Time-frequency energy analysis of chaotic mechanical system affected by additive impulses

1.1. Control of nonlinear mechanical systems

Control of the inverted pendulum system remains as a challenging task as it is highly unstable, highly non-linear, and nonminimum phase system [2]. The system is widely applied experimentally as an example of a nonlinear mechanical system to demonstrate chaotic behavior upon harmonic excitation. Recent studies on chaos in inverted flexible pendulum with tip mass were demonstrated by Gorade et al. [3] introduced a new mechanical approach for modeling the system of an elastic inverted pendulum on a cart with significant tip mass having multiple dynamic equilibria, with the mathematical model is framed using Euler-Lagrange analysis. Donaire et al. in [4] performed the control of underactuated nonlinear mechanical systems using energy shaping procedure without solving the partial differential equations (PDEs). In this energy shaping procedure, the stabilization of the systems by shaping the energy function and preserving the system structure’s kinetic energy is applied. The application of energy shaping control on an inverted flexible pendulum fixed to a cart is further studied by Gandhi et al. [5] in 2016. This paper considers an ultra-flexible inverted pendulum fixed on a cart and a new nonlinear energy shaping controller to keep the pendulum at the upward position. The design is based on a model, obtained via constrained Lagrange formulation, which previously has been validated.

1.2. Time-frequency energy analysis

Experimental chaotic time series data from chaotic systems was analyzed using the orthogonal wavelet transform [6]. The wavelet analysis gives useful information about the system through the energy concentration at specific wavelet levels. In other areas, time-frequency-energy analysis on non-stationary signals are applied in [7]. A new algorithm for efficient and precise time-frequency-energy (TFE) analysis of signals, named intrinsic time-scale decomposition (ITD) was introduced. This method overcomes the limitations of classical methods. (e.g. Fourier transform or wavelet transform based). In [8], an algorithm for the detection of single and multiple tracks in spectrograms are developed. A new potential energy formulation has been introduced to utilize the structural and intensity information to increase detection rates. Signal processing using time-frequency energy method are presented in [1] for ground vibration induced by metro. Through frequency band recognition and energy characterization, the vibration signal induced by non-metro aspects was effectively separated.

1.3. Parseval’s theorem of signal energy

In signal processing, signal $x\left(t\right)$ is often characterized by the term signal energy $E_s$ or signal power. The description of a signal can be represented in two different ways, time domain and Fourier (frequency) representation. The energy and power of a signal expressed in time domain are equal when the same signal is described in the frequency domain. According to Parseval’s theorem, the equations that relates signal energy, $E_s$ in both representation can be calculated through the integral as:

Assuming $e$ and $E$ are the fields of impulse in time and frequency domain respectively, with $E$ is the Fourier transformation of $e$. Therefore in Eq. (1), Parseval’s theorem states that signal energy calculated in time domain is equal to signal energy calculated in frequency domain. In Eq. (2), Parseval’s theorem is stated in the form of energy conservation where power-time $t$ is in unit of Watt and energy-spectrum $f$ is in unit of Watt/Hz²:

In this contribution, the Parseval’s theorem is applied in the energy calculation by evaluating the energy carried by impulses both in the time and frequency domain.

3.1. Analysis of time-frequency energy

In the time-frequency transformed illustration of the energy (Fig. 2), the left figure represents the power spectral density function by Fourier transform while the bottom figure shows the time history of flexible beam jumping event in chaotic vibration. The time history is obtained from jumping event and the frequency domain analysis curve of the amplitude represents the energy of the whole vibration signal. The blue to red bar on the right represents the relatively low and large energy concentration of the time-frequency energy distribution. The time-frequency energy represents the effect of time axis and frequency axis along with the quantity of energy by different colors, as in the figure. The energy is observed mainly distributed in the domain by its lower and higher position.

Therefore, this analysis represents three-dimensional aspects of the signal of the chaotic vibration; the time behavior, the frequency ranges, and the energy distribution. To characterize the behavior of each jumping behavior, time-frequency energy analysis on the energy distribution between jumping states is performed.

3.2. State transition model

Techniques of signal processing and time-frequency energy analysis are applied to the experimental data from jumping events of the flexible beam. From the characterization of the data, the energy distribution behavior between jumping states can be modeled as a state model, shown in Fig. 3. The systems are also characterized by the range of frequencies to which they respond.

This model shows the behavioral change described by relations between energy changes in specific frequency bands, analyzed from the jumping events of the flexible beam during chaotic vibration. In Fig. 3 L, R, and C denote the left, right, and center equilibrium of jumping behavior, respectively. Based on the state model, the state transition appears between L-R-L and L-C-L equilibrium. According to this experimentally-obtained model [10], a suitable impulse event control of the system is newly introduced. Impulses in frequency-specific energy based on this state model are designed and injected into the system to actuate the state transition and control the jumping event. For further details, please refer to [10, 11].

Fig. 2Illustration of time-frequency energy vs. time history and Fourier spectrum

Fig. 3State model established of energy changes in frequency bands

3.3. Analysis of system’s equilibrium shift caused by additive impulses

In Fig. 4(a), a spectrogram of suitably designed impulses, filtered in specific frequency band before being added into the system is shown. This specific frequency band of impulses is based on the state model in Fig. 3, in order to induce the desired jumping behavior. In Fig. 4(b), the result shows that additive impulses designed and injected able to produce equilibrium shift and therefore control the chaotic system behavior. The time response and phase portrait of equilibrium shift from left (L) to center (C) equilibrium are displayed.

From the phase portrait, the beam vibrates to and from passing through two equilibria (left and center) in the steady state. Therefore, the equilibrium region about which vibration occurs depends on the designed impulse and the frequency at which the impulse is injected. It can be stated that the control impulses act as perturbation in terms of the energy distribution between the frequency ranges the system vibrates with, followed by related jumping between the equilibria.