shoe computational domain in Equation (36) is represented by h T on the thrust side and h AT on the antithrust; Equations (37) and (38), respectively. f ( is the guide shoe face profile and d ( is the deformation on the guide shoe due to thermal and mechanical load.
Since the generated film pressure is relatively low, that is below 30 MPa, it is not be sufficient to cause substantial mechanical deformation, therefore the deformation on the lubricated guide system surfaces is neglected. 21,23 The boundary conditions expressed in Equation (39) while the exit boundary conditions are defined by Equation (40). The set of boundary conditions expressed by Equation (40) is approximated by setting all negative pressure encountered during the computation process to zero, ignoring cavitation effects. 9,21 p ( secondary displacements e t , e b and film thickness h T and h AT p ( In mixed lubrication regime, there is a possibility of the contiguous surfaces coming into contact when inadequate film pressure is generated. In such instances, the resulting hydrodynamic force is insufficient to support the applied forces. The asperity pressure is calculated depending on the ratio of the local film thickness to the composite surface roughness (λ) referred to as the Stribeck's parameter. If the calculated value of λ is less than a critical value λ c , asperity pressure P asp is computed and added to the oil film pressure. In this article, λ c is taken as 4.0. 10 The asperity pressure is computed as shown in Equation (41) 26,27 : The compound elastic modulus E ′ and the statistical function F 5∕2 are approximated using the flowing expressions: In Equation (43), υ 1 and υ 2 are the Poisson's ratios for the contiguous surfaces 1 and 2. After calculating the oil film and the asperity pressure distributions over the guide shoe computational domain, the forces F c and F cf and the corresponding moments M c , and M cf on both thrust and antithrust sides are obtained using the following expressions: In Equations (44)-(47), A and A T are the areas of computational domains for the thrust and antithrust sides, respectively. The asperity coefficient of friction is μ f and w is the half width of the guide shoe.

The piston skirt-liner
Analysis of the piston skirt-liner system is similar to that of the crosshead system. Here, Equations (36), (41), (42), and (43) are applied to the piston-skirt system, with a change in local coordinates α and y 2 for the piston skirt circumferential and axial directions, respectively. The circumferential coordinate α = x 2 ∕R, where R is the radius of the piston and x 2 is the circumference of the piston skirt. The instantaneous film thickness of the piston skirt-liner system is approximated as 29 : (48) Figure 5 shows the piston skirt-liner system. The boundary conditions applied to solve for the pressure distribution over the piston skirt surface are: The film pressure at the top and bottom edges of the piston skirt and outside the piston skirt-liner bearing areas defined by α 1 and α 2 are set to zero using Equations (49) and (50), respectively. The exit boundary conditions are defined by Equation (51). The bearing angles α 1 and α 2 are both taken as 46 o . 10 Solving the Reynolds equation for lubrication film forces and considering asperity effects on the piston skirt-liner system yields the forces over piston skirt F p and F pf and the corresponding moments M p and M pf .
The estimation of the average shear stress between the lubricated surfaces is obtained using the expression given by Equation (56), where f , fs , and fp are shear stress factors. 27,28

NUMERICAL SOLUTION
The equations of motion for the coupled system, Equation (33), and all the expressions for the forces and moments on the right-hand side of Equation (33), constitute an initial value problem for a pair of nonlinear second order differential equations in e t , e b , e tp , and e bp . Similar initial value problems have previously been solved with Runge Kutta method, 23 Adams-Bashforth-Moulton (multi-step) methods, Newton Raphson method, [3][4][5] Quasi Newton methods (Broyden), 7,29 and modified extended backward differentiation method. [20][21][22]30 In this article, ode15s is used to solve the stiff nonlinear initial value problem. At each time step, the Reynold's equation is solved for the piston skirt-liner and crosshead guide systems to obtain the oil film pressure which is integrated over the lubricated surface to obtain the pressure forces, corresponding moments, and friction characteristics of contiguous surfaces. Equation (36) is discretized by finite difference method 30,31 and the resulting linear equations for the nodes on the computation domain are solved iteratively by Gauss-Seidel iterative method with successive over relaxation. The pressure convergence is obtained as: where k is the iteration number, m and n represent the number of the nodes along the width and the length on the computation domain, respectively.
For simplicity and to save on simulation time for the transient process, some assumptions have been made in this study. The deformations due to the oil-film pressure on the piston skirt-liner as well as on crosshead guide rails-guide systems are neglected. The lubricant rheology is neglected; therefore, the lubrication oil density and dynamic viscosity are assumed to be constant. 7,8 Fully flooded boundary conditions are assumed in solving the Reynolds equation, 11,23 hence the effects of cavitation are ignored.
The stiff nonlinear second order ordinary differential equation is transformed into equivalent first order . The resulting equations represent an initial value problem y ′ = f(t, y), with initial conditions y(0) = y 0 . The six equivalent first order equations obtained from Equation (33) are presented in Equations (57)-(62).̇e The developed models are applied on a slow speed two-stroke crosshead marine diesel engine, with a rated maximum speed of 142 r/min. The basic parameters of the engine are shown in Table 1. The guide rail and guide shoe root-mean square (rms) surface roughness parameters, σ 1 and σ 2 are respectively taken as 0.40 and 0.5 μm. 21 The compound rms value for the piston skirt and liner surface parameter is taken as 0.781 μm. The measure of asperity gradient for the crosshead guide-guide shoe (σ 1 ∕β 1 ) and piston skirt-liner systems (σ 2 ∕β 2 ) are taken as 0.02 and 0.05, respectively. To determine the asperity pressure, roughness parameters of the crosshead guide system (η 1 β 1 σ 1 ) and piston skirt-liner system ( η 2 β 2 σ 2 ) are taken as 0.05 and 0.01, respectively. The solution for the unknowns involves solving the six stiff nonlinear ordinary differential equations, Equations (57)-(62) over a complete engine cycle. The cycle convergence tolerance is taken as ε tol and ε cali (i = 1,2,3, … ,0.6) represents the calculated error on the six variables, Equations (63)-(68). If the condition ε cali ≤ ε tol is not satisfied after a cycle, the initial conditions are set to the values of the variables at the end of the cycle and the cycle is repeated. In this article, ε tol is taken as 1e−4 which is sufficient to give accurate and reliable results within reasonable solution time.
In this article, nominal engine operating condition refers to the engine running at 129 r/min/75% load with clearances for the crosshead guide and piston skirt-liner systems taken as 0.2 and 0.25 mm, respectively. Figure 6 shows the cylinder pressure curves measured at different running conditions, with TDC taken as 0 • CA.  The resulting hydrodynamic pressure between lubricated surfaces, the friction force as well as friction power loss greatly depend on the entrainment velocity. Moreover, acceleration affects the inertia characteristics of the engine. Figure 7A,B presents a plot of axial velocity and acceleration of the reciprocating parts of the coupled system at the nominal engine operating condition. It is observed that the maximum acceleration is attained at the TDC. This implies that the maximum axial inertia force for the reciprocating components is reached around this region. Owing to the characteristic slow running speed of two stroke engines, the maximum axial speed attained is relatively low, about 11.8 m/s for the engine under study at nominal running conditions.
A mesh of 160 nodes is used along the length of the guide shoe and 80 for its width. The piston skirt is discretized into 100 nodes along the piston skirt and 60 nodes on its circumference. The tribodynamic behavior of the coupled system is first analyzed under nominal engine operating condition. Occurrence of impacts on the crosshead guide-guide shoe system were identified from the tribodynamic behavior of the coupled system. The intensity of the slaps was quantified using slap energy. The influence of different engine running speeds and varying crosshead guide system clearances on the energy transferred to the engine block through the crosshead guides is then investigated.

RESULTS AND DISCUSSION
The predicted results of the lateral displacements and velocities for both the piston group and the crosshead guide shoes are similar to those published in References [20,21]. There are variations noted in the actual values of the trend at different points of the engine cycle. This is because the engine parameters used are of different engines and therefore, the resulting simulation results are different. The trends observed are also consistent with other reported works in literature, where highly transient events are observed at the vicinity of TDC due to fuel injection and subsequent explosion within the cylinder liner, 8,9,33 which further validates this model. The rest of the article is organized as follows: First, the tribodynamics of the model are analyzed at nominal engine running condition, to understand the general trends exhibited by the system. Second, the effects of engine running speed on the tribodynamic characteristics of the piston group and crosshead guide system is investigated and discussed. Third, the model is solved at different clearances on the crosshead guide system to simulate the behavior of the system due to tight clearance as well as enlarged clearances due to wear. These characteristics are related to occurrence and magnitude of impact force on the engine structure by the guide shoes through the guide rails. Fourth, a statement on further research is made and finally a conclusion drawn on the study.

5.1
Tribodynamic characteristics at nominal engine operating conditions Figure 8A shows the lateral forces acting on the crosshead guide system. At TDC, the connecting rod force reaches the peak value and changes direction to point toward the thrust side of the crosshead guide system. This force is balanced by a reaction side force generated by the hydrodynamics and asperity actions between the crosshead guides and the guide shoe.
Since the connecting rod force is supported by two guide shoes, it is balanced by a hydrodynamic twice the magnitude of the side force generated on a single guide shoe. At bottom dead center (BDC), the connecting rod force changes direction and points toward the antithrust side and keeps this orientation throughout the compression phase of the engine cycle. Slap impacts are expected at TDC and BDC. This evident from the change of direction of the connecting rod force which is one of the indicators of slap impact occurrence. 34 Figure 8B shows the side force of the piston skirt-liner system. It should be noted that this force is the reaction of the liner to the piston skirt motion resulting from the connecting rod force and movement of the guide shoes. It is observed that as the connecting rod force changes direction at TDC, the piston group is pushed toward the antithrust side of the liner and hence the reaction force on the liner points toward thrust side. The piston group is then deflected toward the thrust side, leading to the change in direction of the liner reaction force toward the antithrust side. These movements are sustained for about 30 • CA, then the piston group readjusts itself to its original position and hence the side force generated tends to zero. The change of connecting rod force toward the antithrust at BDC side pushes the piston group toward the thrust side of the liner resulting to generation of a reaction side force on the liner toward the antithrust. The piston group readjusts itself and the side force tends to zero once more. The observed changes of direction of the side force for the piston group lead to impacts on the liner. However, the maximum side force on the liner is about 330 N compared to 188KN exerted by crosshead guide shoes. The lateral force from reciprocating group in a crosshead engine is supported by the crosshead guides, not the piston skirt and hence the force difference. Figure 9A shows the traverse displacement at the top edge of the guide shoe at nominal engine operating condition. The connecting rod force attains its maximum in the vicinity of TDC, and hence sufficient hydrodynamic force is generated to support it. The magnitude of the hydrodynamic force depends on the relative entraining velocity, oil film thickness, and squeeze action between the lubricated surfaces. It is observed that at TDC, axial velocity is zero, and therefore the force generated largely depends on the oil film thickness and the squeeze action. In order to generate enough side force, the guide shoe is pushed further toward the crosshead guides on the thrust side. As the axial velocity between the lubricated surfaces increases, the guide shoe is pushed away from the crosshead guides until maximum entraining speed is attained around 70 • CA. The guide shoe maintains this position until around BDC when the entraining velocity of the lubricated surfaces tends to zero. After 180 • CA the guide shoe is pushed further toward the crosshead guides. This action aids in generating enough side force to support the connecting rod force. This continues until the connecting rod force changes direction at BDC. Figure 9B shows the displacement of the bottom edge of the guide shoe. It is observed that when the guide shoe is pushed toward the thrust side of the crosshead guides, it maintains its position with insignificant changes until the connecting rod force changes direction toward the antithrust side. Figure 10 shows the traverse displacement of the piston top edge (A) and bottom edge (B) under nominal engine operating condition. It is observed that both top and bottom piston edges are pushed toward the antithrust side immediately after TDC. When the guide shoes are pushed toward the thrust side of the crosshead guides, due to the bearing coupling, the piston group is pushed toward the opposite direction; antithrust side. The piston group then bounces toward the thrust side before adjusting back to its initial position. At BDC, the piston top and bottom edges are pushed toward the thrust side before recovering their original position for the rest of the engine cycle. The rebounds observed at both TDC and BDC infer occurrence of slap impacts on the liner. Figure 11A shows the traverse velocities of guide shoe top and bottom edges. It is noted that the change of traverse velocities with time is very transient around the TDC and points toward the thrust side. It then decreases to almost zero until the connecting rod force changes direction at BDC to point toward the antithrust side. The velocities change direction toward the antithrust side between 176 and 189 • CA, then diminish to about zero for the rest of the engine cycle. Similar to the traverse displacements, the traverse velocities of the guide shoe edges are driven by the generated hydrodynamic  Figure 11B. The trend of the curves is similar to that of the displacements in Figure 9A,B. At the nominal engine running conditions, the traverse velocities of the guide shoe are higher than those of the piston skirt. This is because in crosshead engines, the connecting rod force is supported by the crosshead guide shoes resulting to higher guide shoe velocities relative to piston skirt velocities. Figure 12A presents the MOFT on the guide shoe at different crank angles. In order to generate enough reaction force to balance the connecting rod force, the guide shoe adjusts to reduce the gap between the guide shoe, and the guide rails. The entraining velocity drops to minimum near TDC and the generation of the oil film force is primarily due to reduction of oil film thickness and squeeze effect. On the antithrust side, an opposite effect is observed where the gap between the crosshead guide rails and the guide shoe increases to its maximum value. At BDC, the connecting rod force changes direction and the gap between the guide shoe and guide rails reduces. Figure 12B shows the squeeze velocity of the guide shoe for the thrust and antithrust sides. As mentioned earlier, the squeeze action on contiguous surfaces plays a key role in generating force required to balance the connecting rod force when the entraining velocity is at its minimum. The maximum energy transfer approach was adopted in identifying impact occurrence. It is observed that at TDC, there is an abrupt rise in the squeeze velocity on both thrust and antithrust sides to support the connecting rod force. The squeeze velocities on both sides diminish shortly after TDC and are observed again around BDC as the piston group changes reciprocation direction. The squeeze velocities for the thrust and antitrust sides act in opposite directions. It is observed that a high magnitude is attained in the vicinity of TDC, acting toward the thrust side than in the opposite direction. Similarly, at BDC the squeeze velocity on the antithrust side is higher than that on the thrust side. This difference is mainly because of the difference in magnitude of side force separately generated on the two sides. The squeeze velocity is a key indicator of occurrence of transient impacts 35 and hence a relatively severe slap impact would occur on the thrust side compared to the antithrust side. Figure 13A shows the friction force of a single guide shoe at nominal engine operating condition. To support the connecting rod force during the expansion stroke of the engine cycle, the gap between the crosshead guides and the guide shoe reduces. This leads to an increase in the friction force according to Equation (56). During the compression stroke, the connecting rod force is smaller in magnitude, hence an increase in oil film thickness and consequently, reduced friction force. Figure 13B shows the resultant power loss for a single guide shoe. It is observed that the power loss is higher during the expansion stroke compared to the compression stroke and follows the trend of the friction force. Figure 14 shows the friction force and friction power loss over the piston skirt. It is observed that the friction force over the piston skirt is low compared to that on the guide shoe and that its magnitude is approximately equal on both the expansion and compression strokes. The piston skirt does not support the connecting rod force in a two-stroke crosshead engine. Therefore, the friction force generated is primarily due to piston group secondary motions. Similar observation is made on the friction power loss over the piston skirt, Figure 14B.

5.2
Influence of engine running speed/load on engine tribodynamic behavior Impacts due to secondary motion of the piston can be identified by analyzing the lateral forces trends; quasi-static and transient, the minimum oil film thickness, lubricant squeeze film, and impact energy. 35,36 This section investigates the effects of engine speed on the tribodynamic model behavior of the piston and guide shoe in relation to impacts exerted on the crosshead guides. Figure 15A shows the hydrodynamic reaction force on a single guide shoe at different engine running speeds. It is noted that this force increases with increase in engine running speed from 47.26 KN at 89 rev/min to 102 KN at 142 rev/min in the vicinity of the TDC. There is a slight increase in the oil film force on the piston skirt, with a maximum of 389.7 N recorded at 142 rev/min, (see Figure 15B). The piston-skirt system and the crosshead guide-guide system are coupled and hence dynamics of either system affects the other. Figure 16 shows the traverse displacement of the guide shoe edges; (a) top edge, (b) bottom edge. As the engine running speed increases, there is an increase in the traverse displacements. However, the changes are minimal. Increase in connecting rod force pushes the guide shoes further toward the rails, resulting to increased traverse displacements around TDC. The values of piston displacements e tp with increase in speed are 30, 33, 36, and 38 μm at 89, 114, 129, and 142 rev/min, respectively. Similarly, e tp increases with increase in speed as; 28, 30, 33, and 35 μm, respectively as shown in Figure 17A,B. It is also noted that the changes in the traverse displacements are notable around the regions of maximum reciprocating speeds. This is due to increase in inertia which makes the system more responsive to inertia effects as the combustion force diminishes after TDC. The changes appear more pronounced on the piston displacements than on the guide shoes, since the piston has a larger reciprocating mass than the guide shoe. Figure 18A shows thaṫe t increases with increase in engine running speed around TDC pointing toward the thrust side as: 0.06, 0.08, 0.10, and 0.12 m/s at 89, 114, 129, and 142 rev/min, respectively. From Figure 18B it is observed thaṫe b increases from 0.05 m/s at 89 rev/min to 0.11 m/s at 142 rev/min. Energy through a structure is transmitted through force which is approximated by F e = mv ∕ Δt , where m is mass, v is velocity, and Δt is change in time. 23 Therefore, the structural impact on the guide rails/engine block can be related to the velocity of the impacting masses. Based on this, the increase in secondary motion velocities of the guide shoes and the piston group is an indication of increase in impact energy on the engine structure. Figure 19 shows that botḣe tp anḋe bp increase with increase in engine running speed, however, the maximum secondary velocities attained are 0.014 and 0.011 m/s at 142 rev/min, which are much lower compared to the guide shoes, pointing to a lower impact energy on the liner by the piston.   Figure 22 shows the guide shoe friction characteristics. Both friction force and friction powerloss increase with increase in engine speed to support the resultant connecting rod force. As connecting rod force increases, the guide shoe is pushed toward the guide rails, leading to a reduction in oil film thickness. This, inturn increases the friction force over the lubricated surfaces according to Equation (

F I G U R E 24
Guide shoe impact energy 160. 5, 198.7, and 227.8 N at 89, 114, 129, and 142 rev/min, respectively. The maximum power loss obtained is 2854 W at 142 rev/min. Figure 23 shows the piston skirt friction characteristics. It is observed that the friction characteristics increase with increase in engine running speed and the calculated forces are approximately equal on both expansion and compression strokes. Compared to the forces obtained on crosshead guides, the maximum friction forces on the piston skirt are much lower; a value of 41.1 N at 65 • CA and a speed of 142 rev/min. From these observations, it is important for the engine operation to be optimized to provide sufficient power at optimal speeds to avoid excessive power loss through friction. Figure 24 shows the impact energy for a single guide shoe over an engine cycle. The total impact energy is obtained by summing both linear and angular kinetic energy components of the guide shoes, similar to piston motion in a cylinder liner. 23,37 The impact energy from a single guide shoe onto the engine structure is given by; E = 0.5m c .̇e 2 c + 0.5I c .γ 2 c , where E is the energy due to secondary motion,̇e c anḋγ c are the lateral and tilting velocities, and I c is the guide shoe moment of inertia referenced at the crosshead pin. Work done is energy transferred to a body by a force acting on that body. In addition, work done can be related to the kinetic energy by the work-energy theorem. Therefore, the impact on the engine structure relates to the net kinetic energy by the crosshead guides through an impact force. The total kinetic energy is used to determine the magnitude of the impact on the engine structure. The impact observed is highly transient and occurs immediately after TDC, with it's maximum value attained at 4 • CA. At the engine running speeds considered, the energy transferred onto the engine structure due to the secondary motion of a single crosshead guide is; 0.02, 0.05, 0.08, and 0.10 J at 89, 114, 129, and 142 rev/min, respectively. This points to an increase in structural excitation with increase in running speeds.

5.3
Influence of clearance between the crosshead guide rails and guide shoes In this section, the influence of crosshead guide-guide shoe clearance on the tribodynamic behavior of the coupled system is investigated, at a speed of 129 r/min and 75% load with the piston skirt-liner clearance maintained at 0.25 mm. This models the increase in clearance due to wear on the crosshead guide system and investigates its effects on the coupled system with a view of relating it to impact occurrence and its intensity on the engine structure. Three crosshead guide system clearances are considered: 0.1 mm models a tight clearance, 0.2 mm is the normal clearance while 0.25 mm represents an enlarged loose clearance. Figure 25 shows the traverse displacements e t and e b . As the clearances increase, the displacements increase to 0.22 and 0.25 mm for e t and e b , respectively toward the thrust side for 0.25 mm clearance. Another interesting observation is that, the increase in guide system clearance results to a considerable increase in the piston skirt top and bottom lateral displacements as shown in Figure 26A,B. Therefore, the effects of a worn-out crosshead guide system influence the impact characteristics of piston skirt-liner system and the intensity of noise and vibration on the liner. Figures 27 shows the traverse velocitieṡe t anḋe b . It is observed that the traverse velocities increase with increase in clearance on the crosshead guide system, notably at the dead centers. For the clearances considered,̇e t values obtained are 0.020, 0.1008, and 0.1564 m/s for clearances of 0.1, 0.2, and 0.25 mm. Similarly,̇e b values are; 0.019, 0.098, and 0.1745 m/s for the respective clearances. When the clearance is changed from 0.2 to 0.25 mm, it is observed that the approximate increases iṅe t anḋe b are approximately 55% and 78%, respectively. This is an indicator of increased structural impact by the crosshead guides. The change in clearances on the crosshead guide system has a minimal effect oṅe tp anḋe bp as shown in Figure 28.  Figure 30 shows the MOFT on the thrust and antithrust sides of the crosshead guide system at varying clearances. It is observed that MOFT remains fairly constant with variation in clearance in the expansion stroke. In the compression stroke, MOFT increases with increase in clearance. A contrary observation is made on the antithrust side, after the connecting rod changes direction at TDC and points toward the antithrust side, hence a higher side force is generated to support the connecting rod force. Figure 31 shows the hydrodynamic forces for the crosshead guides system. At the different clearances, the guide shoes readjust to generate enough hydrodynamic force to support the applied external load. There is a noticeable, but small increase in the piston skirt hydrodynamic force around the dead centers. This is because the dynamics of the piston group respond to the dynamic changes on the components of the coupled system. Figure 32 shows the friction characteristics of force on the guide shoe at the different clearances. It is observed that the friction force slightly increases with decrease in clearance. During the expansion stroke, the friction forces are; 212. 5, 199.5, and 196.9 N at 0.1, 0.2, and 0.25 mm, respectively about 49 • CA. The power loss is 2407 W at 0.1 mm, 2259 W at 0.2 mm, and 2225 N at 0.25 mm. Since the side force is constant over the range of clearances considered, any change in friction force is mainly due to the changes in the film thickness due to changes in the gap between the lubricated surfaces. Hence, it follows that a decrease in clearance increases the friction force and the resulting friction power loss of the guide shoe. Figure 33 shows the friction characteristics of the piston skirt. Minimal changes in the piston skirt friction force and friction power loss are recorded when the clearance between the piston skirt and the liner is kept fairly constant, and relatively small changes in the side force on the piston skirt. Figure 34 shows the influence of guide system clearance on the impact energy on the engine structure form a single guide shoe. It is observed that the impact energy increases with increase in clearance, especially at the dead centers. Around TDC, it increases from 0.000397 J at 0.1 mm clearance, 0.078 J at 0.2 mm, and 0.184 J at 0.25 mm on the thrust side. The position of impact is around 4 • CA on the thrust side. Hence, the intensity of noise and vibration emitted from the engine due to guide shoe dynamics increases with increase in the clearances.

FURTHER RESEARCH
This work forms a basis for relating the tribodynamics of the piston group and crosshead guides for two stroke marine diesel engines at different operating conditions with the vibration response of engine block. The impact characteristics present a numerical tool for understanding the structural excitation that influences the vibration, noise, and harshness on the engine. This is important in developing a nonintrusive condition monitoring tool for monitoring wear on piston-liner and crosshead guide systems. Further research will focus on developing a numerical model to correlate the impact forces from tribodynamics of the piston group and crosshead guide shoes to vibration responses on the engine block.

CONCLUSION
In this article, a tribodynamic analysis of a coupled system of piston group and crosshead guide system is solved to predict occurrence and intensity of transient impact on the engine structure from guides shoes. The tribodynamic characteristics are related to impact energy at different engine speeds and clearances of the guide system. The main tribodynamic characteristics investigated are the displacements, velocities, MOFT and squeeze velocities, as well as friction characteristics. The transient impact energy is determined by considering both linear and angular characteristics of the coupled system of guide and piston group systems. The following is observed: 1. The secondary displacements and velocities increase with increase in speed of engine and clearance of the crosshead guide system and are most evident around TDC. 2. The friction force and frictional power loss on the guides shoes and piston skirt increase with increase in engine running speed. 3. There is a marginal increase in friction characteristics on the crosshead guide system with decrease in clearance.
However, change in clearance has no effect on the friction characteristics of the piston skirt. 4. The impact energy increases with increase in both engine speed and clearance of the guide system and it is observed that the maximum impact energy for all the cases considered occurs at about 4 • CA. At the speeds considered, the impact energy obtained is 0.025, 0.050, 0.078, and 0.1023 J at 89, 114, 129, and 142 rev/min, respectively. Increasing the crosshead guide clearances increases the impact energy. For clearances of 0.1, 0.2, and 0.25 mm, the impact energy obtained is 0.000397, 0.078, and 0.185 J, respectively. This shows that the system is sensitive to both engine running speed and guide system clearances and hence, it can be used for condition monitoring of increased clearances of the guide system due to wear. This forms a basis for relating the impact energy to surface vibration responses measured on the surface of the engine at different engine running conditions. Lateral displacement of crosshead pin at pin center R c Crank radius R s R c sin θ G pr Mass weight of piston rod G c Mass weight of a single guide shoe G cp Mass weight of crosshead pin F g Cylinder pressure force F l Reaction on the connecting rod small end F c Guide shoe side force F Lateral connecting rod force on crosshead pin F cf Guide shoe friction force F p Piston side force F pf Piston skirt friction force M c Guide shoe moment due to side force

APPENDIX
The coefficients M ij in Equation (33) M 32 = −m p ( 1 − L p ∕2 + R p + L pr L p + L pr + R p ) ( L p ∕2 + R p + L pr ) − m pr ( 1 − L pr ∕2 + R p L p + L pr + R p ) ( L pr ∕2 + R p ) + I T L p + L pr + R p . (A9)