Estimation of Rotational FRFs via Cancellation Methods Nomenclature ij ( l, k ) ij M. Reza shory, Semnan university, Iran Email: mashoori@semnan.ac.ir exact ccelerance measured accelerance when two mechanical elements are attached to the structure at DOFs l and k ir, ir ccelerance of points i and j and in rotational direction θ. here i z and j refer to the points and R refers to the rotational DOF r jk the modal constant for mode r and DOFs j and k bsract Rotational FRFs are present in 75% of the whole FRF matrix. Their availability is therefore of extreme importance in the calculations related to the applications of the modal testing such as coupling, structural modification and model updating. However, there are neither practical means of applying moment excitation, nor accurate rotational transducers for measuring the moment and rotational response of the structure. For these reasons most of the modal analysis techniques tend to be developed avoiding as much as possible the knowledge of rotational FRFs, or minimizing the effects of not knowing them. In this article an indirect method is suggested to overcome the practical difficulties of measuring angular motion and applying excitation moments. In this technique the inertial effects of an added object to the test structure are cancelled in order to evaluate the unmeasured FRFs. The technique has revealed some difficulties in practice, as the algorithm tends to be very sensitive to noise. 1 Introduction In [1,,] correction methods were introduced in order to cancel the effects of the transducers and suspension springs on the measured FRFs. Then, in [4,5], the same approach was used in order to generate the translational FRFs from measurements on just a single column of the FRF matrix. In this paper a method is presented for generating the rotational FRFs of the type displacement/moment and rotation/moment by exciting the structure at translational DOFs and by modifying the test structure at particular points. The problem is that we still need to measure rotational responses. The T-block technique have already been used to generate the rotational FRFs by modifying the test structure for theoretical models [6]. The finite difference technique itself produces rotational FRFs related to a point on the structure [7]. In this paper we use the finite difference technique and correction methods in order to develop a method for generating the rotational FRFs related to two different points on the structure. The aim is to eliminate the need to apply moment excitation. Instead, the structure is excited at translational DOFs and is modified by the mass and the rotational inertia of the attached objects at desired points.
Generation of the rotational FRFs by modifying the test structure.1 Measurement technique Consider the structure shown in Figure 1. 1. Obtain, 11 and 1 using the finite difference technique.. Obtain, and 4 using the finite difference technique. 44. Measure. 1 (,4) 4. ttach an object (with a known mass and moment of inertia) at point b and obtain 1 using the finite difference technique. ttach another object (instead of the first object) at point b with the same mass as that of the first object but with a different moment of inertia (,4) (,4) (,4) and measure 1 (here is different from 1 1 ). (1,) 5. ttach an object (with a known mass and rotational inertia) at point a and obtain 4 using the finite difference technique. ttach another object (instead of the first object) at point a with the same mass as that of the first object but with a different moment of inertia (1,) (1,) (1,) and measure 4 (here is different from 4 ). 4 Figure 1: Generation of rotational FRFs using the finite difference technique Figure shows different stages of the measurement using this measurement technique. It should be noted that any method of generating the rotational FRFs is more successful the fewer computation stages it has. The reason is that, owing to the presence of noise and other systematic errors in the measurements, the results of the computations in each step become less reliable. Thus, if a FRF can be obtained from direct measurement, the risk of possible errors decreases. Moreover, it is assumed that the attached accelerometer is so light that its mass-loading effect is negligible. The above-mentioned measurement technique is based on the minimum possible number of computation stages, and on the step-by-step correction approach. This means that the masses (m) of the attached objects are the same although their moments of inertia (I) are different.. Generation of the rotational FRFs The procedure for generating all of the FRFs from the measured FRFs listed in the previous section is: 1. In equation (1,). 4 4 = 4 1/ I +
Figure : Measurement stages for obtaining rotational FRFs is known because, 11 and 1 are known: ( 1 ) = () 1/ m + 11 For two different objects with the same mass (m) but different moments of inertia: ( I 1 and I ) and 4. can be obtained from equation. 4 Then can be obtained from : 14 1. 14 4 = 4 () 1/ m + 11, 4 and 1 are known and 11 was obtained from the last stage and therefore 4 is 14 obtainable.. In equation () () (,4) () 41. 4 1 = 1 (4) () 1/ I + () is known because 44, and 4 are known: 44 () ( 4 ) 44 = 44 (5) 1/ m + For two different objects with the same mass (m) but different moments of inertia: ( I 1 and () () () I ) and 1. can be obtained from equation (4). 41 4 Then can be obtained from : () 1. 1 = 1 (6) 1/ m + 44
(), 1 and 1 are known and was obtained from the last stage and therefore is 1 obtainable.. To obtain 4 we have:. 1 1 = (7) 1/ m + 11 (),, 1 and 1 are known. Therefore 11 is obtainable. From step 1 1. was 4 obtained. So is obtainable. 4.. 1 14 1 14 4 = 4 4 = 4 + (8) 1/ m + 11 1/ m + 11 ll of the elements on the right hand side of equation (8) are known; therefore 4 is obtainable. Numerical case study theoretical model of a beam was considered for the numerical validation of the method. Table 1 shows the parameters of the beam. The model of the beam was made in the same manner as was explained in [1]. Table 1: Parameters of the beam Length 0.7 m Mass of density 7800 Kg / m Height 0.0048 m Mass.59 Kg Thickness 0.065 m Modulus of elasticity.1 e11 ( N / m ) Number of elements 56 Figure shows the result of the FEM analysis of the third elastic mode of the beam which is at 74 Hz. Points 9 and were chosen to determine the rotational FRFs using the method presented in this chapter. In order to estimate the rotational responses using the finite difference technique, point 10 was chosen to be close to point 9, and point 4 close to point. It should be mentioned that the distance between the points is important in the results of the finite difference technique. However, the aim of this work has not been to find the optimum distance between the points; this distance was chosen based on the work presented in [7]. Figure : Third mode of the free-free beam
Two different additive elements were chosen which have the same masses but different moments of inertia. The parameters of these elements are: Element 1: mass=0.04 Kg, moment of inertia =5.104 e-5 Kg.m ; Element : mass=0.04 Kg, moment of inertia = 9.47 e-5 Kg.m ; Elements 1 and defined above are assumed to behave as rigid bodies. The simulated measurements were conducted by computing the FRFs following the measurement technique stated in section.1. The FRFs were generated according to the procedure stated in section.. Figure 4 compares the exact 9, (here R refers to the rotational direction R θ at point z ), and generated 9,. The two FRFs coincide around the resonance area although R some part of the generated FRF shows a considerable difference from the exact FRF at the frequency that the resonance of the modified beam exists. These differences are due to the approximate nature of the method of the finite difference technique for estimating the rotational FRFs. 4 Experimental case study simple beam was used for the experimental validation of the method presented in this paper. The beam was made of mild steel, and had dimensions 6.5 0.48 70 cm. Figure 5 shows the experimental set-up for the hammer test of the beam. The beam was discretised into 57 points and was tested in a free-free configuration, suspended by soft elastic cords with the stiffness of 410 N/m attached to the beam at points 6 and 5. Figure shows the theoretical mode shape of the third mode of such a beam. s shown in this figure, points 6 and 5 are translational nodes, and as a result are suitable for the attachment of the suspension springs to fulfil the free-free condition in this mode. Moreover, this figure shows that points 9 and are far from nodal points (rotational and translational); and consequently are suitable for applying the method presented in this paper. Therefore, points 9 and were chosen in order to generate all of the FRFs in the region of the third mode of the beam. To estimate the rotational FRFs using the finite difference technique, points 10 and 4, which are close to points 9 and, were considered for the measurement. The beam was excited by a hammer at its neutral axis and the response was measured by a light accelerometer of mass 0.004 kg. When the response was measured at one of the points 9 or 10 or or 4, three dummy masses, each of 0.004 Kg, were attached to the beam at three other points to avoid reciprocity problems. The extra elements which were due to be added to the structure at points 9 and were chosen by trial and error. fter trying different elements, two cylindrical elements with the same masses but different moments of inertia were chosen. The properties of these elements are: Element 1: Shape: Cylinder ; Diameter = 0.007 m; Length = 0.08 m; Material: Steel;Mass = 0.04 Kg; Computed moment of inertia = 6 e-5 Kg.m. Element : Shape: Cylinder ; Diameter = 0.006 m; Length = 0.1088 m; Material: Steel; Mass = 0.04 Kg; Computed moment of inertia =. e -5 Kg.m. Here, the second moment of inertia quoted is with respect to the diameter of the cylinder at one of its ends namely the place where it is attached to the structure. s the elements have
the same mass, the difference between the two FRFs of the modified beam is related to the moment of inertia of the added elements. It can thus be concluded that the chosen elements are suitable for the test. Moreover, the measured FRFs showed sufficient changes around the resonance of the beam, which is the most important area in the test. The additive elements are long with respect to their cross section area, as a result of which two problems may arise. Firstly, the elements may not exhibit rigid body behaviour in all of the frequency points. Secondly, the addition of these long elements may result in nonlinear behaviour of the modified structure due to large deflections of the elements. Both of these problems should be checked before applying the method. s noise pollutes the result of the computations, all of the FRFs present in the computations were regenerated in the frequency range of 60-90 Hz using ICTS program. The result of the regeneration was carefully controlled to match the raw data. ll the required FRFs were measured following to the measurement technique stated in section.1 and the rotational FRFs were generated following to the procedure stated in section.. Figure 6 shows the two computed rotational FRFs (the FRFs were regenerated using ICTS), 9 R,. However, a comparison between these two computed FRFs in R Figure 6 shows a slight difference. One way of assessing the success of the method is to conduct an internal check of the generated FRFs using their modal constants. In principle, the following equations should hold between the modal constants of the measured FRFs of the beam: ( )( ) 9,9R 9,R 9R,R = (9) 99 ( )( ) 9R, R, 9R,R = (10), where is the modal constant for mode r and coordinates j and k. On this basis the r jk required FRFs for equations (9) were regenerated and their modal constants derived using the ICTS program. The result of the calculation for the right hand side of equation (9) is 5 e8 compared with5.5 e8 for the left hand side. For equation (10) the result of the right hand side of equation is 5.9 e8 comparing to 5.5 e8 for the left hand side. The results of this procedure show that the two sides of equations (9) and (10) above are close together, demonstrating that the method is valid and can be used for estimation of the rotational FRFs around the resonance area of the structure. However, the approximate nature of the finite difference technique and the experimental errors (systematic and random) present in the measurement detract from the reliability of the generated FRFs. 6 References [1] Silva, J. M. M., Maia, N. M. M. and Ribeiro,. M. R., : Some pplications of Coupling/Uncoupling Techniques in Structural Dynamics, Part 1 : Solving the Mass Cancellation Problem. Proceedings of the 15 th International Modal nalysis Conference, pp. 141-149, 1997.
[] shory, M. R., Correction of Mass-loading Effects of Transducers and Suspension Effects in Modal Testing, Proceeding of the 16 th International Modal nalysis Conference, pp. 815-88, 1998. [] shory, M. R. " The Effect of Suspension Springs On Test Structures in Modal Testing", Tenth International Congress on Sound and Vibration, Sweden July 00. [4] Silva, J. M. M., Maia, N. M. M. and Ribeiro,. M. R., : : Some pplications of Coupling/Uncoupling Techniques in Structural Dynamics, Part II : Generation of the Whole FRF Matrix from measurements on a Single Column-The Mass Uncouplig Method (MUM). Proceedings of the 15 th International Modal nalysis Conference, pp. 141-149, 1997. [5] shory, M. R. and Ewins, D.J., Generation of the Whole FRF Matrix from Measurements on one column, Proceeding of the 16 th International Modal nalysis Conference, pp. 800-814, 1998. [6] Silva, J. M. M., Maia, N. M. M. and Ribeiro,. M. R., : Some pplications of Coupling/Uncoupling Techniques in Structural Dynamics, Part III : Estimation of Rotational Frequency Response Functions Using MUM). Proceedings of the 15 th International Modal nalysis Conference, pp. 145-146, 1997. [7] Duarte, M. L. M., : Experimentally-Derived Structural Models for Use in Further Dynamic nalysis PhD Thesis, Imperial College, University of London, 1996. Figure 5 : Experimental set-up for the generation of RDOFs on the beam
Figure 4 : Comparison of the exact, R 9, (solid) and generated R 9, (dashed) Figure 6 : Two regenerated FRFs of 9 R, R