Impressed Force Research Articles

Theory of Vibration Isolation of a System with Two Degrees of Freedom : 2nd Report, The Case Where a Harmonic Force Acts on the Mass Located Far from the Supporting Foundation

This paper presents expressions and computational graphs to determine the optimum tuning and damping parameters for a linear system with two degrees of freedom. The mass m1 is connected to an immovable foundation through a spring k1 and a dashpot c1. An impressed harmonic force acts on the sass m2. Either of the masses (named main mass) should be isolated from the vibration originating from the impressed force; whereas, the other mass functions as a dynamic absorber, The optimization criterion is minimizing the maximum displacement of the main mass. The optimum design parameters can be formulated when c1=0. If c1≠0, and tested on a vibratory model. The experiment makes it clear that the theory is very useful for the isolation of actual mechanical vibrations.

Experimental study of fluctuations in materials

Mechanical behaviour is decided by the structure and kinetics of defects in materials. External forces play an important role in determining the growth, decay and motion of defects. In addition there is an inherent fluctuation in the various microscopic characteristics of the system as the latter acts as a ‘heat bath’. A disturbance that is set up in the system is affected by these fluctuations. The response of the system to external forces can be related to the behaviour of the system due to intrinsic fluctuations in the absence of impressed forces. This is the basis of relation between study of fluctuations in the system and various relaxation phenomena observed. It is proposed to discuss certain features of this subject in relation to various material properties.

Development of the flow field of a point force in an infinite fluid

By means of similarity principles an analytical solution is constructed for the development of the linear flow field due to the instantaneous application of a constant point force in an infinite liquid. If the force is applied at the origin O and if ν denotes distance from O, ν denotes the coefficient of kinematic viscosity of the fluid and t the time from the application of the force, the solution constructed exhibits the following features. Initially the flow field set up has a dipole structure with centre at O and axis along the direction of the impressed force. At a station r this dipole structure persists so long as 4νt [Lt ] r2. In an axial cross-section the field lines form two sets of closed loops about two stagnation points in the equatorial plane. The stagnation points occur at r = 1.76(νt)½ and thus propagate to infinity with speed 0.88(ν/t)½. The steady state is reached algebraically.

Generality and Nomological Form

It is usually supposed that the ‘fundamental’ laws of nature must be general, i.e. must essentially begin with a universal quantifier (Hempel and Oppenheim 1965, p. 272). Other ‘derivative’ laws may not satisfy this requirement but only because they are the logical or mathematical consequences of a set of fundamental laws. As it stands, this requirement is either too strong or trivial. Consider the Newtonian gravitational law: where FG(x, y, t) is the impressed gravitational force on x due to y at t, ms(x) is the mass of x, d(x, y, t) is the displacement of the center of mass of x from the center of mass of y at t, and G is the universal gravitational constant. (For simplicity, we ignore the vector nature of FG). What is the role of the constant “G”? Reflection shows that it is actually a disguised existential quantifier, or, more precisely, a Skolem function replacement for one. Rendering Newtonian mechanics in a parsimonious language, we could drop the “G” and rewrite (1) as The constant “G” could then be defined, if desired, as the unique r satisfying (2) (the propriety of this definition is guaranteed by (2)). Such a theory would have exactly the same deductive consequences as (1). But (2) has an initial existential quantifier essentially.

Generic bifurcations of forced oscillations of integrable mechanical systems

Although integrable Hamiltonian systems are non-generic (Robinson (13)) they have some importance in classical mechanics, e.g. the two-body problem, a free rigid body not subject to a gravitational field, the Toda lattice (Moser(12)). Arnol'd (cf. (2) and (1), appendix 26) proved that under quite general conditions action-angle coordinates can be introduced. Accordingly we consider a fixed system with n degrees of freedom in the standard formwhere H0 is a smooth function of the action variables Li only and I ∈ I and I is an open set of n-tuples of positive reals. now subjected to a periodic impressed force, the resulting system, , being determined by a small, periodic perturbation of the energy functionwhere e is a small positive real and K(c, I; α0, α1, …, an) is a smooth function of parameters c ranging over a smooth r-manifold E, I ∈ I, and K has period 2π in each of the angles αi, i.e. (α0, …, αn) ∈ Tn (n-torus). The purpose of this paper is to define the forms of bifurcation of oscillations of the perturbed system (K):

Parametric phenomena in synchronization of Thomson systems. I

It is shown that under conditions of a weak external impressed force the oscillation in the locking range of a self-excited oscillator is compounded of self-oscillations and forced oscillations with the same frequency but different phases, the self- and forced oscillations acting one upon the other.

Forced Vibration of Damped Circular and Annular Membranes

The response of circular and annular membranes to sinusoidal vibration is described theoretically. In contrast to prior investigations, emphasis is placed here on the forced vibration of membranes with damping, which is accounted for by writing the membrane tension as a complex quantity. The membranes, of outer radius a, are driven symmetrically as follows: by a ring force of arbitrary radius; by a force that is uniformly distributed within a concentric circle, also of arbitrary radius; or by a central “point” force. The point force can be visualized as the limiting case of the ring- or distributed-force excitation when the radius of application μa—μ being some numerical multiplier ⩽1.0—approaches zero. As this limit is approached, both the force transmitted to the membrane boundary and the transfer impedance become independent of the value of μ; however, the expressions for driving-point impedance approach a common value that is proportional to [loge μ]−1. Since, in practice, the impressed force cannot be applied at a point of zero radius, μ can realistically be assigned a small value (μ≠0) for which the logarithmic term remains finite and for which meaningful calculations of driving-point impedance can be made. In the example considered, the area over which the point force is applied is 10−6 times smaller than the membrane area (μ=0.001). The physical significance of graphical results that show the frequency dependence of impedance and transmissibility is discussed, and the effect of mass loading the membrane is described. Simple expressions are given from which the membrane damping factor can be deduced if the transmissibility across the membrane is measured at resonance.

A Proposed Mechanics for the Investigation of Surface Runoff From Small Watersheds: 1, Development

The difficulties of predicting runoff rates and volumes owing to the inadequacy of existing functional forms of the prediction equation and a lack of analytic definitions for the pertinent parameters related to the runoff process are discussed. An analogy is established between the path of water on the watershed surface and the phenomenon of Brownian motion. Probabilistic approaches are used to model a watershed surface and to obtain a stochastic surface impressed force, which acts upon a fluid particle and is found to be a statistically homogeneous function of position. Analogously to Brownian motion theory, a Langevin‐type equation containing the stochastic surface impressed force is postulated as a model for fluid particle motion on an impermeable watershed surface. The influence of surface roughness on the amount of energy available to induce and perpetuate fluid motion on a surface is considered by examining the rate of available energy degradation for a particle collection undergoing random motion induced by surface‐impressed forces.

Coupled Vibration of Geared Systems

In the design of geared rotating systems in engineering practice a variety of configurations are used, and their torsional vibration has been extensively studied. Detailed analyses of the modes and frequencies of vibration of these systems have been given by Ker Wilson and Nestorides. When the rotating system is rigidly supported, standard methods may be used to obtain the characteristics of free vibration and the response to impressed forces, from which the transmissibility of the gear-box may be calculated. However, there are many applications in which the gearbox is flexibly mounted. The simple example of a geared engine with a flexibly mounted crankcase has been considered by Den Hartog and Butterfield. In some epicyclic gear-box assemblies, elastic elements are introduced between the reactive element and the gear-case to increase the capacity for absorbing impulsive loads and as a means of varying the natural frequencies of vibration'. Harmonic forces acting on the rotors excite coupled vibration of the rotating system and the supporting frame, and Ker Wilson has shown how the natural frequencies and modes of the combined system may be determined by Holzer-type tabular methods. This involves a trial-and-error technique which may sometimes be tedious. Vibration of geared systems may also be excited by the motion of the frame supporting the gear-box, and an analysis of the response is necessary in the determination of dynamic loads.

Television in the Service of the School in Austria

The greater part of psychoanalytical literature about television as a phenomenon of our century, stresses generally the impressive force of television. It affects the habits of man, determines the rhythm of his life and his attitude towards life. This being so, judgements are preconceived and so man's power of unbiased decision is weakened. Television has a personal intimate style of presentation and effects thus in the viewer a readiness of identification that outstrips by far that of the film experience.

Quantum Theory of the Forced Harmonic Oscillator

A unified treatment by the methods of the operator mechanics is given of the quantum theory of the harmonic oscillator subjected to an applied force which is a given, but arbitrary, function of time. Few of the results are new, and most have appeared before scattered among papers devoted to a variety of topics and using a variety of mathematical techniques. Emphasis is placed on numerical methods for the evaluation of the relevant transition probabilities, and on the equality, under certain conditions, of the classical and quantum values of the average energy transfer to an oscillator by an impressed force. Generating functions for the transition probabilities are derived. Some numerical results are presented for illustration.

The Propagation of Cracks and the Energy of Elastic Deformation

Abstract The Griffith-model of brittle fracture of elastic solids and the model by Irwin and Orowan for the brittle fracture of elastic-plastic solids predict the propagation of cracks on the basis of energy supplied by the work of externally impressed forces and by the change of strain energy. Previous discussions have neglected the three-dimensional viewpoint and the body forces. Since the Irwin method of fracture-strength analysis has become of increased interest, especially with respect to rotor fracture, a general analysis of energy supply is presented. The analysis uses Clapeyron’s theorem and Betti’s reciprocal theorem. One of the results is that the energy supplied for crack extension equals the strain energy of the difference of the two stress fields before and after crack extension.

Dependence of Dynamic Properties of Rubber Vibration-Absorber on Initial Strain

The vibration test of the rubber samples is performed with a forced resonance vibrator. The sample is circular cylinder (10φ×60), both edge planes of which are cemented to the platens. The variations of dynamic modulus Ed (appearent value) and internal friction ηω with the initial strain (0∼30% in tension side) are investigated under constant temperature, frequency and impressed force. The following results are obtained. 1. Both Ed and ηω decrease with the increase of initial strain. 2. However, the ratio ηω/Ed remains unaffected with the change in initial strain. 3. The ratio of the static modulus to Ed is independent of the initial strain. 4. The larger the stiffness of the sample, the larger the rate of the decrease of Ed to the increase of initial strain.

Application of the Fourier Integral in Circuit Theory and Circuit Problems

The straightforward application of the Fourier integral to determine the response of a linear invariable circuit to an arbitrary impressed force is reviewed. When a Fourier integral representation of the impressed force exists and the system starts from rest, the problem is routine. Extensions to cases such as impulses, step functions, and ramp functions, for which the Fourier integral along the real axis does not converge, may be made by suitable changes in the path of integration. Boundary conditions other than zero stored energy can be included by introducing equivalent driving forces to produce the same effect as the initial voltages and currents. Noise voltages and currents in general can not be represented as Fourier integrals over all time. Representation over a finite time with a subsequent limiting process can be used to define a spectral density or mean-square amplitude in an elementary frequency interval. Spectral densities of network responses to a specified noise source are then calculable by multiplying the squared absolute value of the appropriate transfer function versus frequency by the spectral density function of the source. Knowledge of spectral densities can be used to design optimum circuits for separation of signal and noise.

Noto pri devigata movado de lineara oscilatoro

Frequency characteristics of a linear oscillator or a seismometer, whose equation of free oscillation is\(\ddot x + 2\varepsilon \dot x\)+n2x=0, are usually represented by takingh(=e/n) as parameters. In this case, however, the independent variable is the frequency of impressed force or displacement from outside on the oscillator. But, we often encounter those cases, where the frequencies from outside are constant, and the frequency of the oscillator or pick-up is to be changed, or the several oscillators with various frequencies are to be used. Then, of course,h cannot be taken as parameters, as they vary with the oscillator's frequencyn. The author here calculated the amplitude- and phase-characteristics for the latter case, taking\(\bar h\)(=e/n) as parameters and represented them in thick lines in the figures together with the ordinary ones in thin lines.

Flat Spring With Large Deflections

Abstract This paper gives an analysis of a flat spring initially curved and inserted in the apparatus of which it is an element in a buckled condition. The unstressed spring is shaped in the arc of a circle, and it is shown that then, and only then, the problem may be reduced to that of an initially straight bar by adding to the actually impressed force a couple which, if acting alone, produces the initial curved shape. On this basis the paper ascertains the various possible shapes of the buckled spring and the forces associated with them. It is shown that these must have at least one point of inflection of the elastic line. More particularly, there are two shapes, each with a single point of inflection, yet differing widely in their properties. In one of them there is a range of instability while the other is stable throughout. Finally, it is shown how by a suitable variation of parameter shapes two or more points of inflection may be obtained.

The Mechanism of Gravitational Force

Although it has been known since the time of Newton, nearly three centuries ago, that bodies attract each other according to the inverse-square law, no explanation as to how gravitation operates has been accepted. However, with our present knowledge of matter it would seem that some explanation should be possible. Keeping in mind Newton's first law of motion; namely, “Every body continues in its state of rest or uniform motion in a straight line unless compelled to alter that state by impressed force,” let us consider, for the purpose of illustration, an atom within which the motion is produced by the vibration of the nucleus in whole or in part, or by the electron as it rotates on its axis or as it travels in its orbit around the nucleus. The direction of rotation or revolution of the particles within the atom is apparently unimportant since the nucleus of the atom may rotate with or in opposition to the motion of the electron as is evidenced by the position of the spectrum lines.

Network analyzer solution of the equivalent circuits for elastic structures

The equivalent circuits of elastic structures developed in a companion paper are numerically checked for two known frames. Both structures are also measured on the A-C Network Analyzer for the case of known impressed forces. In each case the measured values checked very satisfactorily with the calculated values, showing that existing analyzers can be used to good advantage to determine the forces and displacements in statically indeterminate structures.

Oscillations in Certain Nonlinear Driven Systems

The development of the theory of nonlinear oscillating systems has given rise to some rather intractable nonlinear differential equations. The formerly used isoclyne method of solution is inadequate for the solution of the equation of the nonlinear system producing relaxation oscillations under the influence of an impressed periodic force. In this paper, using the specific example of a triode oscillator with impressed sinusoidal electromotive force, differential analyzer solutions of the governing diferential equation, are given. A brief study of the phenomena of automatic synchronization and frequency demultiplication is made on the basis of the nonlinear theory.

Physical Nature of Certain of the Vibrating Elements of the Internal Ear

THE sensation resulting in the human subject from a change of phase of (180°) occurring in the course of a continuous musical tone has been the subject of a number of earlier publications1. Under certain conditions, the sensation has been found to resemble the beat produced by two pure tones slightly out of unison, and has been described as a ” phase-change beat”2. As stated by Hartridge1, it is demanded by the Helmholtz resonance theory that the physiological event which corresponds to such a phase-change beat must be a transient arrest of the resonant elements of the internal ear brought about by the opposition of the impressed forces following the change of phase to the after-swings enforced by resonance.