Structural modification. Part 1: rotational receptances

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Abstract

The inverse problem of assigning natural frequencies and antiresonances by a modification to the stiffness, mass and damping of a structure is addressed. Very simple modifications such as the addition of masses and grounded springs can be easily accommodated and require the measurement of translational receptances at the connection coordinates. Realistic modifications of practical usefulness, such as a modification by an added beam, require the measurement of rotational as well as translational receptances. Such data are difficult to obtain because of the practical problems of applying a pure moment. One method, the so-called ‘T-block’ approach, has received considerable attention in the literature, but the accompanying problem of ill-conditioning has not been fully addressed until now. The T-block is attached to the structure at the modification point, so that a force applied to the T-block generates a moment together with a force at the connection point between the T-block and the parent structure. Forces and linear displacements measured on the T-block, together with a mass and stiffness model of the T-block itself, allow the problem to be cast as a special case of excitation by multiple inputs. The resulting equations are generally ill-conditioned, but can be regularized by using a small number of independent measurements. The methodology and signal processing techniques required to estimate the rotational receptances are described. An experimental example is used to demonstrate the practical application of the method.

Introduction

Receptances obtained by the processing of data from translational sensors have been readily available for more than two decades and modern modal tests are regularly carried out using hundreds of data channels. Throughout the 1980s and 1990s the need for rotational receptances intensified for a variety of reasons including the development of procedures for the correlation and updating of finite element models, but most importantly for structural modification and substructure coupling. A typical case of structural modification is where the dynamic behaviour of a system modified by an added mass or a grounded spring is determined from measurements on the system in its unmodified condition. Simple modifications such as these are entirely described in translational coordinates and require the measurement of translational receptance data only. If a modification is to include a rotational inertia or a rotational stiffness then measured rotational receptances are needed from the unmodified system.

The problem of determining rotational receptances can of course be separated into the two sub-problems of (i) measuring rotational motion and (ii) exciting the structure with a moment and measuring it. The first subproblem is the easiest and many papers on rotational measurements concentrate on this aspect. Attempts have been made to apply pure moment excitation, but this is very difficult to achieve in practice. The alternative is to apply a force, which simultaneously imparts a moment, but this is problematic too. Both aspects of rotational receptance estimation are beset with ill-conditioning problems as will be explained in what follows.

Smith [1] used two electromagnetic shakers in a configuration capable of applying a couple by two equal and opposite forces applied to a specially designed fixture. Requirements were described to control the amplification of random experimental errors. Sanderson and Fredo [2] and Sanderson [3] considered bias errors caused by the fixture. These arise from errors in the moment applied to the structure and unwanted excitation of the rotational motion. The former can be corrected by rotational inertia cancellation and the latter depends not only on the design of the moment-excitation fixture but also on the structure itself. Sanderson and Fredo [2] and Sanderson [3] considered fixtures in the form of a T-block and an I-block. Petersson [4] used two giant magnetostrictive alloy rods, excited out of phase with each other to produce a couple on a small T-block, assumed to be a pure moment. Gibbs and Petersson [5] adapted the principle of magnetorestrictive rods to produce a smaller moment actuator to demonstrate the importance of rotational vibrations to structure-born sound. The methodology, together with the necessary equations, required to estimate the moment mobility using the magnetorestrictive actuator are given in detail in Ref. [6]. Su and Gibbs [7] used two-shaker configurations and the magnetostrictive actuator. They concluded that the main drawback of the former stems from the mismatching of forces from the two shakers. A lightweight electromagnetic moment shaker is described in a research report [8] on work carried out under EU contract BRPR-CT97-540. Trethewey and Sommer [9] presented a different way of generating a pure moment by utilizing the couple created by the centrifugal forces of eccentric masses attached to two symmetrically connected rotating wheels. In order to cancel an undesirable moment in an orthogonal direction they added a second identical counter-rotating system of masses on top of the previous one. Bokelberg et al. [10], [11] and Stanbridge and Ewins [12] described the measurement of rotational responses by laser sensors and their implementation.

The method described by Ewins and Sainsbury [13], Ewins and Gleeson [14], and Ewins and Silva [15] is amongst the earliest attempts at the measurement of rotational receptances. Using a rigid attachment, such as the T-block, they showed that the full matrix of receptances could be expressed in terms of the measured translational receptances, a coordinate transformation matrix and the mass matrix of the attachment. Cheng and Qu [16] and Qu et al. [17] used a similar approach, but based upon the use of a rigid ‘L’-shaped attachment. Silva et al. [18] proposed a method for the recovery of the rotation-moment receptance from a single column of the receptance matrix corresponding to an applied force. The mass and inertia of the T-block were eliminated by using a substructure coupling technique. Yasuda et al. [19] and Kanda et al. [20] also used a rigid attachment but their studies were concentrated on the measurement of rotational displacements and did not extend to the determination of rotational receptances. Duarte and Ewins [21] considered the transformation matrices for closely spaced accelerometers. Yoshimura and Hosoya [22] determined the rotational receptances by solving an overdetermined least-squares problem using measured time-domain data from a rigid T-block. All methods using rigid fixtures lead to poorly conditioned equations for two reasons: (1) Rotational motion is given by the small difference between translations from accelerometers on the rigid fixture. (2) The excitation points on the rigid fixture are close in comparison to the wavelengths of the modes excited in the structure and therefore the receptances excited from close points on the attachment are very similar. Dong and McConnel [23] relaxed the requirement of a rigid fixture. They determined the full inertance matrix by using a finite element inertance matrix of the T-block together with inertances measured by an instrument cluster (including rotational accelerometers) on the T-block itself.

Ewins [24] showed that the complete matrix of receptances could be determined from measurements of a single row or column, which seems to suggest that the most difficult rotational point receptances are recoverable from other measurements. However Ewins went on to point out the impracticability of his result, since in most cases the measured frequencies are restricted to a range less than the range of natural frequencies of the system and in that case the residuals due to the out-of-range (higher) modes must be included. O’Callahan et al. [25], [26] and Avitabile et al. [27], [28] used a finite element model to expand the set of measured translational degrees-of-freedom thereby producing an estimate of the unmeasured rotations.

In this paper a multiple-input, multiple-output H1 estimator is described for rotational receptances using an elastic T-block. The T-block may be quite large since its mass and elasticity properties are included in a finite element model, which allows the displacements and forces (and moments) at the connection point with the main structure to be determined from measurements at accessible points on the T-block where accelerometers and force sensors are located. The resulting estimates are the receptances of the structure under test without any mass, inertia or stiffness effects of the T-block, which are conveniently removed by the T-block finite element model. The use of a large T-block helps with the conditioning of the resulting equations. In addition, the equations may be regularized using independent measurements taken with the T-block removed. The measured data are rich in the natural frequencies of the system with the T-block attached and a further benefit of the regularization is that it tends to counteract this effect. The final result is a multiple-input, multiple-output regularized H1 estimator, which may be used to determine the full frequency response function matrix at the connection point.

Section snippets

Theory

The T-block is connected to the parent structure as shown in Fig. 1 and receptances, including rotational receptances, are required at the connection point 0. The analysis is presented for the case of vibration in a plane so that the point and transfer receptances at 0 form a 3×3 matrix. The generalization to vibration in three dimensions is a straightforward extension to the theory presented but the equations become more complicated. In that case, a T2-block is used having perpendicular arms

Finite element model of the T-block

The T-block is a simple structure which we assume can be accurately modelled using finite elements and is represented in Eq. (1) by the dynamic stiffness matrix B˜00(ω)B˜02(ω)B˜20(ω)B˜22(ω).Subscripts 0 and 2 represent the coordinates of the connection point and the accelerometer measurement coordinates respectively. The finite element model should include nodes at these coordinates. Finite element nodes at coordinates not included in x0 and x2 are unmeasured and therefore need to be eliminated

Experimental arrangement

The experimental rig shown in Fig. 3 takes the form of a portal frame with one leg missing. In the companion paper [33] the sectional properties of the missing leg are determined by an inverse method in order to assign the natural frequencies of the complete system. The initially missing leg is represented by a single Euler–Bernoulli beam element and added to the measured receptances (taken with the leg missing) in order to predict the receptances of the complete portal frame. The predicted

Regularization

Most previous researchers assume the T-block to be rigid in the frequency range of interest, which means that it must be very much smaller than the test structure. Then the problem of estimating rotational frequency responses is ill-conditioned for two reasons. Firstly, the rotation is generally obtained from the small difference of two much larger translational displacements measured at the ends of the T-block arms. Secondly, the receptances at the two arm-ends will be very similar in the

Results

Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10 show the estimated receptances obtained by excitation separately at each of the four points shown in Fig. 1. Twenty sequences of time dimain data (n=20) were used at each shaker location. In Fig. 5, Fig. 6, Fig. 7 a set of receptances measured independently using linear accelerometers at the connection point 0, without the T-block, are overlaid on the estimated receptance curves. It can be seen from Figs. 5 to 7 that the estimated results for the

Conclusions

The full 3×3 matrix of receptances for in-plane motion of a beam structure has been determined by a method that uses an elastic T-block. The method requires linear displacement and force measurements at a number of points on the T-block together with a finite element model of the T-block itself. In theory there is no restriction on the size of the T-block and this assists with conditioning of the resulting equations. However the data are rich in the natural frequencies of the system including

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