The response of a vibrating structure as a function of structural parameters

doi:10.1016/S0022-460X(75)80009-5

Journal of Sound and Vibration

Volume 38, Issue 2, 22 January 1975, Pages 255-266

https://doi.org/10.1016/S0022-460X(75)80009-5 Get rights and content

The response of a structure excited by an external oscillatory force is examined for variations in certain structural parameters of the system. This is done with a view to manipulating the structure in order to achieve a desired response. The variation of the response with one structural parameter is seen to be simple, and the effect of changing in addition the forcing frequency is illustrated. When two structural parameters are considered, it is seen that a desired response may or may not be attainable, as defined by a “feasible response region”. The application to practical cases is discussed briefly.

References (6)

A. Simpson
Eigenvalue and vector sensitivities in Kron's method
Journal of Sound and Vibration
(1973)
K.W. Newman
Predicting the effects of parameter changes in a dynamical system
RAE T.R. 69142 (A.R.C. 32597)
(1969, (July))
A.H. Vincent
A note on the properties of the variation of structural response with respect to a single structural parameter when plotted in the complex plane
Westland Helicopters Ltd. Report GEN/DYN/RES/01OR
(1973, (September))

There are more references available in the full text version of this article.

Cited by (33)

The applicability of the Moebius transformation to mechanical and acoustic impedance modifications
2009, Journal of Sound and Vibration
The Moebius transformation maps straight lines or circles in one complex domain into straight lines or circles in another complex plane. This paper will demonstrate that the acoustic response will trace a circle in the complex plane for straight line or circular modifications to mechanical or acoustical impedance. This is due to the fact that the equations relating the acoustic response to the modification are in a form consistent with the Moebius transformation. This is demonstrated for series and parallel mechanical and acoustic impedances. The principles presented in this paper for the case of mechanical impedance are in essence equivalent to what has been termed the generalized Vincent circle by other authors. This paper shows that the principle is valid for multiple excitation problems while also showing the applicability of the principle to acoustic impedance. The key qualifications for applying the Moebius transformation to mechanical and acoustic impedances is that the problem should be linear, and the impedance modification should be in one coordinate direction and at one position. The principle is shown to be valuable for understanding the effect of acoustic impedance modifications in waveguides. The principle is illustrated for several examples including a point excited plate, a construction cab, and an acoustic waveguide.
Finite element synthesis of structural or acoustic receptances in view of practical design applications
2008, Journal of Sound and Vibration
The present study addresses the modal synthesis of dynamic responses or receptances, which can be considered as a major issue in vibratory or acoustic design. A new method termed “method of orthocomplement” offers to increase the accuracy of modal superposition. Two apparently independent domains are considered: first, the numerical control of round-off errors in finite-dimensional models; second, the regularization of modal boundary discontinuities in infinite-dimensional formulations. Concerning the finite-dimensional systems, using the proposed method in conjunction with pseudo-inversion theorems, proves that “inertia relief corrections” remedy modal truncation problems in singular floating systems, just as static corrections do for simply supported systems. The proof is based on hitherto unpublished expressions of inertial contributions in terms of orthogonal projectors. It is worth noting that such expressions present by themselves some technical interest. Modal formulae being delivered in closed form, a thorough analysis of errors becomes possible. Special “spectrograms” are introduced to appraise the convergence of ordinary or modified modal sums. When applied to infinite-dimensional modal boundary discontinuities, the method of orthocomplement eliminates the Gibbs oscillations that affect the ordinary spectral sums, and produces the same corrected formulae as it did for finite-dimensional examples. Effective solutions for the computation of vibratory or acoustic boundary receptances, are developed in that way. The paper once again illustrates how orthogonal projectors can contribute to the determination of static corrections. Future applications to coupling problems, model reduction and modal synthesis are briefly presented at the end of the text.
The generalized Vincent circle in vibration suppression
2006, Journal of Sound and Vibration
In 1972 A. H. Vincent, the then Chief Dynamicist at Westland Helicopters, discovered that when a structure excited at point p with a constant frequency is modified, for example by the addition of a spring between two points r and s, then the response at another point q traces a circle when plotted in the complex plane as the spring stiffness is varied from minus infinity to plus infinity. This discovery, although apparently little known today, has many useful applications some of which are described in papers by various authors appearing in the 1970s and early 1980s. Vincent's discovery is in fact a particular example of the bilinear transformation due to August Ferdinand Moebius (1790–1868). In this paper, the Vincent circle method is generalized for the case of any straight-line modification in the complex plane, typically $z = k + i ω c - ω^{2} m$ , where $c = α (k - ω^{2} m) + β$ . A new method for the visualization of Vincent circle results, including the case of multiple modifications is also presented.
Inverse eigenvalue problems in vibration absorption: Passive modification and active control
2006, Mechanical Systems and Signal Processing
The abiding problem of vibration absorption has occupied engineering scientists for over a century and there remain abundant examples of the need for vibration suppression in many industries. For example, in the automotive industry the resolution of noise, vibration and harshness (NVH) problems is of extreme importance to customer satisfaction. In rotorcraft it is vital to avoid resonance close to the blade passing speed and its harmonics. An objective of the greatest importance, and extremely difficult to achieve, is the isolation of the pilot's seat in a helicopter. It is presently impossible to achieve the objectives of vibration absorption in these industries at the design stage because of limitations inherent in finite element models. Therefore, it is necessary to develop techniques whereby the dynamic of the system (possibly a car or a helicopter) can be adjusted after it has been built. There are two main approaches: structural modification by passive elements and active control. The state of art of the mathematical theory of vibration absorption is presented and illustrated for the benefit of the reader with numerous simple examples.
Use of optimization in helicopter vibration control by structural modification
1981, Journal of Sound and Vibration
The application of a mathematical optimization process to helicopter vibration control by structural modification is described. Attention is focused on the reduction of vibration in the crew area. With stiffness parameters as design variables, use is made of forced vibration response circles to identify the parameters most effective in controlling the response in the crew area, thereby reducing the number of available design variables to a tractable size. The problem of reducing vibration is then cast as a non-linear programming problem and a sequential unconstrained minimization technique incorporating an algorithm based on the methods of Davidon, Fletcher and Powell is used to determine the precise values of the parameters. The method is applied to a simple two-dimensional beam-element helicopter fuselage model and the results discussed. Although the model is too simple for useful deductions of practical significance to be made in the strictly engineering sense, the exercise does demonstrate what can and cannot be done in controlling vibration by using an optimization routine.
Methods and application of structural modelling from measured structural frequency response data
1980, Journal of Sound and Vibration
A structure with a number of resonances is difficult to analyze theoretically but may be investigated from measured data. By exciting a structure at one point and measuring the frequency response at a number of positions it is possible to construct a mathematical model of the structure. The model is valuable because it enables unmeasured frequency response functions to be predicted. Methods of mathematical modelling are reviewed and a widely applicable procedure is proposed. By modelling two separate components of a structure from measured data it is possible to obtain an estimate of the subsequent motion and power flow through the two components when coupled. A critical examination is made of the use of mathematical modelling for the determination of mass and stiffness distribution and for the prediction of the response of coupled structures.

View all citing articles on Scopus

View full text

The response of a vibrating structure as a function of structural parameters

Journal of Sound and Vibration

Predicting the effects of parameter changes in a dynamical system

RAE T.R. 69142 (A.R.C. 32597)

A note on the properties of the variation of structural response with respect to a single structural parameter when plotted in the complex plane

Westland Helicopters Ltd. Report GEN/DYN/RES/01OR