In a previous paper (1) the free and forced vibrations of systems of one degree of freedom with hereditary damping characteristics were discussed. In the present paper the classical equations of motion for elastic media are extended on the basis of the general linear stress-strain law involving hereditary damping. These equations are applied to the case of free radial vibrations of a sphere. Furthermore, the free vibrations of strings, the free transverse vibrations of beams, and the free vibrations of rectangular and circular membranes are studied under the assumption of hereditary damping.

Both free vibrations and forced motions due to crosswinds may create important problems in the design of pipe lines supported above ground. An analytic investigation, based on simple beam theory, shows that the flow of fluid in such a pipe line has no beneficial effect upon the vibrations. The fluid velocity causes a dynamic coupling of the simple modes of vibration so that the normal modes of vibration are of complex shape with 90-deg out-of-phase components. The solution is presented for free vibrations and for steady-state forced vibrations, and it is shown that large amplitudes may be developed if the amount of damping is too small. It is shown that at low fluid velocities there is negligible effect upon the vibration of the pipe line, and at a certain high critical velocity the fluid flow causes a dynamic instability. The present analysis revises the conclusions which appeared in an earlier publication.

In this paper it is shown that if the hysteresis loop for a material has a particular shape the damping can be considered adequately by multiplying the modulus of elasticity of the material by the complex number e 2 bi where 2 b is called the complex damping factor. For small values of b it is shown that both for free and forced vibrations of a simple spring-mass system the motion in the case of complex damping is the same as in the case of viscous damping, with b = c/c cr , except that in the steady-state case the phase angles are slightly different. Also, it is shown how complex damping may be applied to cases of forced vibrations of uniform rods and beams. The greatest advantage of using complex damping, however, is in numerical calculations of forced vibrations of engine crankshafts, airplane wings, and other types of structures; and for such calculations it already has been extensively used.

This is a study of the vibrations of thin elastic shells freely suspended in a compressible fluid medium. The effect of the fluid reaction on the dynamic characteristics and, in particular, on the natural frequencies is investigated for cylindrical and spherical shells. The dynamic configuration of such shells undergoing forced vibration and the associated radiation of sound are determined. The problem is analyzed by means of the classical methods of the theory of mechanical vibrations; the Lagrange equations for the system are derived, the fluid reaction being introduced in the form of generalized forces. From the boundary condition that the normal shell deflection be equal to the normal fluid-particle displacement at the shell surface, and introducing the concept of acoustic impedance, it is shown that the fluid reaction is equivalent to an accession to the inertia of the shell and to a damping force. Numerical examples show that the effect of the fluid reaction on the dynamic characteristics of a shell may be of such magnitude as to render valueless calculations neglecting it.

A machine, consisting of rotating tuned pendulums free to oscillate in planes containing the axis of rotation, is proposed as a dynamic absorber for linear vibrations. The influences of size, exactness of tuning, and damping are investigated and curves for evaluating the effectiveness of the machine are shown.

The effectiveness of the viscously damped vibration absorber is presented for the case in which the magnitude of the periodic exciting force acting upon the main system is proportional to the square of its frequency. Dimensionless expressions for the amplitudes of the main mass and absorber mass and for their phase relationships are derived as functions of frequency for three cases, namely, one in which the absorber is tuned to the natural frequency of the main system, one in which the absorber is tuned for maximum effectiveness over a wide range of forcing frequencies, and one in which the absorber is coupled to the main system by a viscous fluid only (the viscous Lanchester damper). The influence of main-system damping upon the amplitude of vibration of the main mass is shown for each case. Diagrams are presented showing the optimum damping, the maximum amplitude of the main mass, and the maximum relative amplitude between the main mass and absorber mass, as functions of the mass ratio. The performance of the absorber when applied to the system having velocity-squared excitation is compared with its performance when applied to the system having constant exciting force, published previously (1, 2). The tuning and damping for optimum performance are found to be different in the two cases. A model absorber with controllable tuning and damping, constructed for experimental work, is described and experimental data are presented for the case of most favorable tuning.

The authors have developed a true dynamic analogy which has been used with the Cal Tech electric-analog computer for the rapid and accurate solution of both steady-state and transient beam problems. This analogy has been found well suited to the study of beams having several coupled degrees of freedom, including torsion, simple bending, and bending in a plane. Damping and effects such as rotary inertia may be handled readily. The analogy may also be used in the study of systems involving combined beams and “lumped-constant” elements.

The problem relates to the transient vibration of a symmetrical, continuous, simply supported two-span beam which is traversed by a constant force moving with constant velocity. The beam is of slender proportions, flexure alone being considered. Damping is zero, and there is no mass associated with the moving force. Exact theoretical solutions for bending stress have been derived in general form. They consist of three infinite series, each related to one of three time eras as follows: ( a ) Where force is crossing first span; ( b ) is crossing second span; ( c ) has left the beam. Each term of a series is related to a natural mode of vibration. Quantitative theoretical studies show the variation in individual terms of the series, and also in summations of the first five terms, as the traversing velocity is varied. A mechanical model with electrical recording of stress was employed to obtain a more complete quantitative solution than was feasible analytically. The agreement between theory and experiment was reasonably good. Large magnifications of stress (of the order of 2.5) were found in the neighborhood of resonance with the fundamental mode.

The forced and resonant-vibration characteristics of a torsional system are easily enough determined by the well-known Holzer methods, in the case where the only major damping influences available in the system are those of the prime-mover itself, or of the propeller, in the case of a propeller-propulsion system. Where a damper exists within the system, the classic solution confines itself to the determination of the resonant -vibration characteristics under optimum damping conditions, and utilizes the principle of energy balance in conjunction with an equivalent system. The problem of forced vibration in a system having a damper is not susceptible, however, to such simple treatment regardless of whether or not the damping is optimum. This paper outlines an exact method, whereby both the forced- and the resonant-vibration characteristics of a viscously damped system may be calculated for any existing damping condition, optimum or otherwise, and which utilizes the ever-popular Holzer form.

An alternating force (for example, due to unbalance in a locomotive driving apparatus) moves across a symmetrical two-span beam (or equivalent bridge) with uniform velocity. The stress-time equations for the three time eras of the problem have been derived by classical methods. The experimental part of the investigation employs a mechanical system with wire resistance strain gages. Resonances from the first through the sixth mode have been investigated. There is good agreement between theory and experiment in so far as number and location of resonances are concerned. The discrepancy in amplitude is to be expected due to the omission of damping in the theory. The most important observation is the “multiplicity of resonances” associated with “each” natural mode.

Free vibrations and forced motions due to cross winds may both create important problems in the design of pipe lines supported above ground. An analytic investigation, based on simple beam theory, shows that the flow of fluid in such a pipe line produces marked damping tendencies and thus may reduce the severity of loading encountered. The time dependence of the fundamental mode of a simply supported pipe line is calculated for a number of mass-flow rates, the damping being observed to increase rapidly and the frequency to remain nearly constant over the practically important range. A method is outlined for studying forced vibration, higher modes, and other end conditions. Finally the problem is discussed in terms of traveling waves on an “infinite” unsupported pipe line.

During acceleration of a machine through a critical speed, displacements and stresses may attain large values. For linear systems F. M. Lewis (1), and J. G. Baker (2) have studied different aspects of this problem, the former analytically, the latter by mathematical-machine methods. This paper presents a generalized solution obtained on the mechanical analyzer (3) for the effect of an accelerated sine-wave force on a spring-mass system having linear plus cubic elasticity and linear damping. The range of accelerations covered applies to electrically driven reciprocating machines and to other rapid starting units. Some interesting effects of nonlinearity on displacements and forces are pointed out.

A mechanical-analogy-type analyzer is described which is of relatively simple construction being limited to single-degree-of-freedom problems. Whithin this limitation solutions may be obtained for systems which include various types of nonlinear elasticity and of nonlinear damping. Included is a generalized solution obtained on the analyzer giving in dimensionless form the maximum displacements and forces in a system having nonlinear (linear plus cubic) elasticity and linear damping caused by a force pulse of constant magnitude and finite duration. The bearing of the results on the starting torques in nonlinear systems is indicated.

In this paper a study is made of the problem of the central impact of a mass on a simply supported beam on an elastic foundation with considerations of internal and external damping. The differential equation for the forced vibration of the beam is developed. It is solved for the case in which the force is a function of time and is concentrated at the center of the beam. Formulas are obtained for the deflections. An expression is developed for the coefficient of restitution which is essential in determining the deflections and the strains. Criteria are devised for determining the cases in which the beam may be considered as a single-degree-of-freedom system when damping and an elastic foundation are considered. The importance of these criteria is discussed. A numerical example illustrating the theory developed in the paper is worked out in detail. Results of computations for several numerical solutions are given in tabular form.