Vibratory phenomena have always surrounded human life. The need for more knowledge and domain of such phenomena increases more and more, especially in the modern society where the human-machine integration becomes closer day after day. In that context, this work deals with the development and practical implementation of a hybrid (passive-active/adaptive) vibration control system over a metallic beam excited by a broadband signal and under variable temperature, between 5 and 35°C. Since temperature variations affect directly and considerably the performance of the passive control system, composed of a viscoelastic dynamic vibration neutralizer (also called a viscoelastic dynamic vibration absorber), the associative strategy of using an active-adaptive vibration control system (based on a feedforward approach with the use of the FXLMS algorithm) working together with the passive one has shown to be a good option to compensate the neutralizer loss of performance and generally maintain the extended overall level of vibration control. As an additional gain, the association of both vibration control systems (passive and active-adaptive) has improved the attenuation of vibration levels. Some key steps matured over years of research on this experimental setup are presented in this paper.
In passive vibration control, the successful use of dynamic vibration neutralizers (also called dynamic vibration absorbers) is long recognized [
The interest in hybrid (passive + active/semiactive) vibration control systems has increased fast in the last decades [
A passive vibration control system (PVCS) usually contributes robustness, while an active vibration control system (AVCS) contributes flexibility. Thus, the association in parallel (i.e., in the same frequency range) of these two systems permits the design of a very high performance hybrid vibration control system (HVCS), with robustness, flexibility, and reliability.
In general, passive vibration control techniques are based on fixed modifications of stiffness, mass, and/or damping characteristics of the mechanical system of concern [
On the other hand, fully active vibration control techniques are based on strategies of destructive interference [
In a feedback control approach, the controller works with information on the vibration of the mechanical system whereas, in feedforward control strategies, information on the disturbance is of primary interest. In the latter, the capability of adapting the controller transfer functions, according to some predefined criteria, is usually present, allowing any modifications on the excitation and/or in the mechanical system to be accounted for [
In this work, a hybrid vibration control system (HVCS) is introduced and discussed, designed to attenuate vibrations in a metallic beam under broadband excitation and variable temperature. It combines a viscoelastic dynamic vibration neutralizer (VDVN), of passive nature, with components of fully active vibration control in an adaptive feedforward control approach (presented in [
It is shown that this association in parallel of a passive vibration control system (PVCS) with an active vibration control system (AVCS) handles successfully the matter of detuning, given the adaptive nature of the AVCS. Besides, it also extends and maintains, in most cases, the overall level of vibration attenuation, over a broad frequency range. To the authors’ knowledge, no such system has ever been clearly presented as done herein.
In the current application, the passive vibration control system (PVCS) consists of a viscoelastic dynamic vibration neutralizer (VDVN). As seen in Figure
Schema of a (a) conventional (b) viscoelastic dynamic vibration neutralizer applied to a primary system.
It should be stressed that the use of the above DVN transforms the primary mechanical system (plant) into a two-degree-of-freedom system, the natural frequencies of which are different from those of the primary and the secondary systems alone, according to designer’s interests. The damping of the DVN is related to its frequency band of action. The design parameters must be well chosen in order to attain a good control system.
In the case of a viscoelastic neutralizer (VDVN), its characteristics of stiffness and damping are determined by the viscoelastic material employed in the device [
The stiffness
Dynamic shear modulus and loss factor as functions of (a) frequency and (b) temperature.
The single degree-of-freedom VDVN is characterized by its antiresonant frequency,
The VDVN can be designed to work, in an optimal way, in the so-called “transition band,” where the material loss factor is maximum (see Figure
The above optimization process, carried out by nonlinear techniques, corresponds to the reduction of the frequency response function amplitudes of the primary system in the chosen range. It is highlighted that the VDVN is modeled in the optimization process by the concept of generalized equivalent parameters. By that concept, the compound system (primary + secondary) is described only in terms of the generalized coordinates of the primary system. The design process is fully described in De Espíndola et al. [
The PVCS (VDVN), as part of the HVCS investigated in this work, is illustrated in Figure
Schema of the employed hybrid vibration control system.
An active vibration control system (AVCS) can be implemented in different ways, regarding the architecture, control algorithm, types of sensors and actuators, available signals, and operational conditions. As previously mentioned, two basic control approaches can be listed: feedback and feedforward [
Generally speaking, the basic components of a feedforward control system are a sensor, which captures the input (reference) signal related to the primary excitation, a controller (or control unit), which processes the input signal according to a given algorithm and generates the output (control) signal, and an actuator, which applies the output signal in the system to be controlled. There should also be a second sensor, which captures the net response (error) signal. In the current case, the controller consists primarily of a digital finite impulse response (FIR) filter of adaptive nature. The feedforward approach usually requires the use of adaptive digital filters in order to track closely what happens to the system (plant) and generate an efficacious control signal of destructive interference.
Finite impulse response (FIR) filters have, as opposed to infinite impulse response (IIR) filters, the important characteristic of stability; they have no poles in their transfer function. The popularity of FIR filters can also be justified by the fact that the corresponding optimal coefficients can be obtained by an objective function of quadratic nature, with a unique point of minimum. This point can be found in a prompt way through an adaptive algorithm. In the case of IIR filters, the objective function may have several local points of minimum and the algorithm may converge to any of those local points of minimum, instead of the global minimum [
The FIR filter output (control) signal vector
The AVCS, as part of the investigated HVCS, is displayed in Figure
The concept and the fundamental importance of feedback and secondary paths to the success of a practical implementation of an AVCS must be clear. The feedback path estimation filter
According to the design theory of Wiener filters (which are of stochastic origin), the optimal coefficients for FIR filters result from the minimization of the mean square error (MSE), or
In adaptive filtering, as is the current case, the optimal coefficients can be obtained iteratively by LMS (least mean square) algorithms, regarded as practical solutions to Wiener filters [
Equation (
It is observed in Figure
The advantages of LMS algorithms are simplicity, reliability, and robustness. On the other hand, a considerable disadvantage is the high dependence of the input signal power spectral density (PSD), so that the more uniform the input signal PSD, the faster the convergence.
A MIMO (multiple input multiple output) application of a very similar AVCS is implemented in an aircraft fuselage section by Marra et al. [
Another aircraft fuselage section is the plant used by Griffin et al. [
Zeng et al. [
The above PVCS and AVCS can be combined to form a hybrid vibration control system (HVCS). Apart from dealing with the matter of detuning of a viscoelastic device, an HVCS like this, schematically shown in Figure
The system to be controlled (primary mechanical system, or plant) is a 0.5 kg simply supported steel beam, the dimensions of which are 930 × 23 × 3 mm. A schematic view of the beam is shown in Figure
Beam schematic view.
A previously performed theoretical and experimental modal analysis on the beam provided complete information about the dynamic behavior of the beam at all positions. It led to the selection of the modes and the frequency range of interest and also to the selection of the point in the beam where to locate both the VDVN and the control shaker. The 4th, 5th, and 6th vibration modes of the beam are chosen because they allow the design of a VDVN with a relatively small amount of mass and compact viscoelastic elements, resulting in a device of reduced size. As shown further (Figure
The applied excitation signal is a white noise filtered to a band of 200 to 430 Hz. This signal, displayed in the time and frequency domains in Figure
Excitation signal (filtered white noise, 200–430 Hz).
FRF (
The PVCS, a viscoelastic DVN with a steel lump of mass of 0.0335 kg and two elements of butyl rubber 45 Shore A, is optimally designed to act in the frequency band of 190 to 440 Hz and at the temperature of 25°C [
VDVN drawings (all dimensions are in millimeters).
Viscoelastic material properties of the butyl rubber.
The controller is based on an Analog Devices board, the EZ-KIT Lite ADSP 21161N, on which the FXLMS algorithm is implemented. Before running the control tests, identification routines for the secondary and feedback paths have to be performed, in order to have the FXLMS algorithm fully implemented, as portrayed in Figure
It should be clear, however, that the main FIR filter,
As observed in Coan Jr. [
FXLMS optimal parameters for active-adaptive vibration control.
|
|
|
|
|
---|---|---|---|---|
300 | 500 | 500 | 0,9999995 | 1,50 |
In Table
As previously mentioned, the digital filters intend to represent the impulsive response of a specific system (for instance, the plant, the secondary path, or the feedback path). So, the more information is acquired by the digital filter about the impulsive response of the system of concern, the higher is the fidelity of such a digital model, that is, the better is the quality of identification.
Figure
Portion of system impulsive response acquired with a given digital filter size.
Experimental setup inside the temperature chamber.
The experimental setup, which allows all the investigations regarding the vibration control systems, is displayed in Figure
In a first stage, for the beam alone, the ratio of the acceleration (at a point of the beam) to the disturbing force is computed, in the frequency domain. That ratio is denominated herein a response to force ratio (RFR). After installing the VDVN and maintaining exactly the same primary excitation (disturbance) applied before, it is possible to evaluate the PVCS attenuation performance by computing a response to force ratio (RFR) of the compound system (beam + VDVN).
An explanation should be given at this point. A response to force ratio (RFR) is a ratio between the response at a point in the system of concern and the disturbing force at the same or at other points, in the frequency domain [
After exclusively testing the PVCS, the AVCS is turned on, so to act on the compound system (beam + VDVN), and its additional contribution is evaluated by an RFR of the global system (beam + VDVN + AVCS), still keeping the same disturbance. All those tests are run at room temperature (around 25°C, the optimal VDVN design temperature).
The next stage is to test the adaptability of the main digital filter of the AVCS, which is carried out by varying the temperature under which the experimental setup is, inside the temperature chamber, as seen in Figure
Those adaptability tests are performed at the temperatures of 5, 15, 25, and 35°C. At each test temperature, before taking the measurements, a 30-minute interval is considered to allow the temperature of the viscoelastic elements of the VDVN to stabilize. All the RFRs are computed between the error sensor location point (point 1) and the disturbance actuator location point (point 19). Those locations are shown in Figures
The attenuations obtained in each test condition are calculated according to (
In the above equation,
The corresponding results are presented below.
The results are as follows.
Figure
Passive vibration control system (viscoelastic DVN) performance.
It is expected that the VDVN performance varies markedly with the temperature, due to the behavior of the viscoelastic material (see Figures
VDVN performance (vibration attenuation), broadband analysis.
Test temperature [°C] | RMS broadband attenuation [dB] |
---|---|
5 | 7,9 |
15 | 9,1 |
25 | 9,5 |
35 | 8,9 |
Considering the frequency and temperature ranges of the current application (200 to 430 Hz, and 5 to 35°C, resp.) and the frequency and temperature dependencies of a typical viscoelastic material (particularly detailed in Figure
Focusing on the resonance peaks only, it is observed that the greatest attenuations, regarding each mode of vibration, do not occur at 25°C, as recorded in Table
VDVN performance (vibration attenuation), resonance peak analysis.
Test temperature [°C] | Resonance peak attenuation (4th mode) [dB] | Resonance peak attenuation (5th mode) [dB] | Resonance peak attenuation (6th mode) [dB] |
---|---|---|---|
5 | 2,0 | 8,0 | 19,0 |
15 | 6,0 | 11,0 | 20,0 |
25 | 6,0 | 16,0 | 19,5 |
35 | 14,0 | 18,0 | 17,0 |
The results obtained with the PVCS and the HVCS (comprising both passive and the active-adaptive vibration control systems) are shown in Figure
PVCS and HVCS performances at (a)
The marked control action at the resonance peaks after the AVCS is turned on can be explained as follows. The control signal generation is based on an online plant identification method and this makes the highest energy frequency components of the system response (the resonance frequencies) to be carried to the main filter coefficients. Thus, the control signal is generated with the same characteristics; that is, the control signal has the same highest energy frequency components as the plant response. Hence, when the control signal is reintroduced into the system (see Figure
Table
PVCS and HVCS performance (vibration attenuation related to the uncontrolled beam vibration levels), broadband analysis.
Temperature | Control system | ||
---|---|---|---|
Passive [dB] | Hybrid [dB] | Difference (HVCS to PVCS) [dB] | |
5°C | 7.9 | 16.8 | 8.9 |
15°C | 9.1 | 19.4 | 10.3 |
25°C | 9.5 | 19.4 | 9.9 |
35°C | 8.9 | 19.9 | 11.0 |
PVCS and HVCS performance (vibration attenuation related to the uncontrolled beam vibrations levels), resonance peak analysis.
Temperature [°C] | Vibration control system | Vibration mode | ||
---|---|---|---|---|
4th [dB] | 5th [dB] | 6th [dB] | ||
5 |
|
2,3 | 7,7 | 19,3 |
|
0,0 | 25,6 | 34,1 | |
|
||||
15 |
|
6,2 | 10,8 | 20,0 |
|
11,5 | 24,6 | 32,2 | |
|
||||
25 |
|
6,2 | 15,5 | 19,5 |
|
13,3 | 24,3 | 29,1 | |
|
||||
35 |
|
14,0 | 18,6 | 17,0 |
|
17,8 | 24,7 | 30,6 |
Tables
The behaviors of the vibration control systems are also monitored through the error sensor signal, in the time domain, as displayed in Figure
Error sensor time history.
The adaptation capability introduced by the AVCS, as part of the HVSC, can be inferred from the above results. However, in order to make this point even more clear and help to visualize how the FXLMS algorithm handles the main filter coefficients to generate the best control signal at each test condition, the values of the main filter coefficients are traced at each test temperature. The results are presented in Figure
Adaptability of main filter coefficients.
The vibration attenuation performance and robustness of the PVCS (VDVN), as well as its expected changes in dynamic behavior under frequency and temperature variations, are verified once more. That adds to the long and successful chain of broadband applications in which the same technique for VDVN design is employed [
The adaptive behavior of the AVCS becomes well characterized when there are changes in the compound system (metallic beam + VDVN) due to the variation in temperature, confirming and extending some previous efforts of tonal nature [
It is worth mentioning that the adaptive capability is very important to compensate not only system modifications but also some modeling errors or inaccuracies in design, thereby improving the applicability of the AVCS. It is understood that improvements on the quality and effectiveness of the FXLMS algorithm will be achieved if the “static” secondary and feedback filters become adaptive as well.
It is also pointed out that the damping characteristics of the mechanical system under control, altered by the insertion of the VDVN, are an important factor for the order (number of coefficients) of the employed FIR filters and for the FXLMS effectiveness. In fact, when a VDVN is inserted in a mechanical system, this system becomes more damped, with impulsive response functions of smaller nonzero length. As the AVCS is applied to this more damped plant (beam with VDVN), the FIR filters can be of reduced order. That has a direct impact on the convergence of the employed LMS algorithm.
Finally, the HVCS, comprising both the above PVCS and AVCS, reveals itself as a powerful alternative in vibration control, in which the advantages of each particular system can be well associated in order to have an adaptable, flexible, and robust global control system, of reduced cost. This association also contributes to a more reliable control system, where the responsibility is shared between its component systems and some overall vibration control level can still be maintained in case of failure of any of those systems.
The authors declare that they have no competing interests.
The first author wishes to express his gratitude to the Federal University of Santa Catarina, Brazil, and Federal University of Paraná, Brazil, for all the support over the development of this work. The second author acknowledges the financial support of CNPq. A special acknowledgement must be expressed for the relevant contributions of Dr. Orlando Jose Tobias (“in memoriam”) to this work.