Experimental Mechanics

, Volume 48, Issue 5, pp 683–692

A Simple Closed-Loop Active Control of Electrodynamic Shakers by Acceleration Power Spectral Density for Environmental Vibration Tests

Authors

    • UFRGS, Universidade Federal do Rio Grande do Sul, Programa de Pós Graduação em Engenharia Mecânica
  • D. dos Santos Gaspareto
    • UERGS, Universidade Estadual do Rio Grande do Sul
  • F. de Souza Ferreira
    • UFRGS, Universidade Federal do Rio Grande do Sul, Programa de Pós Graduação em Engenharia Mecânica
  • C. A. K. Thomas
    • UFRGS, Universidade Federal do Rio Grande do Sul, Programa de Pós Graduação em Engenharia Mecânica
BRIEF TECHNICAL NOTE

DOI: 10.1007/s11340-008-9134-4

Cite this article as:
Gomes, H.M., dos Santos Gaspareto, D., de Souza Ferreira, F. et al. Exp Mech (2008) 48: 683. doi:10.1007/s11340-008-9134-4

Abstract

This work presents the main results of a simple closed-loop active control for an electrodynamic shaker in order to generate acceleration Power Spectral Densities (PSD) according to prescribed Standards used in environmental vibration tests. The main idea is to start generating acceleration pseudo-signals obeying the prescribed Power Spectral Density and then to acquire acceleration data from the electrodynamic shaker’s table behaviour. So the Power Spectral Density of the acquired acceleration is computed and compared with the required PSD and then the time-varying pseudo-acceleration is updated to reflect this corrected PSD. It was noticed that for piecewise narrow bands frequencies, the electrodynamic shaker acceleration behaves near linearly, both in frequency and voltage, for the input signals. A code in AgilentVee 7.5 software to acquire, send and process signals for the active control in a closed-loop scheme was developed. The used A/D D/A hardware was a single PC sound card with specific characteristics. The control could be accomplished sending and acquiring at the same time with a range of input/output of ±1.5 V with 16 bits of resolution, at 48 kHz and assistance of an external sound amplifier.

Keywords

Active controlElectrodynamic shakerPower spectral densitySound card

Introduction

There is a general trend on increasing the reliability of several devices (mainly those with critical components) which are tested and used at adverse conditions. This is accomplished by vibration tests because this phenomenon affects the durability and serviceability of such devices. The main purpose of this work is the development of a simple, cheaper and reliable system to control an electrodynamic shaker with relative accuracy through a closed loop and then use it to perform environmental vibration tests. The main control variable is the shaker’s table acceleration. The function to be minimized is the difference, in frequency domain, between the shaker’s acceleration Power Spectral Density and a Standard Acceleration Power Spectral Density.

The developed system makes it possible to perform tests, such as: mechanical fatigue strength, evaluation of resonant frequencies (by sine sweeps or random vibrations), replication of acquired acceleration’s field signals, impact strength, shock, fall and general modal analysis. Obviously, these tests are bounded as in amplitude as in frequency by the electromechanical behaviour of the system (shaker and the structure under test).

Electrodynamic Exciter Systems (Shakers)

A shaker is any device that applies vibration on test pieces (alternate acceleration with known frequencies) in a controlled way. The main work operation principle may be based on combustion engines, hydraulic engines, pneumatic or electrodynamic ones (this work).

The electrodynamic shaker works based on the electromagnetic forces generated by two interacting magnetic fields generated by coils. One of them has its own magnetic field proportional to the applied voltage (armature coil), the other has a static magnetic field (field coil). There is a fixed armature for the field coil (exciter base) and another acting armature coupled (through a spider) to a table that supports the test specimen where the acceleration is measured for the active control. Several Standards specify this acceleration measured on the shaker table [2, 35, 9, 16] instead of the acceleration on the piece under test, mainly because this acceleration represents an experimental acceleration measurement.

Figure 1 shows two sketches for typical electrodynamic shakers. Figure 1(b) is similar to the shaker used in this work.
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Fig. 1

Sketches for typical electrodynamic shakers: (a) shaker supported by simple foundation base, (b) shaker supported laterally and by a foundation base

As stated by McConnell [6], when a vibration exciter (with a heavily damped foundation case) is used to excite an ungrounded structure (like in the present work), the system can be modelled as a two degree of freedom (DOF). One of the degrees of freedom simulates the shaker behaviour and the other the test specimen. In this case there is an interaction between the exciter and the structure under test (SUT) which depends on both dynamical characteristics, so two resonance frequencies will show up. It is worth it to say that these resonances are test system resonances (shaker and SUT acting together) and not structure under test resonances. However, for small test pieces, such as in the case of electronic components, this interaction may vanish as indicated by McConnell [6]. Besides, the transmissibility acceleration ratio between exciter table and SUT is independent of the vibration exciter’s dynamic characteristics. Anyway, it is evident that measuring either the interface force or the interface acceleration is a good idea, since the vibration exciters effects are eliminated.

Short Bibliographical Review

M. Fujita and Y. Uchiyama [1, 11] achieved the electrodynamic shaker robust control by means of the acceleration and displacement control of the shaker’s table. The work presented the acceleration and displacement control of an electrodynamic shaker that could not be controlled by standard control methods. Two control variables were used in order to control a broad frequency band. The acceleration control was used on frequency bands where the response signals were larger than the noise. The displacement control was used for lower frequency bands. Both methods were coupled in a series way. The control softwares were projected using the μ Synthesis Method with two degrees of freedom in order to improve the system transient response. Finally the authors tested the developed control on an electrodynamic shaker and, at the end, some comparisons were drawn between the proposed control system and a standard single loop system.

Cavaliere and Cavalca [10] presented an implementation of one wave vibration test. This implementation was based on the LabView software acting as a signal generator. The main purpose was to generate vibrations with several wave shapes.

Salehzadeh-Nobari, Chambers, Green, Goodfellow and Smith [12] presented an adaptive control on the frequency domain. The authors stated that the correct choice of the control technique is dependent on the system modelling assumptions and that, using available high level software tools, the adaptive inverse control technique shows to be suitable mainly in situations where non-linear behaviours are presented as in vibration tests. The focus of the paper is the design of controllers by means of the “Teaching Company Scheme” and the impact of this development on the “Ling Dynamic Systems” (LDS), a manufacture company of industrial vibration test machines. The work includes some short description of the vibration systems and the uses in industry followed by explanations concerned with the choice of the algorithm used and the main steps for the implementation on specific hardware.

Flora and Gründling [13] presented an interesting work where it is shown the development of digital acceleration controllers for sinusoidal vibration tests using an switching-mode AC power source-ACPS with an electrodynamic shaker. The used electrodynamic shaker was of the same manufacturer, model and load capacity than that used in this work. The proposed scheme is based on two control loop interactions: one for the acceleration regulation and the other one for the ACPS voltage output. It is used an algorithm for robust control (Robust Model Reference Adaptive Algorithm-RMRAC) in order to reduce the effects caused by the variations on the system and to reduce the harmonic and resonant vibrations on the test piece. The acceleration controller is improved with a feed-forward and a Robust Controller. The experimental results have shown that the proposed system is able to attain a good performance in standard frequency bands of 20 Hz to 1 kHz.

Moutinho et al. [14] described the implementation of an active control system applied to a vibration table used for seismic vibration tests on reduced scale test specimens. According to the authors the use of such control system is intended to impose an a priori movement to the shaker’s table, which is strongly dependent on the interaction between the table and the test specimen as well as random phenomena associated to experimental tests. That paper describes the working principle and implementation of the proposed active control developed in LabView and compares the efficiency with those evaluated using two different control strategies: the first one using a Proportional Differential and Integral controller (PID) and a second one using a predictive controller. The controller was implemented in the following equipment: a vibration platform, a unidirectional electrodynamic exciter APS, model 400 by ELECTRO-SEIS and an adaptive aluminium table for model support. The total mobilized mass limit was 12.32 kg. The PID controller was less efficient than the predictive controller in the evaluation of the desired measured response (particularly in sudden movement variations), as much as in maximum amplitude load control. However, it does not require a numerical model to generate the control action necessary in the predictive controller.

Requirements for Vibration Equipment and Tests

A modern vibration system is composed of three main devices: a vibration exciter, an amplifier and a control system. The vibration exciter transforms the electrical energy on vibration movements applied to the test specimen, often providing ways to study the damage evolution while in normal operation. The amplifier transfers the power from the line source and transforms it on electrical signals of desired frequency and amplitude. The controller monitors the vibration test, the output exciter’s table acceleration and supplies the adequate input signal for the amplifier.

The vibration amplifier requires some features that are not necessary in common applications. In addition, for an adequate load ratio, moving the vibration exciter with the right force requires the amplifier to dissipate and absorb reactive loads from alternating levels of impedance. The amplifier distortion is a severe subject since the distortion itself can easily harm the test specimen. So the amplifier distortion should be kept below 3% or less than 2% on critical regions where harmonic vibrations could excite shaker’s fundamental frequency. Nowadays these requirements are easily fulfilled by several audio amplifiers. As indicated by McConell [6], there are two ways of operation of a shaker amplifier: the voltage mode and current mode. The last one has advantages over the first due to less pronounced force dropout that occurs at the SUT’s ungrounded natural frequency. In fatigue tests, for instance, the current mode of operation shows better performance, however to evaluate SUT’s natural frequencies and damping, this mode will just allow to measure the global system behaviour instead of SUT’s behaviour.

The main tasks of a modern shaker arise from the several types of controller systems available. These tasks could be grouped into two classes: periodical vibration and random vibration.

To submit devices to periodic vibration is the main cause of failure. The simulation of such periodic disturbances by laboratory sine excitation has been frequent for failure or durability tests as well as diagnosis tests. The simulation of in service periodic vibration is explained since the damage in objects by sine vibration is due to magnified effects of resonance.

The natural frequencies of the test specimens should be evaluated in order to perform fixed frequency vibration tests. These tests are performed for each of the natural frequencies of the test specimen in turn. Such tests could be used to evaluate the fatigue strength when subjected to resonance frequencies, to evaluate the structural damping ratio and investigate equipment noise sources, for instance, appliances or other structures. The devices used for vibration tests on fixed sine frequency are quite simple since the input signal to be amplified could be obtained, for example, by a simple signal generator. Some auxiliary equipment is also useful like those for vibration amplitude control, devices for coil displacements and acceleration measurements and devices for quickly move the shaker’s table in order to simulate pulse damped vibrations.

The sine sweep vibration test could be used to adequately simulate the damage growth (often due to unbalancing). Since the frequency is swept through a large band frequency, most of the resonant vibration modes can be excited and evaluated according to its severity. These tests are used for terrestrial vehicles, vessels and aircrafts since these tests sweep each frequency sequentially at a low cost.

The random vibration induced by jet engines related to top aircraft flying has demanded new design criteria and new test procedures to assure the reliability of devices in such an environment. When the random vibration is measured on field this signal can be used as boundary parameter to define reliability criteria to be reach. Sometimes the actual acceleration signal of a working environment is not the most suitable way to characterize this vibration and latter be used on a shaker or exciter system in an accelerated test. In addition there are other intervening factors that show these signals will not probably be reproduced again since there is presented some level of randomness. One of the best ways to detail this type of signal is by the Acceleration Power Spectral Density (PSD) obtained from an actual acceleration. This PSD is defined by frequency bands of the whole interested frequency domain. So it is provided plots of Acceleration PSD and frequencies.

There are many Standards dealing with the dynamic tests (each company or supplier specifies its own minimum requirements). However it is notice that there is a global trend in using Acceleration Power Spectral Densities as a standard for random vibration tests, mainly in environmental vibration tests.

The Acceleration Power Spectral Density can be evaluated by the Fourier Transform of the desired auto-correlation signal. In other words:
$$S_{\text{a}} \left( \omega \right) = \frac{1}{{2\pi }}\int\limits_{ - \infty }^\infty {R_{\text{a}} \left( \tau \right)} \operatorname{e} ^{ - i\omega \tau } \operatorname{d} \tau $$
(1)
where Ra(τ) is the acceleration auto-correlation function, i is the imaginary number, ω is the frequency in rad/s units and τ is a time separation. Ra(τ) may also be evaluated by:
$$R_{\text{a}} \left( \tau \right) = E\left[ {a\left( t \right)a\left( {t - \tau } \right)} \right] = \mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{r = 1}^N {a_{\text{r}} \left( t \right)a_{\text{r}} \left( {t + \tau } \right)} $$
(2)
where the symbol E[.] represents the mean value and N means the sample size. One important characteristic of the auto-correlation function for zero mean processes is that [8]:
\(R_{\text{a}} \left( 0 \right) = \sigma _{\text{a}}^2 \), and so it is also valid the relation:
$$\sigma _{\text{a}}^2 = \int\limits_{ - \infty }^\infty {S_{\text{a}} } \left( \omega \right)\operatorname{d} \omega $$
(3)
In other words, the acceleration process variance is the area above the Spectral Density, or the value of Ra(τ) on origin.
The Standards generally specifies Power Spectral Densities on its one-sided form, i.e., they are specified of positive frequencies bands. As stated by equation (1), the Power Spectral densities definition uses infinity upper and lower limits, so the One-sided Power Spectral Densities are defined as indicated by equation (4):
$$G_{\text{a}} \left( \omega \right) = 2S_{\text{a}} \left( \omega \right)$$
(4)
For the pseudo-acceleration signal generation it could be used sine series as indicated by equation (5):
$$a\left( t \right) = \sum\limits_{i = 1}^N {\sqrt {2S_{\text{a}} \left( {\omega _i } \right)\Delta \omega } } \sin \left( {\omega _i t + \varphi _i } \right)$$
(5)
where the Power Spectral Density was divided by N frequency interval Δω, and the phase angles ϕi are random values for each centre frequency band ωi, so the process is supposed to be a superposition of sine periodical processes [15]. Figure 2 represents a sketch for this generation.
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Fig. 2

Generation of pseudo-acceleration waveforms

In order to accomplish the correct acquisition of the acceleration PSD, some care is necessary in the frequency resolution selection, sampling rate and total time for acquisition. Specifically, for a given Standard PSD with lower and upper frequency band limits, the total time should be enough to acquire not less than three cycles (as used in this study) of the lowest frequency component and the sampling rate should be enough to discretize the highest frequency component on ten or more points (as used in this study).

Depending on the Standard PSD shape (smooth or composed by connected lines), the number of points used to discretize the Standard PSD and generate the pseudo-acceleration signals might vary. In this study it was used the same number defined by the acquired acceleration PSD, so it is possible to compare point by point the Acquired and the Standard PSD or to use frequency bands (octave, 1/3 octave, etc.).

Design Methodology and Experimental Testing

Experimental Apparatus

The used equipments will be briefly described. The used Shaker was the model St. 5000/300 by TIRA Vibration Test Systems, Germany. According to the equipment data sheet it has an effective working frequency band ranging from 20 Hz to 5 kHz. It has a maximum displacement of ±6.0 mm. The maximum applied load is 2,940 N. The total device mobilized mass, which includes table, armature and fastening devices, is 8.5 kg. The maximum allowed test specimen mass (including the mobilized mass is 60.0 kg. This shaker has an internal electrical resistance of 4 Ω, and it was designed to a maximum power consumption of 500 W (RMS).

The used Signal Amplifier used was the model DBK-6000 by WATTSOM. It has two amplified channels. The output power level by channel is 375 W (AC: 230 V) and class AB. The data sheet presents the amplifier frequency response ranging from 15 Hz to 40 kHz. It has a total harmonic distortion plus noise (THD + N) in 8.0 Ω less than 0.05% in the frequency range of 20 Hz to 1 kHz and a THD + N less than 0.1% from 20 Hz to 20 kΩ. The input impedance (unbalanced) is 20.0 kΩ, and the Signal to Noise ratio (S/N-without weighting) is greater than 90 dB.

The on-board Sound Card used was the Realtek ALC650/AC′97 by Realtek. Its Signal to Noise ratio in the corresponding frequency range is greater than 90 dB. It has six channels, which uses 16 bits resolution for AD and 20 bits resolution for DA. The input and output limits are ±1.5 V and the working frequency range varies from 20 Hz to 48 kHz for input signals and from 20 Hz to 8,000 kHz for output. Figure 3 shows the experimental setup with the used equipments.
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Fig. 3

Scheme for the experimental setup

The used software for acquisition, processing and control was the AgilentVee 7.5 [7, 8].

The used accelerometer was of the low cost MEM type (capacitive model ADXL 250) produced by the Analog Devices [17]. It has a maximum output voltage of 2.5 V for 500 m/s2 (50 g) acceleration. It measures ±500 m/s2 on two orthogonal axes and has sensitivity of S = 76.0 mV/ g. Figure 4 shows a typical frequency response for the ADXL 250 accelerometer. Figure 5 shows the way the equipment is connected.
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Fig. 4

Typical output response vs. frequency of ADXL 150/250

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Fig. 5

Block diagram of electrodynamic controlled test system

Design Methodology

Most of the vibration test Standard defines the vibration experiment by an Acceleration Power Spectral Density. The unit of such variable is stated as (m/s2)2/Hz, versus frequency (see Fig. 6). Based on the measured acceleration, one can plot the Power Spectral Density and thus compare the total energy of the signal generated, for example, by a shaker and the Standard. These comparisons may be done by spectral bands or by octave bands. In this way, it is possible to evaluate a correction weight for each band frequency of the shaker output in order to match the desired PSD.
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Fig. 6

Plot for Acceleration Power Spectral Density × frequency [2], the letters represents service severity conditions

The area above the Power Spectral Density and the frequency axis defines the RMS acceleration value. This parameter is important since it this variable is used by the test Standard as a check parameter for the generation. In the above Fig. 6, the severity curve A has an acceleration RMS value greater than the severity curve B.

The method used in this paper is based on a closed loop scheme, so the acceleration is measured by an accelerometer on the shaker’s table at the same time it is compared with the desired acceleration. It is also compared the one-sided Power Spectral Density. If these values are different for the frequency band, the developed software weights this band in order to compensate this behaviour. It is accomplished iteratively acquiring and sending voltage signals till the discrepancy are negligible for each of the frequency bands (Fig. 7).
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Fig. 7

Flowchart of the generated program in AgilentVee

The used frequency band follows the well known procedure to work with octave or 1/3 of octave bands in order to quickly sweep the frequency range. This implementation was based on a specific Standard that requires the generation of PSD in the 10 Hz−1 kHz range.

After some time interval, the developed system behaves as an open loop system and continues to generate signals from the last converged iteration. Figure 7 shows the flowchart for the generated program in AgilentVee 7.5 [7, 8].

In order to estimate mean characteristics of used shaker it was evaluated the following response curves: acceleration vs. frequency, velocity vs. frequency, displacements vs. frequency, and acceleration vs. input voltage. It was noticed that the relation input voltage vs. acceleration follows a linear trend, as indicated by Fig. 9. The shaker’s acceleration vs. frequency curve follows a non-linear trend. This last result was expected since the input power was kept constant for changing frequencies.

Preliminary Results

Analyzing the obtained response curves, it was noticed that the system could be initially supposed to behave linearly from 100 to 1,000 Hz for acceleration response. The iterative software could handle differences originated by this assumption weighting the frequency bands appropriately. Figure 8 indicates typical relationships between acceleration and frequency with the same input voltage.
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Fig. 8

Typical acceleration vs. frequency curves for the shaker

Using Fig. 8 and Fig. 9 it is possible evaluate parameters for voltage transformation to acceleration and vice versa. These parameters were used as a starting point in the first iterations of the software since there were no exact values. After some iteration the system gathers information and evaluate weights to correct the output signals so the respective curves become useless.
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Fig. 9

Typical plot for amplifier voltage output versus measured acceleration on shaker’s table (without test specimen)

The software was developed based on the AgilentVee 7.5 platform. Figure 10 shows the programme’s user interface. On the left column there are fields corresponding to number of channels, total time of acquisition, Sample frequency, A/D number of bits, input and output ranges, input/output gains and sensor sensitivity. In the centre of the interface there are two plots. The upper one corresponds to the adjusted acceleration PSD that are being sent to the shaker and the bottom one is the acceleration PSD of the acquired and desired acceleration PSD. In the upper right corner there is a selection box with several Standard acceleration PSD and just below there is a selection box for the send/acquire mode of operation or only send mode of operation. At the top there are boxes that allow specifying file names paths to write data and weights of the adjusted PSD. And finally, at the left bottom corner there are three boxes related to the RMS of the acceleration signal, RMS and mean value of the acquired signal.
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Fig. 10

Acceleration PSD for single sine at 500 Hz and acceleration amplitude of 1 m/s2 (0.707 m/s2 RMS)

The software shows a graphical interface for the user with PSD of the measured acceleration and Standard PSD. It also shows RMS of acceleration, evaluated weighting factors at each iteration step (Fig. 7). The weighting factors are recorded on files along iterations and later used for sending the corrected desired PSD to the exciters. The values of the last cycle are used for this purpose.

For a single frequency vibration test at 500 Hz and acceleration amplitude of 1 m/s2 (0.707 m/s2 RMS), the following plot (Fig. 10) shows the experimental results at the 10th cycle of iteration with the corrected one-sided acceleration PSD (upper white plot) and the acquired one-sided acceleration PSD (inferior plot, with the desired PSD in white and the measured PSD in gray). The RMS acceleration value is showed at the bottom left corner with a value of 0.7062 m/s2. Figure 11 shows the acquired acceleration signal on shaker’s table with a structure under test (SUT) of 1.0 kg.
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Fig. 11

Acquired acceleration on shaker’s table (m/s2) for single sine at 500 Hz and acceleration amplitude of 1 m/s2 (experimental RMS value of 0.7304 m/s2)

Figure 12 shows the 15 h iteration for the corrected one-sided acceleration PSD to take into account non-linear behaviour of the shaker (upper white plot) and the acquired one-sided acceleration PSD (bottom plot with the Standard in white and the measured in grey). The RMS acceleration value is showed at the bottom left corner with value of 13.99 m/s2. Figure 13 shows the acquired acceleration signal on shaker’s table with a structure under test (SUT) of 1.0 kg.
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Fig. 12

Acceleration PSD for random vibration test with severity level 3, target RMS acceleration of 13.9 m/s2 [9]

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Fig. 13

Acquired acceleration on shaker’s table (m/s2) for random vibration test with severity level 3 (experimental RMS acceleration of 13.98 m/s2, [9])

In this case, some differences were noticed between the acquired and the target PSD although the RMS acceleration is quite similar. This happens due to the target spectrum discretization for the signal generation since this methodology applies equation (5). Thus, some frequencies will not be present on the generated signal and other frequencies will have its P.S.D. magnitude increased by the contributions from neighbour’s frequencies. So the response of the electrodynamic shaker will not contain some frequencies. This can be outlined by a refined discretization of the P.S.D. that will increase the computational time and effort.

Conclusion

The conducted experiments have shown that with low cost and effort it was possible to develop a closed loop active acceleration control of an electrodynamic shaker. The operation of the system starts by acquiring acceleration signals from the shaker’s table due to the sent input voltage. Later, the system behaves as a single open loop just sending weighted voltage signals to the shaker. The results and software used in this paper can be used on other shakers with different load capacities since the PSD comparisons and weighting correction accounts for non-linearities present in the system. This approach avoids difficulties related to the traditional control in evaluating system parameters, like stiffness, damping, mass, transfer functions, etc. and its implementation is very simple and straightforward. The displacements control, which was neither implemented nor investigated here, could be used to improve the system performance, mainly for low frequencies. However, the cost would be higher than the one of the proposed strategy. This could improve some noticed differences between generated and desired acceleration PSD in above figures.

Acknowledgements

The authors wish acknowledge the GMAp laboratory for the shaker and equipments used in this work. The authors also thank for the Brazilian Councils CNPq and CAPES for the financial support and involved scholarships.

Copyright information

© Society for Experimental Mechanics 2008