Support Systems for Developing System Models

Transcription

1 Support Systems for Developing System Models Hasan G. Pasha, Karan Kohli, Randall J. Allemang, David L. Brown and Allyn W. Phillips University of Cincinnati Structural Dynamics Research Lab (UC-SDRL), Cincinnati, Ohio, USA Abstract. From a system modeling perspective, it is important to develop an accurate mathematical model of a structure. Such a model can be built with measured frequency response functions obtained from a modal test. The test structure can be subjected to free, constrained or operating boundary conditions. Setting up a structure with either constrained or operating boundary conditions is impractical/prohibitive. It is easier to approach free boundary condition in the lab using soft supports, such as foam, shock cords, air suspension, etc. A good support system should have the least intrusion. It should not introduce nonlinearity or damping. The effectiveness of a good support system is quantified by the extent of separation of the rigid body modes from the elastic modes and a high degree of uncoupling of the dynamics of support system and the structural modes. The influence of various boundary conditions and soft support systems chosen for modeling a lightweight structures, such as Formula SAE racecar space frame, a rectangular steel plate structure and a circular aluminum plate are discussed. Keywords. Support Systems, Modal Testing 1. Introduction In a lab environment it is tough to simulate unconstrained or completely constrained boundary conditions. However, it is relatively easier to approach unconstrained boundary conditions by supporting the test structure on a soft support system such as foam, shock cords or air rides. The primary goal of a support system is to insure that it does not add any stiffness or damping to the test structure, which could affect the dynamic characteristics of the test structure. For a pseudo unconstrained structure (nearly free-free), the rigid body modes should be observed close to 0 Hz and well separated from the elastic modes. From a system modeling perspective, it is important to develop an accurate mathematical model of a structure. Such a model can be built from measured frequency response functions in a modal test. However, in the process of setting up and exciting the test structure, acquiring and processing the data, several errors can creep in. As mentioned earlier, it would be ideal to test a completely unconstrained structure. However, since the structure has to be supported in some manner to be able to conduct the testing, it is extremely important to analyze the effectiveness and suitability of support systems for conducting modal testing on a particular structure. The primary requirement is that the support system should be non-intrusive, in that it should not introduce any additional mass or stiffness to the test structure. It should also not introduce nonlinearity or damping. In addition, the dynamics of the support system should be isolated from the dynamics of the test structure. It is also desirable to be able to build an inexpensive support system without the need for sophisticated gadgets where possible. The following sections focus on reviewing the basic guidelines for selecting support systems for conducting modal tests in a lab environment. The various supports available for modal testing are briefly discussed. The results obtained with different support systems while testing structures such as, a Formula SAE racecar space frame, a rectangular steel plate and a circular aluminum plate, are presented. During the setup and testing, relatively inexpensive supports were used that yielded good quality data. 2. Guidelines for selecting support systems There are some general guidelines in selecting support systems Separation of elastic modes from rigid body modes. The rigid body modes must be well separated from the elastic modes. A frequency separation ratio (ratio of the lowest elastic mode frequency to that of the highest rigid body mode) of 10 is highly preferable, but it is hard to achieve. While conducting a modal test, a frequency separation ratio of at least 3 to 5 is recommended and used by test engineers [1]. The first few modes that are close to the rigid body modes are affected the most due to a stiff support system.

2 2.2. Location of supports. The location of the supports in relation to the mode shapes of the structure can impact the results significantly. Ideally, if the structure is supported on its nodal points, the supports have negligible support structure interaction. Due to the modal distribution, this is not possible in general. A compromise on the location of supports should be reached. If the modes of interest are identified, the supports can be located at their nodal points Isolation of test structure from the support system and external noise sources. The dynamics of the support system can interact with the test structure. For instance, the support system can act as a vibration absorber if its frequency is in the vicinity of any of the elastic modes of the structure for that particular mode. In addition, vibration from external noise sources can be transmitted to the test structure through the support system. During the test setup, the response of the support system should be examined to ascertain that the support motion is negligible. 3. Types of support systems Various supports are available to support structures while performing modal testing. The decision to select the support system is driven by the weight of the structure, ease of mounting the test structure and to a certain extent the cost and availability of the components. A direct relationship is evident between the support stiffness and the modal frequencies. Generally, modal frequencies increase when supported by stiffer components like springs as compared to softer support systems made of foam. Commonly used support system components are shown in Fig Shock cords. The shock cords are flexible and simple, which makes them suitable to suspend a light weight test structure. Shock cords are available in various sizes and stiffness. The test structure is generally suspended using shock cords of negligible mass and stiffness as compared to the test structure. Shock cords practically contribute negligible stiffness in the transverse direction. Thus the test structure is very weakly constrained in the transverse direction, which is ideal for measuring the elastic modes of interest. The position of shock cords generally has negligible influence on modal frequencies but can affect the damping estimates Soft foam. Soft foam is relatively inexpensive and can be used to support test structures in a modal test. It is helpful in isolating the test structure the influence of external noise sources. In comparison to shock cords, which are generally considered weightless, foam could add more damping to structure. It is difficult to predict the damping effects. Foam supports can also have more mass loading effect as compared to shock cords. However, in the case of shaker tests, foam supports are considered advantageous to isolate the shaker excitation of a common shaker-structure base such as the floor Coil springs. Coil springs can be used to support test structures. Springs are available in various sizes and stiffness. They are relatively stiffer than other typically used supports. They can also cause clicking noise or rattle, which could be transmitted to the test structure Air springs. Air springs are typically used in modal testing of heavy structures, such as auto-bodies. They offer flexibility in terms of adjusting the support stiffness and typically result in very low rigid body frequencies, in the order of 3 Hz. They offer relatively higher degree of isolation compared to other types of supports. They are compact and light weight. Racquet balls and tennis balls could be used as air springs for supporting lightweight structures. 4. Experimental examples Relatively light weight structures, such as Formula SAE racecar space frame, a rectangular steel plate and a circular aluminum plate were tested with various support systems and configurations. The results obtained are discussed in the following sections. Soft support systems including foam, bubble wrap, springs, shock cords and air springs were used to simulate unconstrained boundary condition. The constrained boundary condition was also simulated for a structure. The effectiveness of the support systems is evaluated using the mode indicator functions, mode shape animations and consistency diagrams.

3 Figure 1: Support systems used in Modal Testing 4.1. Formula SAE Racecar Space Frame. The car space frame is a welded steel structure containing various cross members, weighing about 55 lb. The objective of the modal test was to estimate the static torsional stiffness of the frame using driving-point and cross-point FRF measurements at the four shock absorber connection points. This method to estimate the static torsional stiffness is highly sensitive to the quality of the FRF measurements. Any localized stiffness could yield incorrect estimates for the static torsional stiffness value. The frame was was supported in two different configurations to simulate the unconstrained boundary condition. In the first configuration it was suspended using 2 garage door extension springs and in the other configuration it was rested on soft foam blocks as shown in Fig. 2(a) and Fig. 2(b) respectively. (a) Car frame suspended using springs (b) Car frame supported on soft foam blocks Figure 2: Formula SAE Racecar Space Frame supports In the spring suspension configuration, there were two modes observed in the Hz frequency range. However, in the foam support configuration there was only one mode observed in the frequency range, as evident from the CMIF comparison plot shown in Fig. 3. This suggests that there is a dynamic coupling of the spring support system with the frame in the suspended configuration. The supports act as a vibration absorber at the first mode and

4 the peak splits in the case of spring supports. In addition, small peaks are noticeable in the spring support system (blue curves in Fig. 3). It was also determined that the roof was excited due to an unbalanced ventilation fan. The external noise was transmitted to the car frame through the spring supports. However, the foam blocks damped out the noise and isolated the frame. The data acquired with soft foam supports was chosen to estimate the static torsional stiffness for the frame as it offered better isolation of the frame from the supports and external noise source. Figure 3: Comparison of CMIF plots for Car Frame Supports 4.2. Rectangular Steel Plate. A rectangular steel plate structure of dimensions 34 x22.5 x.25 and weight of about 52.5 lb was tested. The goal of the modal test was to obtain a modal model of the plate to calibrate the FE model of the rectangular plate. The rectangular plate was tested in various configurations with different support systems Unconstrained boundary condition testing. To simulate the unconstrained boundary condition, the plate was supported on soft foam, bubble wrap, shock cords and air springs. A limited impact test was performed to evaluate the quality of data obtained with different supports. The soft foam and bubble wrap supports did not yield good quality data. The relative error of the first elastic mode in these cases was too high. As a result, these configurations were not considered for further testing. The shock cords offered a high frequency separation ratio of between the elastic modes and the rigid body modes, which is highly preferable. The stiffness of the cords was about 0.59 lbf /in. However, the supports act as a vibration absorber for the third mode (II torsion mode at around 92 Hz) as evident from the consistency diagram shown in Fig. 7. Air springs have been traditionally used for supporting airplanes and auto bodies while conducting modal tests. Smaller sized basketballs had been successfully used for supporting sail planes in previous tests conducted by UCSDRL. The test structures typically tested in the lab as part of projects are relatively very light weight compared to airplanes or auto bodies. The goal is to come up with an inexpensive support system that could support light weight structures in a lab environment. In order to support lighter structures, such as the rectangular plate, stress balls, tennis balls and racquet balls were considered. Tennis balls are typically filled with compressed air (1 psi) and were relatively stiffer compared to stress balls and racquet balls. The stress balls tend to deform a lot and created a large contact patch (3 4 diameter) on the plate, which could introduce localized damping into the plate. Racquet balls

5 seemed to be optimum supports for the plate. With a relatively smaller contact patch (less than an 1 diameter), there were no concerns of additional damping from the balls that could affect the quality of data obtained. It was also reasonably easier to control the location of the balls without gluing them down to the plate. In comparison to the shock cords, the racquet ball supports provided a lower frequency separation ratio. However, the plate structure was well isolated from the supports. This configuration provided the best quality data. The plate was supported with the racquet balls supported close to the nodal lines of the first 4 elastic modes (shown in Fig. 4). Figure 4: Node lines for first 4 elastic modes of the Rectangular plate Table 1. Rigid body modes of the Rectangular plate Excitation Support Description Freq(Hz) Frequency Separation Ratio Impact Impact Shaker Shock Cords Racquet Balls Racquet Balls Bounce 1.4 Pitch 1.54 Roll 2.46 Roll 5.5 Bounce 8.16 Pitch 9.37 Roll 5.38 Bounce 8.05 Pitch

6 (a) Rectangular Plate Suspended by Shock Cords (b) Rectangular Plate Supported on Racquet Balls Figure 5: Rectangular plate supports (a) Roll Mode (X axis rotation) (b) Pitch Mode (Y axis rotation) (c) Bounce Mode (Z axis translation) Figure 6: Rigid body modes of the Rectangular plate Table 2. Comparison of Modal Frequencies (Relative Error %) for various support systems used for testing Rectangular Plate FE Model (Hz) Bubble Wrap (Hz) Foam (Hz) Shock cord (Hz) Stress Balls (Hz) Tennis Balls (Hz) Racquet Balls (Hz) (3.33) (1.42) (2.41) (1.36) (2.26) (1.32) (1.55) (1.10) (2.76) (1.72) (2.36) (0.75) (1.98) (0.80) (1.45) (1.13) (4.35) (2.41) (2.84) (0.63) (2.30) (0.54) (1.63) (1.18) (1.69) (0.80) (0.62) (1.54) (0.87) (0.43) (0.59) (0.59) (5.86) (1.78) (0.72) (1.65) (0.69) (0.74) (0.50) (0.46) (1.98) (1.16) (0.48) (1.56) (0.76) (0.43) (0.32) (0.23)

7 Figure 7: Rectangular plate suspended using Shock Cords Consistency Diagram Constrained boundary condition testing. The plate was also tested in a constrained boundary configuration. The plate was clamped to ground at two locations near one of the edges using steel spacers, which in turn were grounded to a huge isolated mass as shown in Fig. 8. (a) Constrained plate setup (b) Clamping supports and location Figure 8: Constrained plate In order to evaluate the effectiveness of the constrained boundary conditions employed in the Measured Constrained structure, the CMIF of the measured data was compared with that of the modeled data. The data from an unconstrained boundary modal test with driving-point FRF measurements at locations corresponding to the clamping locations was obtained. Impedance modeling technique was applied to ground the clamping locations using very stiff springs to obtain a Modeled Constrained system. Clamping constrains rotation of the plate in addition to

8 Figure 9: Comparison of CMIF plots: Constraints modeled using Impedance modeling vs measured constraining the vertical motion. This is evident from Fig. 9, the Measured Constrained system is considerably stiffer than the Modeled Constrained system Circular Aluminum Plate. A circular aluminum plate structure, shown in Fig. 10(a), of diameter 30, thickness.25 and weight of about 17.5 lb was tested. The goal of the modal test was to perform a modal correlation of the impact test based modal model with the FE modal model. Thirty uniaxial accelerometers were mounted on the plate and thirty driving-point locations were impacted. The circular plate was supported on 3 coil springs placed about 10.5 away from the center and 120 apart as shown in Fig. 10(b). In order to prevent rattle, the springs were glued to the plate. (a) Circular plate with accelerometers (b) Support system Figure 10: Circular Plate The glue material is viscoelastic. As a result, there is some localized damping introduced to the structure at the support points. At the 565 Hz and 770 Hz modes, there is considerable motion at the support points as shown in Fig. 11(a) and Fig. 11(b) respectively. Due to local damping at the support points, these modes are slightly complex. This was evident by observing the mode shape animations and by examining the Modal Phase Collinearity (MPC)

9 criterion displayed in the Consistency Diagram (Fig. 12). The MPC criterion for real modes is close to 1; however, for the 565 Hz and 770 Hz modes the values were.643 and.538 respectively. (a) Mode at 565 Hz (b) Mode at 770 Hz Figure 11: Circular plate Complex Modes Figure 12: Circular plate Consistency Diagram

10 5. Conclusions It is easier to simulate unconstrained boundary condition in the lab when compared to clamped or operating boundary conditions. Certain guidelines should be followed while selecting support systems for modal testing: (1) The frequency separation ratio of 10 is highly preferred for a support system. However, it should be at least in the range of 3 to 5. (2) Based on the purpose of the modal test, the supports should be positioned at or near the node line of the modes of interest if possible. (3) While support system that offers higher frequency separation ratio is more desirable, special care should be taken to insure that the structure is isolated from the support system and external noise sources. The first few elastic modes are affected the most due to a stiff support system. Typically soft foam, shock cords and springs are used to support lightweight structures in the lab. Air springs are typically used to support heavier structures, such as auto-bodies and airplanes. Racquet balls could be used as air springs for supporting light weight structures. The mode indicator functions, mode shape animations and consistency diagrams aid in determining the effectiveness of a support system. In the case of rectangular plate structure it was evident that though shock cords provided the desirable frequency separation ratio greater than 10, they were dynamically coupled with the test structure. Racquet balls could be used as inexpensive and effective support systems for supporting lightweight structures. For the rectangular steel plate structure, they offered a frequency separation ratio of about 4.5 and a high degree of dynamic isolation. References [1] J.A. Wolf Jr., The Influence of Mounting Stiffness on Frequencies Measured in a Vibration Test, SAE Paper , Society of Automotive Engineers, Inc., [2] T.G. Carne, D.T. Griffith and M.E. Casias, Support Conditions for Experimental Modal Analysis, Sound and Vibration, June [3] S. Badshah, On Modal Testing and Analysis of Aluminum Foam: Experimental Setup and Approach, International Journal of Emerging Technology and Advanced Engineering, Volume 3, Issue 4, April 2013 [4] Literature review - The Fundamental of modal Testing, Application Note [5] R. Blevins, Formulas for Natural Frequency and Mode Shapes [6] B.J. Schwarz, M.H. Richardson, Experimental Modal Analysis, CSI Reliability Week, Orlando, FL, October, 1999