Finite Element Analysis of a Rocker Arm

Finite Element Analysis of a Rocker Arm

1. Component Description

1.1.            Component Function

The rocker arm is an oscillating,  two-arm lever that provides a means of actuating the valves in the combustion chamber of an internal combustion engine. It translates the radial motion of the profile of the cam lobe through a fulcrum into linear motion for opening and closing the intake and exhaust valves.  It also provides a means of multiplying the lift ratio.

During operation, the rocker arm experience stresses and undergo deflection. Severe rocker arm deflection causes inefficient engine performance, and often results in metal fatigue leading to increased wear and friction in the valve train and eventually engine failure.

Figure 1: Diesel engine valve train showing rocker arm (1)

1.2.            Component Geometry

The rocker arm is a two-arm lever that pivots about a fulcrum. One end is connected to the push rod which rests over cams on the camshaft, while the other acts on the spring-loaded valve stem and pivoted on the rocker shaft.

Figure 2:(From left) 3-D model of the rocker arm and the 2-D geometry

1.3.            Service Loading

The rocker arm is subjected to compressive load at the fulcrum on the rocker shaft during the opening and closing of the valves.  The forces acting at the valve end (F_E) include; the gas back pressure on the valve, the spring force and the force due to valve acceleration. There is also the load at the cam end, (F_C) which is transmitted to the rocker arm through the push rod.

Figure 3: Free Body Diagram of Rocker arm depicting the location of the acting forces: F_E and F_C

2. Methodology (154)
   1.          Two-Dimensional Idealization

In a rocker arm, the constraint at the pivot hole and the applied loads at both ends, all act in a plane, parallel to the cross-section plane of the rocker arm.

To estimate the stress concentration of the geometry, it is idealized as a shell and shaped in the profile of the planar cross section of the rocker arm under the assumption that the rocker arm is completely solid with only the holes drilled through as seen in Figure 4. The two-dimensional idealization makes it easier to deal with and the main stresses are obtained with reasonable accuracy.

Figure 4: 2D geometrical Idealization using Autodesk Inventor

2.2.            Loading, Boundary Conditions and Constraints

The rocker arm has two major loads applied on either end respectively:

• Force due to exhaust valve loading; F_E
• Force due to cam action through the push rod; F_C

The main forces will be calculated based on the Diesel Engine specifications below.

Table 1: Diesel Engine Specification (2)

Total load on valve: $P_t=P_g+w=\left(\left(\frac{\pi }{4}\right)\times d_1^2\times P_c\right)+w=502.4N$

Initial spring force: $F_s$

=   $\left(\left(\frac{\pi }{4}\right)\times d_1^2\times P_c\right)–w$

= 24.249 N

Force due to valve acceleration with valve weight: $F_a=\left(m\times a\right)–w=140.38N$

Maximum load on the rocker arm for exhaust valve:

${\mathit{F}}_{\mathit{e}}=P+F_s+F_a+(P_1+\left(k\times h\right))=\mathit{1133}\mathit{.}\mathit{87}\mathit{N}$

Load on push rod side of the rocker arm: ${\mathit{F}}_{\mathit{c}}=F_e\times \mathit{Rocker\; arm\; ratio}=\mathit{1859}\mathit{.}\mathit{54}\mathit{N}$

A pinned constraint is applied on the pivoting hole in all degrees of freedom.

3. Finite Element Representation
   1.          Modelling and Justification of Shell type

The material for the rocker arm; carbon steel is assumed to be homogeneous and isotropic with details in Table 3.

Table 3: Carbon Steel Material Properties

Plane stress idealization is assumed because the thickness in the z-direction is smaller than the length in the x-y plane. Given the 2D geometry, it is idealized as a plane stress model and shell elements must be used. The shape of the element can either be triangular or quadrilateral and the element order is either linear or parabolic.

Table 4: Comparison of Element type and order

The element type chosen for the meshing is the second order quadrilateral element. This was due to the accuracy and consistency of the von Mises stress values obtained when the Force F_E is applied on the rocker arm in comparison to other element types. Therefore, it has proven to be the best for this geometry.

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To accurately represent the pinned pivot constraint, a pinned, rigid body connector was applied at the centre point and connected to the pivot hole so all the nodes at the hole can better represent the motion.  For each loading scenario, one hole must have a fixed constraint the other has the load applied for the solution to be solved. This can be seen in Figure 5 and Figure 6.

Figure 6: Mesh diagram for F_E loading scenario

4. Contour Plots

After the finite element model was solved, the maximum Von-Mises stress was recorded. The finite element contour plot for each loading scenario can be seen in Figure 7 and Figure 8.

Figure 7: F_E loading scenario

Figure 8: F_C loading scenario

When Force F_C was applied, it predicted the largest maximum von-Mises stress compared to when Force F_E was applied. Hence, F_C is the worst-case loading scenario and the stress obtained from this scenario will be evaluated.

4.1.            Justification of results parameter

The von Mises stress is chosen as the main result parameter to be evaluated. This is required to evaluate the failure rate of the rocker arm, since it is made of a ductile material and is largely likely to fail as a result of shearing action. The von Mises stress yield criterion predicts failure for a ductile material, when the von Mises stress becomes equal to the yield strength.

The contour plot predicts the maximum von Mises stress value as 203.3MPa, occurring at the neck of the rocker arm.  This value is within the Yield strength for carbon steel of 415MPa.

5. Mesh Convergence

In order to explore the effects of mesh density on the prediction from the finite element model, several finite element models were generated, varying the number of elements from 39 (coarsest mesh) to 12069 (finest mesh) from element sizes ranging from 0.2mm to 20.5 mm. The von Mises stress from the worst-case scenario were calculated and compared, to study the influence of mesh density on the stresses predicted, as shown below.

Table 5: Effect of mesh density

From Figure 9, the stress converges to the maximum possible value of 203.3 MPa.

6. Conclusion

In this case, them model produced acceptable results. The maximum stress was 203.3 MPa and was localized mostly and the neck of the rocker arm which is where the stress was expected to be most concentrated. The convergence of the results shows that the result is not significantly dependent on the choice of discretization and is an indication of the validity of the accuracy of the prediction. Also, the result obtained under both loading conditions, indicating the same location (the neck of the rocker arm) as where the maximum von Mises stress is predicted and where failure is likely to occur provides further validation of the result. However, the component was assumed to be completely solid except for the holes. In reality, the 3D geometry (Figure 2) is a better representation of the component and the aforementioned assumption is a relatively poor one. Apart from the applied loads, the component would be subjected to thermal stresses. Furthermore, though plane stress was assumed, the thickness of 19 mm was relatively large compared to the length of 50.8 mm. To validate the model, plane strain assumptions may be made during the next analysis in which a displacement-controlled test can be used to identify the strain that causes failure.

7. References

1.) Diesel Engine Valve Train. Nuclear Power Training. [Online] [Cited: November 14th, 2018 at 21:34.] http://nuclearpowertraining.tpub.com/h1018v1/css/Figure-10-Diesel-Engine-Valve-Train-31.htm.

2.) DESIGN AND STATIC STRUCTURAL ANALYSIS OF ROCKER ARM IN I.C. Bacha, Sachin et al. s.l. : INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH, 2018, INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH, p. 9.