Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
A Benchmark Test Structure for Experimental Dynamic Substructuring
P.L.C. van der Valk; J.B. van Wuijckhuijse; D. de Klerk Delft University of Technology, Faculty of Mechanical, Maritime and Materials Engineering Department of Precision and Microsystem Engineering, section Engineering Dynamics Mekelweg 2, 2628CD, Delft, The Netherlands
[email protected]
Abstract In this paper a benchmark test structure for experimental dynamic substructuring is presented. The benchmark is a simple structure designed to gain insight into difficulties experienced in experimental dynamic substructuring (DS). First a brief introduction of dynamic substructuring is presented, followed by a summary of current bottlenecks in experimental DS. From these difficulties a set of requirements for the benchmark is formulated. Thereafter the design is presented and its numerical model is validated by a measurement on the fabricated benchmark. Finally a DS analysis is performed on the benchmark structure to show it’s ability to quickly verify or falsify a DS analysis.
1
INTRODUCTION
Dynamic substructuring is a very useful tool in structural dynamics and is becoming increasingly popular in the engineering society. Dividing large structures into smaller substructures has several advantages in structural dynamic analysis; first and foremost it allows the coupling of numerical substructures with experimentally obtained components in order to compute the dynamic behavior of the total system. Another big advantage is that the dynamic behavior of systems that would otherwise be too large, complex or time consuming to measure and/or simulate can be determined. More benefits of (experimental) dynamic substructuring are described in [5]. There are three domains in which dynamic substructuring can be performed; the ‘physical’ domain (using physical DoF u), the modal domain (using reduced substructures) and the frequency domain (using FRFs). In addition two types of DS can be identified; numerical dynamic substructuring and experimental dynamic substructuring. In a numerical DS analysis, only numerical substructures are coupled, these substructures can either be full or reduced FEM models (using CMS methods [1, 8, 2, 13]). Numerical DS is already well developed and accepted within the engineering community; for instance Guyan reduction and the method of Craig-Bampton are already integrated in many FE packages. Experimental DS involves both numerical substructures and substructures obtained through measurements (’experimental’ substructures). This is done in either the modal domain using CMS methods [11] or in the frequency domain using FRF coupling methods [5, 9, 3]. Experimental DS was first developed in the 1980’s and is still an interesting research field: There remain difficulties which one could encounter when performing an experimental DS analysis, these are described in section 2. There was a need for a simple and versatile benchmark test structure to investigate these difficulties, but also other phenomena (such as non-linearity, high damping, etc.) which could be present in a (sub)structure. The goal of this paper is to present this benchmark test structure for experimental dynamic substructuring. The design will be shown in Sec. 3. The validation of the FEM model will be done in Sec. 4, where also the results of the first DS analysis will be presented.
2
DIFFICULTIES IN EXPERIMENTAL DYNAMIC SUBSTRUCTURING
When performing an experimental DS analysis, there are some important issues that have to be dealt with in order to achieve accurate results. If these difficulties are not properly dealt with, significant errors might be present in the coupled system representation. All these issues are caused by the fact that it is virtually impossible to (properly) measure all properties of the different substructures. The different methods require their own steps, each step with their own difficulties. For a coupling analysis with CMS methods, a modal analysis has to be done in order to acquire the first m modes of the subsystem. When coupling techniques in the frequency domain are used, one couples directly with the obtained FRF’s,
hence no modal analysis is required [5]. A brief overview of the different issues often encountered in dynamic substructuring is given below.
Truncation errors A problem encountered when using CMS for experimental DS is modal truncation. Since only the first m modes of the (sub)structure are extracted from the measurements, the information from the higher modes is lost. This leads to a stiffer system with less degrees of freedom to deform in. The result can be improved by adding residual flexibility, this is one deflection shape approximating all higher modes. Even though shifts in resonance frequencies of the coupled system and a better result of the DS analysis can be expected, an error will still be made.
Modal Analysis Modal analysis might not always be possible, since the structure could, for instance, be non-linear, have high damping or have a very high modal density. In such cases, direct coupling using the measured FRF’s can be more appropriate.
Rigid Body Modes Rigid body mode information is essential in coupling (unrestrained) substructures using CMS. If the rigid body modes are not included, the coupling of substructures will give erroneous results. Since the structure will always move in a combination of rigid and flexible modes, this effects the entire frequency range. This is only a problem when using CMS, since measured FRF’s will always contain all the rigid body mode information.
Rotational Degrees of Freedom One of the classical problems encountered in experimental DS is the measurement and excitation of the rotational degrees of freedom. The rotational information is essential in order to obtain the full receptence matrix (1). Due to the coupling between the rotational and translational DoF, both have to be determined to give an accurate representation of the interface, neglecting this will give erroneous results on the coupled system representation, see also [6, 4]. ·
ut uθ
¸
· =
Ytt Yθt
Ytθ Yθθ
¸·
ft fθ
¸ (1)
Displacement vectors are denoted by u, force vectors by f and the transfer functions by Y . Translational information is denoted by the subscript t and rotational information by the subscript θ. There are different approaches to obtain rotational information. One can either try to measure them or reconstruct the rotational information from measured translational data. One way to do the latter is to assume that the interface has only six rigid body motions, which can be reconstructed from a minimum of six DOF on three nodes (see [16]). This method will show good results if the assumption of the rigid interface is valid.
Continuity of the interface In practical applications all interface connections are continuous surfaces, when usually only a limited number of points are measured. The continuous behavior of the interface is than approximated from this discrete number of measurement points, thus creating a truncated description of the interface motions.
Dynamics of joints Usually connections are either modeled as rigid or with linear flexible joints (see [9, 10]). In engineering practice, however, it shows that a lot of connections show non-linear dynamic behavior. An example of this non -linear behavior is the dry friction which occurs between bolted parts.
Experimental errors As the name experimental dynamic substructuring already implies, experimentally acquired data is used to describe the coupled system. It is obvious that measurement errors, can directly affect the accuracy of the DS analysis. Numerous errors can be made while performing measurements; a selection is briefly discussed below.
• Measurement noise: Since some random measurement noise is practically unavoidable, measures have to be taken to reduce it as much as possible. One effective way of minimizing random noise is to average over a large set
of measurements. The effect of random noise is described in [7]; • Sensor positioning and alignment: Anti resonances can be very sensitive to the exact location of the excitation; • Unmeasured side forces from the stinger: By misalignment or due to bending of the stinger, unmeasured side forces are introduced into the system, thus falsifying the FRF estimates; • Added mass effect: Mass will locally be added due to attached measurement equipment. This mass loading effect will alter the FRF’s; • Signal processing errors: Several sorts of signal processing errors could occur, for example due to leakage and errors from converting signals from analog to digital; • Local non-linear behavior in the (sub)structure: This is for example due to frequency dependent behavior (e.g. rubber) or dry friction between bolted connections; • Influence of the suspension: Measurements of lightly damped systems can be heavily affected due to damping of the suspension. It is also possible that the lower eigenfrequencies will shift due to the suspension stiffness.
3
DESIGN OF THE BENCHMARK STRUCTURE
From section 2 it can be concluded that there are two major challenges in improving the accuracy of the experimental DS analysis. On the one hand it is to reduce the experimental errors as much as possible in order measure the‘true’ behavior. On the other hand it is to identify and take into account all the different phenomena which occur in the total structure. Taking into account all the difficulties described in section 2 as well as the fact that structure should be very versatile, a set of requirements for the benchmark structure has been formulated. This set will be presented in section 3.1. In section 3.2 the design is presented and the design choices are explained.
3.1
Set of requirements
Connecting different substructures and elements Since the benchmark structure should be able to connect to different substructures but also different kinds of passive and active components (e.g. springs, actuators), a broad range of possible connections should be available. These interfaces have to be able to accommodate different types of connections, which also have to be detachable without causing damage to benchmark setup. From these requirements and [15] the following set of connection methods are chosen: bolted, blind riveted, glued, soldered and clamped connections.
Eigenfrequencies In order to make sure that the flexible modes of the system are well separated from its rigid modes, the first flexible mode of the system should not be below 50 Hz. Since a modal analysis of the benchmark has to be possible, it must have very low damping and eigenfrequencies should be well spaced.
Obtaining rotational information In sec. 2 the importance of rotational information is discussed. In the design there should be a section which can be assumed rigid or can be made rigid, to enable use of the EMPC method for determining rotations. In this method the interface is assumed to be a rigid section with six DoF (three translational and three rotational), which are constructed by measuring the translational DoF on three points [16].
Reproducibility An important feature of the setup is the reproducibility of results. Materials that show temperature or environmental dependent behavior or are susceptible to aging should be avoided (i.e. rubber). It is also important that the setup itself is reproducible, meaning the benchmarks should differ no more than 0.1 mm from each other.
Cost In order to allow ’destructive’ testing (e.g. welded connections) or modifications to the benchmark, the cost per piece should be kept low, therefore one can choose to order them in batches.
From these requirements it can be concluded that the DS benchmark should be a relative simple setup which shows linear behavior, so it can easily be measured and modeled. It should also be versatile and adaptive so all the phenomena mentioned above can be investigated.
3.2
Final design
Figure 1 shows the design of the benchmark structure.
Figure 1: Benchmark structure
The benchmark system is a simple but versatile structure because it is accessible in three dimensions and expandable by using multiple structures. All connections issued in section 3.1 are applicable using the pre-drilled holes and geometric features. The versatility of the structure lies not only in the fact that a single structure is in many ways adaptable and expandable with new elements and can be measured and excited in many directions, but also that multiple benchmark structures can be combined to form different geometries. Two examples are shown in figure 2.
(a)
(b)
Figure 2: a and b show two possible combinations
In figure 2a two structures are connected in such a way that a cavity forms between them, in which items can be placed (e.g. springs, dampers). The dynamic effects of these components on the experimental DS analysis than can be investigated easily since the dynamics of the main body (in this case two benchmark structures) are well known dynamically. In figure 2b another possible configuration using two benchmark structures is shown. The most important features of the benchmark are:
• A first eigenfrequency at 61.3 Hz, assuring that the flexible modes are well separated from the rigid body modes; • Well separated eigenfrequencies up to 1 kHz, with a minimum of 3 Hz separation, so all the modes can easily be found by performing a modal analysis; • Double (but distinctive) modes, due to the single symmetric geometry. By attaching other structures the setup can be made totally unsymmetrical or double/triple symmetrical; • Sections can locally be ”rigified” by attaching a rigid part to the plate, thus allowing for the implicit measurement of rotations using multiple point connections; • Linear behavior and very little damping (e.g. damping ratios less than 0,01% and mainly due to the suspension). The simplicity of the structure however is a certainty since it is made out of plain 5 mm stainless steel and can be manufactured by use of laser cutting and machinal bending. Since the whole process is governed by numerical controls the reproducibility is very high and also non expensive, enabling destructive testing at little cost. The accuracy of the manufacturing process lies within 0.1mm.
4
MEASUREMENTS
In this section two different measurements will be handled. In section 4.1 the model will be validated by modal analysis on the structure. In section 4.3 a first DS analysis will be performed by attaching a simple mass to the benchmark structure and the result will be compared to a measurement.
4.1
Model validation
In this section the numerical model of the benchmark structure will be validated by comparing it with a measurement on the structure. In order to validate the model, a few properties of the material used (stainless steel AISI 304) are given: • Isotropic material model; • Young’s modulus, E = 200 GPa; • Poisson’s ratio, ν = 0.29; • Density, ρ = 8000 kg/m3 . The first 50 eigenfrequencies and eigenmodes of the model were obtained by FEM analysis in COMSOL using the CADmodel as seen in figure 1. The measurement was performed using Laser Doppler Vibrometry (LDV). The benefit of this measuring method is that no additional mass loading effects of sensors (accelerometers) is seen in the measurement results. In figure 3 the measurement setup is displayed. In order to validate the FEM model of the benchmark, a roving hammer test was performed on the structure using a grid of 45 points. While measuring velocity at one point, the system was excited at all the 45 points in the z-direction. In order to isolate the setup from environmental disturbances, it was suspended using low-stiffness elastic bands. Averaging of the measurements was applied to reduce the random noise on the measurement results. The driving point (DP) FRF is shown in figure 4, where it is compared to the equivalent FRF synthesized from the modes obtained from the FEM model. Eigenfrequencies, eigenmodes and modal damping were obtained by modal analysis from the measurements. A Modal Assurance Criterion (MAC) analysis was performed between the measured modes and the modes obtained from the FEM model. The results are shown in table 1, together with the measured eigenfrequencies and the difference between the measured and computed eigenfrequencies. From table 1 it can be seen that within the frequency range of interest, the deviation of the eigenfrequencies of the FE analysis with respect to the measurement are within 1.4%. The MAC analysis, a more detailed and objective analysis, showed that of 17 modes, 13 have a correlation of 99% or higher. All modes have a correlation of 93% or higher.
Figure 3: Measurement setup
Measurement DP FRF Model DP FRF
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Figure 4: DP FRF of model and experiment
4.2
Rigidity
In section 2 the importance of rotational information was discussed. In order to use the EMPC method (and assume parts of the benchmark as rigid), the rigidity of the design was checked [16]. The DoF on the interface are projected onto the rigid body modes and compared to the original interface DoF (2).
Mode 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
feig,meas. (Hz) 61.3 71.3 113 189 200 237 278 370 410 539 590 648 651 900 906 1067
feig,FE vs. feig,meas. (%) -1.4% -0.7% -1.7% -0.4% -0.9% -0.5% 0.3% 0.1% 0.1% -1.4% -0.9% 0.0% 0.0% -0.7% -0.7% -0.9%
MAC (FE vs. Meas.) (%) 100 100 100 100 99 99 99 100 100 99 99 99 100 96 93 99
Table 1: Model validation results
° ° °R(RT R)−1 RT uc ° rigidness = 100% kuc k
(2)
The most rigid part of the benchmark was found to be as situated in figure 5a. Figure 5b shows the rigidity of this part over the desired frequency range.
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Rigidness (%)
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Figure 5: Most rigid section (a); rigidity (b)
Up to 400 Hz the rigidity is higher than 97%. Between 400 Hz and 650 Hz dips can be found down to 93%. From 650 Hz up to 1 kHz, the rigidity is again higher than 97%.
4.3
Numerical DS analysis
After validation of the model a first numerical DS analysis was performed and compared to a measurement on the assembled system (figure 6). After attaching a simple mass, a roving hammer test was performed using the same setup as in section 4.1. The mass was attached to the benchmark structure with 4 bolts (M5, 12.9) torqued to 9 Nm (figure 6), according to [12]. Since the bolted connections will only locally enforce compatibility between the substructures, a diameter around the bolt centerline has been computed in which the substructures are assumed to be rigidly connected. This diameter was determined to be 13.3 mm [14]. The FEM models are coupled at these interfaces and the first fifty eigenfrequencies
Figure 6: Benchmark structure (substructure 1) with added mass (substructure 2)
and eigenmodes were determined. From these eigenmodes and -frequencies a driving point FRF has been synthesized, in figure 7 this synthesized FRF is compared to the driving point FRF from the measurement. In table 2 the eigenfrequencies computed from the FEM model are compared to the measured eigenfrequencies.
measurement DP FRF substructered model DP FRF
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Figure 7: NDS, DP FRF of model and measurement
In figure 7, the small peaks at 195 and 655 Hz are also eigenfrequencies. Their amplitude is small due to the driving point measurement being at a modal node of that specific mode and therefore the laser vibrometer practically cannot detect the motion. An interesting observation is that the eigenfrequency at 535 Hz is highly dependent on the radius of the interfaces. A bigger radius will mean a stiffer connection to the (rigid) mass, this will lead to ”stiffning” of the mid-section of the structure. Since mid-section of the structure will highly influence the 10th eigenmode, its eigenfrequency will shift upwards for larger interface radii. In other words, a larger interface area will result into a higher (local) stiffness and thus a higher eigenfrequency. In figure 8 the influence of the coupling radii on the 10th eigenfrequency are shown.
Mode 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
feig,meas. (Hz) 59.7 79.4 108 183 195 242 276 367 412 535 620 637 654 900 905 1020
feig,FE vs. feig,meas. (%) 0.2% 0.l% -2.3% -1.0% -1.5% -1.1% -0.7% -0.9% -0.2% -0.3% 0.7% -1.5% -0.5% -1.7% -1.8% 3.4%
Table 2: DS results
1 node/bolt r= 3.9 mm Measurement DP FRF r= 4.6 mm r= 4.9 mm r= 6.9 mm
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Figure 8: DP FRF’s for different radius (r) of joint stiffness area per bolt
5
CONCLUSION
From the measurements the following conclusions can be drawn: • The FEM model of the DS benchmark was validated by measurements and showed exceptionally good results, with MAC values higher that 93% and a maximum difference of 1.4% between the measured and calculated eigenfrequencies. This means the FEM model is accurate in the frequency range of interest (40 Hz - 1000 Hz) and can therefore be used as a valuable research tool. • Because the rigidness of the most rigid section of the benchmark displayed in figure 5 remains well above 93% for the complete frequency range of interest, the section can be assumed rigid. This allows for methods to ‘measure’ the rotations (eg. EMPC). • A numerical substructuring analysis was performed and compared with a measurement of the coupled system (4.3). It shows a very good correlation up to 600 Hz and allows for ”tuning” the bolt connection model. The differences at higher frequencies could be due to the assumed rigid bolt connections between the DS benchmark and the attached mass. At higher frequencies non-linear behaviour due to dry friction on the interface could become more dominant.
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