CWT:

Continuous wavelet transform

DOF:

Degree of freedom

Dmeyer:

Discrete wavelet Meyer Filter

EMD:

Empirical mode decompostion

FFT:

Fast Fourier transform

Fs:

Sampling frequency

FT:

Fourier transfomr

FSST:

Short-time Fourier transform-based synchrosqueezing transform

HT:

Hilbert transform

HHT:

Hilbert-Huang transform

HSA:

Hilbert spectral analysis

MEMS:

Micro-electro-mechanical systems

SE:

Spectral entropy

SK:

Spectral kurtosis

SST:

Synchrosqueezed transform

STFT:

Short-time Fourier transform

TFA:

Time-frequency analysis

TRF:

Time-frequency representation

WSST:

Wavelet-based synchrosqueezing transform

WVD:

Wigner–Ville distribution

1 Introduction

With the spread of microcomputer-based and, more recently, open platform-based data acquisition systems [27, 46, 70, 83, 109,110,111], beyond the use of MEMS sensors to measure mechanical measurements [24, 45, 108], these works generally use the Arduino microcontroller and Raspberry Pi single-board computers with sensors such as accelerometer, gyroscope, ultrasound, and infrared, to instrument vibrations in mechanical systems. These tools are available to a wide range of students, professionals, and young researchers in various areas of knowledge. On the other hand, while in dedicated systems all the signal processing was done by proprietary software or hardware, without major user intervention and/or knowledge, in generic or open platform-based data acquisition systems it is the user who implement the computational routines that calculate the spectra and other parameters of the acquired signals. One of the objectives of this tutorial is to present time-frequency analysis techniques for the processing of digitized signals in a microcomputer with special attention to the application in linear and nonlinear mechanical systems, particularly to mechanical vibration signals.

Spectral analysis techniques based on Fourier transform through FFT are efficient for analysis of signals with stationary characteristics; however, many random processes are non-stationary, causing frequencies to change over time. Thus, the signal misses the periodic characteristic, one of the basic assumption for the use of the Fourier transform [2, 21, 71].

Furthermore, systems excited by non-ideal power sources, such as numerous motors, may exhibit some nonlinear phenomena in their response, such as the Sommerfeld effect and the jump phenomenon. These phenomena occur due to nonlinear interactions between energy sources and elastic systems. In the occurrence of these phenomena, frequency jumps occur in the system response and the increase and/or decrease of the vibration amplitude. Processing signals of this nature need more specific techniques, as there are large concentrations of energy in the time-frequency domain that eventually hide frequencies with less energy.

According to Addison [2], various signals from machines and mechanical components have non-stationary characteristics, such as stopping and starting signals from electric motors. Stop and start signals are particularly interesting because they capture the richer frequency spectrum and effectively store all the spectral information of the signal [107, 121]. In these cases, the Fourier transform becomes inadequate for the frequency domain signal representation. In stationary signal analysis, the concept of amplitude and frequency spectrum is generally intuitive [90]; for non-stationary operation, it is more difficult to determine the amplitude or frequency of the signal, since these quantities may vary over time, so it is helpful to introduce the concept of instantaneous amplitude and frequency. These concepts are widely used through the Hilbert transform [30, 90]. To deal with these issues, time and frequency domain representation methods are often combined into a single representation, which presents how the spectral content of a signal evolves over time and is a more appropriate tool for non-stationary signal analysis [28, 30, 86]. It is emphasized, for example, that the characterization of failures in a mechanical system is made by observing changes in the signal frequency spectrum [107]. Also, the feature extraction of images and neurophysiological and speech signals can be realized through the characterization of the signal energy, in a simple and precise way [33]. The time-frequency decomposition [2] maps a 1-D signal to a 2-D time and frequency image and provides information about local frequency variations. The literature presents a large number of methods for this purpose, categorized mainly as linear or quadratic methods [28, 36, 114]. The linear methods include notably STFT and CWT, while the quadratic methods, such as spectrogram, scalogram, and WVD, that treat the signal-associated power spectral density [18]. A detailed and rigorous study on the subject can be found at [11, 28].

One of the techniques used in non-stationary signal analysis is the short-time Fourier transform (STFT). In this case, the signal is placed in two dimensions, time and frequency. It is known that in the analysis made from STFT there is a compromise between time and frequency. Signal analysis gives information on when and which frequencies vary. However, this information is limited to the size of the window, which once set will be the same for all frequencies [28]. In [73], the basic concept of STFT is used but the size of the window is fixed in the frequency domain instead of the time domain. This approach is simpler than similar existing methods, such as adaptive STFT and multi-resolution STFT; it emphasizes that the method proposed by the authors requires neither a band-pass filter banks nor the evaluation of local signal characteristics of adaptive techniques.

Another widely used technique for signal analysis in non-stationary and transient operations is the Wigner-Ville distribution (WVD), which is part of a group of integral transforms called bilinear. This technique was the first attempt to perform a joint time and frequency analysis [72]. The bilinear Wigner-Ville distribution achieves better time-frequency domain joint resolution compared to any linear transform; however, it suffers from a cross-term interference problem, which does not represent any signal information, i.e., the WVD of two signals is not the sum of their individual WVDs [72].

In this context of time-frequency analysis is proposed the wavelet transform in order to overcome the window limitation of the STFT formulation, and overcome the interference problems of the Wigner-Ville distribution and deficiencies of other methods based on integral transforms [21, 107, 127]. An important feature of the Wavelet transform is that the frequency resolution varies in proportion to the center frequency variation. For the past two decades, the Wavelet transform has been successfully used in the most diverse areas of scientific knowledge and especially in mechanical system applications, as well as many other fields of engineering, with proven success in non-stationary signal analysis. The Wavelet transform is applied in several studies with relative success. This can be seen in the work of Yan and Gao [127], where they demonstrate various techniques based on the wavelet transform for fault diagnosis in rotating machines. Several other works use techniques based on Wavelet transform for fault detection in mechanical systems [16, 34, 90, 105, 107, 127, 130], in structural dynamics and structural health [4, 19, 38, 44, 49, 68, 92], and in nonlinear dynamics applications [50, 85, 102,103,104, 112, 124, 125]. In [85] was studied the nonlinear dynamics of a shape memory oscillator subjected to ideal and non-ideal excitations. The numerical results show that the CWT scale index can successfully detect the system behavior when the signal presents chaotic behavior.

Another emergent technique for time-frequency analysis with wide application in mechanical systems is the Hilbert-Huang transform (HHT), this technique is a way of decompose a signal into so-called intrinsic mode functions together with a trend and obtaining data from instantaneous frequency. The method is designed to work well with non-stationary and nonlinear data [39,40,41]. In contrast to other common transforms, such as the Fourier transform, HHT is a similar algorithm and has an empirical approach that can be applied to a dataset rather than a theoretical tool. The HHT is based on the empirical mode decomposition (EMD) and Hilbert transform (HT); thus, it is an empirical analysis method and its expansion base is adaptive, so that it can produce physically significant results in the analysis of nonlinear and non-stationary signals. The advantage of being an adaptive method comes at a price: the difficulty of a theoretical foundation [41]. HHT has wide application in mechanical systems analysis such as failure analysis in rotating systems [15, 47, 56, 116, 128], parameter identification [3, 51, 61, 77, 88], and nonlinear dynamics [13, 14, 17, 32, 42, 63, 80].

Another technique widely used in TFA is spectral kurtosis (SK), which has been introduced as a statistical tool that can indicate not only non-Gaussian components in a signal but also their frequency domain locations. It was initially used as a complement to the power spectral density and demonstrated its efficiency in problems related to the transient detection in noisy signals. The literature presents successful applications in vibration-based condition monitoring with the use of SK [5, 118]. Several application examples of the use of SK can be seen in [35, 57, 64, 66, 89, 113]. A rigorous study can be found at [93].

New time-frequency analysis techniques have been proposed in recent years. An emerging technique is the synchrosqueezed transform. Known limitations, such as tradeoffs between time and frequency resolution, can be overcome by alternative techniques that extract instantaneous modal components, as presented in the synchrosqueezed transforms. The EMD of a signal into components that are well separated in the time-frequency domain allows the reconstruction of these components [6, 79, 123]. In particular, the work presented in [6] provides an overview of time-frequency reassignment and synchronization techniques used in multicomponent signals, covering theoretical history and applications, and attempts to explain how synchrosqueezing can be viewed as a special case of rebuilding the reassignment enable mode. Synchrosqueezing transform-based methods are actually an extension of the CWT that incorporates empirical mode decomposition elements and frequency reassignment techniques. This new tool produces a better defined time-frequency representation, allowing instantaneous frequency identification to highlight individual components.

Wavelet-based synchrosqueezing transform (WSST) is a time-frequency analysis method proposed by Daubechies and Maes in 1996 [22]. In 2011, Daubechies et al. [23] propose a combination method through wavelet analysis and EMD relocation method. By introducing a precise mathematical definition for a class of functions that can be seen as an overlay of a fairly small number of roughly harmonic components and proving that our method actually succeeds in decomposing arbitrary functions in this class. The anti-noise capability and time-frequency resolution of the WSST are enhanced based on the wavelet transform (WT). WSST maintains the advantages of EMD and CWT. WSST is adaptive like EMD and does not depend on the mother wavelet. The problem of mixing modes is significantly improved. Nowadays, WSST has been deeply applied to mechanical systems with applications as parameter estimation [53, 71, 74, 75, 84, 119, 120, 129], Rotor Rub-Impact Fault Diagnosis [65, 101, 117], structural dynamics [62, 71, 84], and characterization of chaotic behavior in several nonlinear systems [69, 96, 104, 117, 126]. In [37], a new stochastic chaotic secure communication scheme based on the WSST algorithm is proposed.

About other recent works on SST , we can mention the demodulation transform-based SST method (also called instantaneous frequency-embedded-SST) was introduced in [48, 58, 115] to change the instantaneous frequency (IF) of signal so that more accurate IF estimation and component recovery (mode retrieval) can be achieved. Adaptive FSST and adaptive WSST were proposed and analyzed in [60, 67, 94] and [9, 12, 59] respectively. In these papers, the window width of the continuous wavelet for WSST and the window function for FSST are time-varying. The results presented in the literature demonstrate these methods outperform the conventional WSST and FSST.

In addition, for the improvement of current machines, nonlinear investigation of their dynamics is necessary and this point is especially important when it comes to rotating machines. It is observed that when it is considered that a nonlinear motor exciting a structure has limited power and its rotation depends on the response of the structure to which it is coupled, the resultant electromechanical system presents a much richer dynamic than the linear case, being possible to identify the most diverse nonlinear effects [7, 8, 20, 76].

A very notable phenomenon is the Sommerfeld effect, which occurs in systems that are at some point excited at their resonant frequencies by a non-ideal power source. In this phenomenon, the power source cannot transform all incoming energy into useful work and transforms a fraction of this energy into mechanical vibration [20, 31]. Thus, the Sommerfeld effect causes large vibration amplitudes to the system, which can damage the structure due to the high stresses generated.

However, it is known that algorithms based on the Fourier transform are not indicated for nonlinear signal analysis, so for nonlinear phenomenon identification in mechanical systems it is necessary to use signal processing techniques applicable to nonlinear signals with good time and frequency domain resolution.

In particular, this tutorial consists of presenting applications of time-frequency analysis in synthetic and experimental signals of mechanical systems, but the techniques presented can also be used in the most diverse applications and also in the numerical solution of differential equations. The independent variable is generally assumed to be the time, but nothing prevents to be a spatial variable or any other quantity.

This paper is divided into some sections for better organization of the topics. Section 2 presents the methodologies used for the time-frequency representations realized in this paper. Section 2.1 presents the short-time Fourier transform, Section 2.2 introduces the wavelet transform, Section 2.3 discusses the Hibert-Huang transform, and Section 2.4 discusses the syncrosqueezed transforms. Section 3 presents the case studies covered by this paper, being divided into synthetic test signals, Section 3.1, and experimental test signals, Section 3.2; altogether, four case studies were performed in this paper. Section 4 presents the results obtained and Section 5 presents the conclusions and final remarks.

2 Time-Frequency Representation Methodology

2.1 Short-Time Fourier Transform

STFT is the most commonly used method for studying non-stationary and transient signals due to its simplicity and computational power. The basic idea of the method is to divide the signal into several short time blocks that are separated or overlapped and then perform the FT on each block. This is done by multiplying the signal analyzed by a family of models obtained through time translation and frequency modulation of a window function [90]. In (1), we present the expression of STFT where the signal x(t) is previously “windowed” by a g(t) function, around a τ time.

$$ STFT(\tau,f) = {\int}^{\infty}_{-\infty} x(t)g(t-\tau)e^{-i2\pi ft}dt $$
(1)

The magnitude spectrum, spectrogram, can be obtained by:

$$ S(\tau,f) = \left| {\int}^{\infty}_{-\infty} x(t)g(t-\tau)e^{-i2\pi ft}dt \right|^{2} $$
(2)

There is a trade-off between time and frequency in the STFT. In [1], a study is carried out comparing the efficiency of the STFT in separating the components of a signal for different window lengths. The signal analysis gives information on when and which frequencies vary. However, this information is limited to the window length, which once set will be the same for all frequencies. A more generic formulation can be found in [86] and a more detailed and mathematically rigorous formulation in [6, 28].

2.2 Continuous Wavelet Transform

The CWT was proposed in order to overcome the limitation of the window length in STFT. The wavelet transform uses a variable window, where the resolutions vary along the time-frequency plane, so as to obtain all the information contained in the signal [2]. Equation (3) presents the wavelet transform in the continuous form (CWT).

$$ W(a,b) = {\int}_{-\infty}^{\infty} x(t) \bar\psi_{a,b}(t)dt $$
(3)

where,

$$ \bar\psi_{a,b}(t) = \frac{1}{\sqrt(a)}\bar\psi \left( \frac{t-b}{a}\right) $$
(4)

In (4), the term \(\bar \psi (t)\) is the prototype of windows, known in wavelet theory as the mother wavelet. The factor\( \frac {1}{\sqrt {a}}\) ensures the normalization of energy to any scale. This analysis determines the correlation of the x(t) signal through translations τ and changes of scales, for a given wavelet mother. Criteria for choosing the mother wavelet are found in [107]. In mechanical vibration analysis, Daubechies and Meyer discrete wavelets are successfully used [105,106,107]

2.3 Hilbert-Huang Transform

According to Huang [39], Hilbert-Huang transform (HHT) is a TFA method that can be divided into two steps: In the first, there is the EMD process, which allows an adaptive set of base functions to be obtained, and the second, through the HSA, allows obtaining a time-frequency domain representation by calculating the Hilbert transform of each of the functions obtained by EMD analysis.

The two main steps of HHT are [81]:

  1. 1.

    Apply EMD to decompose a time domain signal into n intrinsic mode functions ci corresponding to different intrinsic time scales as follows:

    $$ u(t) = \sum\limits_{i=1}^{n} c_{i}(t) + r_{n} $$
    (5)

    the term rn in (5) denotes the residue.

  2. 2.

    Apply the Hilbert transform and calculate the time-dependent frequency and amplitude Ai of each term ci.

    The intrinsic mode function is a function that satisfies two conditions:

    • The number of extremes and the number of zero crossings must be the same or different from one in all data.

    • The envelope defined by the local minimum and maximum must be symmetrical, and therefore the average value of the maximum envelope and the minimum envelope is 0 at any given point. Once the extremes are identified, all local maxima are interpolated by a cubic spline as the upper envelope. The procedure for the local minimums to produce the lower envelope is then repeated. The average of the upper and lower envelopes is designated as m11, and the first intrinsic function c11 is given by:

      $$ c_{11} = u(t) - m_{11} $$
      (6)

The sifting process has to be repeated more times. In the k th sifting process, c1k− 1 is treated as the data, then [81].

$$ c_{1k} = c_{1k-1}-m_{1k} $$
(7)

were k = 2,...,K. The process is repeated until all local maxima are positive, all local minima are negative, and the waves are almost symmetrical, so c1k is accepted as c1. One systematic method for ending the iteration is to limit the Dv deviation calculated from the two consecutive sifting results as follows:

$$ D_{v} = \sqrt{\frac{{\sum}_{i=1}{N}(c_{1k}(t_{i})-c_{1k}(ti))^{2}} {{\sum}_{i=1}^{N} c_{1k-1}^{2}(t_{i})}} $$
(8)

In (8), ti = iδt and Nδt = T denote the sampled period. Once you get c1, then set the residue r1, treat r1 as the new data, and repeat the steps in (6) and (7) as follows:

$$ \begin{array}{@{}rcl@{}} c_{21} &=& r_{1} - m_{21}, \quad r_{1} \equiv u(t) - c_{1} \\ c_{2k} &=& c_{2k-1} - m_{2k} \end{array} $$
(9)

\(k=2,\dots , K\). After obtaining the term c2, the residue r2 is defined and the steps are repeated as in (10).

$$ \begin{array}{@{}rcl@{}} c_{31}&=&r_{2} - m_{31}, \quad r_{2} \equiv u(t)-c_{1}-c_{2} \\ c_{3k}&=&c_{3k-1}-m_{3k}, \quad k=2,\dots,K \end{array} $$
(10)

After determining all the terms ci(t), the Hilbert transform can be applied to get di(t) from each term ci. Thus, it is defined:

$$ \begin{array}{@{}rcl@{}} z_{i} (t)&=&c_{i} (t)+jd_{i} (t)=A_{i} e^{j\theta_{i}} \\ A_{i}&=&\sqrt{{c_{i}^{2}}+{d_{i}^{2}}}, \quad \theta_{i} (t)= \frac{\tan^{-1}d_{i}}{c_{i}},\quad \omega_{i} = \frac{d\theta_{i}}{dt} \end{array} $$
(11)

Replacing ci(t) in (5) with zi(t) in (11) and neglecting rn yield:

$$ u(t) = Real \left( \sum\limits_{i=1}^{n} A_{i} (t) e^{j \theta (t)_{i}} \right) $$
(12)

Where

$$ \theta(t)_{i} = {{\int}_{0}^{t}} \omega_{i} (t)dt = arctan \left( \frac{d_{i}}{c_{i}} \right) $$
(13)

HHT is an adaptive method and intrinsic mode functions are generally physical because the characteristic scales are physical. Since distorted harmonics with time-dependent frequencies and amplitudes are allowed in data decomposition, spurious harmonics are not required to represent non-stationary and nonlinear signals [40, 81]. In addition, EMD essentially acts as an adaptive filter bank. A more detailed and mathematically rigorous formulation can be found in [40, 43].

2.4 Synchrosqueezed Transform

STFT and CWT are the main approaches for simultaneously decomposing a signal into time and frequency components. Known limitations, such as trade-offs between time and frequency resolution, can be overcome by alternative techniques that extract instantaneous modal components. EMD aims to break down a signal into components that are well separated in the time-frequency plane, allowing the reconstruction of these components [6]. On the other hand, a recently proposed method called synchrosqueezing transform (SST) is an extension of the wavelet transform that incorporates empirical mode decomposition elements and frequency reassignment techniques. This new tool produces a well-defined time-frequency representation, allowing the identification of instantaneous frequencies in non-stationary signals to highlight individual components [6, 123].

The SST was initially proposed for the Wavelet transform [23] and then later extended to the STFT [6]. In fact, it corresponds to a nonlinear operator that emphasizes the time-frequency representation of a signal by combining the location and dispersion properties of the reassignment methods with the invertibility property of the linear time-frequency representations [23].

2.4.1 Wavelet-Based Synchrosqueezing Transform

Wavelet-based synchrosqueezing transform, proposed by Daubechies et al. [23], is basically composed of three steps. The first step is to calculate CWT according to Eq. (3). In the second step, a preliminary ωs(a,b) frequency is obtained from the oscillatory behavior of Ws(a,b) in a, such that:

$$ \omega_{s}(a,b) = -i \left( W_{s} (a,b)\right)^{-1} \frac{\partial}{\partial b} W_{s} (a,b) $$
(14)

In the third step, the time scale plane is transformed to the time-frequency plane. Each value of Ws(a,b) is reassigned to (a,ωl), where ωl denotes the frequency that is closest to the preliminary frequency of the original (discrete) point ωs(a,b). This operation is presented in Eq. (13) [74]:

$$ T_{s}(\omega_{l},b) = \left( {\Delta} \omega \right)^{-1} \underset{a_{k}:| \omega(a_{k},b)-\omega_{l}| \leq {\Delta}\omega/2}{\sum} W_{s}(a_{k},b)a_{k}^{-3/2} \left( {\Delta} a\right)_{k} $$
(15)

In Eq. (14), Δω is the width of each frequency bin Δω = ωl − ωl− 1 and equivalently for Δb.

SWT can obtain a high-resolution time-frequency spectrum by compressing (reassigning) the CWT result. However, when the amplitude of the high-frequency components of a signal is low, it is difficult to identify the components in the CWT spectrum or the SST spectrum that is based on the CWT result. In contrast to CWT, the SWT transform is able to more efficiently display the high-frequency and low-amplitude components of a signal and perform a lossless reverse transformation, but the ST resolution is not yet satisfactory [23, 84].

Originally proposed in the case of wavelet, the SST was extended similarly to the STFT context, known as STFT-based SST. This method was first proposed and studied in [100, 122]. A mathematically more rigorous formulation can be found in [6, 79, 123].

3 Case Studies

3.1 Synthetic Test Signals

3.1.1 Case 1: Multicomponent Signal Analysis

In this first application was studied multicomponent signals, that is, wave modulation overlays that appear in many physical systems, especially in mechanical systems. To facilitate replicability and comparison with other methods, a set of two synthetic multicomponent signals is initially considered, with 100,001 samples, sampling frequency (Fs) equal to 1000 Hz, and duration of 10 s. The signals with the addition of + 30 dB of additive white Gaussian noise.

The first signal consists of two linear chirps to show the linear rate of change of frequency as a function of time. The first chirp has f0 = 50 Hz, f1 = 400 Hz. Since the second chirp has f0 = 400 Hz, f1 = 50 Hz, this model is also known as cross chirp signal (Fig. 1a) [11, 97].

The second signal consists of the sum of three linear chirps, a model known as divergent chirp signal (Fig. 1b) [91], each showing the linear rate of change of frequency as a function of time. The first component has f0 = 0 Hz, f1 = 400 Hz. The second component has f0 = 0 Hz, f1 = 200 Hz, while the third component has f0 = 0 Hz and f1 = 100 Hz.

3.1.2 Case 2: 1DOF Syntentic Mechanical System

The resonance phenomenon is commonly found in nature. In electrical and mechanical systems, it occurs when two different energy storage devices have equal (but opposite signals) impedances. During resonance, energy is passed back and forth between the two energy storage devices. The signal shown in Fig. 5a is from a 1DOF mechanical system with resonant frequency wn = 39,789 Hz, being excited by variable sinusoidal (0 − 160 Hz) for 5 s. The signal has Fs = 1000 Hz.

3.2 Experimental Test Signals

The measured signals obtained by the most diverse sensors during dynamic events generally contain nonlinear, non-stationary, and noisy properties. In this work was utilized experimentally obtained acceleration signals from two mechanical systems with 1 and 3 DOF, which have their experimental and data acquisition procedure carefully described in [102, 111].

3.2.1 Case 3: Experimental 1DOF System

The system under study with 1DOF [102] consists of a portico with 400 mm width and 300 mm height. Its sides are made of ASTM A-36 steel and the lower and upper floors are made of polypropylene (Nitapro\({\circledR }\)). The steel plates that make up the sides have dimensions of 76.2 × 300 mm and 1.75 mm thickness, while the polypropylene plates have dimensions 76.2 × 400 mm and 15 mm thickness. The lower floor is considered rigid and fixed to the ground. At the top of the portico, there is an unbalanced DC motor with a mass of 6.59 g and a radius of 15 mm. The Sommerfeld effect occurs due to a nonlinear interaction between the motor speed and the dynamic response of the portico. In this system, the Sommerfeld effect occurs at its second natural frequency, 45 Hz. A detailed description of the experimental apparatus can be found at [102].

Signals are obtained through an accelerometer attached to the side of the portico at its upper end. A Type 4508 Brüel & Kjær accelerometer was used, which can work in a temperature range of − 54 to 121 ∘C and is capable of measuring frequencies from 0.3 to 8 kHz. Utilized were a Brüel & Kjær signal conditioner and the National Instruments USB-6251 signal acquisition system [102]. The signals were acquired with a sampling frequency of 1000 Hz and 32,768 points, totaling an acquisition time of 32,768 s. These signals are transient signals of the motor starting with a small permanent at the end, where the motor speed is varied from 0 to 20 Hz where there is no Sommerfeld effect (Fig. 7a), and 0 to 65 Hz where the Sommerfeld effect is present (Fig. 7b).

3.2.2 Case 4: Experimental 3DOF System

The system under study [102, 111] consists of a three degree of freedom (3 DOF) shearbuilding structure, which has a width of 400 mm and a height of 900 mm. Its sides are made of ASTM A-36 steel and the floors are made of polypropylene (Nitapro\({\circledR }\)). The steel plates that compose the sides have dimensions of 76.2 × 300 mm and 1.75 mm of thickness, while the polypropylene plates have dimensions 76.2 × 400 mm, 15 mm of thickness, and are equally spaced 300 mm apart. The lower floor is considered rigid and fixed to the ground. At the top of the shearbuilding, there is an unbalanced DC motor with a mass of 6.59 g and a radius of 15 mm. The Sommerfeld effect occurs due to a nonlinear interaction between engine speed and shearbuilding dynamic response. In this system, the Sommerfeld effect occurs at 45 Hz [102].

Signals are obtained through an accelerometer attached to the side of the portico at its upper end. A Type 4508 Brüel & Kjær accelerometer was used, which can work in a temperature range of − 54 to 121 ∘C and is capable of measuring frequencies from 0.3 to 8 kHz. Utilized were a Brüel & Kjær signal conditioner and the National Instruments USB-6251 signal acquisition system [102]. The signals were acquired with a sampling frequency of 1000 Hz and 32,768 points, totaling an acquisition time of 32,768 s. These signals are transient signals of the motor starting with a small permanent at the end, where the motor speed is varied from 0 to 20 Hz where there is no Sommerfeld effect (Fig. 11a), and 0 to 65 Hz where the Sommerfeld effect is present (Fig. 11b).

4 Results

In this section are presented the results of the case studies presented in Section 3, presenting the temporal signals, obtained either numerically or experimentally, and their respective time-frequency representations, which are described in Section 2.

4.1 Synthetic Test Signals

4.1.1 Case 1: Multicomponent Signal Analysis

Figure 1 presents the analysis of the chirps described in Section 3 (Fig. 1a and b) using STFT using a 256-sample Hann window (Fig. 1c and d) and through CWT using Bump wavelet (Fig. 1e and f). The representation through STFT was adequate for the analysis of both chirps, and their spectral contents are well represented as shown in Fig. 1c and d.

Fig. 1
figure 1

Time-Frequency Analysis by means of the STFT and CWT: a Cross Chirp signal, b Divergent Chirp signal, c STFT of cross chirp d STFT of divergent chirp e CWT of cross chirp signal and f CWT of divergente chirp signal

Full size image

Both analyses, STFT and CWT, have good time-frequency domain representation and correctly show the linear rate of change in frequency of the two chirps as a function of time. Note in the STFT analysis that Gaussian noise is also well represented in the analysis of both chirps. The codes for STFT Matlab\({\circledR }\) implementations can be found at [10, 11, 28], in Python [26, 52, 54].

Figure 2a and b show the analysis realized by means of the CWT using the Meyer wavelet, while Fig. 2c and d show the analysis realized with the WPT using the DMeyer filter. In this CWT analysis, using the Meyer wavelet, we can observe the good representation of the spectral content and less energy dispersion in the time-frequency representation.

Fig. 2
figure 2

Time-frequency analysis using CWT and WPT: a CWT of cross chirp signal, b CWT of divergent chirp signal, c WPT of cross chirp, and d WPT of divergent chirp

Full size image

In the WPT analysis was utilized the DMeyer filter and the decomposition was done until the 8th level. This level already guarantees an adequate resolution in the time-frequency representation in this specific case. The DMeyer filter is known to be especially efficient for chirping and transient analysis [107], as observed in the results obtained (Fig. 2c and d). CWT and WPT Matlab\({\circledR }\) implementation codes can be found at [10, 87, 90, 95, 98]. Python can be found in [29, 54, 55, 78, 82].

Figure 3 presents the analyses made through HHT, using cubic spline interpolation [40]. The use of HHT allows high resolution in the time and frequency domain, as shown in Fig. 3a and b, but the HHT method is very sensitive to noise. For HHT implementation in Python, the PYHHT library can be used [25].

Fig. 3
figure 3

Time-frequency analysis by means of the HHT: a HHT of the cross chirp signal, b HHT of the divergent chirp signal

Full size image

Figure 4 shows the analysis made through FSST (Fig. 4a and b), and analysis through WSST (Fig. 4c and d). It is apparent in both synchrosqueezed transform-based analyses that the energy dispersion in the time-frequency plane is much smaller than in the other cases, demonstrating excellent resolution in the time-frequency domain. Matlab\({\circledR }\) implementations can be found at [99].

Fig. 4
figure 4

Time-frequency analysis using FSST and SWT: a FSST of cross chirp signal, b FSST of divergent chirp signal, c WSST of cross chirp, and d WSST of divergent chirp

Full size image

4.2 Case 2: 1DOF syntentic Mechanical System

Figure 5 presents the analysis of the 1 DOF mechanical system described in Section 3. The analysis performed through STFT is presented in Fig. 5b and c using CWT with the bump wavelet. In Fig. 5d, the result obtained through the Meyer wavelet is presented. In STFT analysis, a Hann window with 256 sample length was used. In all cases the resonant frequency, fn = 39,789 Hz, is well characterized; however, the CWT analyses present the energy more concentrated in the system resonant frequency on the time-frequency domain, which makes the characterization most effective.

Fig. 5
figure 5

Time-frequency analysis using STFT and CWT: a 1DOF syntentic mechanical system signal, b STFT 1DOF syntentic mechanical system, c CWT using bump wavelet 1DOF syntentic mechanical system, d CWT using Meyer wavelet 1DOF syntentic mechanical

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Figure 6 presents the analysis of the 1DOF mechanical system through the application of WPT, HHT, FSST and WSST. In WPT analysis, Fig. 6a, it is possible to characterize the resonant frequency of the system, and even with a significant energy dispersion in the time-frequency domain due to the excitation signal, its characterization is still quite proper. The HHT analysis clearly characterizes the resonant frequency. Figure 6c and d are performed the analysis via FSST and WSST. Again it is possible to observe in both analyzes using the synchrosqueezed transform techniques that the energy dispersion in the time-frequency plane is much smaller than in the other cases, with excellent resolution in the time and frequency domain. The use of the WSST method proved to be much more appropriate.

Fig. 6
figure 6

Time-frequency analysis 1DOF syntentic mechanical system using WPT, HHT, FSST, and WSST: a WPT using Dmeyer filter 1DOF syntentic mechanical system signal, b HHT 1DOF syntentic mechanical system, c FSST 1DOF syntentic mechanical system, and d WSST using Meyer wavelet 1DOF syntentic mechanical

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4.3 Experimental Test Signals

4.3.1 Case 3: 1DOF Experimental Mechanical System

In this section, the experimental signals presented in Fig. 7 come from a portal frame mechanical system, which has a degree of freedom, briefly described in Section 3 and detailed in [102]. The objective of this analysis is to characterize the resonance phenomenon and the Sommerfeld effect in the time and frequency domains.

Fig. 7
figure 7

Time-frequency analysis using STFT: a 1DOF experimental mechanical system signal without Sommerfeld effect, b 1DOF experimental mechanical system signal with Sommerfeld effect, c STFT 1DOF experimental mechanical system signal without Sommerfeld effect, d STFT 1DOF experimental mechanical system signal with Sommerfeld effect

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Figure 7 also presents the system time-frequency analyses by applying STFT (Fig. 7c and d).

Figure 7c shows the presence of several harmonics in the signal, but the most energetic presence related to the motor acceleration signal is highlighted, in addition to the presence of the system resonant frequency, in a point where there is the highest concentration of energy. Already in Fig. 7d it is possible to observe that besides the presence of the harmonics and the first natural frequency of the system, there is also a greater concentration and dispersion of energy in the time-frequency plane, coming from Sommerfeld effect and jump phenomena. Despite the dispersion of energy in the time-frequency spectrum in both analyses performed with the STFT, it is noted the very energetic presence of the first natural frequency. In Fig. 7c at about 9 s, there is a large energy concentration near the frequency of 11 Hz and in Fig. 7c there is a concentration of energy at the same frequency, now at 4 s, in addition to a high-energy concentration at a frequency of 45 Hz between 15 and 19 s due to the resonance capture phenomenon and energy dispersion before the frequency jumps to 60 Hz at 19 s due to the jump phenomena.

The analysis presented in Fig. 8a and b deals with the application of CWT using the Morlet wavelet to signals. Additionally, it is also analyzed through the use of WPT using Dmey filter in Fig. 8a and b. In the CWT analysis in Fig. 8a, the resonant frequency is clearly identified as in the STFT analysis at 9 s with a value of approximately 11 Hz as well as the entire operating regime of the motor. In Fig. 8b, there is no spectrum dispersion as shown in [102], but the characteristic jump phenomenon of the Sommerfeld Effect is clearly observed, with energy concentration at 45 Hz between 14 and 19 s with a jump to the frequency of 60 Hz at 19 s. Already the analysis made through WPT shows in Fig. 8c the presence of harmonic frequencies in the signal, but the frequencies related to the resonant frequency of the system and the operating regime of the motor are clearly more energetic. It is also noted that the packet related to the first natural frequency is more energetic than the others. Figure 8d shows that the first natural frequency of the system does not have relevant energy in the analysis, while a dispersion of energy over the time-frequency plane is highlighted, with significant concentration of energy in the packet related to the frequency of 45 Hz, which characterizes the Sommerfeld effect and the jump phenomena.

Fig. 8
figure 8

Time-frequency analysis using CWT and WPT: a CWT 1DOF experimental mechanical system signal without Sommerfeld effect, b CWT 1DOF experimental mechanical system signal without Sommerfeld effect, c WPT 1DOF experimental mechanical system signal without Sommerfeld effect, d WPT 1DOF experimental mechanical system signal with Sommerfeld effect

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Figure 9 presents time-frequency analyses using HHT; the results can be viewed in Fig. 9a and b. High resolution in the frequency domain is observed. The resonance phenomenon is well characterized in Fig. 9a at 9 s, as well as the Sommerfeld effect in Fig. 9b at 19 s of analysis.

Fig. 9
figure 9

Time-frequency analysis via HHT: a HHT 1DOF experimental mechanical system signal without Sommerfeld effect, b HHT 1DOF experimental mechanical system signal with Sommerfeld effect

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Finally, Fig. 10 presents the analysis based on the synchrosqueezed transform methods, using FSST and WSST. The results of the 1 DOF Experimental mechanical system analysis without sommerfeld effect signal are presented in Fig. 10a and c. The results of the analysis of the system with Sommerfeld effect are presented in Fig. 10b and d. In both cases, the resonance phenomenon and the Sommerfeld effect are well characterized with good energy concentration in the time-frequency distribution, most notably the presence of the 11 Hz frequency in the analysis without the Sommerfeld effect while the jump phenomenon is well characterized in both methods. Both techniques are quite suitable for the proposed problem type, highlighting that the analysis performed by WSST presents higher energy concentration, which allows a better identification of the frequencies present in the signal.

Fig. 10
figure 10

Time-frequency analysis using synchrosqueezed transform: a FSST 1DOF experimental mechanical system signal without Sommerfeld effect, b FSST 1DOF experimental mechanical system signal without Sommerfeld effect, and c WSST 1DOF Experimental mechanical system signal without Sommerfeld effect and d WSST 1DOF experimental mechanical system signal with Sommerfeld effect

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4.4 Case 4: 3DOF Experimental Mechanical System Signal

In this section, the experimental signals shown in Fig. 11a and b are derived from a shearbuilding mechanical system with a 3-DOF, previously described in Section 3 and detailed in [102, 111].

Fig. 11
figure 11

Time-frequency analysis using STFT: a 3DOF experimental mechanical signal system without Sommerfeld effect, b 3DOF experimental mechanical signal system with Sommerfeld effect, c STFT 3DOF experimental mechanical signal system without Sommerfeld effect, and d STFT 3DOF Experimental mechanical system signal with Sommerfeld effect

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Figure 11c and d present the system time-frequency analysis through the application of STFT. Figure 11c shows a large presence of noise in the analysis, but even so the system signal has a reasonable representation; it is possible to identify the moment of occurrence of the resonance of the third mode of vibration of the system, at 10 s with the value of 20 Hz, which is the most energetic mode of the system. However, due to the large energy dispersion in the time-frequency plane, it is not possible to identify the other resonant frequencies of the system. Figure 11d presents the analysis of the signal with the Sommerfeld effect. Again, the noise representation is clear, as is the system’s response to engine operation. The third resonant frequency of the system stands out, at approximately 6 s with a value of 20 Hz, but the first two are not distinguished. In addition, the Sommerfeld effect is clearly noted due to the energy concentration between 15 and 20 s at the frequency of 45 Hz and the dispersion of power and the sudden increase in the frequency of the signal to 60 Hz at 20 s.

Figure 12 presents the results based on the wavelet transform by applying CWT to the system (Fig. 12a and b), and WPT (Fig. 12c and d). In the case without the Sommerfeld effect, the signal energy is concentrated in the third natural frequency of the system, 20 Hz at 15 s, due to its greater amplitude. In the case of the presence of the Sommerfeld effect, the signal energy is divided with the frequency in which the resonance capture phenomenon occurs, 45 Hz between 15 and 20 s, so both frequencies are highlighted in the analysis. WPT analysis using the DMeyer filter shows a lower concentration of energy in the frequencies mentioned above, distributing the energy throughout the signal, enabling to identify not only the third resonant frequency and the Sommerfeld effect, but also all the variation of the signal frequency over time.

Fig. 12
figure 12

Time-frequency analysis using CWT and WPT: a CWT 3DOF experimental mechanical system signal without Sommerfeld effect, b CWT 3DOF experimental mechanical system signal with Sommerfeld effect, c WPT 3DOF experimental mechanical system signal without Sommerfeld effect, and d WPT 3DOF experimental mechanical system signal with Sommerfeld effect

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Figure 13 presents the time-frequency analysis of the time series of the 3DOF experimental mechanical system through the use of HHT; the results can be viewed in Fig. 13a and b. In the case analysis without the Sommerfeld effect, the system’s third resonant frequency, 20 Hz at 10 s, and the operating range of the motor are characterized. In the case with the Sommerfeld effect, we can also observe the third resonant frequency of the system, 20 Hz at 6 s, and all the phenomena related to the Sommerfeld effect, which occur at a frequency of 45 Hz at 20 s.

Fig. 13
figure 13

Time-frequency analysis via HHT: a 3DOF HHT experimental mechanical system signal without Sommerfeld effect, b 3DOF HHT experimental mechanical system signal with Sommerfeld effect

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Figure 14 presents the analysis based on the synchrosqueezed transform methods through the use of FSST and WSST. The results of the 3DOF experimental mechanical system analysis without Sommerfeld effect signal are presented in Fig. 14a and b. The results of the analysis of the system with Sommerfeld effect are presented in Fig. 14c and d. Again, there is less energy dispersion by the time-frequency plane, and it is possible to identify the second and third resonant frequencies of the system in the case without the Sommerfeld effect, respectively 13 Hz at 10 s and 20 HZ at 15 s, occurring subtly in the analysis with FSST and more clearly in the case of WSST. Even in the case that the Sommerfeld effect is present, it is possible to identify the second and third resonant frequencies of the system, respectively, 13 Hz at 4 s and 20 Hz at 6 s, and the effects due to the nonlinearities of the system, which occur between 16 and 20 s at the frequency of 45 Hz and 60 Hz, which demonstrates the low-energy dispersion capacity of the SST, which allows the representation of less energetic frequencies with excellent resolution.

Fig. 14
figure 14

Time-Frequency Analysis Using synchrosqueezed transform: a FSST 3DOF Experimental mechanical system signal without Sommerfeld effect, b FSST 3DOF Experimental mechanical system signal with Sommerfeld effect, c WSST 3DOF Experimental mechanical system signal without Sommerfeld effect and d WSST 3DOF Experimental mechanical system signal with Sommerfeld effect

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5 Conclusions

In this paper was investigated several current techniques for non-stationary signal processing, particularly in the study of mechanical vibrations, evaluating their efficiency in separating frequency components, such as in the evaluation of case studies involving chirps, and in identifying frequencies with different energy levels. The STFT is applicable to the studied signals, but presents results with low resolution in the time-frequency plane and it has large energy dispersion, which makes it difficult to identify lower energy frequencies. It may also have resolution issues because of the fixed size of the window.

Wavelet transform-based techniques, CWT and WPT, present good resolution in the time-frequency plane, but CWT still obfuscates other components when there are high-energy concentrations at some signal frequency. WPT, with the use of Dmey filter and decomposition to the 8th level, presented better results in these cases, being possible to distinguish several signal components. Applications of wavelet transform-based techniques to non-stationary signals have demonstrated the ability of this technique to monitor signal frequency variations and to detect short-term transients with excellent time-frequency localization, far exceeding the limitations presented by the techniques based on the Fourier transform.

The application of HHT to the studied signals was also satisfactory, but in the steady state of the motor in the experimental results its resolution was not so great, being not possible to have a value for the signal frequency in this period. In this application, the method was very sensitive to noise.

Finally, the results obtained through the SSTs were excellent, presenting an optimal time-frequency resolution and minimum energy dispersion. Although both SST techniques have shown good results, it is observed that those obtained through WSST have better resolution than FSST, being possible to identify the signal frequency at any time. Thus, the results show that SSTs present better results in the proposed application, with special emphasis on WSST, which presents better resolution in the time-frequency domain.

The results showed the benefits through this approach for structural dynamics analysis and non-stationary operation problems, where the use of time-frequency analysis techniques was quite adequate and easy to apply. The results presented in this paper indicate the great potential of these tools for the characterization of nonlinearities in mechanical systems, especially the use of WPT and WSST.