^{1}

^{1}

^{2}

^{2}

^{3}

^{4}

^{1}

^{2}

^{3}

^{4}

Instantaneous rotational speed measurement of the engine is crucial in routine inspection and maintenance of an automobile engine. Since the contact measurement of rotational speed is not always available, the vibration measurement has been used for noncontact rotational speed estimation methods. Unfortunately, the accuracy of the noncontact estimation methods by analyzing engine vibration frequency is not satisfactory due to the influence of noise and the strong nonstationary characteristic of the vibration signal. To overcome these problems, based on the multiple matching synchrosqueezing transform (MSST) (MMSST, improved MSST with multiple squeeze operations), a novel noncontact method is proposed to accurately estimate the instantaneous rotational speed of automobile engine in this paper. Firstly, a MMSST is proposed to process the vibration signal to obtain a concentrated time-frequency (TF) representation. Secondly, the instantaneous frequency (IF) detection algorithm is employed to extract the fundamental frequency from the TF result. Finally, the rotational speed of the engine is calculated according to the relationship between the fundamental frequency and rotational speed. Results from numerical simulations and test on real engine have proven that the proposed method can obtain much higher frequency resolution and more precise IF estimation of the engine vibration signal and more accurate rotational speed estimation result compared with the MSST method. Furthermore, the proposed method is verified to have a stronger noise robustness and can provide satisfactory estimation results for engine vibration signal containing nonlinear frequency-modulated components.

The engine is an essential part of an automobile, and its quality can comprehensively reflect the performance of the automobile’s dynamic. The rotational speed of the engine is a key parameter in the automobile safety inspection and emission test. For example, when the automobile is idling, the unstable engine speed indicates that the engine needs to be repaired to reduce exhaust emissions. Therefore, it is important to measure the engine speed stably and accurately. At present, the measurement methods of the engine rotational speed can be divided into two types (i.e., the contact-type and noncontact-type). The contact-type measurement methods need to install a toothed or notched encoder disc which rotates synchronously, at the end of the crankshaft of the automobile [

In general, a normally functioning engine generates a regular vibration which is closely related to the rotational speed [

Recently, time-frequency (TF) analysis (TFA) methods have attracted considerable attention in instantaneous frequency (IF) estimation for instantaneous rotational speed estimation [

In contrast, since SST reassigns the TF coefficients in the frequency direction, it retains the ability to reconstruct signals from the TF result. It has been proved that the SST result is equivalent to the ideal TF representation when addressing a purely harmonic signal [

In this paper, a multiple matching synchrosqueezing transform (MMSST) is proposed to obtain a more concentrated TF result and improve the IF estimation accuracy of nonlinear strong frequency-modulated components. By introducing multiple squeeze operations in MSST, MMSST iterates on the IF estimator of MSST to obtain a more accurate IF estimator which is close to the signal true IF and generates the concentrated TF result by reassigning the TF coefficients in the frequency direction. In order to achieve noncontact rotational speed measurement of engine, an accurate noncontact engine rotational speed estimation method that consists of a proposed MMSST and the IF detection algorithm [

When an engine is running, its power is transmitted and transformed through the crank and connecting rod mechanism. The excitation loads acting on the crankshaft of an internal combustion engine are mainly derived from the driving torque due to gas forces generated within the cylinders. In a single cylinder, the driving torque produced by the gas pressure can be expressed as [

For a four-stroke engine with

For an automobile engine with stroke number

As seen from Equation (

The STFT of

Restricted by the Heisenberg uncertainty principle, the TF result is blurry around the IF position, which means the TF resolution along with time and frequency directions cannot become arbitrarily small at the same time. The SST method is proposed to serve as a postprocessing method of STFT; it squeezes these diffused TF coefficients into the correct IF position and its formula is given as follows:

The unbiased IF estimator of SST in Equation (

In the MSST method, the group delay (GD) estimator

Because the MIF estimator incorporates the information of the original IF estimator in (

In MSST, the MIF estimator is constructed based on the IF structure of chirp signal, which solves the defect of the IF estimator in SST processing time-varying signal, and improves the resolution of the TF result. For chirp signal or signals that can be locally approximated as chirp signal, MSST can get a more concentrated TF result than SST. However, for nonlinear frequency-modulated signals that do not meet the above requirements, i.e., signals that have strongly time-varying IF, with high order, MSST will still obtain a blurry TF result like SST. The engine vibration signal in real world usually owns a strongly time-varying IF and cannot be locally approximated as chirp signal while the automobile is speeding up or slowing down. Therefore, a novel TFA method with good performance in processing strongly time-varying signal is needed to analyze engine vibration signal for rotational speed estimation.

MMSST can be viewed as a right combination of MSST and multiple squeezing technique. Although MSST cannot get an ideal TF result for strong frequency-modulated signals that have time-varying IF with high order, it can obtain an unbiased MIF estimator for chirp signals. Based on the result in (

It can be seen that the MMSST (

According to the above calculation, the proposed MMSST is easy to implement by executing multiple squeeze operations. The entire procedure of MMSST is summarized in Algorithm

1: Choose the window function g and iteration number

2: Calculate MSST

3: Let

4: if

5: for

6:

7:

8: end for

9: end if

10: Let

11:

12:

13: Output

The multiple squeeze operations can entirely reduce MIF estimation error in analyzing nonlinear strong frequency-modulated signal and improve the resolution of TF representation. Since MMSST reassigns the coefficients only in the frequency direction and no information missing, the signal reconstruction is still achievable by summing the synchrosqueezing coefficients in the frequency domain as follows:

An effective detection algorithm is described in [

Combining the proposed MMSST and the IF detection algorithm, the proposed MMSST-based method for rotational speed estimation of an automobile engine in this paper consists of the following steps:

Data acquisition and preprocessing: collect the vibration signals of the engine cylinder through the accelerator sensor, and a low-pass filter is applied to restrain the high-frequency interference

TF representation calculation: use the proposed MMSST to obtain the TF representation of the vibration signal, and the implementation of the MMSST is described in detail in Algorithm

Rotational speed estimation: utilize the IF detection algorithm to extract the fundamental frequency of the vibration signal from the TF representation obtained by step (2), and then, the rotational speed of engine is calculated through Equation (

In this section, we focus on the comparisons between the proposed TFA method and other TFA methods in addressing complex signals, for instance, noisy signal and strongly time-varying signal. The comparisons mainly focus on the TF energy concentration and the performance of the IF estimation. According to the characteristics of speed variating engine, the simulation signal is modeled as

The TF representations generated by SST, MSST, and MMSST are shown in Figures

TF result of signal (

To evaluate the energy concentration of different TFA methods quantitatively, in this paper, the Renyi entropy [

The Renyi entropy of TF result by three TFA methods (

TFA | SST | MSST | MMSST |
---|---|---|---|

Renyi entropy | 10.9527 | 9.877 | 8.7512 |

The Renyi entropy of TF result under different SNR.

The IF of the signal is an important feature. It includes essential information about the analyzed object, for example, the rotational speed of an engine. Therefore, we have tested the IF estimation performance of the MMSST method under different noise levels. The signal (

Figure

The MRE of the detected IF under different SNR.

Since MMSST introduces multiple squeeze operations on MSST to improve the frequency resolution of TF representation, the running time is bound to increase with the increase of iteration times. The efficiency of TFA method is essential in a real-time application. Therefore, we must take both a frequency resolution and a time of running into account to find a suitable iteration number for MMSST.

The simulated signal (

The Renyi entropy of TFR and the running time concerning the number of iterations (

In this section, to evaluate the accuracy of MMSST-based method in engine rotational speed estimation, a test experiment is carried out in an automobile inspection station, and the tested automobile is randomly selected The tested engine is a four-cylinder four-stroke diesel engine. The acceleration sensor is assembled in the form of a magnetic suction probe, and it is attached on the engine hood of the tested automobile, and the sample rate is set to 512 Hz.

The TF representations of the engine vibration signal generated by SST, MSST, and MMSST are shown in Figures

The TF results of engine vibration signal and the test results of rotational speed estimation for the applied methods: (a) SST and (b) SST-based estimation result, (c) MSST and (d) MSST-based estimation result, and (e) MMSST and (f) MMSST-based estimation result.

The relative error of the rotational speed estimation method: (a) SST-based estimation result, (b) MSST-based estimation result, and (c) MMSST-based estimation result.

The Renyi entropy and MRE of three TFA-based methods.

Methods | SST-based | MSST-based | MMSST-based |
---|---|---|---|

Renyi entropy | 17.9104 | 18.0259 | 17.6883 |

MRE | 0.0071 | 0.0077 | 0.0051 |

Based on the TF result, we can obtain the fundamental frequency (related to the engine rotational speed) by using the IF detection algorithm and calculate the rotational speed according to the relationship between speed and frequency. The results of rotational speed estimation by the proposed method, SST-based and MSST-based, are displayed in Figures

In the end, the relative error of the rotational speed estimation results obtained by SST-based, MSST-based, and the proposed method are presented in Figure

In this paper, an accurate method for estimating the automobile engine rotational speed based on vibration signals has been proposed. The proposed method consists of a proposed MMSST and the IF detection algorithm. The proposed MMSST is an improvement of the MSST, which overcomes the shortcoming that MSST cannot provide an accurate IF estimator for nonlinear strong frequency-modulated signals by introducing multiple squeeze operations in MSST. Therefore, MMSST could obtain a TF representation with higher frequency resolution for automobile engine signals. The IF detection algorithm is used to extract the IF from the TF representation. As compared to the commonly used contact methods for measuring the engine rotational speed, the proposed method does not require any installation of encoder disks and only needs to use the proposed algorithm to analyze the engine vibration signal.

Results from numerical simulations and experiment on real engine have proven that the proposed method could provide accurate IF estimation results and more concentrated TF result. The numerical simulation also shows that the proposed MMSST can obtain a more robust TF result even in a high noise environment compared with MSST and SST. In addition, the results of experiment on real engine show that the proposed method has the lowest MRE value at 0.0051, compared with 0.0077 in MSST-based method and 0.0071 in SST-based method. Meanwhile, the proposed method can accurately estimate the speed when the engine is working in a steady state or under conditions of large fluctuations in speed, which verifies the proposed MMSST can characterize correctly the strongly time-varying feature of considered signal. The proposed method does not require complex sensor equipment and can conveniently get the accurate rotational speed estimation results; it is suitable for routine inspection and maintenance of an automobile engine.

The data of this study can be obtained from the authors of this article (e-mail:

The authors declare there are no conflicts of interest.

This work was partially supported by the National Natural Science Foundation of China under Grant 61771190 and Natural Science Foundation of Hunan Province under Grant 2019JJ20001.