The potential for use of cubic nonlinear systems in internal combustion engine drivetrains

. Modern-day approaches to reducing the emissions of combustion engines lead to unconventional strategies which include downspeeding, downsizing, cylinder deactivation. A common characteristic of these strategies is a shift in torque excitation components of the combustion engine. This excitation shift results in an increase in vibrations of the drivetrain and the risks associated with it. The presented article aims to investigate the elementary characteristics of the pure cubic nonlinear system and to identify the potential for its use in mechanical systems.


Introduction
The requirements and constraints on modern mechanical systems are becoming more and more demanding, which leads to new approaches to dynamic optimisation [1]. This dynamic optimisation is often targeted on vibration reduction and its goals stem mainly from legislation and health protection initiatives [2]. The focus of research in this area is to push the vibration levels further below legal limits, thereby improving interactions in the man-machine-environment system [3]. At present, it is not possible to eliminate vibrations completely, however, it is possible to reduce them to an acceptable level [4,5,6]. The mechanical systems at the centre of attention are predominantly systems with internal combustion (IC) engines [7]. The combustion cycle in IC engines is a significant source of vibrations [8,9]. Furthermore, new environmental requirements are applied to modern combustion engines [10]. On the one hand, the result is a positive effect of emission reduction [11]; however, on the other hand, an adverse effect on vibration excitation content [12,13]. Presently, several approaches to reducing torsional vibrations in mechanical drivetrains with IC engines are implemented. Widely used are various types of dual mass flywheels [14,15,16,17]. It has been demonstrated that it is possible to use pneumatic tuners with variable torsional stiffness to achieve active tuning of torsional vibration [18,19,20,21,22,23]. Research in this area has been focused on the use of Nonlinear energy sink (NES) [24] and Target energy transfer (TET) [25] with the aim of developing a tuning device with quasi-zero torsional stiffness [26]. These research avenues require novel approach to solving analytical models. It is common practice in engineering that complicated analytical models are simplifiedfor instance nonlinear systems are often linearised around the operating point [27]. While linearisation can be accurate enough in come cases, as demonstrated in [28,29,30], there are applications which require nonlinear models, because linearisation is not accurate enough, or not possible at all. A very common example of nonlinear systems often subject to analysis [31,32,33], is a system governed by the Duffing equation, whereby both linear and nonlinear terms are present in the restoring force. Our attention in this article is aimed at pure cubic nonlinear systems. This type of system has been described in the literature [32] both theoretically and computationally. Our approach is differentwe aim to outline a simplified approach to analysis using dimensional analysis and computer simulations to understand the key traits of behaviour of the cubic system. The discussion is then aimed at identifying the key characteristics of the nonlinear cubic system and to assess their advantages and disadvantages for implementation in mechanical systems.

Definitions and conventions
In this article, no concrete solution is presented, and the discussion is applicable to both linear and rotational coordinates (such as x or θ). Therefore, generalised coordinates q, q̇, q̈ shall be used to denote displacement, velocity, and acceleration, respectively. The equation of motion of cubic system with linear damping and force excitation is: mq̈+ λq̇+ μq = f ext Symbols m, λ, μ denote the inertia, damping and stiffness (∂ q 2 V) as the properties of the cubic system, where V(q) is potential energy. Stiffness of linear system has conventional label k.

Potential energy of cubic system
Traditionally, the theory of small vibration uses linearization to idealise a vibrating system to linear restoring force and hence quadratic potential V ∝ q 2 as shown in Figure 1a. The system in the focus of our study has quartic potential V ∝ q 4 . The shape of the potential well is displayed in Figure 1b.
Quartic potential gives rise to cubic restoring force, hence the system is termed cubic.
Physically, the cubic system can be realised by means of a special "cubic spring", by a combination of linear springs which are tuned to zero restoring force around equilibrium, or more intuitively, one can think of the cubic system as a point mass (or ball) moving under gravity inside valley which is a scaled version of the potential well of Figure 1b. That is because the gravitational potential is given by V = mgz, and therefore if the valley has shape z ∝ x 4 , the restoring force due to gravity will be cubic with respect to displacement x. Unlike many other forms of potentials, the cubic restoring force due to quartic potential cannot be linearised around equilibrium, and the theory of small-amplitude linear vibrations cannot be used. The fact that cubic systems cannot be linearised has profound implications. In linear systems, superposition can be used as a powerful tool to calculate the output of a system to (1) a superposition of inputs. Bode plots and transfer functions come useful because any input can be decomposed into simpler constituent parts using Fourier transform. However, that is not true of the cubic system. The knowledge of the response of the cubic system to various individual inputs does not imply that the response to the sum of those inputs is easily obtained.

Natural frequency of free vibrations in cubic system
It was recognized that one of the most interesting features of nonlinear vibrating system is that the natural frequency is not constant but depends on the initial conditions (initial displacement and velocity). It was therefore sought to calculate or estimate the frequency of free vibration of the cubic system. The first attempt at the problem was dimensional analysis. Since the governing equation of the system is known (Eq. (1) without the RHS term), it is straightforward to determine which parameters have to be included in the problem. Let us first consider the case of free vibration subject to initial displacement 0 starting from rest (Ẋ0 = 0) and in the absence of damping (λ = 0). The relevant variables (4) and dimensions (3) are: Thus, according to Buckingham's Pi Theorem, there remains only 4-3=1 non-dimensional group, which has to be constant. Mass can be eliminated between and μ, length between 0 and μ and time between μ and ω . Therefore, the dimensionless group governing free vibration is ω = ω n 0 √ μ Note that we are not a priori enforcing a specific function (such as sine or cosine). Rather, the only assumption is that the response is periodic with period , such that by convention ω n = 2π . Several simulations were carried out with various system properties and initial conditions to check whether non-dimensional group ω is, indeed, constant and to determine the value of this constant. The simulations were written in Julia programming language using Runge-Kutta fourth-order integration method. Output time sequence was frequency-transformed using Fast Fourier Transform. The output and its frequency spectrum are displayed in Figure 2. The value of non-dimensional constant ω was determined at approximately 0.847 with a moderate dependence on the length of simulation. The reason is presumably that the number of samples affects the resolution of the FFT, thereby resulting in a slight shift in the peak value. Secondly, non-zero initial velocity ̇0 will be allowed in the initial conditions. If we decided to follow the approach of dimensional analysis, one extra non-dimensional group would have to appear in the problem. Instead, we use physical insight to determine the natural frequency of free vibration. Although the system is nonlinear, it is nevertheless conservativei.e. without damping, energy is conserved. The total energy in the system is: Therefore, if the system is started with non-zero initial displacement 0 and velocity ̇0 , the effective initial displacement is given by: This effectively transforms the initial kinetic energy to additional potential energy, allowing simple calculation of the frequency of free vibration using the non-dimensional constant ω presented above.
Results of simulations confirm that the conversion to effective initial displacement and estimate of natural frequency are correct. Graphs are very similar to those shown in Figure  2 and were omitted for brevity. It should be noted that it is possible to seek the non-dimensional constant in terms of the initial velocity ̇0 , in which case arbitrary initial conditions 0 and ̇0 would need to be transformed to effective initial velocity ̇ via a similar equation as Eq. (3)

High and low frequency regimes in force-driven cubic system
When cubic system with linear damping is driven by external force, the equation of motion has the form given in Eq. (1). In a similar fashion to linear systems, one could identify various frequency regimes in this system. Low-frequency regime is chararcterised by small phase-shift between input excitation and we expect that the system response should follow the input up to an amplification factor: In this low-frequency regime, the cubic spring is engaged and its stiffness μ determines the amplification factor between the input and output.
High-frequency regime is characterised by a 180 ∘ phase shift between input excitation and the system output. This is because at high frequencies, the dominant term on the LHS of Eq. (1) is the inertia ̈, and therefore the system behaviour is linear. In such linear system, any periodic excitation can be represented as a summation, or equivalently, nonperiodic excitation can be represented in integral form: Then the response is given by a superposition of responses to individual components: (for discrete case) which is the classic -40dB/decade decay on the Bode plot.
In this high-frequency regime, the cubic spring is not engaged, but the mass moves predominantly in the plateau surrounding the equilibrium position. The question that arises is how to properly frame the frequency rangeswhat is the cutoff below which the system is in low-frequency regime and what is the lowest bound of the high-frequency regime. The matter of resonance is more complex in nonlinear systems, however, we make an attempt to apply the same physical argument that applies to linear systems. In linear systems, resonance occurs when the frequency of excitation approaches the natural frequency of the systemthat is the frequency of free vibration, which in linear systems, is solely a function of system properties ( , ). In the cubic system under consideration, it was shown that the frequency of free vibration is also a function of amplitude. Therefore, it is necessary to convert the excitation force amplitude into response amplitude. We restrict ourselves to excitation whose waveform is periodic and is increasing during one half of period and decreasing during the other half, however, the shape need not be precisely sinusoidal. Under quasi-static loading, the amplitude of response can be estimated as: where is excitation amplitude. From amplitude , frequency of free vibration can be easily calculated from system properties and μ using procedure outlined in section 2.3. Then, the frequency of excitation can be compared to the frequency of free vibration. If the excitation frequency is much higher than the natural frequency corresponding to the given excitation amplitude, then the system operates in high-frequency regime. Conversely, if the excitation frequency is much lower, the system operates in low-frequency regime. As mentioned, the matter of resonance in cubic systems is quite complex and additionally, damping parameter λ needs to be taken into account. However, the procedure outlined above can be used as a first estimate of the working regime, as long as the excitation and natural frequencies are an order of magnitude apart. The results of simulations in low-and high-frequency range are shown in Figure 3. Simulations were carried out for values: = 1, μ = 1, = 1. In low-frequency regime, ω = 0.1ω , and in high-frequency regime ω = 10ω , where ω is the frequency of free vibration calculated as outlined above. System behaviour in high-frequency range is as expected. Both input and amplified output are displayed in Fig. 3b. There is a 180 ∘ phase shift between the input and output. In the low-frequency range, unexpected behaviour is observed. (Excitation is sinusoidal and is not shown in the figure for clarity.) We expected that the output should have the same form as input up to an amplification factor. However, that is not the case even for very low frequencies. Since there is zero restoring force in the close neighbourhood of the equilibrium position = 0, a very small excitation causes the mass to accelerate away from the origin so that it "overshoots" the input. The strength of subsequent oscillation depends on damping λ. With enough damping, one can minimise or elliminate these oscillations as shown by the second line in Fig. 3a. Ignoring these oscillations, when we compare the envelope of the output to a sine wave, we notice that it is more "square". One of the reasons for the discrepancy between the expected and observed behaviour is that we associated excitation with displacement control, in which case at low frequencies the load would be adjusted to match the prescribed displacement and the output would, indeed, follow the input. However, in this simulation, load (not displacement) is prescribed. Even at very low frequencies, system response always overshoots the presctibed load since there is zero restoring force around the equilibrium. In terms of energy, the contrast between linear and cubic system is the scaling of potential energy. For instance, when the driving force acts to displace the system to = 0.1, the potential energy of linear system scales as k 2 , hence is on the order of 0.01 , but for the cubic system the scaling is μ 4 , therefore the potential energy in the cubic system is on the order of 0.0001μ. The power input from excitation force is therefore predominantly captured in kinetic energy, increasing velocity rapidly, which is the reason why the response of the cubic system overshoots the input force and subsequent oscillation occurs. Further insight can be obtained from phase portrait of this motion. This is shown in Figure 4a. The phase portrait shows velocity ̇ plotted against displacement . The appearance of this graph suggests that there are two attractors in the phase plotswhich corresponds to the squareness of the system response.

System behaviour around resonance
The behaviour of cubic system with excitation around its resonant frequency is rather complicated. System response with low damping is sensitive to initial conditions as demonstrated in Figure 4b. System properties in the simulation are = 1, μ = 1, λ = 0.05. Excitation frequency is ω = ω and amplitude is = 1, where ω was calculated as outlined in section 2.4. The sensitivity to initial conditions is evident in Figure 4b. In run 1, the system was started with ( = 0) = 0,̇( = 0) = 0.2802551 and in run 2, the initial velocity was modified very slightly: ( = 0) = 0,̇( = 0) = 0.2802552. There is a qualitative difference in output waveform, even though damping is present and enough time was allowed for the establishment of steady state. It was also found out empirically, that when damping is increased above λ = 0.05, both initial conditions produce virtually identical steady state. This shows that damping has an important role in determining the response of cubic system around resonance. One of the most notable differences between linear and cubic systems is that in the absence of damping, the output amplitude does not grow to infinity as is the case in linear systems. Instead, the amplitude has a finite value, but the response appears to be chaotic with extreme sensitivity to the initial conditions.

Discussion
Several key features of the cubic system emerge from the presented investigation. The frequency of free vibrationi.e. natural frequencywas investigated and it was found that it depends on the initial conditions. This is in contrast to linear systems with quadratic potential, in which the natural frequency is constant and only a function of system properties.
Although it appears to be a complication, this phenomenon could find its use. In the future, we would like to investigate the possibility of using the cubic system as an antiresonant tuner coupled to linear system. It may be possible to use the non-constant natural frequency to make the tuner absorb energy over wider frequency range than linear tuned mass dampers. Furthermore, a major difference between linear systems and cubic system is that in the absence of damping, the response of cubic system to excitation at resonant frequency does not grow infinitely. Instead, the input oscillations are broken up into lower and higher frequencies.
Presently, it appears that this could be a source of unwanted rattle in a mechanical system. However, there might exist some application for the multi-frequency response.
It was noted that in high-frequency regime, the cubic system behaves linearly, for the spring is not strained. The response to excitation force decays with increasing frequency following the −40dB/decade decay. This is a desirable operating condition; however, it is not always achievable. One drawback is that in powertrains, load needs to be transferred. Thus, the spring needs to be strained. The proposed solution is to tune the powertrain to non-zero force, yet with negative stiffness. Therefore, the potential well would lose symmetry with the addition of a linear term. Additionally, the feasibility of operation of cubic system in the high-frequency regime has to be compared with equivalent linear system. If linear system is designed with a heavy flywheel or low stiffness, resonance is shifted to lower frequencies, thereby extending the high-frequency regime. It would only be beneficial to operate a cubic system in highfrequency regime over linear system if such a design reduced the mass of flywheel necessary, thereby reducing the power needed to accelerate the total powertrain inertia. That said, the cubic system has the ability to shift the frequency range of operating regimes by changing excitation amplitude, unlike the linear system. In the low-frequency regime, when the system is excited by force applied to the mass, oscillation is observed when the system response overshoots the excitation. These oscillations appear to be detrimental in a mechanical system as there are likely source of rattle. However, they can be reduced or eliminated by increased damping. As a result, the response has a cleaner waveform, and compared with the input sine wave, it is more rectangular. Presently, we are unaware of an existing application of this shape-changing capability of the cubic system. We discovered that this is capability is not unique to cubic system. Instead, systems with 5 th -and 7 th -power nonlinearity make the input sine wave even more square. The significant drawback of operating the cubic system in the lowfrequency regime is that high damping has to be used to eliminate the discussed oscillation, which wastes input power. A proposed route to bypass this limitation is to investigate how non-dissipative "damping" mechanisms could be used.
Another difference between the presented cubic system and linear system is noted. In linear systems, one can change operating regimes only by changing the excitation frequency, since the resonant frequency is fixed. However, in cubic system, the resonant frequency is also a function of amplitude, therefore at a fixed excitation frequency, changes in excitation amplitude have the capability to shift operating regimes between high-and low-frequency regimes. This property could be the basis of the variable-frequency antiresonant tuner as outlined above.

Conclusion
Several features of the cubic system were identified. The natural frequency is not constant, but it varies with excitation force amplitude. In force-driven system, frequency bounds for high-and low-frequency regimes were estimated. The cubic system behaves linearly in the high-frequency regime. In the low-frequency regime, the output overshoots force input and oscillation occurs, which can be attenuated by damping. Output waveform is a more square version of the input sine wave. In the absence of damping, the response is finite even at resonant frequencya phenomenon which is due to the superlinear restoring force. The observation that natural frequency of free vibration is a function of initial conditions and that resonant frequency depends on the forcing amplitude could become the basis of an antiresonant tuner. System operation can move between frequency regimes not only by changing excitation frequency, but also by changing excitation amplitude. This phenomenon could be used in tuning IC engine powertrains. In low-frequency range, the cubic system transforms sinusoidal input force into a more rectangular version, which appears promising for modulation of mechanical signals beyond IC engines.