improving the vibratory behavior of a flexible...
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IMPROVING THE VIBRATORY BEHAVIOR OF A FLEXIBLE PLATE USING COMBINED EXPERIMENTAL MEASUREMENTS AND F.E. MODELLING
F. Mabrouki1 M. Thomas1 A.A. Lakis2
1 Department of Mechanical Engineering, École de technologie supérieure, Montreal (QC), Canada 2 Department of Mechanical Engineering, École Polytechnique, Montreal (QC), Canada
ABSTRACT In this paper, we will be combining experimental modal analysis and finite-element modeling to re-design a base plate of a mechanical system made up of steel sheets in order to reduce its level of vibrations, which are responsible for undesirable noises. The new model to be designed must be longer than the current one, and it was agreed with our industrial partner that the solution to be proposed must not generate additional production costs, i.e., we must only modify the shapes of the base plate to rigidify it, without adding any damping materials. To come up with a new design for the base plate having the same defects, it was necessary to study the vibratory behavior of the current model in order to know the causes of the high vibratory level. We thus performed experimental modal and harmonic analyses of the entire system and of some of its parts separately, in order to progressively build and update a reliable finite-element model. After observing the mode shapes of the base plate, we came up with four possible new designs. The finite-element model of each new design has undergone both a modal and a harmonic analysis. Some assumptions were made when simulating the multi-frequency excitation force produced by the electric motor on the finite-element model. Using the results of these simulations, we could classify the various models according to a specific criterion: the best conception was useful as the basis for the design of a final model, because we noticed that certain forms or certain present on the other models contribute to a reduction of the amplitude of the vibrations. The final model comprises a synthesis of the strong points of four previously designed models. The modal and harmonic analysis carried out on the finite-element model of the last design showed that it presented a better vibratory behavior compared with the others. 1. INTRODUCTION
An experimental analysis of the structure vibration generated by a motor revealed a repetitive multi-frequency harmonic force excitation that was not easy to simulate using the finite-element method. Consequently, it was necessary to develop a new strategy in order calculate the response of that model to a multi-frequency excitation by using the harmonic module. The finite-element software used for simulations does not offer a tool to simulate a multi-frequency excitation with different damping values at each frequency, and thus it was necessary to make some assumptions on the different modes of the structure under study, and
to use the tool available on the finite-element software such as to create or simulate the excitation produced by the electric motor.
2. SYSTEM PRESENTATION
2.1. Presentation of the system’s design
The actual base plate (Figure 1) is produced from a steel sheet 0.71 mm thick, and features three large-size square embosses on each side, as well as a large oblique surface in the middle which extends to two-thirds of the base plate, and ends with two horizontal cascading embosses. The oblique surface is connected to the three embosses (on each side) by two oblique side surfaces. All these embosses, with their different shapes, are used to rigidify the base plate and to enable it to receive some devices, such as an electric motor. A special form is pressed on the side part of the base plate, on which is encased a plastic part used as the housing for the motor’s fan, which comprising a part of the air exhaust system. All the parts are made from pressed steel sheets of varying thickness.
Figure 1: The base plate
2.2. Presentation of the electric motor
The electric motor is an AC current 0.25 HP asynchronous motor with a rotating speed of 30 H. Its construction is very simple and its U-shaped frame is composed of two steel plates of different thicknesses that are assembled. The electric motor is supported by two bearings on two opposite circular holes applied to its frame. The following photo shows the electric motor:
Figure 2: The electric motor
3. DYNAMIC STUDY OF THE BASE PLATE 3.1. Building and simulation of the finite-element model of the base plate
The construction of the finite-element model of the base plate took several steps. The comparison of the results of its numerical simulation (natural frequencies and mode shapes) to those of the experimental modal analysis enabled us to readjust it gradually. We tried to adopt the most simplified possible representation of the base plate geometry by withdrawing some details whose absence does not influence the exactitude of the model, such as the small holes intended to receive the screws or the curves resulting from pressing operations. We chose boundary conditions that best simulated the frame of the base plate. Figure 3 shows the finite-element model of the base plate:
Figure 3: The finite-element model of the base plate The simulation for a modal analysis was carried out in order to calculate the natural frequencies and mode shapes within a 250 Hz range. The results will be compared with those obtained through the experimental modal analysis.
3.2. Experimental modal analysis of the base plate
In order to validate the theoretical model, it is necessary to measure the modal characteristics of the structure since damping cannot be predicted. Consequently, an experimental modal analysis of the base plate dissociated from the whole system was conducted. The results of this step will be compared with those of the numerical modal analysis of the base plate discretized by a finite-element method in order to update it. The impact technique was used. We used a two-channel acquisition board with an accelerometer and an impact hammer instrumented with a force transducer. A sufficient number of points covering all the horizontal parts of the base plate (77 points) were considered in order to allow a precise drawing of mode shapes. Figure 4 shows the amplitude of a transfer function between the acceleration and the force used to extract the natural frequencies of the base plate.
Figure 4: Natural frequencies of the base plate
3.3. Comparing results and mode shapes
Table 1 contains the natural frequencies and damping ratios compared to those (natural frequencies only) obtained by measurements.
Table 1: Natural frequencies and damping ratios of the base plate
Order Natural Frequencies
(Hz) Measurements
Natural Frequencies (Hz)
F.E. model
Error (%) Damping ratio
1 28 35 -22 % 0.048 2 60 57 5 % 0.017 3 91 90 1 % 0.036 4 102 110 -73 % 0.019 5 120 119 1 % 0.013 6 133 131 2 % 0.011 7 152 160 - 6 % 0.008 8 170 189 -11 % 0.005 9 192 194 -1 % 0.009 10 211 209 1 % 0.009
The correspondence between the results obtained by measurements and those obtained through the numerical simulation of the finite-element model of the base plate was established by comparing the mode shapes. Figures 5 and 6 show the comparison between some mode shapes. It is important to note that for the
experimental modal analysis, we only took measurements on the horizontal part of the base plate.
Experimental Numerical
Figure 5: Experimental and finite-element mode shapes of the second mode
Experimental Numerical
Figure 6: Experimental and finite-element mode shapes of the third mode
4. STUDY OF THE ELECTRIC MOTOR After having studied the modal behavior of the base plate, the next step consisted in studying the vibratory behavior generated by the electric motor, which is the source of the excitation that the base plate undergoes. It is thus appropriate to identify its characteristics in order to determine their incidence on the form of the excitation.
4.1. Modal and harmonic analysis of the electric motor
The vibratory characteristics of the electric motor have a direct incidence on the form of the excitation, i.e., on the excited frequencies and the magnitude of the vibrations transmitted to the base plate. The electric motor was attached rigidly via its fixing plate, and vibration measurements were taken on two points very close to the two bearings of the motor. In order to understand the vibratory behavior of the electric motor, we carried out both a harmonic and a modal analysis. Both analyses were
made within the 200 Hz range. We were interested in the vibratory characteristics specific to the construction of the motor, such as its rotational speed and its harmonics and the pulsated couple at 120 Hz.
4.1.1. Results of the harmonic analysis
Figure 7 shows the magnitude of the speed vibration in mm/s measured near the drive shaft in the horizontal direction. We can observe the high magnitude at 30 Hz, which is the rotational frequency of the motor and at all its harmonics, especially at 120 Hz, which characterizes the pulsated couple.
Figure 7: Magnitude of the vibration near the drive shaft, measured in a horizontal direction
4.1.2. Results of the modal analysis
The modal analysis was aimed at determining the natural frequencies of the motor in order to know whether some of them coincide with its rotational frequency or with one of its harmonics, since that can result in the amplification of the amplitude of the excitation at this frequency. As with the harmonic analysis, we took measurements near the two bearings in the horizontal and vertical directions. Figure 8 shows the FRF used to determine the natural frequencies.
Figure 8: Natural frequencies of the electric motor Table 2 recapitulates all the natural frequencies of the motor.
Table 2: natural frequencies of the electric motor
Order Natural Frequency (Hz) 1 17 2 21 3 41 4 60 5 88 6 110 7 128 8 153 9 166
We can see that the electric motor has a natural frequency (60 Hz) that is close to the first harmonic of its rotational frequency. The amplification of the excitation at 60 Hz is extremely probable. This will be confirmed during the experimental harmonic analysis of the base plate.
4.2. The finite-element model of the motor
In order to be able to model the effect that the electric motor has on the base plate, we created a simplified finite-element model of the unit formed by the motor and its U-shaped support. This model takes into account the physical characteristics of the
electric motor as the mass and the moments of inertia (Ixx and Iyy) as well as the mode of attachment of the support with the base plate. The electric motor is modeled by two coaxial beams of equal lengths joining the two opposite surfaces of the support. Each beam has a diameter equal to that of the driving shaft of the motor. A concentrated mass with the values of the motor’s mass (3.855 kg) and those of both moments of inertia Ixx (4.14 kg m2) and Iyy (1.01 kg m2) is added to the joining point of the two beams. This model will be simulated for harmonic or modal analysis in order to update it by comparing the simulation results to those of the experimental modal and harmonic analyses already made. Figure 9 shows the meshed finite-element model.
Figure 9: The finite-element model of the electric motor
5. STUDY OF THE WHOLE SYSTEM
5.1. Experimental modal analysis In this step we carry out an experimental modal analysis of the base plate assembled to the other parts (the electric motor, the two lateral panels, the back and the front parts). The results of this step will be used to update the finite-element model of the whole system. We used the same hardware as for the base plate to take the measurements on the same number of points (77 points) within the same frequency range. Figure 10 shows the magnitudes of some transfer functions used to extract the natural frequencies of the base plate:
Figure 10: Natural frequencies of the whole system measured on the base plate.
The measured natural frequencies and damping ratios will be compared to those obtained through the numerical simulation of the finite-element model.
5.2. The finite-element model
To numerically simulate the base plate attached to the system to which it belongs, we used the same base plate model shown above, and added the other part (lateral panels, back and front parts); the finite-element model of the electric motor was added by creating dynamic coupling to join its support to the base plate. Figure 11 shows the finite-element model of the whole system.
5.3. Comparing results and mode shapes Table 3 contains the natural frequencies and damping ratios compared to those (natural frequencies only) obtained through the numerical simulation of the finite-element model.
Figure 11: The finite-element model of the whole system Table 3: Natural frequencies and damping ratios of the whole system measured on the base
plate.
Order Natural Frequencies (Hz)
Measurements
Natural Frequencies (Hz)
F.E. model
Error (%) Damping ratio
1 13 14 -5 % 0.027 2 24 21 13 % 0.017 3 29 33 -13 % 0.015 4 37 41 -13 % 0.015 5 54 60 -12 % 0.009 6 99 103 -4 % 0.005 7 108 115 -6 % 0.007 8 135 142 -5 % 0.009 9 157 161 -2 % 0.005 10 198 204 -3 % 0.003 11 208 208 0 % 0.003 12 229 233 -2 % 0.005
The correspondence between the results obtained by measurements and those obtained through the numerical simulation of the finite-element model of the base plate attached to the system was established by comparing the mode shapes. Figures 12 and 13 show the comparison between some mode shapes. It is important to note that for the experimental modal analysis, we only took measurements on the horizontal part of the base plate.
Experimental Numerical
Figure 12: Experimental and finite-element mode shapes of the sixth mode
Experimental Numerical
Figure 13: Experimental and finite-element mode shapes of the twelfth mode
6. THE NEW DESIGNS It was agreed with our industrial partner that the solution to be proposed must not generate additional production costs, i.e., we must only modify the shapes of the base plate to rigidify it without adding any damping materials.
6.1. Weak points in the current design (points to change) Both studies of the base plate (separately and attached to the whole system) allowed us to discover the weak points of the current design. To improve the vibratory amplitude, it was necessary to change the sizes and layout of some surfaces and to add new reinforcements. We therefore decided to remove the two oblique side surfaces and create other forms to rigidify the base plate. It was thus necessary to create two longitudinal ribs. We also decided to suppress the large oblique surface at the middle of the base plate and to replace it with small separated plane surfaces. It
was similarly important to reinforce all embosses because we noticed large vibration magnitudes at their centers. Figure 14 shows the weak points on the current base plate:
Figure 14: Weak points of the current base plate 6.2. Presentation of the four new designs
We created four possible designs of a new base plate, each of which comprises different geometrical details in order to allow possible improvements in its vibratory behavior. The following are brief descriptions of the construction of each design:
• The first model to be evaluated presents a long version of the existing base-
plate, but with 4 horizontal embosses on each side (Figure 15). This model will simply enable us to see the effect of lengthening the existing model.
• In the second version, we added braces to reinforce all the horizontal embosses and the oblique surface in the middle section of the base plate (Figure 16). Because we had already noticed by observing the mode shapes of the current model that these surfaces present maximum amplitudes (in the form of bellies) for some excitation frequencies of the electric motor, we put just 3 embosses on each side.
• The third design presents a significant change compared to the current model (Figure 17). We replaced the two side oblique surfaces with two longitudinal ribs as a solution for rigidifying the base plate. The oblique surface at the middle of the base plate was replaced by small plane surface separated by
Oblique surface
Oblique side surface
Horizontal emboss
transversal ribs. We put four horizontal embosses on each side of the base plate.
• The fourth design was directly inspired by the third one (Figure 18). We connected the successive horizontal embosses with a bridge .The purpose of these bridges will be to limit the bending vibrations.
Figures 15 to 18 show the four evaluated designs.
Figure 15: The long version Figure 16: The long version with braces
Figure 17: The new design Figure 18: The new design with low bridges
6.3. Simulation of the four designs and results (determination of the dynamic forces)
For each model studied, the numerical simulation was carried out by assembling the base plate to the other parts (motor, front, back and side panels). We performed a modal analysis to compute the mode shapes at the frequencies close to the rotational frequency of the motor and to its harmonics. We developed a strategy involving the use of a harmonic analysis to determine the best model of the four by computing the displacement brought for each one compared to the current one. The comparison of the models was based on the observance of the lowest RMS value of the response to the harmonic excitation produced by the electric motor. In order to achieve our aim, it was necessary to find a way to simulate the response of the finite-element models of these different designs due to a multi-frequency excitation force. Since FE harmonic analysis does not allow the application of more that one frequency at a time, we developed a new strategy for computing the RMS value at each plate location. The values of the excitation dynamic forces were estimated at the motor’s rotational frequency and at its harmonics by comparing the response obtained at the structural point showing the maximum RMS value as measured on the base plate with the calculated response (from the finite-element model of the current base plate) due to an excitation with a constant arbitrary force at the same point. Figure 19 shows the measured response at the point having the maximum amplitude on the base plate.
Figure 19: Spectrum of the measured response on the point having the maximum
amplitude on the base plate
We can see on the curve above that the amplitude at 60 Hz is greater than that at 30 Hz, which is due to the natural frequency of the electric motor at 60 Hz. The analysis
allowed us to establish the force spectrum. The estimated dynamic forces used for computation depend on the location of the measurement and on modal damping. Each force at each frequency was applied to the structure one at a time in order to compute the response at each excitation frequency. All the responses computed at each frequency were combined in the frequency domain in order to estimate the RMS value at each location and for each design. Table 4 recapitulates the dynamic forces used in the numerical simulations.
Table 4: Dynamic forces to be used in numerical simulations
Frequency (Hz) 30 60 90 120 150 210 Damping ratio 0.015 0.009 0.005 0.009 0.005 0.003 Measured amplitudes (microns) 14 18 5 4 0.5 0.2
Dynamic forces (N) 39 5 2 4 0.2 17
We did not take into account the 180 Hz frequency because the amplitudes observed at this frequency (in all the spectra of the measured responses) were very low. For the numerical simulation, we assumed the modes of the different models to be independent of each other, i.e., the dynamic force used at a given frequency does not excite another mode.
Once the simulations were done, we took measurements of the harmonic response on the points marked by black dots on the model drawings (see Figures 15 to 18). These points were determined after observing the mode shapes of each model at the frequencies close to the motor’s excitation frequencies. We then looked for the locations showing the maximum response amplitudes by comparing the RMS values to the responses measured on points belonging to the same model (The ‘maximum amplitude’ locations are marked with red circles in Figures 15 to 18). Using the results of the numerical simulations, we can classify the various models according to the criterion referred above. Because we noticed that certain forms or certain details present on the other models contribute to reduce the amplitude of the vibrations, the final model comprises a synthesis of the strong points of the four models previously designed. The modal and harmonic analysis carried out on the finite-element model of the last design showed that it presented a better vibratory behavior as compared to the other designs. Table 5 recapitulates the results of the numerical simulations as well as the improvement in the vibratory behavior of each model as compared to the current one.
Table 5: Comparison of results
Model Maximal RMS values (microns) Improvement (% ) initial model 28 Long version 35 -24 % Long version with braces 33 -16 % Model with low bridges 25 12 % Model without bridges 23 18 % Final design 22 21 %
Figure 20 shows the final design
Figure 20: The final conception
7. CONCLUSION
We applied combined experimental modal analysis and finite-element modeling to build and update a finite-element model of a base plate of a mechanical system made up of flexible steel sheets without increasing the production cost and without adding any extra components. Some pieces of the system were subject to some measurements in order to identify the effects of their characteristics on their vibratory behaviors. As an F.E harmonic module is not well-suited for analyzing response to complex excitation frequency spectra, which is usually encountered in machinery vibration, a new strategy was developed in order to use commercial FE software. As excitation force amplitudes were unknown, we used the results of a harmonic analysis computed at each excitation frequency, one at a time, and at the location showing the maximum RMS vibration, and compared each response with the results of the experimental harmonic analysis in order to estimate the
dynamic forces at each frequency as produced by the electric motor. Each response computed at each frequency was combined in the frequency domain in order to obtain the RMS value. This procedure was applied to the four designs in order to compare their resulting displacements as possible alternatives to the current one. We can see that the vibratory behaviors of a long version and of one with braces do not represent any improvement compared to that of the existing model because these two versions are only long versions of the existing one. We can however note that the addition of the braces improves the vibratory behavior. Designing a base plate with small embosses and two ribs crossing the base plate longitudinally improves the vibratory behavior of the base plate by 21% by reducing the amplitude from 28 microns to 22 microns; however, the simple fact of adding more reinforcements (low height bridges between embosses) did not have the desired effect.
ACKNOWLEDGMENTS
The authors thank the CRSNG and FQRNT programs for their financial support. REFERENCES
1. Dascotte E, Linking FEA with test, Sound and vibration, April 2004, pp 12-17 2. Ewins D.J.1984 Modal testing: Theory and practice. Letchworth, England: Research
Studies Press. 3. Hopkins R. N., Carne T.G., Dohrmann C.R., Nelson C.F. and O’gorman C.C., Combining
test-based and finite element-based models in NASTRAN, Sound and vibration, April 2004, pp 18-21.
4. Sayer R.J., Finite element analysis-numerical tool for machinery vibration analysis, Sound and vibration, May 2004.
5. Thomas M, February 2003, Fiabilité, maintenance prédictive et vibrations de machines, Éditions ETS, 616 pages.
BIOGRAPHY Ø Marc Thomas has been a professor of Mechanical Engineering at the École de
technologie supérieure (Montreal) for the last 12 years (www.etsmtl.ca). He is president of the ACVM (Quebec Chapter) and the author of the book: Fiabilité, maintenance predictive et vibrations de machines. He had previously acquired a wide industrial experience as the group leader at the Centre de recherche industrielle du Québec (CRIQ), for 11 years.
Ø Aouni.A. Lakis is professor of Mechanical Engineering at the École
Polytechnique (Montreal) and a leader in fluid-structure interaction and time frequency analysis.
Ø Farid Mabrouki is a Master’s student at the École de technologie supérieure
(Montreal).