*6.2. Reference Stroke*

A sinusoidal curve is used to compare the trajectories. The trajectory can be described analytically as follows:

$$\chi = -\frac{\varkappa\_{peak}}{2} \cdot \cos(\varphi) + \frac{\varkappa\_{peak}}{2} \tag{24}$$

$$v = \frac{\varkappa\_{peak} \cdot \frac{d\varphi}{dt}}{2} \cdot \sin(\varphi) \tag{25}$$

$$a = \frac{\varkappa\_{\rm peak} \cdot \frac{d\varphi^2}{dt}}{2} \cdot \cos(\varphi) + \frac{\varkappa\_{\rm peak}}{2} \cdot \frac{d^2\varphi}{dt^2} \cdot \sin(\omega t) \tag{26}$$

The first and second derivatives of the stroke with respect to time result in the velocity and acceleration. For the four strokes, it is assumed that the stroke amplitude *xpeak* and the virtual angular velocity *ω* do not change (Assumption: *ω* = *const*.). The used reference trajectory is shown in Figure 9.

**Figure 9.** Sinusoidal trajectory for the four strokes with the parameters *xpeak* = 0.03 m and *ω* = 80 rads depending on time.

As can be seen here, the time intervals for the TDC, BDC and the stroke amplitude are constant and the change in the piston position is very symmetrical. This trajectory is used in the following as a reference trajectory in order to show the changes in the piston position.

#### *6.3. Time Displacement of the Dead Center (Trajectory A)*

First, a function has to be determined with which the duration of one stroke can be changed without changing the period duration of the four strokes. The following individual stroke duration (*Ti*) and their setting parameters (Δ*Ti*,*rel*) are defined depending on the period time *TP*:

$$T\_1 = \Delta T\_{1,rel} \cdot T\_P \tag{27}$$

$$T\_2 = \Delta T\_{2,rel} \cdot T\_P \tag{28}$$

$$T\_{\ $} = \Delta T\_{\$ ,rel} \cdot T\_P \tag{29}$$

$$T\_4 = T\_P - T\_1 - T\_2 - T\_3 \tag{30}$$

To change the duration of a stroke, the average angular speed of one stroke must be adjusted. In addition, the transition between the cycles must take place without discontinuities in the acceleration. This is achieved in that the initial angular speed *<sup>ω</sup>Begin* is identical to the final speed *ωEnd* and the average speed *ωP* over the four cycles:

$$
\omega\_{\text{Begin}} = \omega\_{\text{End}} = \omega\_P = \frac{4\pi}{T\_P}.\tag{31}
$$

These conditions can be met with various functions. A sinusoidal change was chosen to implement the time displacement. The mean value of a half sine wave can be calculated as follows:

$$\overline{\omega}\_{i} = \frac{1}{\pi} \int\_{0}^{\pi} \hat{\omega}\_{i} \cdot \sin(\varphi) d\varphi = \frac{2}{\pi} \,\hat{\omega}\_{i} = \frac{\pi}{T\_{i}}.\tag{32}$$

In order to meet the boundary conditions, the sine function has to be shifted by *ωP*:

$$
\omega = \frac{\pi}{2} \cdot (\overline{\omega}\_{\bar{i}} - \omega\_P) \cdot \sin(t \,\, \overline{\omega}\_{\bar{i}}) + \omega\_P. \tag{33}
$$

The integration of Equation (33) results in the angle *ϕ*, which can be used in Equations (24) and (25) in order to obtain an analytical description of the speed and acceleration. The trajectory for the dead center shift is shown in Figure 10.

**Figure 10.** Comparison between the reference curve and dead center shift in the operating point *ω* = 80 rads , *xpeak* = 0.03 m, *T*1 = 0.25 · *Tp*, *T*2 = 0.225 · *Tp*, *T*3 = 0.275 · *Tp* and *ω* = 80 rads , *xpeak* = 0.03 m, *T*1 = 0.25 · *Tp*, *T*2 = 0.2 · *Tp*, *T*3 = 0.3 · *Tp*.

With the advanced features, it is possible to vary the period of three cycles. The period of the fourth cycle is calculated from the specified period for the entire cycle.

#### *6.4. Compression and Extension of the Stroke (Trajectory B)*

Furthermore, it was investigated how the variation of the piston holding time affects the combustion process. For this, it was necessary to develop a function with which it should be possible to compress or stretch the stroke curve without changing the temporal position of the dead centers. The piston stroke trajectory can be varied by changing the virtual angular velocity during a cycle. The position of the dead centers can be kept constant if the mean angular velocity remains the same. A sinusoidal modulation was also chosen here:

$$
\Delta \omega = \Delta \omega\_{\overline{1}\overline{1}} \cdot \sin(2\varrho) + \omega\_{\overline{P}} \tag{34}
$$

where Δ*ωTi*. is an adjustable parameter which corresponds to the amplitude of the superimposed sinusoidal oscillation from the i-th stroke. The frequency is twice as high so that the mean value is not changed within one stroke. A combination of the method for shifting the dead centers is also possible. All you have to do is replace *ωP* with *ωi*. The trajectory for a stroke compression and extension is shown in Figure 11.

**Figure 11.** Comparison between the reference curve and compression and extension in the operating point *ω* = 80 rads , *xpeak* = 0.03 m, Δ*ωT*1 = 0 rads , Δ*ωT*2 = 0 rads , Δ*ωT*3 = 40 rads , Δ*ωT*4 = 0 rads and *ω* = 80 rads , *xpeak* = 0.03 m, Δ*ωT*1 = 0 rads , Δ*ωT*2 = 0 rads , Δ*ωT*3 = −40 rads , Δ*ωT*4 = 0 rads .

It can be seen in Figure 11 that the piston stroke trajectory can be changed significantly without changing the position of the dead centers. It can also be seen here that the acceleration has no points of discontinuity.

#### *6.5. Variation of the Stroke Amplitude (Trajectory C)*

The adjustment of the stroke amplitude is another interesting degree of freedom for modulating the trajectory. The position of the top dead center was left constant and the bottom dead center was shifted. The piston stroke has already been described analytically with Equation (24). If *xpeak* is changed here, it is possible to vary the target trajectory. This change in stroke should also take place in such a way that there are no discontinuities in the acceleration. Various functions can be used for this. A sinusoidal change in the stroke amplitude was decided:

$$\alpha\_{\text{peak}} = -\frac{\pounds\_1 - \pounds\_2}{2} \cdot \sin\left(\frac{\varphi}{2}\right) + \frac{\pounds\_1 - \pounds\_2}{2}. \tag{35}$$

where *x*ˆ2 corresponds to the stroke amplitude at the bottom dead center between the intake and compression stroke and *x*ˆ1 is the stroke amplitude at the bottom dead center between the expansion and exhaust stroke. The trajectory is shown in Figure 12.

**Figure 12.** Comparison between the reference curve and stroke amplitude variation in the operating point *ω* = 80 rad/s, *x*ˆ1 = 0.03 m, *x*ˆ2 = 0.033 m and ω = 80 rad/s, *x*ˆ1 = 0.03 m, *x*ˆ2 = 0.036 m.

As can be seen here, the different stroke amplitudes can be set separately from one another. In addition, no steps in acceleration can be seen here either, so practical implementation is possible.

#### *6.6. Combination of Proposed Trajectories (Trajectory D)*

As the last type of variation, it is shown that it is possible to combine the mentioned trajectory generation methods. It was decided to adjust the position of the TDC and to increase the acceleration of the piston after the TDC. The trajectories can be calculated as a combination of the methods from Sections 6.3 and 6.4. The trajectory is shown in Figure 13.

**Figure 13.** Comparison between the reference curve and stroke amplitude variation in the operating point *ω* = 80 rad/s, *xpeak* = 0.03 m, *T*2 = 0.225 · *Tp*, *T*3 = 0.275 · *Tp*, Δ*ωT*3 = 40 rads and *ω* = 80 rad/s, *xpeak* = 0.03 m, *T*2 = 0.2 · *Tp*, *T*3 = 0.3 · *Tp*, Δ*ωT*3 = 40 rads.
