MODAL v8n1 - Predicting the Frequency Response Function ... - Scholar

Modal Analysis: the International Journal of Analytical and Experimental Modal Analysis v 8 n 1 Jan 1993 p 55-62

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by P. Salvini, Universita di Roma "Tor Vergata" and A. Sestieri, Universita di

Roma "La Sapienza"

ABSTRACT A method is developed to predict the dynamic behavior of a structure from experimental FRF data of the same

system subjected to different constraints. In particular it is required that the new structure undergoes more restrained

conditions, but any type of ideal constraint, involving either translational or rotational degrees of freedom, can be

accounted for. Among several interesting applications. the method can be used to overcome typical experimental

drawbacks on rigid tested structures and to estimate untestable FRF t erms of constrained systems.

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experimental results are provided to show the consistency of the method and the possible range of applications.

Several problems exist in testing particular structures under given boundary conditions. Recently Barney, et al[l J considered the problem of designing a support device, capable of separating the rigid body modes of an unrestrained flexible structure from the structure's first flexural modes. In order to avoid the difficulty of testing the structure in free-free conditions, the algorithm developed in Ref. [ 1] identifies the free-free features of the considered structure from those of a multiply constrained system by means of a

direct procedure which uses a force measure at lhc boundary points. However, not only free-free structures present critical testing conditions. The measurement of any restrained system is often troublesome, too, and the experimental results differ significantly from the theoretical ones. In fact: •

experiments o n rigidly constrained structures (e.g., clamped-clamped beams) produce sometimes unacceptable excitation conditions, as double peaks in impact excitation;

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ideal rigid boundary conditions are very hard to obtain, so that the response of the tested structure is highly affected by the modal behavior of the supporting system;

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the nature of the real constraints are generally very dissimilar from the designed ones, yielding

inevitably impredictable results. In order to avoid the above fixes, the behavior of the actual system can be predicted from experiments

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performed on the same structure, though subjected to unrestrained or less restrained conditions. This

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P ietm Salvini, Dipartimento di Ingegneria M accanica, U niversira di Rvma ''Tor Vergata. , ViaE. Carnevale- 00173 R oma,Italy. A /do St·stieri (SEM memherJ, Profes:;or, Dipartimento di Meccanica e Aeronautica, Utzi\'ersitit di Rnma "La Sapienza", Via Eudossiana 18- 00184 Roma, Italy.

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An earlier version o f this paper was to have heen presenred at rhe 9'h lnlernational Modal Analysis Conferena. Florence,Italy, April

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Final manuscripT received:

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procedure is simple when accomplished on theoretical discrete systems, e.g., obtained by finite elements, provided that the mass and stiffness matrices of the unrestrained system are known. When, on the contrary

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the less constrained structure is identified from its experimentally determined frequency response function

{FRF), other procedures can be followed to determine the new FRF. The method o f general constraints [2]

provides the FRF of a structure, after introduction of any set of linear constraints, and some applications are being developed

[3,4]. In this paper a new procedure is presented, which is in principle more simple

than the previous one. The method may have interesting engineering applications. •

the forced response of the structure subjected to external forces.

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It may be helpful to model the actual constraints acting on a tested structure, when the boundary conditions cannot be easily identified, e.g., when dealing with non-ideal constraints.

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It can be used to check the effect of adding constraints on the modal behavior of the system or on

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[t represents an effective way to obtain an optimal constraint location, when different solutions are

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available for the designer.

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Moreover, it basically represents a predictive method of structural modification that can be

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employed in developing an optimization procedure without requiring any modal identification on

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the original data. Modal identification, in fact, can lead to erroneous estimates of structural

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modifications, as it was shown by several authors in the last decade (see, e.g., Elliot and Mitchell

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[5] and Braun and Ram [6]).

Finally, the method can be advantageous!y applied to estimate the FRF of coupled structures when

the related substructure parameters are quantities difficult to determine experimentally. Any type of ideal constraint, involving either translational or rotational degrees of freedom, can be considered. The method is developed here for any possible increment of constraints. It only requires that the FRF matrix of the origina1 system be measured at the points where further constraints must be applied, along the whole set of degrees of freedom affected by the constraints.

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The input -output relation for a linear, time invariant dynamic system can be expressed in the frequency

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I ,r(m)) includes both linear ( x) and angular ( 8) accelerations and {f} includes both forces (F) and moments (M); consequently the elements of the [H) matrix (FRF) involve translational as well as rotational

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degrees of freedom (DOFs ). The experimental evaluation of rotational FRF elements is not very accurate

because rotational accelerometers are only recently becoming feasible and there is not sufficient experience with them. Furthermore, lumped moments are not easily applicable. Different solutions have

[7,8,9]. Among them, the use of a finite difference scheme involving translational measurements [8] and modal curve fitting [7] are, up to now, valuable techniques been proposed to compute the rotational terms

used to compute the translational-rotational ( 8 /F, .X /M) and rotational-rotational ( 8/M)tenns respectively, .�

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at least for beam-type structures. Based on these algorithms, these authors have recently developed a

technique for predicting the assembled structure behavior, considered the sensitivity of the computed FRF

clements to the finite difference spacing and the importance of low and high residuals on the accuracy of resu Its f I 0]. In order to derive the FRF matrix of a structure subjected to additional constraints from the knowledge of the FRF of a less restrained structure, let us consider a system having a+ c DOFs, where c DOFs have to be constrained. A DOF constraint is meant here as a condition of null acceleration. Equation ( 1 ), written

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January

1993

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for the original system, can be partitioned as follows

{Xa} [Haa ] [Hat'] {J�} {xc} - [Hca] [Hu.] {Jc.}

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The constraint condition on the c DOFs implies

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