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PRESENTATION OF THE ECL BENCHMARK
Mechanical Systems and Signal Processing (2003) 17(1), 195–202 doi:10.1006/mssp.2002.1560, available online at http://www.idealibrary.com on
PRESENTATION OF THE ECL BENCHMARK F. Thouverez Ecole Centrale de Lyon, Lyon, France. E-mail: [email protected] (Received 1 October 2002, accepted in revised form 1 October 2002) Many different non-linear identification methods have been developed so far. Most of them proceed from various experimental or numerical approaches. Inside the cost action F3 project, one common benchmark was chosen to test several identification methods in order to compare their efficiencies and their specificities. To do so, a large amount of data (time data, frequency data) has been collected and proposed to researchers. The benchmark, which has been defined, involves one single local non-linearity coupled with a simple linear structure. # 2003 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
The benchmark proposed by the Ecole Centrale de Lyon (ECL) consists of a clamped beam with a local non-linearity at the end. Two different set-ups were built : the first one at ECL [Fig. 1(a)] and the second one at the University of Li"ege (ULg) [Fig. 1(b)]. The coupling between a linear flexible structure and a local non-linearity is a classical problem in the industry. Indeed, most links like solders, screws, bushings, and dampers are often the main source of non-linear effects in the dynamical response of the whole structure. The non-linear identification techniques have to be able to estimate such non-linear components. In order to test the efficiency of these procedures, we built-up a simple test set-up avoiding this way any complex geometrical description problem, but including a local non-linearity. The choice of a simple non-linear structure permits a clear understanding of the non-linear phenomena. It gives us also a good experimental comparing data for the development of efficient numerical models.
2. DESCRIPTION OF THE STRUCTURE
The test rig is composed of two parts as illustrated in Fig. 2. The first part (denoted A) which is the main component of the structure is a steel beam and the second part (denoted C) is a thin beam which enables the non-linear behaviour. The main beam is assumed to be clamped at one of its end. 2.1. THE ECL SET-UP [FIG. 2(A)] In the ECL set-up, the thin beam (part C) is clamped at one end and linked to the beam at the other end through a coupling element (denoted B). Part B was designed to avoid any slipping between both beams. For this purpose, pins have been realised and included in the element on both sides. The same locking solution was applied to the clamping. Screws maintain the whole assembly. The mechanical and geometrical characteristics of the different parts are given in Table 1. 0888–3270/03/+$35.00/0
# 2003 Elsevier Science Ltd. All rights reserved.
196
F. THOUVEREZ
Figure 1. ECL benchmark: (a) the ECL set-up; (b) the ULg set-up.
2.2. THE ULG SET-UP [FIG. 2(B)] In the ULG set-up, the thin beam (part C) is clamped at one end. There is no coupling element as for the ECL set-up but the thin beam is clamped into the main beam using screws. The mechanical and geometrical characteristics of this set-up are given in Table 2.
3. THE LOCAL NON-LINEARITY
The non-linear behaviour of the structure may be enabled by the thin beam when large displacements occur during the test. In the ECL set-up, the assembly is pre-loaded with an axial force. The value of this preload is unknown but it determines the linear and non-linear characteristics of the element. A theoretical calculation demonstrates that the non-linearity is cubic (Duffing spring) and comes from the non-linear deformation tensor. Indeed, the non-linear part of the strain is no more negligible due to a coupling between the axial deformation and the transversal one. Some quadratic effects could also be taken into account, assuming that the gravity has some influence on the initial position of the structure. In the contribution of Bellizi, a theoretical development is proposed to compute analytically the linear and non-linear components of this joint.
PRESENTATION OF THE ECL BENCHMARK
197
Figure 2. (a) Local non-linearity and driving point (ECL set-up). (b) Local non-linearity (ULg set-up).
In the ULg set-up, the assembly is not pre-loaded with an axial force. Due to the thin beam (part C), the effect of gravity is not negligible. Gravity is at the origin of a static deflection of the two beams. This deflection imposes a non-negligible prestress in the thin
198
F. THOUVEREZ
Table 1 ECL Set-up characteristics Young’s modulus (Pa) Part A Part B Part C
Poisson ratio
Length (mm)
Thickness (mm)
Width (mm)
Mass (gr)
0.35 0.35 0.35
593 40 57
14 20 0.5
14 30 30
907 187 6.7
2.1 E11 2.1 E11 2.1 E11
Table 2 ULG set-up characteristics Young’s modulus (Pa) Part A Part C
Poisson ratio
Length (mm)
Thickness (mm)
Width (mm)
Mass (gr)
0.35 0.35
700 40
14 0.5
14 14
1070 2.2
2.1 E11 2.1 E11
Table 3 Sensors characteristics Sensor C0 C4 C3 C2 C1
Position from clamping (mm) 63 47 220 420 605
Type
Reference
Force Acceleration Acceleration Acceleration Acceleration
BK 8200 ENDEVCO ENDEVCO ENDEVCO ENDEVCO
Total weight (g) Modal Modal Modal Modal
61-100 61-100 61-100 61-100
ICP ICP ICP ICP
21 8 8 8 8
beam part. In order to reduce its influence, a set-up was built in which the thin beam is vertical and the hammer excites the structure in a horizontal plane.
4. EXPERIMENTAL DATA
4.1. THE ECL SET-UP In order to collect useful data for the different identification methods, the structure was instrumented with four accelerometers and one force sensor. The type and the location of the sensors are given in Table 3. The whole structure is excited by an electromagnetic shaker located at position C0 (at 63 mm from the clamping) close to the sensor C4, which allows to measure the co-located transfer function of the system. The position of the shaker was chosen in order to minimise the influence of the dynamical response of the structure and therefore facilitate the control of the force level. 4.1.1. Frequency data measurement The first kind of measurement carried out was up and down swept sine tests. These data correspond to an harmonic excitation whose level is controlled. During this test only the
199
PRESENTATION OF THE ECL BENCHMARK
Table 4 Force levels for frequency data Level no.
Force (N)
1 2 3 4 5
2 3 5 9 11
Figure 3. Frequency response of channel C1 (test 5), x-axis: frequency (Hz), y-axis: transfer function ((m/s2)/N).
first harmonic of the response is measured. The frequency range of excitation was 8– 500 Hz with a sampling frequency equal to 250 mHz. Between each frequency step a period of 10 cycles is expected to suppress the transient response, and 10 cycles are used to estimate the amplitude and the phase of the response. Several input levels were used to obtain the up and down frequency responses (Table 4): In Fig. 3, a jump phenomenon associated to the first mode can be observed. Even for the smallest level of force this jump is still present. Figure 4 shows the evolution of this jump vs the level of excitation. In Fig. 5, it can be noticed that the second mode does not exhibit any jump but slightly moves down as if a softening effect was present. The hardening phenomenon observed for the first mode is consistent with the non-linear characteristics of the thin beam. On the other hand, the evolution of the second mode is probably due to a phenomenon of another kind.
200
F. THOUVEREZ
Figure 4. Frequency response of channel C1 (tests 1 and 5) near the first mode, x-axis: frequency (Hz), y-axis: transfer function ((m/s2)/N).
Figure 5. Frequency response of channel C1 (tests 1 and 5) near the second mode, x-axis: frequency (Hz), yaxis: transfer function ((m/s2)/N).
PRESENTATION OF THE ECL BENCHMARK
201
Table 5 Tension levels for time data Level no.
Tension (V)
1 2 3 4 5
1 4 6 8 10
Figure 6. Output power spectral density of channel C1 (tests 1–5), x-axis: frequency (Hz), y-axis: acceleration (m/s2).
4.1.2. Time data measurement Random excitations were also used to obtain time data. Those tests were carried out to collect large continuous time blocks. A first set of data was then collected, including 202 752 samples per channel (five channels in total). Another measurement with a reduced number of sensors was made, providing 342 016 samples per channel (for sensors C0, C1, C4). The measurements were realised without controlling the level of excitation as in the frequency tests. The random level was adjusted only by means of the knob of the power amplifier driving the shaker. The actual force applied to the structure was measured by the force sensor. Different tests were obtained by changing the applied tension from 1 to 10 V (Table 5). The sampling rate of those data was equal to 1024 Hz and no window was applied. The frequency range of the white-noise excitation was 0 to 1024 Hz. The frequency response and the power spectral density (PSD) of the measurement were also computed. As can be
202
F. THOUVEREZ
Figure 7. Input power spectral density (tests 1–5), x-axis: frequency (Hz), y-axis: acceleration (m/s2).
seen in Fig. 6, the first mode behaves as if the system is hardening and the second mode as if it is softening. These remarks are in agreement with the behaviour observed in the frequency domain. Figure 7 shows the PSD evolution which is not flat due to the dynamical effect of the structure response. Note also the presence of some harmonics (especially at twice the frequency of the first resonance) in the frequency responses. 4.2. THE ULG SET-UP In order to collect data for a time-domain identification method, seven accelerometers evenly spaced on the beam were used to measure the response. The structure was excited with a hammer impact at coordinate number 6 and the resulting force was measured. The time response of the system was computed over a time period of 1.34 s with a sampling frequency of 5120 Hz.
5. CONCLUSION
The tests carried out on this benchmark gave us a large number of data in both the time and frequency domains. The overall quality of data sets collected allows to tackle precisely the non-linear behaviour of the test set-up. These measurements have shown a good coherence between them. Indeed, we observed two main phenomena on these different sets of data. The first one is a hardening effect of the first mode, as predicted by the theory. This effect on the overall dynamics of the beam is very important it leads to an increase of more than 30% of the first resonance. The second one is associated to a softening of the second mode whose effects are yet less important on dynamics.
PRESENTATION OF THE ECL BENCHMARK F. Thouverez Ecole Centrale de Lyon, Lyon, France. E-mail: [email protected] (Received 1 October 2002, accepted in revised form 1 October 2002) Many different non-linear identification methods have been developed so far. Most of them proceed from various experimental or numerical approaches. Inside the cost action F3 project, one common benchmark was chosen to test several identification methods in order to compare their efficiencies and their specificities. To do so, a large amount of data (time data, frequency data) has been collected and proposed to researchers. The benchmark, which has been defined, involves one single local non-linearity coupled with a simple linear structure. # 2003 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
The benchmark proposed by the Ecole Centrale de Lyon (ECL) consists of a clamped beam with a local non-linearity at the end. Two different set-ups were built : the first one at ECL [Fig. 1(a)] and the second one at the University of Li"ege (ULg) [Fig. 1(b)]. The coupling between a linear flexible structure and a local non-linearity is a classical problem in the industry. Indeed, most links like solders, screws, bushings, and dampers are often the main source of non-linear effects in the dynamical response of the whole structure. The non-linear identification techniques have to be able to estimate such non-linear components. In order to test the efficiency of these procedures, we built-up a simple test set-up avoiding this way any complex geometrical description problem, but including a local non-linearity. The choice of a simple non-linear structure permits a clear understanding of the non-linear phenomena. It gives us also a good experimental comparing data for the development of efficient numerical models.
2. DESCRIPTION OF THE STRUCTURE
The test rig is composed of two parts as illustrated in Fig. 2. The first part (denoted A) which is the main component of the structure is a steel beam and the second part (denoted C) is a thin beam which enables the non-linear behaviour. The main beam is assumed to be clamped at one of its end. 2.1. THE ECL SET-UP [FIG. 2(A)] In the ECL set-up, the thin beam (part C) is clamped at one end and linked to the beam at the other end through a coupling element (denoted B). Part B was designed to avoid any slipping between both beams. For this purpose, pins have been realised and included in the element on both sides. The same locking solution was applied to the clamping. Screws maintain the whole assembly. The mechanical and geometrical characteristics of the different parts are given in Table 1. 0888–3270/03/+$35.00/0
# 2003 Elsevier Science Ltd. All rights reserved.
196
F. THOUVEREZ
Figure 1. ECL benchmark: (a) the ECL set-up; (b) the ULg set-up.
2.2. THE ULG SET-UP [FIG. 2(B)] In the ULG set-up, the thin beam (part C) is clamped at one end. There is no coupling element as for the ECL set-up but the thin beam is clamped into the main beam using screws. The mechanical and geometrical characteristics of this set-up are given in Table 2.
3. THE LOCAL NON-LINEARITY
The non-linear behaviour of the structure may be enabled by the thin beam when large displacements occur during the test. In the ECL set-up, the assembly is pre-loaded with an axial force. The value of this preload is unknown but it determines the linear and non-linear characteristics of the element. A theoretical calculation demonstrates that the non-linearity is cubic (Duffing spring) and comes from the non-linear deformation tensor. Indeed, the non-linear part of the strain is no more negligible due to a coupling between the axial deformation and the transversal one. Some quadratic effects could also be taken into account, assuming that the gravity has some influence on the initial position of the structure. In the contribution of Bellizi, a theoretical development is proposed to compute analytically the linear and non-linear components of this joint.
PRESENTATION OF THE ECL BENCHMARK
197
Figure 2. (a) Local non-linearity and driving point (ECL set-up). (b) Local non-linearity (ULg set-up).
In the ULg set-up, the assembly is not pre-loaded with an axial force. Due to the thin beam (part C), the effect of gravity is not negligible. Gravity is at the origin of a static deflection of the two beams. This deflection imposes a non-negligible prestress in the thin
198
F. THOUVEREZ
Table 1 ECL Set-up characteristics Young’s modulus (Pa) Part A Part B Part C
Poisson ratio
Length (mm)
Thickness (mm)
Width (mm)
Mass (gr)
0.35 0.35 0.35
593 40 57
14 20 0.5
14 30 30
907 187 6.7
2.1 E11 2.1 E11 2.1 E11
Table 2 ULG set-up characteristics Young’s modulus (Pa) Part A Part C
Poisson ratio
Length (mm)
Thickness (mm)
Width (mm)
Mass (gr)
0.35 0.35
700 40
14 0.5
14 14
1070 2.2
2.1 E11 2.1 E11
Table 3 Sensors characteristics Sensor C0 C4 C3 C2 C1
Position from clamping (mm) 63 47 220 420 605
Type
Reference
Force Acceleration Acceleration Acceleration Acceleration
BK 8200 ENDEVCO ENDEVCO ENDEVCO ENDEVCO
Total weight (g) Modal Modal Modal Modal
61-100 61-100 61-100 61-100
ICP ICP ICP ICP
21 8 8 8 8
beam part. In order to reduce its influence, a set-up was built in which the thin beam is vertical and the hammer excites the structure in a horizontal plane.
4. EXPERIMENTAL DATA
4.1. THE ECL SET-UP In order to collect useful data for the different identification methods, the structure was instrumented with four accelerometers and one force sensor. The type and the location of the sensors are given in Table 3. The whole structure is excited by an electromagnetic shaker located at position C0 (at 63 mm from the clamping) close to the sensor C4, which allows to measure the co-located transfer function of the system. The position of the shaker was chosen in order to minimise the influence of the dynamical response of the structure and therefore facilitate the control of the force level. 4.1.1. Frequency data measurement The first kind of measurement carried out was up and down swept sine tests. These data correspond to an harmonic excitation whose level is controlled. During this test only the
199
PRESENTATION OF THE ECL BENCHMARK
Table 4 Force levels for frequency data Level no.
Force (N)
1 2 3 4 5
2 3 5 9 11
Figure 3. Frequency response of channel C1 (test 5), x-axis: frequency (Hz), y-axis: transfer function ((m/s2)/N).
first harmonic of the response is measured. The frequency range of excitation was 8– 500 Hz with a sampling frequency equal to 250 mHz. Between each frequency step a period of 10 cycles is expected to suppress the transient response, and 10 cycles are used to estimate the amplitude and the phase of the response. Several input levels were used to obtain the up and down frequency responses (Table 4): In Fig. 3, a jump phenomenon associated to the first mode can be observed. Even for the smallest level of force this jump is still present. Figure 4 shows the evolution of this jump vs the level of excitation. In Fig. 5, it can be noticed that the second mode does not exhibit any jump but slightly moves down as if a softening effect was present. The hardening phenomenon observed for the first mode is consistent with the non-linear characteristics of the thin beam. On the other hand, the evolution of the second mode is probably due to a phenomenon of another kind.
200
F. THOUVEREZ
Figure 4. Frequency response of channel C1 (tests 1 and 5) near the first mode, x-axis: frequency (Hz), y-axis: transfer function ((m/s2)/N).
Figure 5. Frequency response of channel C1 (tests 1 and 5) near the second mode, x-axis: frequency (Hz), yaxis: transfer function ((m/s2)/N).
PRESENTATION OF THE ECL BENCHMARK
201
Table 5 Tension levels for time data Level no.
Tension (V)
1 2 3 4 5
1 4 6 8 10
Figure 6. Output power spectral density of channel C1 (tests 1–5), x-axis: frequency (Hz), y-axis: acceleration (m/s2).
4.1.2. Time data measurement Random excitations were also used to obtain time data. Those tests were carried out to collect large continuous time blocks. A first set of data was then collected, including 202 752 samples per channel (five channels in total). Another measurement with a reduced number of sensors was made, providing 342 016 samples per channel (for sensors C0, C1, C4). The measurements were realised without controlling the level of excitation as in the frequency tests. The random level was adjusted only by means of the knob of the power amplifier driving the shaker. The actual force applied to the structure was measured by the force sensor. Different tests were obtained by changing the applied tension from 1 to 10 V (Table 5). The sampling rate of those data was equal to 1024 Hz and no window was applied. The frequency range of the white-noise excitation was 0 to 1024 Hz. The frequency response and the power spectral density (PSD) of the measurement were also computed. As can be
202
F. THOUVEREZ
Figure 7. Input power spectral density (tests 1–5), x-axis: frequency (Hz), y-axis: acceleration (m/s2).
seen in Fig. 6, the first mode behaves as if the system is hardening and the second mode as if it is softening. These remarks are in agreement with the behaviour observed in the frequency domain. Figure 7 shows the PSD evolution which is not flat due to the dynamical effect of the structure response. Note also the presence of some harmonics (especially at twice the frequency of the first resonance) in the frequency responses. 4.2. THE ULG SET-UP In order to collect data for a time-domain identification method, seven accelerometers evenly spaced on the beam were used to measure the response. The structure was excited with a hammer impact at coordinate number 6 and the resulting force was measured. The time response of the system was computed over a time period of 1.34 s with a sampling frequency of 5120 Hz.
5. CONCLUSION
The tests carried out on this benchmark gave us a large number of data in both the time and frequency domains. The overall quality of data sets collected allows to tackle precisely the non-linear behaviour of the test set-up. These measurements have shown a good coherence between them. Indeed, we observed two main phenomena on these different sets of data. The first one is a hardening effect of the first mode, as predicted by the theory. This effect on the overall dynamics of the beam is very important it leads to an increase of more than 30% of the first resonance. The second one is associated to a softening of the second mode whose effects are yet less important on dynamics.