Vibration Control and Actuation of Large-Scale Systems

Vibration Control and Actuation of Large-Scale Systems gives a systematically and self-contained description of the many facets of envisaging, designing, implementing, or experimentally exploring advanced vibration control systems. The book is devoted to the development of mathematical methodologies for vibration analysis and control problems of large-scale systems, including structural dynamics, vehicle dynamics and wind turbines, for example. The research problems addressed in each chapter are well motivated, with numerical and simulation results given in each chapter that reflect best engineering practice.

• Provides a series of the latest results in vibration control, structural control, actuation, component failures, and more
• Gives numerical and simulation results to reflect best engineering practice
• Presents recent advances of theory, technological aspects, and applications of advanced control methodologies in vibration control

• Language: English
• Publisher: Academic Press
• Release date: May 20, 2020
• ISBN: 9780128211984

Book preview

Australia

Preface

Hamid Reza Karimi, Milan

Vibration is a phenomenon that affects systems such as robot manipulators, bridges, buildings, towers, vehicles, and aircrafts. The protection of these large-scale systems against the harmful effects of vibration has become a major field of research in recent years. In the literature of vibration control of mechanical, electrical, or hydraulic systems, different damping systems, mainly passive, active, and semiactive damping systems, have been proposed and successfully applied to tackle the vibration problem. One critical characteristic common to most of these actuators is that they, in general, exhibit nonlinear dynamical behaviour and thus complex control techniques must be employed for an appropriate performance. To this aim, this book unifies existing and emerging concepts concerning advanced vibration control methodologies and actuations of large-scale systems toward practical applications, such as vehicles, buildings, wind turbines. The book may be useful for researchers in control systems, mechatronics, mathematics, mechanics, and alike.

The book consists of 13 chapters, which are organized as separate contributions and listed according to the order of the list of contents as follows:

In Chapter 1, Biernat’s contribution dealt with frequency-frequency analysis (F-F analysis) of the vibration spectrum. The vibration spectrum was transformed into an image that revealed the occurring frequency components as well as their harmonics and side bands. F-F images showed a unique approach to track changes occurring in the vibration process during the operation of electromechanical systems. In Chapter 2, Ji studied the implicit resonances in the forced response of nonautonomous time-delayed nonlinear systems in the vicinity of Hopf bifurcations. In Chapter 3, Abeykoon et al. analyzed a force (torque) sensorless vibration suppression methodology for externally applied vibrations on actuators. In Chapter 4, Tliba et al. investigated the problem of vibration damping for a thin beam, with an Euler-Bernoulli configuration, using output feedback controller based on delayed proportional actions. In Chapter 5, Siami et al. proposed some methods for vibration protection of statues and cultural heritage objects against earthquakes and ambient vibrations. In Chapter 6, Palacios-Quiñonero et al. proposed an iterative linear matrix inequality computational procedure for designing high-performance static output feedback controllers for vibration control of multistory buildings equipped with a distributed set of interstory actuators. In Chapter 7, Pan et al. proposed the finite-time state and output feedback control issues for vehicle active suspension systems subjected to unknown external disturbance. In Chapter 8, Huang et al. proposed robust adaptive parameter estimation and control method for half-car active suspension systems in the presence of uncertainties and nonlinearities. In Chapter 9, Ren et al. proposed an observer-based finite-time robust H∞ vibration control problem for half-car active suspension systems with actuator parametric uncertainties and nonlinearities. In Chapter 10, Tu et al. presented a negative stiffness magnetic spring to install on a seat suspension system to achieve high-static-low-dynamic stiffness characteristic and such a system can also be semiactively controlled to reduce vibration responses via an electromagnetic damper. In Chapter 11, Xiong and Chang addressed the robust fault-tolerant H∞ control problem of semiactive suspension systems with quantization. In Chapter 12, Zhao et al. studied the effects of bending moments on the dynamic response of a planetary three-stage gearbox used in an indirect-drive wind turbine under different load conditions. Finally, in Chapter 13, Damaziak et al. presented a design approach for a structure of a small wind turbine with respect to control its response in the frequency domain.

Finally, I would like to express appreciation to all contributors for their excellent contributions to this book.

Chapter 1

Analysis of vibration signals

Adam Biernat    Electrical Department, Institute of Control and Industrial Electronics (ISEP), Warsaw University of Technology, Warsaw, Poland

Abstract

Unbalanced forces of mechanical and electromagnetic origin are the cause of time-varying displacements of elements of a given electromechanical system. Displacements arise as a result of the propagation of disturbances by a number of paths through elements of the mechanical system characterized by specific resonance properties, nonlinearity caused by clearance, discontinuity and anisotropy of the medium, and modulating properties related to the cyclicity of changes in the mutual position of system elements. Identifying primary unbalanced forces based on the recorded vibration signal requires an analysis of the vibration spectrum over the widest possible frequency range. The proposed solution is frequency-frequency analysis (F-F analysis) of the vibration spectrum. The vibration spectrum is transformed into an image that reveals the occurring frequency components as well as their harmonics and side bands. Image analysis makes possible to identify all the frequency components of the spectrum, their interrelationships, and also reveals a possible failure to meet the condition of process parameters stability—amplitude and frequency of the primary disturbance. F-F images showing a unique individuality allow to track changes occurring in the vibration process during the operation of electromechanical systems.

Keywords

Electromechanical systems; Image analysis; Spectral analysis; Vibration analysis

Outline

1Introduction

2Vibration process

3Frequency-frequency analysis

4Analysis of vibration process based on F-F image

4.1Analysis of vibrations caused by a unbalanced forces of kinematic origin

4.2Analysis of vibrations caused by unbalanced forces of electromagnetic origin

5Conclusions

References

1 Introduction

Unbalanced forces occurring in electromechanical systems are the cause of time-varying displacements (vibrations) of elements of a given mechanical system (structure). Displacements arise as a result of the propagation of disturbation caused by the occurrence of an unbalanced force by a number of paths through elements of the mechanical system characterized by specific resonance properties, nonlinearity caused by the clearance, discontinuity and anisotropy of the medium, and modulating properties associated with the cyclicity of changes in the mutual position of system elements. The amplitude and shape of the time course of displacements (determined by the content of frequency components of the displacement spectrum) depends on the place of measurement (due to availability, displacement of external surfaces is usually measured). In consequence, as a result of adding up the propagating distortions, mechanical vibrations occurring at a given point of the surface are determined by the mechanical properties of the given structure and the place of origin of unbalanced forces.

2 Vibration process

The general (presented in a modified form) relationship describing the vibrations occurring at a given point of the surface, approximating the complicated process of disturbance propagation, must take into account both the path and direction of propagation, resonance properties, modulating properties, and the influence of external and random factors [1]:

   (1.1)

where xξ(t,p) is the vibration signal in position p, xs(t) is the signal representing the primary disturbance, xd(t) is the function dependent on the path and direction of propagation of the disturbation, xwm(t) is the function describing all factors modulating the disturbance signal, xwn(t) is the function describing all nonmodulating external factors, h(t) is the function describing the resonance properties of system components, n(t) is the function representing random disturbances.

Primary disturbances may in general be purely mechanical distortions, distortions caused by unbalanced forces in the medium where the electromagnetic field is propagated and electromagnetic torque ripples appearing when the electromechanical energy converter is coupled with another mechanical or electromechanical device. The relationship reveals the complicated vibration structure that is specific to a given point on the surface of the electromechanical system under consideration. It should be emphasized that modulating factors as well as resonance properties of system components have a significant impact on the frequency range of the spectrum of generated vibrations. Thus, identifying primary unbalanced forces based on the recorded vibration signal requires an analysis of the vibration spectrum over the widest possible range of frequencies. Therefore, it is important to ensure a wide frequency range of the signal while maintaining proper frequency resolution. In the case of recording a discrete signal, this requires a high sampling frequency and a sufficiently long measurement time. When the condition of the stability of basic process parameters is not met (this problem concerns most technical implementations), there are problems with identification (especially in the range of higher frequencies) of the frequency components of the vibration spectrum. The proposed solution is frequency-frequency analysis (F-F analysis) of the vibration spectrum. The vibration spectrum is transformed into an image that reveals the occurring frequency components as well as their harmonics and side bands. Image analysis makes possible to identify all the frequency components of the spectrum, their interrelationships, and also reveals a possible failure to meet the condition of process parameters stability—amplitude and frequency of the primary disturbance.

3 Frequency-frequency analysis

The idea of creating an F-F image [2] is presented in Fig. 1.1. The real part of the discrete frequency spectrum (DFT) of the vibration signal after removing the element representing the constant component is treated as a vector S = [sk], k = 1, 2, … ks, ks = 1/2(T/Δt) is a vector S length, T is a measurement duration, Δt is sampling period of the vibration signal, where sk represents the amplitude of the spectral line at a frequency kΔf, Δf = 1/T is the spectrum frequency resolution. The S vector is transformed into a P table:

   (1.2)

where n = 1, 2, … np, np—row length, m = 1, 2, … mp, mp—column length, in such a way that consecutive fragments of the vector S with the length of np create the next rows of the table P, wherein ks ≥ npmp. The quantity fP = npΔf is to be called the characteristic frequency of the table P. In the F-F image created on the basis of the table P, the amplitude of individual frequency components is represented by the degree of color intensity similar to the case of single-color cartographic maps. The F-F image is organized so that the p1,1 element is in the lower left corner.

Fig. 1.1 The idea of creating an F-F image.

Next, the properties of the P table are discussed.

Property I

The frequency of the spectral line associated with the element of the table P is unambiguously described by the indices n and m:

   (1.3)

so successive elements in the row are distant by a frequency Δf, while successive elements of the column are distant by a characteristic frequency fP.

Property II

Operation (1.2) can be treated as shifts of the m-th fragment of the spectrum in the frequency domain by the frequency fP. This means that in the case of existence of higher harmonics caused by the primary disturbance xs(t), successive parts of the spectrum show a high degree of similarity.

Let's introduce the criterion for choosing the characteristic frequency fP. In this purpose, the concept of F-F image ordering is defined:

The F-F image or its fragment is ordered when the given fundamental frequency component and its higher harmonics form successive elements of the table P column.

Fig. 1.2

Fig. 1.2 Ordering criterion.

illustrates the adopted ordering criterion. Marked elements of the table P (image points) represent the given frequency component fx and its second harmonic 2fx. Image fragment ordering is obtained when fP = fx (left side of the drawing). A fragment of the image that does not meet the criteria of ordering fP ≠ fx is presented on the right. A given vibration process creates an individual pattern in the F-F image. Next, the patterns appearing in the F-F image that represent the basic vibration processes are discussed in turn.

Pattern 1

Single primary disturbance. Spectrum of the vibration process contains the basic and higher harmonics components with the kfa (k = 1, 2, … kh) frequencies. The characteristic frequency fP = fa. The frequency components form successive elements of the column pa,k (indexed by the parameter a) of the table P (Fig. 1.3A).

Fig. 1.3 Pattern 1. The F-F image elements with the kfa frequencies are marked with color. Characteristic frequency fP = fa (A). Characteristic frequency fP = 2fa (B).

In many technical issues, it is convenient to observe the occurrence of even and odd harmonics separately. Separating them in this case requires only adopting the characteristic frequency fP = 2fa (Fig. 1.3B). The pattern consists of two columns, left containing odd harmonics, right containing even harmonics of the vibration process. If both even and odd harmonics are present, the right and left side of the F-F image is characterized by a high degree of similarity.

Pattern 2

Single primary disturbance. The spectrum of the vibration process contains the basic and two-sided bands components with the fb ± kfa, (k = 1, 2, … kh) frequencies resulting from the nonlinear modulation process. The characteristic frequency fP = fa. The frequency components form successive elements of the column pb,k (indexed by the parameter b) of the table P (Fig. 1.4).

Fig. 1.4 Pattern 2. The F-F image elements with the fb ± kfa frequencies components are marked with color. E (fb/fP)—natural part of the ratio fa/fP.

Pattern 3

Single primary disturbance. The spectrum of the vibration process contains the basic and higher harmonics components with the kfa, (k = 1, 2, … kh) frequencies and their two-sided bands with the kfa ± lfb, (l = 1, 2, … lh) frequencies resulting from the nonlinear modulation process. The characteristic frequency fP = fa.. The frequency components form successive elements of the columns pa,k, p(a ± lb),l (indexed by the parameters a and a ± lb) of the table P (Fig. 1.5A). In some cases only side band components are present.

Fig. 1.5 Pattern 3. The F-F image (A) and the modified F-F image (B) elements with the kfa and the kfa ± lfb frequencies components are marked with color.

Often it is convenient to arrange the F-F image so that the characteristic frequency and its higher harmonics are in the image center (Fig. 1.5B). This makes easier to capture the symmetries of image patterns associated with existing side bands.

Note that the relationship (1.3) between the spectral lines frequency and the indices n and m of the table P cells should be modified accordingly.

Pattern 4

Single primary disturbance. As in the case of Pattern 1, the spectrum of the vibration process contains the basic and higher harmonics components with the frequencies lfb (l = 1, 2, … lh). The characteristic frequency fP > fb. The frequency components form elements pn,m of the table P. Starting from pb,1 the diagonal pattern is created (Fig. 1.6).

Fig. 1.6 Pattern 4. The F-F image elements with the lfb frequencies components are marked with color.

The diagonal pattern deviates from the vertical axis by the angle γ depending on the difference in real frequencies f*P and f*b and the scale of the image determined by the ratio md to nd:

   (1.4)

The deviation angle γ of the pattern from the vertical axis can be treated as a measure of the ordering of a given vibration process. The smaller it is, the greater the orderliness is. When γ = 0 there is order due to the characteristic frequency fP and in this case fP = fb.

Pattern 5

Low-level wideband stochastic disturbances. Mechanical construction has a limited frequency range resonance properties. In the spectrum of the noise-like vibration process, no dominant frequency component can be clearly distinguished. Noise-like frequency components form a horizontal pattern with a width determined by the frequency range of resonance vibrations (Fig. 1.7A).

Fig. 1.7 Pattern 5 (A). The F-F image elements with the noise-like frequencies components are marked with color. Frequency fa is optional. Pattern 6 (B). The F-F image elements with the noise-like frequencies and the kfa frequencies components are marked with color.

Pattern 6

Low-level wideband stochastic disturbances and single primary disturbance. The mechanical construction has resonant properties. The spectrum of the deterministic vibration process contains basic and higher harmonics with the kfa (k = 1, 2, … kh) frequencies components, whose amplitude increases significantly within the resonance range (Fig. 1.7B). The characteristic frequency fP = fa. The deterministic frequency components form successive elements of the column pa,k (indexed by the parameters a) of the table P, stochastic frequency components form horizontal pattern (combination of Patterns 5 and 1).

At low levels of deterministic disturbances, the frequency components of the vibrations process are present only close to the resonance range. (Pattern 6a—Fig. 1.8A

Fig. 1.8 Pattern 6a (A). The F-F image elements with the noise-like frequencies and the limited band kfa frequencies components are marked with color. Pattern 6b (B). The F-F image elements with noise-like frequencies and limited band kfa ± fb frequencies components are marked with color.

). In some cases of the modulation process, only side bands frequency components appear in the resonance range (Pattern 6b—Fig. 1.8B).

Property III

The distances between the vertical patterns of the F-F image formed by a given vibration process expressed in frequency units correspond to the fundamental frequencies of the vibration process. The occurrence of higher harmonics in a given vibration process is marked by the length (height) of the pattern that represents them.

Property III implies adopted convention of F-F image axis units: unit of the horizontal axis is frequency, unit of the vertical axis is number indicating the fragment of the S vector.

Property IV

When, for a given np, an ordered image or fragment of an image (pattern) is obtained, one can assume that the frequency fP = Δfnp is the basic frequency of the relevant vibration process.

Property V

The side bands created as a result of modulation process introduce symmetry of image fragments relative to the vertical axis.

Property VI

In case of several simultaneously occurring primary disturbances x1(t), x2(t) … with different fundamental frequencies (first harmonics), each of them will create its own pattern, with the F-F image being the sum of all patterns.

Property VII

The pattern meeting the adopted ordering criterion is obtained only for the disturbance with the frequency fx = fP. Disturbance with other frequencies will create their own patterns that do not meet the order criterion. This property allows to find subtle changes occurring in the vibration process.

Property VIII

There is a possibility of occurrence of patterns suggesting a false frequency of the vibration process, associated with the occurrence of aliases in the same way as in the case of sampling an analogous signal that does not meet the Nyquist criterion. This applies to patterns created by processes with a basic frequency that is not a characteristic frequency.

The intensity of the vibration process is demonstrated by both its frequency range and vibration amplitude. Based on the F-F image, one can create a measure that shows sensitivity to both the amplitude of the vibrations and their frequency range—the average value of the amplitudes of the frequency components and their higher harmonics:

   (1.5)

where Γ is possible operation of changing the amplitude scale.

Measure (1.5) can be organized in the form of a vector containing the average values of columns of the table P (distribution of average vibration amplitude):

   (1.6)

An important feature of the proposed measure thus created is the preservation of information about significant modulation processes.

4 Analysis of vibration process based on F-F image

Next, F-F images of selected vibration processes and their individual features will be discussed.

4.1 Analysis of vibrations caused by a unbalanced forces of kinematic origin

The reason for the occurrence of unbalanced periodic forces of kinematic origin is the limited accuracy with which the elements of the rotating mechanical system are made, as well as their wear and deformation. Among them the unbalance and faulty coupling of rotating elements, the clearances between mutually moving surfaces (an interesting aspect of the occurrence of clearances is the appearance of certain vibration features indicating their chaotic nature), and the wear of rolling and sliding elements of bearings. A number of causes (bearing load, shocks, high operating temperature, defective lubrication, bearing currents) leading to flaking, wiping, and increase in roughness and even local damage are responsible for the wear of the rolling element surface and raceway. In the process of bearing wear, the increase of unevenness of interacting surfaces is responsible for the occurrence of a number of excitations of varying amplitude and frequency. In the initial period when the degree of wear is low, the forces are stochastic in nature with a relatively small amplitude. With increasing wear, local surface defects increase until local damage occurs. Expected basic vibration frequencies caused by damage to individual bearing components—fs (Ball Spin Frequency), fi (Ball Pass Frequency of Inner ring), fo (Ball Pass Frequency of Outer ring), and ft (Fundamental Train Frequency) are determined by bearing geometry and relative speed of both raceways. These frequencies are not always easy to identify due to the complicated signal modulation process. Relations between