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Parametric Study

Parametric studies of the proposed nonlinear energy sink designs

6.3. Parametric Study

The equations of motion for piston equipped with a single pendulum and double pendula NES were derived parametrically in Chapter 5. The parameters defining the NES design are the pendulum mass, stiffness, damping and length. The robustness of the NES performance with engine speed variations was shown in the previous section. Thus, a single speed (3500 rpm) will be exploited in the parametric study. In pendulum oscillations, the natural frequency of the system depends on its mass (π‘šπ‘šπ‘‘π‘‘), torsional stiffness (π‘˜π‘˜π‘‘π‘‘) and length (𝐿𝐿𝑑𝑑). This natural frequency should coincide with the excitation frequencies of the primary system (piston impacts).

Provided the torsional stiffness is linear and external excitations are negligible, the problem is similar to the classic pendulum with small angular oscillations (Figure 6.5). This analogy is exploited to determine the interaction amongst the aforementioned design variables.

(%)

(%) Impact improvement at 𝑒𝑒𝑑𝑑

Impact improvement at 𝑒𝑒𝑏𝑏

Figure 6.5. Pendulum with no external excitation and linearly torsional stiffness at its pivot

The equation of motion for the pendulum of Figure 6.5 is in the form of equation 6.1, i.e.

π‘šπ‘šπ‘‘π‘‘πΏπΏ2π‘‘π‘‘πœƒπœƒΜˆ + π‘šπ‘šπ‘‘π‘‘π‘”π‘”πΏπΏπ‘‘π‘‘sin πœƒπœƒ + π‘˜π‘˜π‘‘π‘‘πœƒπœƒ = 0 (6.1) A simple harmonic solution is selected as Θ = Θ𝐢𝐢cos πœ”πœ”π‘‘π‘‘. The natural frequency of the linear pendulum (small angular oscillations) is given by

πœ”πœ”π‘£π‘£ = �𝑔𝑔

𝐿𝐿𝑑𝑑+ π‘˜π‘˜π‘‘π‘‘

π‘šπ‘šπ‘‘π‘‘πΏπΏ2𝑑𝑑 (6.2)

Although the concept of a single natural frequency does not exist in the case of the nonlinear pendulum (NES), the analogy between the two systems assists with the reduction of design variables for the DoE study. Different combinations of π‘˜π‘˜π‘‘π‘‘, π‘šπ‘šπ‘‘π‘‘ and 𝐿𝐿𝑑𝑑 values can lead to the same natural frequency πœ”πœ”π‘£π‘£ (Equation 6.2). Provided one of these design parameters is assumed as invariable, the same frequency is still achievable through the other two variables. π‘˜π‘˜π‘‘π‘‘ is the nonlinear stiffness coefficient in the current NES concept and its influence on energy transfer is significant. The pendulum mass (π‘šπ‘šπ‘‘π‘‘) should be variable for the DoE study, as the influence of added mass on engine’s performance is of paramount importance. Thus, the pendulum length is assumed as a fixed value. Another significant parameter for the NES performance is the damping coefficient (𝑐𝑐𝑑𝑑) and its influence on energy dissipation. The moments due to torsional stiffness and damping are directly influenced by the angular displacement and velocity of the pendulum. Thus, large angular oscillations of the pendulum produce greater energy transfer

and energy dissipation in the NES. Those oscillations are restricted to the cylinder bore size and piston skirt length. The selected pendulum length (0.05 π‘šπ‘š) allows for large angular oscillations of the pendulum inside the cylinder bore whilst its contact with the piston skirt is avoided. The contact between the NES and cylinder liner is confined to very weak stiffness coefficient values (values below 10 π‘π‘π‘šπ‘š/π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ3) for this specific pendulum length.

The three aforementioned parameters (π‘šπ‘šπ‘‘π‘‘, π‘˜π‘˜π‘‘π‘‘ and 𝑐𝑐𝑑𝑑) are the examined variables of the NES design. A series of parametric studies are carried out, where the nonlinear stiffness coefficient is varied between 10 and 110 π‘π‘π‘šπ‘š/π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ3 (as explained in the description of the NES performance analysis with the engine speed variations) with 5 π‘π‘π‘šπ‘š/π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ3 steps. The NES mass in ungrounded designs should be small compared to the mass of the primary system. Thus, the NES mass ratio is expressed in terms of the proportion of the NES mass over the total mass of the piston and pin (πœ€πœ€ = π‘šπ‘šπ‘‘π‘‘/(π‘šπ‘šπ‘π‘π‘‘π‘‘π‘‘π‘‘+ π‘šπ‘šπ‘π‘π‘‘π‘‘π‘π‘)). The value of πœ€πœ€ is commonly chosen between 0 and 20% for the parametric study of two-degree-of-freedom and multi-degree-of-freedom systems (Vakakis, 2008). As a constraint, the NES mass ratio should not exceed 20% for the current problem and it is varied between 5 - 20% with 2.5% steps.

Finally, the energy dissipation mechanism is a key to the absorption of the energy excess (secondary motion) of the primary system by the absorber. The dissipated energy shows the proportion of wasted energy through the absorber’s damping to the input energy in the piston (Equation 5.62, Chapter 5). Figure 6.6 shows the percentage of dissipated energy through the NES damping element for four NES mass ratios (7.5, 12.5, 17.5 and 20%). The mass ratio has critical influence on the capacity of the nonlinear attachment to passively absorb energy from the piston during a cycle of motion. Up to 50% energy is dissipated through the NES damping element for 20% mass ratio (Figure 6.6-d). This is achieved due to the large angular velocity of the NES, since the energy dissipated energy proportional to the angular velocity (Equation 5.62, Chapter 5). Low values of the stiffness coefficient and large NES mass ratios (inertia) result in larger angular displacement and velocities in the NES. Satisfactory energy dissipation is achieved over wider range of stiffness and damping coefficients as the mass ratio increases.

The effective damping coefficient ranges up to 0.04 π‘π‘π‘šπ‘šπ‘ π‘ /π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ for 20% NES mass ratio (Figure 6.6-d). Since the NES damping should be linear (and small, in order not to prevent the NES oscillations), the maximum value of 0.04 π‘π‘π‘šπ‘šπ‘ π‘ /π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ has been selected for the damping coefficient. 𝑐𝑐𝑑𝑑 is varied from 0 to 0.04 π‘π‘π‘šπ‘šπ‘ π‘ /π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ with 0.005 steps. The results of the parametric study are illustrated in the form of contour plots; the horizontal axis represents the

damping coefficient, whereas the vertical axis signifies the variation of the stiffness coefficient.

The colour map shows the variation of each objective function. Contour plots are obtained for each NES mass ratio. Seven mass ratios have been analysed. In order to prevent unnecessary repetition and for the brevity of the discussion only three cases are presented (7.5, 12.5 and 20%

mass ratios). The results of other mass ratios are presented only if they are required for the clarity of the discussion.

Figure 6.6. Energy dissipation through the NES with the variation of stiffness and damping coefficients. Mass ratios are (a) 7.5%, (b) 12.5%, (c) 17.5% and (d) 20%.

Figure 6.7 illustrates the eccentricity accelerations (severity objective function) for 7.5% mass ratio. This NES mass ratio generally improves eccentricity accelerations at the top and bottom lands of the piston skirt. Positive accelerations are labelled by ATS as the coordinate system points towards the ATS of the cylinder liner. Thus, negative accelerations are towards the TS and they are labelled accordingly. Accelerations are mainly improved for damping coefficients between 0.005 and 0.01 π‘π‘π‘šπ‘šπ‘ π‘ /π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ. Further improvements are present at the top eccentricity for damping coefficients above 0.03 π‘π‘π‘šπ‘šπ‘ π‘ /π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ. These areas are shown as stripes over different

(%) (%)

(%) NES damping coefficient (Nms/rad) (%) NES damping coefficient (Nms/rad)

NES damping coefficient (Nms/rad) NES damping coefficient (Nms/rad)

stiffness coefficient values. The improvement area for lower damping coefficients (about 0.01 π‘π‘π‘šπ‘šπ‘ π‘ /π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ), however, is restricted between stiffness values of 20 and 50 π‘π‘π‘šπ‘š/π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ3 at the ATS of the piston top land. The area of interest at the TS of the piston bottom land is shifted towards damping coefficient value of 0.02 π‘π‘π‘šπ‘šπ‘ π‘ /π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ. The same study is carried out for 12.5% mass ratio and the severity objective functions are presented in Figure 6.8. The areas of improvement are extended over damping coefficient values of 0.01 to 0.02 π‘π‘π‘šπ‘šπ‘ π‘ /π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ. In spite of wider areas of beneficial performance, the overlapping between areas of improved performance (TS of bottom eccentricity) and areas of exacerbated performance (ATS of top eccentricity) restricts the number of scenarios for optimal design. The eccentricity accelerations are consistently improved between damping coefficient values of 0.01 and 0.04 π‘π‘π‘šπ‘šπ‘ π‘ /π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ for 20% mass ratio at the TS (Figure 6.9). The area of improved performance grows wider as the NES mass ratio increases. Simultaneously, the overlapping between areas of improved performance (ATS of piston top land) and areas of exacerbated performance (ATS of piston bottom land) enlarges in Figure 6.9.

Figure 6.7. Variation of the eccentricity acceleration objective function with damping coefficient 𝑐𝑐𝑑𝑑 and stiffness coefficient π‘˜π‘˜π‘‘π‘‘ for π‘šπ‘šπ‘‘π‘‘= 7.5%

(%) (%)

(%) (%)

NES damping coefficient (Nms/rad) NES damping coefficient (Nms/rad)

NES damping coefficient (Nms/rad) NES damping coefficient (Nms/rad)

Figure 6.8. Variation of the eccentricity acceleration objective function with damping coefficient 𝑐𝑐𝑑𝑑 and stiffness coefficient π‘˜π‘˜π‘‘π‘‘ for π‘šπ‘šπ‘‘π‘‘= 12.5%

Figure 6.9. Variation of the eccentricity acceleration objective function with damping coefficient 𝑐𝑐𝑑𝑑 and stiffness coefficient π‘˜π‘˜π‘‘π‘‘ for π‘šπ‘šπ‘‘π‘‘= 20%

(%) (%)

(%) (%)

NES damping coefficient (Nms/rad) NES damping coefficient (Nms/rad)

NES damping coefficient (Nms/rad) NES damping coefficient (Nms/rad)

(%)

(%)

(%)

(%) NES damping coefficient (Nms/rad) NES damping coefficient (Nms/rad)

NES damping coefficient (Nms/rad) NES damping coefficient (Nms/rad)

The second objective function is the number of impacts (NoI). Figure 6.10 illustrates the percentage of improvement in this objective function for 7.5% mass ratio. The areas of improved performance are restricted between damping coefficient values of 0.005 and 0.015 π‘π‘π‘šπ‘šπ‘ π‘ /π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ (similarly to the severity objective function). This area is stretched over a wider range of stiffness coefficients for the bottom eccentricity. The improved areas are shifted towards higher damping coefficient values for 12.5% mass ratio (Figure 6.11). The effective damping coefficients are between 0.01 and 0.025 π‘π‘π‘šπ‘šπ‘ π‘ /π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ . The areas of exacerbated performance overlap with the areas of improved performance for stiffness coefficient values below 40 π‘π‘π‘šπ‘š/π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ3 at the bottom eccentricity. As the NES mass ratio increases to 20% (Figure 6.12), the number of impacts are improved between 0.01 and 0.04 π‘π‘π‘šπ‘šπ‘ π‘ /π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ damping coefficients at the top eccentricity. This area at the bottom eccentricity is covered with a wide area of exacerbated number of impacts. In spite of the larger areas of improved performance, this overlap reduces the number of scenarios for optimal designs as the mass ratio increases.

Regardless of the overlaps with the areas of exacerbated performance, piston impacts can improve about 30 - 50% as the NES mass ratio changes between 7.5 - 20%.

Figure 6.10. Variation of the number of impacts objective function with damping coefficient 𝑐𝑐𝑑𝑑 and stiffness coefficient π‘˜π‘˜π‘‘π‘‘ for π‘šπ‘šπ‘‘π‘‘= 7.5%

(%)

(%) NES damping coefficient (Nms/rad)

NES damping coefficient (Nms/rad) Impact improvement at 𝑒𝑒𝑑𝑑

Impact improvement at 𝑒𝑒𝑏𝑏

Figure 6.11. Variation of the number of impacts objective function with damping coefficient 𝑐𝑐𝑑𝑑 and stiffness coefficient π‘˜π‘˜π‘‘π‘‘ for π‘šπ‘šπ‘‘π‘‘= 12.5%

Figure 6.12. Variation of the number of impacts objective function with damping coefficient 𝑐𝑐𝑑𝑑 and stiffness coefficient π‘˜π‘˜π‘‘π‘‘ for π‘šπ‘šπ‘‘π‘‘= 20%

(%)

(%) NES damping coefficient (Nms/rad)

NES damping coefficient (Nms/rad) Impact improvement at 𝑒𝑒𝑑𝑑

Impact improvement at 𝑒𝑒𝑏𝑏

(%)

(%) Impact improvement at 𝑒𝑒𝑑𝑑

Impact improvement at 𝑒𝑒𝑏𝑏

NES damping coefficient (Nms/rad)

NES damping coefficient (Nms/rad)

The impact severity and number of impacts objective functions are the main criteria to study the NVH performance through piston’s secondary motion. These functions do not take into account the structural deformation. Scenarios in which the structural deformations of the engine components are exacerbated can negatively influence the transferred energy to the engine block in spite of improvement in impact severity. Thus, the transferred energy objective function is exploited to avoid such scenarios. The transferred energy to the cylinder liner is expressed in terms of improvement percentage with respect to the original system’s behaviour without NES (Equation 5.47 in Chapter 5). This objective function is illustrated in Figures 6.13 to 6.15 for 7.5, 12.5 and 20% mass ratios successively. The percentage of improvement drops from about 10% to approximately 0% at the TS of the cylinder liner as the mass ratio increases. This is consistently in the proximity of 0% at the ATS for all mass ratios. The exacerbated areas of performance are confined to zero damping conditions, which are not the concern for NES design. The areas of improved performance are shifted towards higher damping coefficients as the NES mass ratio increases. This effect is more apparent at the bottom eccentricity.

Figure 6.13. Variation of the transferred energy objective function with damping coefficient 𝑐𝑐𝑑𝑑 and stiffness coefficient π‘˜π‘˜π‘‘π‘‘ for π‘šπ‘šπ‘‘π‘‘= 7.5%

(%)

(%) NES damping coefficient (Nms/rad)

NES damping coefficient (Nms/rad)

Figure 6.14. Variation of the transferred energy objective function with damping coefficient 𝑐𝑐𝑑𝑑 and stiffness coefficient π‘˜π‘˜π‘‘π‘‘ for π‘šπ‘šπ‘‘π‘‘= 12.5%

Figure 6.15. Variation of the transferred energy objective function with damping coefficient 𝑐𝑐𝑑𝑑 and stiffness coefficient π‘˜π‘˜π‘‘π‘‘ for π‘šπ‘šπ‘‘π‘‘= 20%

(%)

(%) NES damping coefficient (Nms/rad)

NES damping coefficient (Nms/rad)

(%)

(%) NES damping coefficient (Nms/rad)

NES damping coefficient (Nms/rad)

The aforementioned objective functions are the main criteria for the DoE analysis. The physical constraints of the system, however, should be considered as well. The pendulum can come into contact with the cylinder liner for weak stiffness coefficient values below 10 π‘π‘π‘šπ‘š/π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ3 (as already mentioned). The contour plots of Figure 6.16 show the maximum and minimum amplitude of angular oscillations of the NES (12.5% mass ratio), which are very small for damping coefficient values of 0.025 π‘π‘π‘šπ‘šπ‘ π‘ /π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ and above. On the contrary, the oscillations are severe for zero damping and stiffness coefficient values less than 25 π‘π‘π‘šπ‘š/π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Ÿπ‘Ÿ3. The zero damping conditions are of no interest for the NES design. Thus, the angular oscillations are well confined between 0 and 70Β° (the maximum angle at which the NES impacts the cylinder liner). For the brevity of the discussion, the contour plots for other mass ratios are not presented as their trends are very similar to the case of 12.5% mass ratio.

Figure 6.16. Amplitude of the NES angular oscillations with damping coefficient 𝑐𝑐𝑑𝑑 and stiffness coefficient π‘˜π‘˜π‘‘π‘‘

for π‘šπ‘šπ‘‘π‘‘= 12.5%

NES damping coefficient (Nms/rad)

NES damping coefficient (Nms/rad)

NES maximum oscillation angle (deg)

(deg) NES minimum oscillation angle

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