R.G. White

Vibrational power flow from machines into built-up structures, part I: Introduction and approximate analyses of beam and plate-like foundations

Abstract This is the first of three companion papers concerned with a unifying and novel approach to the vibration isolation of machines and power transmission mechanisms in substructures. The paper presents an introduction to power flow analysis techniques and a study of the simplification of practical structural analysis by use of the frequency response characteristics of an equivalent infinite structure. Beams and plates with force and torque excitation are studied and the resulting near and far field power flow mechanisms are examined. Principal results, which are in the form of very simple formulae, are tabulated and in particular it is shown that torque excitation can often be the dominant substructure power input mechanism from machines at high frequency.

Power flow through machine isolators to resonant and non-resonant beams

Abstract The parameters controlling power transmission from a vibrating machine to the seating structure, via spring-like vibration isolators, are investigated. The low frequency range is considered where the machine moves as a rigid body. It is shown that the finite seating structure can be modelled by an equivalent structure of infinite extent for frequency averaged power transmission calculations. Power transmission to a finite and an infinite beam via a mass and spring in series is measured experimentally and compared with theoretical predictions. The power transmission is measured by two proposed methods; the first involves the real component of the seating impedance, and the second the transfer impedance of the isolator.

The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures part I: Simply supported and clamped-clamped beams

Abstract A method for calculating the non-linear mode shapes and natural frequencies of fully clamped beams at large vibration amplitudes is presented, and results are compared with those of previous studies and of experimental measurements. First, the transverse displacement is assumed to be harmonic and is expanded in the form of a finite series of functions. Then, the non-linear deformation energy is expressed by taking into account the non-linear terms due to the axial strain induced by large deflections. A set of non-linear algebraic equations, which reduces to the classical linear eigenvalue problem when non-linear terms are neglected, is determined through Hamiltons principle. It is also shown that unless a condition is imposed on the contribution of one mode, the solution of this set leads to the linear case. Consequently, in order to obtain a numerical solution for the non-linear problem in the neighbourhood of a given mode, the contribution of this mode is chosen and those of other modes are calculated. In this paper, part I of a series of three papers, this method is applied to obtain the first three non-linear mode shapes of clamped-clamped and simply supported beams. The results obtained corresponding to the fundamental non-linear mode shape are in good agreement with those of a previous theoretical and experimental study. In particular, high values of increase of beam curvatures are noticed near the clamps, causing a highly non-linear increase in bending strain with increasing deflection.

Vibrational power flow from machines into built-up structures, part III: Power flow through isolation systems

Abstract The power flowing through the isolators and into the supporting foundation of a machine is examined by approximating the driving point frequency response function of the foundation. One and two stage isolation of machines with internal force or velocity sources is considered. Two stage isolation is superior to single stage isolation in reducing power flow in those circumstances where the excitation spectra do not cover the two resonances of the system.

The effects of large vibration amplitudes on the fundamental mode shape of a clamped-clamped uniform beam

Abstract The non-linear vibration of a clamped-clamped beam at large displacement amplitudes is examined in this work. Complementary theoretical and experimental studies have been carried out to examine the amplitude dependence of the fundamental mode shape and its derivatives and the spatially-dependent harmonic distortion of the transverse displacement which occurs at large deflections.

Vibrational power flow from machines into built-up structures, part II: Wave propagation and power flow in beam-stiffened plates

Abstract Wave propagation and power flow due to force and torque (moment) excitation has been studied at the driving point and in the far field of an infinite plate with a single line-stiffener. Such a structure excited by forces or torques applied to the beam behaves like an uncoupled beam at the driving point. In the far field, power transmitted by flexural waves in the beam is radiated into the plate whilst power transmitted by torsional waves in the beam is not radiated. The plate carries a cylindrical wave with a strong directivity.

Evaluation of the dynamic characteristics of structures by transient testing

Abstract Transient methods of measuring the frequency response of structures may be used now that the digital computer is established as a means of analysing transient data. The spectral properties of the rapid frequency sweep are well defined and it is shown that this type of forcing function is extremely suitable for transient tests on structures. Methods of deriving the frequency response function from rapid frequency sweep tests are discussed and possible sources of error are investigated. Results from practical tests show how the method may be used to derive the frequency response characteristics of built-up structures.

The effects of large vibration amplitudes on the dynamic strain response of a clamped-clamped beam with consideration of fatigue life

Abstract The effects of large vibration amplitudes on the dynamic strain response, near to the fundamental resonance, of a clamped-clamped, thin beam is examined. Complementary theoretical and experimental studies showed that the harmonic distortion of the induced total strain, due to sinusoidal excitation at mid-span, is mainly due to the axial strain component. A limited set of fatigue experiments showed the considerable decrease in fatigue life, which occurs due to non-linear vibration, compared to that of a cantilevered beam of the same material. The statistical approach to the analysis of non-linear vibration induced by random loading is examined theoretically and experimentally, good correlation being achieved between predicted and measured fatigue lives.

Effects of non-linearity due to large deflections in the resonance testing of structures

Abstract The steady-state response of a single-degree-of-freedom system with a “cubic stiffness” term in the differential equation of motion is examined, this effect in the response of structures being caused by large deflections. The effect of this type of non-linearity in the vector diagram and peak amplitude methods of analysis is examined; the maximum frequency spacing criterion for the identification of natural frequencies used in the vector method is studied in detail. The results are compared with experimental data from a structure.

Data analysis criteria and instrumentation requirements for the transient measurement of mechanical impedance

Abstract The procedure of mechanical impedance measurement by the transient loading technique is considered with particular reference to the accuracy obtained. It is shown that the accuracy is principally affected by three factors: the analogue to digital conversion process, the choice of data record length and hence frequency resolution, and the method of exciter attachment. These effects are investigated theoretically and experimentally and it is shown that extremely accurate measurements can be made if the test conditions are chosen carefully. The method of exciter attachment is investigated in order to reduce the time required for setting up equipment for impedance measurements on structures in the field. It is shown that a simple hand-held exciter yields acceptable test results for typical, practical structures.