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Three-Cylinder Reciprocating Engine
Introduction
A multibody analysis of an engine is often performed considering the individual components as rigid bodies. However, as specific power and engine speed requirements increase together with a need to improve overall efficiency, it is important to understand the critical points in an engine assembly in order to optimize the engine design. In this model, a combination of rigid and flexible body analysis is displayed.
Two types of analyses are shown in this example:
•
A simplified thermodynamic analysis of an air-fuel mixture in an engine cylinder.
•
A multibody dynamics analysis of an engine assembly.
In the thermodynamic analysis, the pressure variation in the combustion chamber due to the compression and combustion of air-fuel mixture is determined. This analysis is performed using the Heat Transfer and Coefficient Form PDE interfaces.
Figure 1: Geometry of the three-cylinder reciprocating engine.
In the multibody analysis, the pressure data obtained from the thermodynamic analysis is used to compute the motion of different components of an engine assembly, the RPM of the engine, and the power output (BHP) of the engine. This analysis is performed using the Multibody Dynamics interface. One of the connecting rods is studied from a component design perspective, and is modeled as a flexible element. The variation of the maximum stress in this component with the crankshaft rotation is analyzed.
The discussions about the thermodynamic study in this model are not required to understand the multibody dynamics modeling, so if you are only interested in the latter, you can skip all descriptions of the first study.
Note: This model requires the Heat Transfer Module if the thermodynamic analysis is performed. For a pure multibody analysis, only the Multibody Dynamics Module is required.
Model Definition
Thermodynamic Analysis
The pressure variation in one of the cylinders is determined during one full revolution of a crankshaft. The geometry modeled in this analysis is shown in Figure 2. It is created from the engine geometry shown in Figure 1 by cutting the 3D geometry though a work-plane (xz-plane), thus generating a planar 2D geometry.
Figure 2: Axisymmetric view of the piston-cylinder assembly. Only the highlighted domain, where the air-fuel mixture is present, is modeled.
This analysis is a simplified form of an actual combustion analysis. The purpose of this analysis is to show how you can determine the cylinder pressure variation required for the multibody analysis. The following assumptions and simplifications are used in this analysis:
•
Only air is considered as a fluid in the combustion chamber.
•
The heat energy, generated by combustion, is added uniformly over the domain.
•
The convection effects in the combustion chamber are neglected.
•
All the equations are solved in the original domain, and the effect of the change in the cylinder volume is accounted for manually in the equations.
The temperature distribution in the air inside the cylinder is modeled using the heat transfer equations. The pressure work is also added to account for the rise in temperature due to the work done by the piston. The pressure distribution in the air is modeled using an ideal gas equation:
where p, m, V, R, and T represent the pressure, mass, current volume, specific gas constant, and temperature respectively.
The current cylinder volume, V, is computed by subtracting the piston swept volume from the initial cylinder volume. The piston displacement, xp, as a function of the crankshaft rotation, θ, can be written as
where xp, rc, l, and θ represent the piston displacement, crank radius, connecting rod length and crank angle, respectively.
During the combustion, it is assumed that a total energy of 600 J is generated in one cycle during the crankshaft rotation from 3° before top dead center (TDC) to 6° after TDC.
Multibody Analysis
In this analysis, the pressure data obtained from the thermodynamic analysis is used to drive the motion of the engine assembly. The engine assembly is shown in Figure 1.
The engine assembly consists of a crankshaft, a flywheel, and three identical sets of cylinders, pistons and connecting rods. Each cylinder is connected to a piston through a prismatic joint and each piston is connected to the top end of a connecting rod through a hinge joint. The bottom end of all three connecting rods are connected to the common crankshaft though hinge joints. The flywheel is mounted on the crankshaft, and this crankshaft-flywheel assembly is supported by journal bearings at both ends.
All the components of the engine are assumed to be rigid, except the central connecting rod which is flexible and uses the material data for structural steel.
All the cylinders are fixed, while the other components are free to move in space. The pressure variation obtained from the thermodynamic analysis, with appropriate phase difference, is applied on the top surface of each piston. A starting torque of 400 Nm is applied on the crankshaft to start the engine until the shaft has rotated 60°. For the first one and a half revolutions of the crankshaft, the engine runs on a no-load condition. After that, an external load proportional to the angular velocity of the crankshaft is applied. Due to this external load, the RPM of the engine slowly reaches a steady state value.
Results and Discussion
Figure 3: P-V diagram for one of the cylinders.
The P-V diagram for one of the cylinders of the engine, as computed from the thermodynamic analysis, is shown in Figure 3. During the compression stroke, the volume of the air-fuel mixture decreases, and hence the pressure increases until the piston reaches close to the top dead center (TDC). At this point, the combustion occurs, which in turn increases the temperature and the pressure of the mixture. Finally, highly pressurized gas in the cylinder pushes the piston back toward the bottom dead center (BDC), thus performing the expansion stroke. During the expansion process, the pressure of the mixture decreases. The enclosed area of this diagram is a measure of the mechanical energy generated over one full cycle of the crankshaft rotation.
Figure 4 shows the variation of the cylinder pressure with the crankshaft rotation. You can clearly observe the compression, combustion and expansion strokes in this curve. The curve is exported and used to prescribe the pressure on the top surface of the piston in the multibody analysis.
The heat energy that was converted into mechanical energy in one full revolution of the crankshaft is also computed. The mechanical energy generated per cycle is approximately 339 J, whereas the input heat energy per cycle is 600 J. This shows that the remaining part of the heat energy is lost in the exhaust gases, so the thermal efficiency of the system is around 57 %.
Figure 4: Variation of the cylinder pressure with the crank rotation.
Figure 5: Displacements of the engine components.
Figure 6: von Mises stress in the flexible connecting rod at 0.16 s.
The displacement of the various components of the engine at t = 0.16 s, computed in the multibody analysis, is shown in Figure 5.
Figure 6 shows the stress distribution in the flexible connecting rod at t = 0.16 s. The maximum stresses occur near the top part of the connecting rod.
Figure 7 displays the time history of the RPM of the engine. The starting torque applied in the beginning of the simulation increases the engine RPM rapidly. After the removal of the starting torque, the RPM increases steadily as there is no external load. Finally after the application of the external load, it approaches a steady-state value close to 2600.
The fluctuations in the engine RPM are caused by the different strokes in a cycle, namely, compression stroke, combustion, and power stroke. During the power stroke, the piston is pushed to accelerate the crankshaft whereas during the compression stroke, the air-fuel mixture in the cylinder is compressed by the inertia of the components. The 120° phase difference between the cylinders makes the angular speed of the crankshaft more uniform. In addition, the flywheel mounted on the crankshaft also helps in making the speed uniform by absorbing the energy during the power stroke and delivering it during the compression stroke.
Figure 7: Time history of the engine RPM.
Figure 8: Mechanical power generated in each cylinder.
Figure 9: Power output (BHP) of the engine.
Figure 8 displays the mechanical power generated in each cylinder. It can be noted that during the compression stroke, the power generated is negative and it suddenly reverts its sign during the combustion, after which the power generated is positive. The time average of the power over a cycle is the net mechanical power generated in one revolution of the crankshaft. The 120° phase shift between the three sets of cylinder-piston can also be seen.
The power output due to the applied external load is shown in Figure 9. It shows that the engine initially runs on a no-load condition. Once an external load is applied, the power output of the engine (BHP) increases and approaches a steady state value close to 50 bhp.
Figure 10 shows the variation of the maximum stress in the flexible connecting rod during its operation. The connecting rod is subjected to a cyclic loading with stresses having a maximum when the piston is at TDC. The values of maximum and minimum stresses in one cycle are close to 160 MPa and 5 MPa respectively. It should be noted that the displayed stress history does not occur in a single point, since it for each time instance shows the maximum equivalent stress in any point in the connecting rod. This curve can thus not form the basis for a fatigue analysis.
Figure 10: Variation of the maximum stress generated in the flexible connecting rod.
Figure 11 shows the forces in the joint between the flexible connecting rod and crankpin. The forces are largest in the z direction and almost zero in the x direction.
These forces obtain their maxima when the piston is near TDC. The change of sign in these forces signify that the flexible connecting rod shifts from a state of compression to a state of tension during its operation. This serves to increase the need for a proper fatigue design.
Figure 11: Forces at the joint between the flexible connecting rod and crankpin.
Notes About the COMSOL Implementation
•
A Joint node can establish a direct connection between Rigid Material nodes. However, for flexible elements, Attachment nodes are needed to define the connection boundaries.
•
Constraint boundary conditions like Prescribed Displacement cannot be used with a Rigid Material node. Hence, the Prescribed Displacement/Rotation node (subnode to Rigid Material) is used to constrain or prescribe the corresponding degrees of freedom.
•
The connections used in the model can be reviewed in the Joints Summary section at the interface settings.
•
The shape function order for flexible components, is by default set to Linear. For better accuracy, you can switch it to Quadratic.
Application Library path: Multibody_Dynamics_Module/Automotive_and_Aerospace/reciprocating_engine
Modeling Instructions
Part 1: Thermodynamic Analysis
The thermodynamic analysis is performed in order to compute the cylinders pressure variation as a function of the crankshaft rotation. This part is optional, and you can skip it and start directly at the second part of the example, the multibody dynamics analysis.
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D Axisymmetric.
2
In the Select Physics tree, select Heat Transfer > Heat Transfer in Fluids (ht).
3
Click Add.
4
In the Select Physics tree, select Mathematics > PDE Interfaces > Coefficient Form PDE (c).
5
Click Add.
6
Click  Study.
7
In the Select Study tree, select General Studies > Time Dependent.
8
Click  Done.
Thermodynamic Analysis
1
In the Model Builder window, click Component 1 (comp1).
2
In the Settings window for Component, type Thermodynamic Analysis in the Label text field.
Geometry 1
Import 1 (imp1)
1
In the Geometry toolbar, click  Import.
2
In the Settings window for Import, locate the Source section.
3
Click  Browse.
4
Browse to the model’s Application Libraries folder and double-click the file reciprocating_engine_2d.mphbin.
5
Click  Import.
6
Click the  Zoom Extents button in the Graphics toolbar.
Use a rectangle function to add heat due to combustion while the crankshaft rotates from 3 deg before TDC to 6 deg after TDC.
Definitions
Rectangle 1 (rect1)
1
In the Home toolbar, click  Functions and choose Local > Rectangle.
2
In the Settings window for Rectangle, locate the Parameters section.
3
In the Lower limit text field, type pi-pi/60.
4
In the Upper limit text field, type pi+pi/30.
5
Click to expand the Smoothing section. In the Size of transition zone text field, type pi/120.
Integration 1 (intop1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
2
Select Domain 2 only.
3
In the Settings window for Integration, locate the Advanced section.
4
Clear the Compute integral in revolved geometry checkbox.
Integration 2 (intop2)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
2
In the Settings window for Integration, locate the Source Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundaries 4, 5, 15, 18, 19, and 27 only.
5
Locate the Advanced section. Clear the Compute integral in revolved geometry checkbox.
Variables 1
1
In the Definitions toolbar, click  Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
rp
40[mm]
m
Radius of piston
rc
40[mm]
m
Radius of crank
l
200[mm]
m
Length of connecting rod
omega
(1000*2*pi/60)[rad/s]
rad/s
Angular velocity of crankshaft
theta
omega*t
rad
Rotation of crankshaft
xp
-rc*cos(theta)+sqrt(l^2-(rc*sin(theta))^2)+rc-l
m
Piston displacement
V0
intop1(2*pi*r)
m³
Initial cylinder volume
V
V0-pi*rp^2*xp
m³
Current cylinder volume
rho0
1.1886[kg/m^3]
kg/m³
Air density at STP
m
rho0*V0
kg
Mass of air
R_air
287[J/kg/K]
J/(kg·K)
Specific gas constant of air
Q
600[J]
J
Heat generated during combustion
Pi
Q*omega/(pi[rad]/20)*rect1(theta)
W
Power input
Po
intop2(p*2*pi*r*d(xp,t)*root.nz)
 
Power output
Add Material
1
In the Materials toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in > Air.
4
Click the Add to Component button in the window toolbar.
5
In the Materials toolbar, click  Add Material to close the Add Material window.
Materials
Air (mat1)
Use the Coefficient Form PDE user interface to model the ideal gas behavior.
Coefficient Form PDE (c)
1
In the Model Builder window, under Thermodynamic Analysis (comp1) click Coefficient Form PDE (c).
2
Select Domain 2 only.
3
In the Settings window for Coefficient Form PDE, locate the Units section.
4
Click  Select Dependent Variable Quantity.
5
In the Physical Quantity dialog, type pressure in the text field.
6
In the tree, select General > Pressure (Pa).
7
Click OK.
8
In the Settings window for Coefficient Form PDE, locate the Units section.
9
Click  Select Source Term Quantity.
10
In the Physical Quantity dialog, type pressure in the text field.
11
In the tree, select General > Pressure (Pa).
12
Click OK.
13
In the Settings window for Coefficient Form PDE, click to expand the Dependent Variables section.
14
In the Field name (Pa) text field, type p.
15
In the Dependent variables (Pa) table, enter the following settings:
p
Coefficient Form PDE 1
1
In the Model Builder window, under Thermodynamic Analysis (comp1) > Coefficient Form PDE (c) click Coefficient Form PDE 1.
2
In the Settings window for Coefficient Form PDE, locate the Diffusion Coefficient section.
3
In the c text field, type 0.
4
Locate the Absorption Coefficient section. In the a text field, type 1.
5
Locate the Damping or Mass Coefficient section. In the da text field, type 0.
6
Locate the Source Term section. In the f text field, type m/V*R_air*T.
Give the atmospheric pressure as the initial value for the absolute cylinder pressure.
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
In the p text field, type 1e5.
Heat Transfer in Fluids (ht)
1
In the Model Builder window, under Thermodynamic Analysis (comp1) click Heat Transfer in Fluids (ht).
2
Select Domain 2 only.
Fluid 1
1
In the Model Builder window, under Thermodynamic Analysis (comp1) > Heat Transfer in Fluids (ht) click Fluid 1.
2
In the Settings window for Fluid, locate the Model Input section.
3
From the pA list, choose User defined. In the associated text field, type p.
Pressure Work 1
In the Physics toolbar, click  Attributes and choose Pressure Work.
Heat Source 1
1
In the Physics toolbar, click  Domains and choose Heat Source.
2
Select Domain 2 only.
3
In the Settings window for Heat Source, locate the Heat Source section.
4
In the Q0 text field, type Pi/V.
Use a coarser mesh, as in this model the solution does not have a spatial dependence. The mesh is only used to compute the cylinder volume.
Mesh 1
1
In the Model Builder window, under Thermodynamic Analysis (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Coarser.
4
Click  Build All.
Study: Thermodynamic Analysis
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, type Study: Thermodynamic Analysis in the Label text field.
Set the time range to solve for one full revolution of the crankshaft.
Step 1: Time Dependent
1
In the Model Builder window, under Study: Thermodynamic Analysis click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type range(0,0.0006,0.06).
4
In the Model Builder window, click Study: Thermodynamic Analysis.
5
In the Settings window for Study, locate the Study Settings section.
6
Clear the Generate default plots checkbox.
7
In the Study toolbar, click  Compute.
Results
Use the following steps to plot the P-V diagram shown in Figure 3.
PV Diagram
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type PV Diagram in the Label text field.
Point Graph 1
1
Right-click PV Diagram and choose Point Graph.
2
Select Point 2 only.
3
In the Settings window for Point Graph, locate the y-Axis Data section.
4
In the Expression text field, type p.
5
From the Unit list, choose bar.
6
Locate the x-Axis Data section. From the Parameter list, choose Expression.
7
In the Expression text field, type V.
8
From the Unit list, choose ml.
9
Click to expand the Coloring and Style section. From the Width list, choose 2.
10
Click to expand the Title section. From the Title type list, choose None.
PV Diagram
1
In the Model Builder window, click PV Diagram.
2
In the Settings window for 1D Plot Group, locate the Plot Settings section.
3
Select the y-axis label checkbox. In the associated text field, type Pressure (bar).
4
Select the x-axis label checkbox. In the associated text field, type Volume (cc).
5
In the PV Diagram toolbar, click  Plot.
Follow these steps to plot the cylinder pressure variation with the crankshaft rotation shown in Figure 4.
Cylinder Pressure
1
Right-click PV Diagram and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Cylinder Pressure in the Label text field.
3
Locate the Plot Settings section. In the x-axis label text field, type Rotation of crankshaft (rad).
Point Graph 1
1
In the Model Builder window, expand the Cylinder Pressure node, then click Point Graph 1.
2
In the Settings window for Point Graph, locate the x-Axis Data section.
3
In the Expression text field, type theta.
4
In the Cylinder Pressure toolbar, click  Plot.
Export the cylinder pressure data to use it in the second part of the model.
Plot 1
1
Right-click Results > Cylinder Pressure > Point Graph 1 and choose Add Plot Data to Export.
2
In the Settings window for Plot, locate the Output section.
3
Click  Browse.
4
Browse to a suitable folder, enter the filename reciprocating_engine_pressure.txt, and then click Save.
5
Click to expand the Advanced section. Clear the Include header checkbox.
6
Clear the Full precision checkbox.
7
Click the Export button to save the file.
Follow these steps to compute the mechanical energy generated per cycle, due to the combustion in one of the cylinders of the engine.
Global Evaluation 1
1
In the Results toolbar, click  Global Evaluation.
2
In the Settings window for Global Evaluation, locate the Data section.
3
From the Time selection list, choose Last.
4
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
timeint(0, 0.06, Po)
J
Mechanical Energy
5
Click  Evaluate.
Table 1
1
Go to the Table 1 window.
The result appears in a table below the Graphics window.
Part 2 — Multibody Analysis
In case you start the modeling here, without having the previous thermodynamics study available, a few instructions need to be modified. An example is that Component 2 should be replaced by Component 1. Such differences should be easy to identify. Since in this case there is only one model and one physics interface, there will be no ambiguities in how to interpret the modeling instructions.
Add Component
In the Model Builder window, right-click the root node and choose Add Component > 3D.
Add Physics
1
In the Home toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
3
In the tree, select Structural Mechanics > Multibody Dynamics (mbd).
4
Find the Physics interfaces in study subsection. In the table, clear the Solve checkbox for Study: Thermodynamic Analysis.
5
Click the Add to Component 2 button in the window toolbar.
6
In the Home toolbar, click  Add Physics to close the Add Physics window.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Time Dependent.
4
Find the Physics interfaces in study subsection. In the table, clear the Solve checkboxes for Heat Transfer in Fluids (ht) and Coefficient Form PDE (c).
5
Click the Add Study button in the window toolbar.
6
In the Home toolbar, click  Add Study to close the Add Study window.
Multibody Analysis
In the Settings window for Component, type Multibody Analysis in the Label text field.
Geometry 2
Import 1 (imp1)
1
In the Geometry toolbar, click  Import.
2
In the Settings window for Import, locate the Source section.
3
Click  Browse.
4
Browse to the model’s Application Libraries folder and double-click the file reciprocating_engine.mphbin.
5
Click  Import.
Make a union of the flywheel and the crankshaft to avoid modeling a fixed joint between them.
Union 1 (uni1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Union.
2
Select the objects imp1(1) and imp1(11) only.
3
In the Settings window for Union, locate the Union section.
4
Clear the Keep interior boundaries checkbox.
Form Union (fin)
1
In the Model Builder window, under Multibody Analysis (comp2) > Geometry 2 click Form Union (fin).
2
In the Settings window for Form Union/Assembly, locate the Form Union/Assembly section.
3
From the Action list, choose Form an assembly.
4
Click  Build Selected.
Disable the analysis of the geometry as the remaining small geometric details can be kept.
5
In the Model Builder window, click Geometry 2.
6
In the Settings window for Geometry, locate the Cleanup section.
7
Clear the Automatic detection of small details checkbox.
For automatic generation of hinge and prismatic joints between different parts, group the identity boundary pairs.
Definitions (comp2)
In the Model Builder window, expand the Multibody Analysis (comp2) > Definitions node.
Identity Boundary Pair 1 (ap1), Identity Boundary Pair 2 (ap2), Identity Boundary Pair 3 (ap3), Identity Boundary Pair 4 (ap4), Identity Boundary Pair 6 (ap6), Identity Boundary Pair 8 (ap8)
1
In the Model Builder window, under Multibody Analysis (comp2) > Definitions, Ctrl-click to select Identity Boundary Pair 1 (ap1), Identity Boundary Pair 2 (ap2), Identity Boundary Pair 3 (ap3), Identity Boundary Pair 4 (ap4), Identity Boundary Pair 6 (ap6), and Identity Boundary Pair 8 (ap8).
2
Right-click and choose Group.
Hinge Joint Pairs
In the Settings window for Group, type Hinge Joint Pairs in the Label text field.
Identity Boundary Pair 5 (ap5), Identity Boundary Pair 7 (ap7), Identity Boundary Pair 9 (ap9)
1
In the Model Builder window, under Multibody Analysis (comp2) > Definitions, Ctrl-click to select Identity Boundary Pair 5 (ap5), Identity Boundary Pair 7 (ap7), and Identity Boundary Pair 9 (ap9).
2
Right-click and choose Group.
Prismatic Joint Pairs
1
In the Settings window for Group, type Prismatic Joint Pairs in the Label text field.
2
Right-click Prismatic Joint Pairs and choose Disable.
Import the cylinder pressure data obtained in the thermodynamic analysis.
Interpolation 1 (int1)
1
In the Definitions toolbar, click  Interpolation.
2
In the Settings window for Interpolation, locate the Definition section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file reciprocating_engine_pressure.txt.
5
In the Function name text field, type p.
6
Locate the Units section. In the Argument table, enter the following settings:
Argument
Unit
t
rad
7
In the Function table, enter the following settings:
Function
Unit
p
bar
Create selections for the top boundaries of each piston, so that the cylinder pressure can be applied.
Cylinder 1
1
In the Definitions toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Geometric Entity Level section.
3
From the Level list, choose Boundary.
4
Locate the Size and Shape section. In the Outer radius text field, type 0.03.
5
In the Bottom distance text field, type -0.081.
6
Locate the Position section. In the x text field, type -0.11.
Cylinder 2
1
Right-click Cylinder 1 and choose Duplicate.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Bottom distance text field, type -0.013.
4
Locate the Position section. In the x text field, type 0.
Cylinder 3
1
Right-click Cylinder 2 and choose Duplicate.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Bottom distance text field, type -0.07.
4
Locate the Position section. In the x text field, type 0.11.
For automatically creating Rigid Material nodes in Multibody Dynamics interface, create a selection of all rigid components. This includes all domains except the second connecting rod, which is the only flexible component of the system.
Rigid Materials
1
In the Definitions toolbar, click  Explicit.
2
Click in the Graphics window and then press Ctrl+A to select all domains.
3
Select Domains 1–5 and 7–10 only.
4
In the Settings window for Explicit, type Rigid Materials in the Label text field.
Add Material
1
In the Materials toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in > Structural steel.
4
Click the Add to Component button in the window toolbar.
5
In the Materials toolbar, click  Add Material to close the Add Material window.
The second connecting rod is the only flexible component and is defined using the default Linear Elastic Material model. Rigid Material nodes on all rigid components of engine can be created automatically from Automated Model Setup section of Multibody Dynamics node.
Multibody Dynamics (mbd)
1
In the Settings window for Multibody Dynamics, locate the Automated Model Setup section.
2
From the Rigid domains selection list, choose Rigid Materials.
3
Click Physics Node Generation in the upper-right corner of the Automated Model Setup section. From the menu, choose Create Rigid Domains.
Crankshaft
1
In the Model Builder window, expand the Rigid Domains (sel1) node, then click Rigid Material 1.
2
In the Settings window for Rigid Material, type Crankshaft in the Label text field.
Reciprocating Engine Components
Rename the other Rigid Material nodes in the group according to the table below:
Rigid Material
Name
Rigid Material 2
Piston 1
Rigid Material 3
Connecting rod 1
Rigid Material 4
Cylinder 1
Rigid Material 5
Piston 2
Rigid Material 6
Cylinder 2
Rigid Material 7
Piston 3
Rigid Material 8
Connecting rod 3
Rigid Material 9
Cylinder 3
Cylinder 1
In the Model Builder window, click Cylinder 1.
Fixed Constraint 1
In the Physics toolbar, click  Attributes and choose Fixed Constraint.
Cylinder 2
In the Model Builder window, under Multibody Analysis (comp2) > Multibody Dynamics (mbd) > Rigid Domains (sel1) click Cylinder 2.
Fixed Constraint 1
In the Physics toolbar, click  Attributes and choose Fixed Constraint.
Cylinder 3
In the Model Builder window, under Multibody Analysis (comp2) > Multibody Dynamics (mbd) > Rigid Domains (sel1) click Cylinder 3.
Fixed Constraint 1
In the Physics toolbar, click  Attributes and choose Fixed Constraint.
Use Create Joints button for automatically creating Attachment nodes and Hinge Joint nodes from Multibody Dynamics Physics.
Hinge Joints
1
The details of Hinge Joint nodes between different components of engine are given in the table below:
Name
Source
Destination
Hinge Joint 1
Crankshaft
Connecting rod 1
Hinge Joint 2
Crankshaft
Connecting rod 2: bottom end
Hinge Joint 3
Crankshaft
Connecting rod 3
Hinge Joint 4
Piston 1
Connecting rod 1
Hinge Joint 5
Piston 2
Connecting rod 2: top end
Hinge Joint 6
Piston 3
Connecting rod 3
Hinge Joint 7
Fixed
Crankshaft
Hinge Joint 8
Fixed
Crankshaft
2
In the Model Builder window, click Multibody Dynamics (mbd).
3
In the Settings window for Multibody Dynamics, locate the Automated Model Setup section.
4
Find the Joint types subsection. From the Planar boundaries list, choose None.
5
From the Spherical boundaries list, choose None.
6
Click Physics Node Generation in the upper-right corner of the Automated Model Setup section. From the menu, choose Create Joints.
Connecting Rod 2: Bottom End
1
In the Model Builder window, expand the Hinge Joints node, then click Attachment 1.
2
In the Settings window for Attachment, type Connecting Rod 2: Bottom End in the Label text field.
3
Locate the Connection Type section. From the list, choose Flexible.
Connecting Rod 2: Top End
1
In the Model Builder window, under Multibody Analysis (comp2) > Multibody Dynamics (mbd) > Hinge Joints click Attachment 2.
2
In the Settings window for Attachment, type Connecting Rod 2: Top End in the Label text field.
3
Locate the Connection Type section. From the list, choose Flexible.
Hinge Joint 7
1
In the Model Builder window, under Multibody Analysis (comp2) > Multibody Dynamics (mbd) > Hinge Joints right-click Hinge Joint 6 and choose Duplicate.
2
In the Settings window for Hinge Joint, locate the Attachment Selection section.
3
From the Source list, choose Fixed.
4
From the Destination list, choose Crankshaft.
5
Locate the Center of Joint section. From the list, choose Centroid of selected entities.
6
From the Entity level list, choose Edge.
Center of Joint: Edge 1
1
In the Model Builder window, click Center of Joint: Edge 1.
2
Select Edges 21 and 24 only.
Hinge Joint 8
In the Model Builder window, under Multibody Analysis (comp2) > Multibody Dynamics (mbd) > Hinge Joints right-click Hinge Joint 7 and choose Duplicate.
Center of Joint: Edge 1
1
In the Model Builder window, expand the Hinge Joint 8 node, then click Center of Joint: Edge 1.
2
Select Edges 297 and 298 only.
For automatically creating Prismatic Joint nodes between cylindrical boundaries, use Prismatic Joint Pairs.
Definitions (comp2)
Hinge Joint Pairs
In the Model Builder window, under Multibody Analysis (comp2) > Definitions right-click Hinge Joint Pairs and choose Disable.
Prismatic Joint Pairs
In the Model Builder window, right-click Prismatic Joint Pairs and choose Enable.
Multibody Dynamics (mbd)
1
In the Model Builder window, under Multibody Analysis (comp2) click Multibody Dynamics (mbd).
2
In the Settings window for Multibody Dynamics, locate the Automated Model Setup section.
3
Find the Joint types subsection. From the Cylindrical boundaries list, choose Prismatic joint.
4
Click Physics Node Generation in the upper-right corner of the Automated Model Setup section. From the menu, choose Create Joints.
Definitions (comp2)
Integration 3 (intop3)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
2
In the Settings window for Integration, locate the Source Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose Cylinder 1.
5
Locate the Advanced section. From the Frame list, choose Material  (X, Y, Z).
Use step functions for smooth application of starting torque and external load.
Step 1 (step1)
1
In the Definitions toolbar, click  More Functions and choose Step.
2
In the Settings window for Step, locate the Parameters section.
3
In the Location text field, type pi/3.
4
In the From text field, type 1.
5
In the To text field, type 0.
6
Click to expand the Smoothing section. In the Size of transition zone text field, type pi/36.
Step 2 (step2)
1
In the Definitions toolbar, click  More Functions and choose Step.
2
In the Settings window for Step, locate the Parameters section.
3
In the Location text field, type 3*pi.
4
Locate the Smoothing section. In the Size of transition zone text field, type pi/18.
Create a maximum operator to compute the maximum stress in the connecting rod.
Maximum 1 (maxop1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Maximum.
2
Select Domain 6 only.
Variables 2
1
In the Definitions toolbar, click  Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
theta
abs(mbd.hgj7.th)
rad
Rotation of crankshaft
theta0
50.25[deg]
rad
Initial rotation of crank 1
N
d(theta,t)*60/(2*pi)
rad/s
RPM of crankshaft
Ti
400[N*m]*step1(theta)
N·m
Starting torque
To
0.5[N*m*s/rad]*d(theta,t)*step2(theta)
N·m
Output torque
p1
p(mod(theta-theta0,2*pi))
Pa
Pressure in cylinder 1
p2
p(mod(theta-theta0+4*pi/3,2*pi))
Pa
Pressure in cylinder 2
p3
p(mod(theta-theta0+2*pi/3,2*pi))
Pa
Pressure in cylinder 3
A
intop3(root.nZ)
m²
Projected area of piston
P1
-p1*A*mbd.prj1.u_t/746[W]
 
Power generated in cylinder 1 (hp)
P2
-p2*A*mbd.prj2.u_t/746[W]
 
Power generated in cylinder 2 (hp)
P3
-p3*A*mbd.prj3.u_t/746[W]
 
Power generated in cylinder 3 (hp)
BHP
To*d(theta,t)/746[W]
rad
Brake horse power
MaxStress_cr
maxop1(mbd.mises)
N/m²
Maximum stress in connecting rod
Use the Applied Moment subnode to apply the starting torque and the external load on the crankshaft.
Multibody Dynamics (mbd)
Crankshaft
In the Model Builder window, under Multibody Analysis (comp2) > Multibody Dynamics (mbd) > Rigid Domains (sel1) click Crankshaft.
Applied Moment 1
1
In the Physics toolbar, click  Attributes and choose Applied Moment.
2
In the Settings window for Applied Moment, locate the Applied Moment section.
3
Specify the M vector as
Ti
x
0
y
0
z
Applied Moment 2
1
Right-click Applied Moment 1 and choose Duplicate.
2
In the Settings window for Applied Moment, locate the Applied Moment section.
3
Specify the M vector as
-To
x
0
y
0
z
Boundary Load 1
1
In the Physics toolbar, click  Boundaries and choose Boundary Load.
2
In the Settings window for Boundary Load, locate the Boundary Selection section.
3
From the Selection list, choose Cylinder 1.
4
Locate the Force section. From the Load type list, choose Pressure.
5
In the p text field, type p1.
Boundary Load 2
1
In the Physics toolbar, click  Boundaries and choose Boundary Load.
2
In the Settings window for Boundary Load, locate the Boundary Selection section.
3
From the Selection list, choose Cylinder 2.
4
Locate the Force section. From the Load type list, choose Pressure.
5
In the p text field, type p2.
Boundary Load 3
1
In the Physics toolbar, click  Boundaries and choose Boundary Load.
2
In the Settings window for Boundary Load, locate the Boundary Selection section.
3
From the Selection list, choose Cylinder 3.
4
Locate the Force section. From the Load type list, choose Pressure.
5
In the p text field, type p3.
Boundary Load 1, Boundary Load 2, Boundary Load 3
1
In the Model Builder window, under Multibody Analysis (comp2) > Multibody Dynamics (mbd), Ctrl-click to select Boundary Load 1, Boundary Load 2, and Boundary Load 3.
2
Right-click and choose Group.
Boundary Loads
In the Settings window for Group, type Boundary Loads in the Label text field.
Mesh 2
1
In the Model Builder window, under Multibody Analysis (comp2) click Mesh 2.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
In the table, clear the Use checkbox for Geometric Analysis, Detail Size.
4
Click  Build All.
Study: Multibody Analysis
In the Settings window for Study, type Study: Multibody Analysis in the Label text field.
Step 1: Time Dependent
1
In the Model Builder window, under Study: Multibody Analysis click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type range(0,4e-4,0.16).
Solution 2 (sol2)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 2 (sol2) node.
3
In the Model Builder window, expand the Study: Multibody Analysis > Solver Configurations > Solution 2 (sol2) > Dependent Variables 1 node, then click comp2.mbd.att1.Fc1x.
4
In the Settings window for State, locate the Scaling section.
5
In the Scale text field, type 1e8*(0.1*0.7011179332388733)^2*100.
6
In the Model Builder window, under Study: Multibody Analysis > Solver Configurations > Solution 2 (sol2) > Dependent Variables 1 click comp2.mbd.att1.Fd1x.
7
In the Settings window for State, locate the Scaling section.
8
In the Scale text field, type 1e8*(0.1*0.7011179332388733)*100.
9
In the Model Builder window, under Study: Multibody Analysis > Solver Configurations > Solution 2 (sol2) > Dependent Variables 1 click comp2.mbd.att2.Fc1x.
10
In the Settings window for State, locate the Scaling section.
11
In the Scale text field, type 1e8*(0.1*0.7011179332388733)^2*100.
12
In the Model Builder window, under Study: Multibody Analysis > Solver Configurations > Solution 2 (sol2) > Dependent Variables 1 click comp2.mbd.att2.Fd1x.
13
In the Settings window for State, locate the Scaling section.
14
In the Scale text field, type 1e8*(0.1*0.7011179332388733)*100.
15
In the Model Builder window, under Study: Multibody Analysis > Solver Configurations > Solution 2 (sol2) click Time-Dependent Solver 1.
16
In the Settings window for Time-Dependent Solver, click to expand the Time Stepping section.
17
From the Steps taken by solver list, choose Free.
18
From the Maximum step constraint list, choose Constant.
19
In the Maximum step text field, type 4e-4.
20
In the Study toolbar, click  Compute.
Results
Displacement (mbd)
The numbering of the datasets and plots may differ from the instructions given below, if you started by directly modeling the multibody analysis.
Follow these instructions to generate the displacement plot shown in Figure 5.
Study: Multibody Analysis/Solution 2 (4) (sol2)
1
In the Model Builder window, expand the Results > Datasets node.
2
Right-click Results > Datasets > Study: Multibody Analysis/Solution 2 (3) (sol2) and choose Duplicate.
Selection
1
In the Results toolbar, click  Attributes and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domains 1–3, 5, 6, 8, and 9 only.
Surface
1
In the Model Builder window, expand the Results > Displacement (mbd) node, then click Surface.
2
In the Settings window for Surface, locate the Data section.
3
From the Dataset list, choose Study: Multibody Analysis/Solution 2 (4) (sol2).
4
From the Solution parameters list, choose From parent.
5
Locate the Expression section. From the Unit list, choose mm.
Volume 1
1
In the Model Builder window, right-click Displacement (mbd) and choose Volume.
2
In the Settings window for Volume, click to expand the Title section.
3
From the Title type list, choose None.
4
Locate the Coloring and Style section. From the Coloring list, choose Uniform.
5
From the Color list, choose Gray.
Selection 1
1
In the Model Builder window, right-click Volume 1 and choose Selection.
2
Select Domains 4, 7, and 10 only.
3
In the Displacement (mbd) toolbar, click  Plot.
Transparency 1
In the Model Builder window, right-click Volume 1 and choose Transparency.
Result Templates
1
In the Results toolbar, click  Result Templates to open the Result Templates window.
2
Go to the Result Templates window.
3
In the tree, select Study: Multibody Analysis/Solution 2 (3) (sol2) > Multibody Dynamics > Applied Loads (mbd) > Boundary Loads (mbd).
4
Click the Add Result Template button in the window toolbar.
5
In the Results toolbar, click  Result Templates to close the Result Templates window.
Results
Boundary Loads (mbd)
In the Model Builder window, expand the Boundary Loads (mbd) node.
Boundary Load 1, Boundary Load 2, Boundary Load 3
1
In the Model Builder window, under Results > Boundary Loads (mbd), Ctrl-click to select Boundary Load 1, Boundary Load 2, and Boundary Load 3.
2
Right-click and choose Copy.
Displacement (mbd)
In the Model Builder window, under Results right-click Displacement (mbd) and choose Paste Multiple Items.
Color Expression
1
In the Model Builder window, expand the Boundary Load 1 node, then click Color Expression.
2
In the Settings window for Color Expression, locate the Coloring and Style section.
3
Clear the Color legend checkbox.
Boundary Load 2
1
In the Model Builder window, under Results > Displacement (mbd) click Boundary Load 2.
2
In the Settings window for Arrow Surface, click to expand the Title section.
3
From the Title type list, choose None.
Boundary Load 3
1
In the Model Builder window, click Boundary Load 3.
2
In the Settings window for Arrow Surface, locate the Title section.
3
From the Title type list, choose None.
4
Click the  Zoom Extents button in the Graphics toolbar.
5
In the Displacement (mbd) toolbar, click  Plot.
Boundary Loads (mbd)
In the Model Builder window, under Results right-click Boundary Loads (mbd) and choose Delete.
Use the following instructions to plot the von-Mises stress in the second connecting rod as shown in Figure 6.
Study: Multibody Analysis/Solution 2 (5) (sol2)
In the Model Builder window, under Results > Datasets right-click Study: Multibody Analysis/Solution 2 (3) (sol2) and choose Duplicate.
Selection
1
In the Results toolbar, click  Attributes and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 6 only.
Stress: Connecting Rod
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, type Stress: Connecting Rod in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study: Multibody Analysis/Solution 2 (5) (sol2).
Surface 1
1
Right-click Stress: Connecting Rod and choose Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Multibody Analysis (comp2) > Multibody Dynamics > Stress > mbd.misesGp - von Mises stress - N/m².
Stress: Connecting Rod
1
In the Model Builder window, click Stress: Connecting Rod.
2
In the Settings window for 3D Plot Group, locate the Plot Settings section.
3
Clear the Plot dataset edges checkbox.
4
Click the  Zoom Extents button in the Graphics toolbar.
5
In the Stress: Connecting Rod toolbar, click  Plot.
Follow these instructions to reproduce the RPM versus crank rotation curve shown in Figure 7.
RPM
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type RPM in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study: Multibody Analysis/Solution 2 (3) (sol2).
Global 1
1
Right-click RPM and choose Global.
2
In the Settings window for Global, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Multibody Analysis (comp2) > Definitions > Variables > N - RPM of crankshaft - rad/s.
3
Click to expand the Legends section. Clear the Show legends checkbox.
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type theta/(2*pi).
6
Click to expand the Coloring and Style section. From the Width list, choose 2.
7
Click to expand the Title section. From the Title type list, choose None.
RPM
1
In the Model Builder window, click RPM.
2
In the Settings window for 1D Plot Group, locate the Plot Settings section.
3
Select the x-axis label checkbox. In the associated text field, type Rotation of crankshaft (cycle).
4
Select the y-axis label checkbox. In the associated text field, type RPM of crankshaft.
5
In the RPM toolbar, click  Plot.
Follow the instructions below to plot the mechanical power generated in each cylinder. The resulting plot should look like the one shown in Figure 8.
Power Generated
1
Right-click RPM and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Power Generated in the Label text field.
3
Locate the Plot Settings section. In the y-axis label text field, type Power generated (hp).
Global 1
1
In the Model Builder window, expand the Power Generated node, then click Global 1.
2
In the Settings window for Global, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Multibody Analysis (comp2) > Definitions > Variables > P1 - Power generated in cylinder 1 (hp) - 1.
3
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Multibody Analysis (comp2) > Definitions > Variables > P2 - Power generated in cylinder 2 (hp) - 1.
4
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Multibody Analysis (comp2) > Definitions > Variables > P3 - Power generated in cylinder 3 (hp) - 1.
5
Locate the Coloring and Style section. Find the Line markers subsection. From the Marker list, choose Cycle.
6
From the Positioning list, choose Interpolated.
7
In the Number text field, type 30.
8
Locate the Legends section. Select the Show legends checkbox.
9
In the Power Generated toolbar, click  Plot.
Power Generated
1
In the Model Builder window, click Power Generated.
2
In the Settings window for 1D Plot Group, locate the Axis section.
3
Select the Manual axis limits checkbox.
4
In the y maximum text field, type 200.
5
In the Power Generated toolbar, click  Plot.
Use the following instructions to plot the brake horse power of the engine. The resulting plot should look like the one shown in Figure 9.
Brake Horse Power
1
In the Model Builder window, right-click RPM and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Brake Horse Power in the Label text field.
3
Locate the Plot Settings section. In the y-axis label text field, type BHP.
Global 1
1
In the Model Builder window, expand the Brake Horse Power node, then click Global 1.
2
In the Settings window for Global, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Multibody Analysis (comp2) > Definitions > Variables > BHP - Brake horse power - rad.
3
In the Brake Horse Power toolbar, click  Plot.
Use the following instructions to reproduce the variation of the maximum stress generated in the second connecting rod shown in Figure 10.
Max Stress: Connecting Rod
1
In the Model Builder window, right-click RPM and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Max Stress: Connecting Rod in the Label text field.
3
Locate the Plot Settings section. In the y-axis label text field, type von Mises stress (MPa).
Global 1
1
In the Model Builder window, expand the Max Stress: Connecting Rod node, then click Global 1.
2
In the Settings window for Global, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Multibody Analysis (comp2) > Definitions > Variables > MaxStress_cr - Maximum stress in connecting rod - N/m².
3
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
MaxStress_cr
MPa
Maximum stress in connecting rod
4
Locate the Title section. From the Title type list, choose Manual.
5
In the Title text area, type Maximum stress in connecting rod.
6
In the Max Stress: Connecting Rod toolbar, click  Plot.
Use the following instructions to plot the forces in the joint between the second connecting rod and the crankpin. The resulting plot should look like the one shown in Figure 11.
Joint Force: Connecting Rod-Crank
1
In the Model Builder window, right-click Power Generated and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Joint Force: Connecting Rod-Crank in the Label text field.
3
Locate the Plot Settings section. In the y-axis label text field, type Joint force (N).
Global 1
1
In the Model Builder window, expand the Joint Force: Connecting Rod-Crank node, then click Global 1.
2
In the Settings window for Global, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Multibody Analysis (comp2) > Multibody Dynamics > Hinge joints > Hinge Joint 2 > Joint force - N > mbd.hgj2.Fx - Joint force, x-component.
3
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Multibody Analysis (comp2) > Multibody Dynamics > Hinge joints > Hinge Joint 2 > Joint force - N > mbd.hgj2.Fy - Joint force, y-component.
4
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Multibody Analysis (comp2) > Multibody Dynamics > Hinge joints > Hinge Joint 2 > Joint force - N > mbd.hgj2.Fz - Joint force, z-component.
5
In the Joint Force: Connecting Rod-Crank toolbar, click  Plot.
Joint Force: Connecting Rod-Crank
1
In the Model Builder window, click Joint Force: Connecting Rod-Crank.
2
In the Settings window for 1D Plot Group, locate the Axis section.
3
In the y minimum text field, type -5000.
4
In the y maximum text field, type 35000.
5
In the Joint Force: Connecting Rod-Crank toolbar, click  Plot.
Use the following instructions to generate an animation of the motion of the different components of the engine.
Animation 1
1
In the Results toolbar, click  Animation and choose Player.
2
In the Settings window for Animation, locate the Scene section.
3
From the Subject list, choose Displacement (mbd).
4
Locate the Frames section. In the Number of frames text field, type 100.
The analyses are now finished. If you open a saved model and want to recompute the solution, then Study: Multibody analysis can be recomputed without making any changes. To run the Study: Thermodynamic analysis, you need to enable the Pressure Work node by right clicking on Component 1 > Heat Transfer in Fluids > Fluid > Pressure Work and then clicking Enable.