An experimental investigation of forced vibrations

The solution of the differential equation y″ + 2Ry′ + n²y = E cos pt is written in a new form which clearly exhibits many important facts thus far overlooked by theoretical and experimental investigators. Writing s = n ? p, and Δn = n ? √n² ? R², it is found: (a) When s ≠ Δn, there are “beats,” and the first “beat” maximum is greater than any later maximum while the first “beat” minimum is less than any later “beat” minimum. The “beat” frequency is $\text{(s ?}\text{Δ}\text{n)}\text{2π}$. (b) When n² ? p² = R², there are no “beats,” and the resultant amplitude grows monotonically from zero to the amplitude of the forced vibration, (c) At resonance, when n = p, we still have maxima which occur with a frequency $\text{Δ}\text{n}\text{2π}$ in a damped system. (d) The absence of “beats” is neither a sufficient nor a necessary condition for resonance in a damped system.In the experimental investigation the upper extremity of a simple pendulum was moved in simple harmonic motion and photographic records obtained of the motion of the pendulum bob. Different degrees of damping were used, ranging from very small to critical.The experimental results are in excellent agreement with theory.     

Steady forced vibrations of single mass systems with symmetrical as well as iinsymmetrical non-linear restoring elements

Part of the theory and all the experiments reported on in this paper were condensed into a four-page discussion or addendum to Professor Den Hartog's original paper on “Amplitudes of Non-Harmonic Vibrations” at the time of its presentation at the June Meeting of the A.S.M.E. in Chicago, 1933. Upon the instigation of Professor Den Hartog and the Editor of the Journal of the Franklin Institute, the original discussion has been brought into the present form.     

On the vibrations of lumped parameter systems governed by differential-algebraic equations

In this paper, we consider the vibratory motions of lumped parameter systems wherein the components of the system cannot be described by constitutive expressions for the force in terms of appropriate kinematical quantities. Such physical systems reduce to a system of differential-algebraic equations, which invariably need to be solved numerically. To illustrate the issues with clarity, we consider a simple system in which the dashpot is assumed to contain a “Bingham” fluid for which one cannot describe the force in the dashpot as a function of the velocity. On the other hand, one can express the velocity as a function of the force.     

On the passive control of vibrations with viscoelastic dynamic absorbers of ordinary and pendulum types

Dynamic vibration absorbers (DVA) provide a cheap and efficient means for vibration abatement in many complex systems, ranging from crankshafts of internal combustion engines, overhead transmission lines, machine casings, structural panels and large turbo machinery sets, to quote a few examples. One can provide a simple classification for them by considering the nature of the resilient material it contains as a form of “spring”: it may be viscous (CDVA), hysteretic (HDVA) or viscoelastic (VDVA). Viscous DVAs are the largely studied devices and one of their most remarkable applications is in mitigating crankshafts torsional vibrations and in very tall buildings. The most well known hysteretic DVA is the Stockbridge damper, largely applied in overhead electric power transmission lines. With modern use of fractional calculus, modelling viscoelastic materials became a routine work. The experimental identification of four fractional parameter models for viscoelastic material has become a standard technique amongst the authors of this work. Modelling viscoelastic materials by four fractional parameters has made advanced analysis of structures and systems where it is applied much more straightforward than it was before. This is true also for structures with VDVA and HDVA attached to it. In this paper it is shown that a hysteretic material model can be derived from a viscoelastic material model based on four fractional parameters. Generalized quantities of ordinary and pendulum type absorbers and for both viscoelastic and hysteretic materials are derived and their nature discussed. The performances of a system with absorbers of viscoelastic and hysteretic nature are compared. Input energy and dissipated energy by the absorbers of both natures and types are computed and compared, using the concept of generalized damping parameter of the absorbers. Conclusions are drawn from the comparisons. One of the ideas behind these computations is to check the validity of some international recommendations for the experimental assessment of Stockbridge dampers, which implicitly neglects the effect of the generalized mass parameter.     

The amplitudes of non-harmonic vibrations

Two new approximate graphical solutions are given for the problem of forced vibration of an undamped single degree of freedom vibrating system with a non-linear spring, whose characteristic is given in the form of a curve. One of these solutions gives much more accurate results than the approximation known so far.     

On a generalization of Kirchhoff's theory of transversal plate vibrations in the vibration problem of steam turbine disks

Introduction: a. The vibration problem in its actual aspects in the steam turbine design; b. The fundamental conception of the solution used in the steam turbine practice as derived from the Rayleigh Ritz method substituted for the original Kirchhoff theory; c. Illustration of the actual problem by an example taken from the one-dimensional theory of vibrations; d. The “Wheel Vibrations” of the disk; e. The blade vibrations with the disk proper entirely at rest; f. Asymptotical representation of the actual (“Disk Blade”) vibrations of the disk; g. Simplifications used in the procedure of frequency computation; h. The disk with heavy rim.     

Boundary optimal control of structural vibrations in an annular plate

A maximum principle is formulated and validated for the vibration control of an annulus plate with the control forces acting on the boundary. In addition, the maximum principle can be applied to plates with multiply connected domains. The performance index is specified as a quadratic functional of displacement and velocity along with a suitable penalty term involving the control forces. Using this index an explicit control law is derived with the help of an adjoint variable satisfying the adjoint differential equation and certain terminal conditions together with the proposed maximum principle. The implementation of the theory is presented and the effectiveness of the boundary control is investigated by a numerical example.     

Complementary formulations in vibrations: bond graph structures and modal transformations

Lumped parameter, undamped vibratory system models are studied starting from a vector bond graph representation which yields a symmetric set of equations of motion in terms of momenta and displacements. Four additional formulations are obtained depending upon the choices of displacement or impulse-momentum degrees of freedom including the classical formulation in terms of mass displacements. Differences in terms of forcing and response variables are found among the alternative formulations and differences in system order are explained. A new form of normal mode equations is developed using first order symmetric variables and a bond graph representation is given. Advantages in the use of the new model analysis for subsystem coupling are discussed.