Showing papers on "Inertia published in 1972"

Effect of Inertia on the Brownian Motion of Rigid Particles in a Viscous Fluid

Abstract: From Boussinesq's work it is known that the frictional resistance of a particle in a viscous, inert fluid, depends on its history. This plays an important part in Brownian motion. The general theory of fluctuations in fluid dynamics leads to explicit expressions for the autocorrelation function G(t) for the random force acting on a Brownian particle and the autocorrelation function φ(t) for its velocity. It is demonstrated that G(t) contains an inertial term which depends on the velocity distribution in the liquid surrounding the particle. The results obtained lead to the correct value of the diffusivity.

Characterization of Equilibrium and Stability in Power Systems

Abstract: The evaluation of the stability of a power system during a transient requires that the dynamics of the subsystems be decomposed into relative and collective motions. This decomposition must establish a distinction between synchronous and frequency equilibria. Such a decomposition is made possible by specifying a system center of angle which is defined as the inertia weighted average of all rotor angles. The angular velocity of the center of angle accurately describes the frequency of the system. An angular coordinate of each rotating element in the system can be specified relative to the center of angle. In terms of these coordinates a simple expression is obtained for the exact transient kinetic energy of the system. A transformation relating center of angle referenced variables to the usual one machine reference allows the simultaneous use of both references each where best suited.

The dynamics of a polymer molecule

Abstract: The different kinds of motion to be expected are classified in terms of the relative magnitudes, expressed in time scales, of inertia, viscosity, barrier energies, and temperature. In particular, and as a comparison for other cases, the idealized case of a strictly inextensible, but otherwise freely flexible chain, is solved in detail. The nonlinear nature of the constraint is shown to lead to the generation of a Fokker-Planck equation. This Fokker-Planck equation shows that there are wave-like solutions to the motion for long wave fluctuations in the conformation of the chain (second sound) in the absence of viscosity. In the high viscosity limit the equation has Rouse-like solutions which correspond to short wave changes in conformation. In general, the equilibrium functions are modified so that if the chain is considered as a series of connected mass points, the energy is kT per point, not 3/2kT, a result which though intuitively obvious, is difficult to establish. This idealized model can be related to several realistic cases but a reinterpretation of the constants is involved.

Engine mounting and energy absorbing frame for a motor vehicle

Abstract: This disclosure relates to an impact absorbing system for a motor vehicle. In the preferred embodiment, the system includes a unique energy absorbing frame and engine mounting construction. The frame has a plurality of plastically deformable energy absorbing sections. A vehicle body is supported on the frame. Break-away engine mounts connect the vehicle''s engine to the frame. When an impact load is exerted against the forward end of the frame, the inertia load of the engine will cause frangible elements in the break-away mounts to fracture and the engine to separate from the frame. With this construction, the inertia load of the engine will not be imposed upon the energy absorbing frame sections whereby those sections need support only the inertia load of the vehicle body.

Analysis of the dynamic response of a rolling string-type tire model to lateral wheel-plane vibrations

Abstract: SUMMARY A theory has been developed for the analysis and prediction of the dynamic frequency response of lateral force and moment acting upon a pneumatic tire when the wheel is moved laterally and swivelled about the vertical axis. The theory establishes the force and moment response of a tire model which consists of a stretched circular string with mass, elastically supported to the wheel-center-plane. The analysis is confined to small deviations from rectilinear motion such that it is permissible to assume that sliding does not occur in the contact area. In this manner, the equations are kept linear. The theory which gives an exact analysis of the dynamic response of the model adopted shows satisfactory qualitative agreement with experiments. The change in the moment response due to tire inertia reduces the tendency to shimmy at higher frequencies and higher speeds. The lateral force response, however, changes in an unfavorable fashion which, for castered wheels, may result in a decrease of the effectiv...

Applications of finite elements in space and time

Abstract: The paper initiates a major extension of the matrix displacement method concerning the analysis of dynamic phenomena in the presence of material and geometric non-linearities. In particular, elasto-plastic behaviour as well as large displacements are taken into account. An iterative procedure of solution of the nonlinear matrix equations is discussed. The application of the theory is described in detail in two examples. The first considers the simple static problem of a rectangular flat strip in a tensile test. The iterative calculation may be carried out for deformations as large as required and shows clearly the necking effect. More ambitious is the second example which demonstrates the non-linear dynamic theory on a cyclindrial deformable billet under the impact by a heavy rigid body. The momentum of the weight and the property of the billet are such that the latter will undergo large plastic deformations. If so required, it is straightforward to incorporate damping and also allow for friction forces on the contacts. The direct applicability of the technique to forging problems is evident. The solution of the dynamic phenomenon is accomplished by extending the discretisation of space also into time. In particular, the inertia forces are taken to vary over a finite time element as a third order polynomial. Exceptional accuracy is achieved by this method.

Two-dimensional sink flow of a stratified fluid contained in a duct

Abstract: A reservoir is assumed to be filled with water which has a linear variation of density with depth. The geometry of the boundaries is simplified to a parallel walled duct with the line sink at the centre of the fluid. The primary focus is on partitioning the flow into distinct flow regimes and predicting the withdrawal-layer thickness as a function of the distance from the sink; the predictions are verified experimentally.For fluids with a Schmidt number of order unity, the withdrawal layer is shown to be composed of distinct regions in each of which a definite force balance prevails. The outer flow, where inertia forces are neglected, changes from a parallel uniform flow upstream to a symmetric self-similar withdrawal layer near the sink. For distances from the sink smaller than a critical distance, dependent on the flow parameters, inertia forces become of equal importance to buoyancy and viscous forces. The equations valid in this inner region are derived. Using the inner limit of the outer flow as the upstream boundary condition, these inner equations are solved approximately for the withdrawal-layer thickness by an integral method. The inner and outer variations of δ, the withdrawal-layer thickness, are combined to yield a composite solution and it is seen that the inclusion of inertia forces yields layers thicker than those obtained from a strict buoyancy-viscous force balance. In terms of the inner variables the only parameter remaining is the Schmidt number.Laboratory experiments were carried out to verify the theoretical conclusions. The observed withdrawal-layer thicknesses were shown to be closely predicted by the integral solution. Furthermore, the data could be represented in terms of the inner variables by a single curve dependent only on the Schmidt number.

Nonuniform Magnetic Field Effects in MHD Slider Bearing

Abstract: A theoretical analysis is made of a slider bearing using an electrically conducting lubricant in the presence of a nonuniform magnetic field applied perpendicularly to the bearing surfaces. In the differential equations inertia terms are retained and the solution is obtained numerically for low Hartmann numbers. The results indicate that the contribution of inertia terms decreases with the increase of Hartmann number and that the nonuniform magnetic field gives higher load capacity than the comparable uniform magnetic field.

Dynamic Nonlinear Elastic Stability of Helicopter Rotor Blades in Hover and in Forward Flight

Abstract: Equations for large coupled flap-lag motion of hingeless elastic helicopter blades are consistently derived. Only torsionally-rigid blades excited by quasi-steady aerodynamic loads are considered. The nonlinear equations of motion in the time and space variables are reduced to a system of coupled nonlinear ordinary differential equations with periodic coefficients, using Galerkin's method for the space variables. The nonlinearities present in the equations are those arising from the inclusion of moderately large deflections in the inertia and aerodynamic loading terms. The resulting system of nonlinear equations has been solved, using an asymptotic expansion procedure in multiple time scales. The stability boundaries, amplitudes of nonlinear response, and conditions for existence of limit cycles are obtained analytically. Thus, the different roles played by the forcing function, parametric excitation, and nonlinear coupling in affecting the solution can be easily identified, and the basic physical mechanism of coupled flap-lag response becomes clear. The effect of forward flight is obtained with the requirement of trimmed flight at fixed values of the thrust coefficient.

Dynamics of neutrally buoyant particles in low reynolds number flows

Abstract: The quasistatic motion of a neutrally buoyant particle suspended in a shearing flow at small Reynolds numbers is reviewed. Special attention is devoted to the translational and rotational motions of a rigid spherical particle suspended in a simple shear flow. An assessment is made of the effects upon these particle velocities arising from the presence of bounding walls, fluid inertia, and an embedded dipole when a uniform external field acts. Motion of the particle when suspended in a Poiseuille flow within a circular tube is also discussed, including radial migration across the streamlines due to fluid inertia effects or translational Brownian motion. The motions of liquid droplets and gas bubbles in uniform and non-uniform simple shear flows are summarized. In contrast with rigid spheres, such particles deform in response to the shear in their proximity. This deformation results in non-linear phenomena (such as radial migration across the streamlines of a Poiseuille flow), despite the linearity of the dynamical equations of motion governing the internal and external motions of the droplet. A general theory is outlined of the motion of a neutrally buoyant, rigid, non-spherical particle suspended in a homogeneous shearing flow. The translational and rotational particle motions are shown to be governed by two intrinsic third-rank tensors that depend only upon the size and shape of the particle. The forms adopted by these tensors for particles possessing various modes of geometric symmetry are discussed, and the results employed to calculate the behavior of axisymmetric bodies of revolution suspended in a simple shearing flow, as well as in various types of elongational flows. The effects of particle inertia and rotary Brownian motion on the rotational motion of the particle are briefly discussed. Modifications of the particle motion arising from inhomogeneities in the shear flow are also outlined.

Low inertia rotor for dynamo electric machines, and method of making the same

Abstract: A hollow insulating cylinder with conductors thereon is placed over an internal metallic tubular support which is supported by an end disk at one end, and open at the other, the open end being flared for stiffness.

Nonlinear wave forces on vertical cylinder

Abstract: The method of diffraction is applied to Stokes' fifth order gravity wave theory to obtain an expression for the velocity potential in the cylindrical polar coordinate system. Based on this expression, the dynamic pressure at the surface of a vertical cylinder and the total horizontal force on the cylinder in the direction of the wave propagation are determined. The wave force is written in an equivalent form of the inertial part of Morison's equation and the effective inertial coefficients are shown to be functions of a single dimensionless quantity. The different parameters involved are plotted or tabulated so that forces on a vertical cylinder due to a nonlinear wave up to fifth order may be easily evaluated. Correlation of the theory with the available experimental data was found to be reasonably good.

On the Calculation of Changes in the Earth's Inertia Tensor due to Faulting

Abstract: Summary Recent interest has arisen in the possibility that changes in the Earth's inertia tensor accompanying earthquakes may provide the excitation for the ‘Chandler’ wobble of the rotation axis. We present a simple procedure for calculating these changes based on the elastic reciprocal theorem and Volterra's formula. In particular, if τ is the shear stress created in the slip direction on a prospective fault surface when a perfect, unfaulted sphere is steadily rotated at unit angular velocity about some axis, then the change in moment of inertia about that same axis is shown to be twice the work of τ when carried through the actual slip displacement which occurs in faulting. It is shown that changes of products of inertia may be computed in a similarly simple way. The method is applied to two homogeneous Earth models, the relation to previous treatments is discussed, and it is noted that inertia changes accompanying less-catastrophic mass movements may be calculated in a similar way. The development in the body of the paper is in the context of classical linear elasticity. An Appendix extends the reciprocal theorem and Volterra's formula, and hence the basis for similar inertia change calculations, to the linearized incremental deformation of self-gravitating, initially stressed elastic systems, such as real-Earth models.

An Automatic Mass-Trim System for Spinning Spacecraft

Abstract: Long-term precision pointing of the spin axis of a spinning satellite may be impossible because attitude control systems cannot counteract the effects of sensor-vehicle misalignments, the motion of the principle axes of inertia, and body-fixed disturbing torques. An automatic mass-trim system (AMTS) consisting of two pairs of movable trim masses is proposed as a means of providing control torques to eliminate errors resulting from these sources. An analysis of the equations of motion of a rigid spinning body with internal moving masses relates the dynamics of mass motion to the attitude motion of the spinning body. Specializing the geometry to a two-pair system allows calculation of the control torque available and the range through which the principal axis set may be moved relative to body-fixed coordinates. A control law for an automatic system used in conjunction with a previously developed spinning-vehicle attitude controller is presented and analyzed using root locus techniques. An analog simulation of the vehicle with trim system verifies the design.

Dependence of helicon wave radial structure on electron inertia.

Abstract: An analytic expression for the dispersion of travelling helicon waves in a uniform non-resistive plasma is used to show the effect of electron inertia on the radial structure of the wave in a cylindrical plasma bounded by perfectly conducting walls.

Active nutation damping in dual-spin spacecraft

Abstract: A dual-spin spacecraft having a de-spun platform is arranged to vary or modulate the torque on the stabilizing member, such as a momentum wheel, by a motor whose speed is varied or modulated in accordance with a signal representing nutation motion to damp or attenutate the nutation motion substantially to zero very rapidly. The nutation motion is sensed by a suitable device such as a horizon sensor, a gyroscope, or an accelerometer. By arranging the mass distribution such that significant cross products of inertia exist, the effects of such products are utilized in a closed loop control system to effect the desired attenuation or damping. A phase shifting network is provided to shift the phase of the cyclic sensor output signal by a predetermined angle whereby optimum damping of the nutation motion is achieved.

Theoretical investigation of the interfacial stability of inviscid fluids in motion, considering surface tension

Abstract: The present work is an analytical study of the stability of interfaces between fluids in motion, special attention being given to the role of surface tension without consideration of viscous effects. A variational approach based upon the principle of minimum free energy, which was first formulated for stagnant fluids, is applied to fluids in motion. This generalization is possible if viscous and inertia effects are unimportant as far as stability is concerned. One stability problem is studied in detail: a gas jet impinging on a free liquid. The analytical results obtained by this variational technique lie within the range of accuracy (15%) of the experimental results for this gas-jet problem. The method is very general and therefore can be applied to quite a number of interface stability problems.

Ball actuated inertia switch

Abstract: Inertia switch with mass freely movable in chamber and energizing switch means upon reception in chamber.

Theory of an experiment in an orbiting space laboratory to determine the gravitational constant

Abstract: The paper analyzes an experiment in an orbiting laboratory to determine the gravitational constantG. A massive sphere, according to a suggestion of L. S. Wilk, is to have three tunnels drilled through it along mutually perpendicular diameters. The sphere either floats in the orbiting laboratory, with its center held fixed by means of external jets issuing from the spacecraft, or is tethered to the spacecraft. In either case it is free to rotate; in the second case this freedom would be achieved by a system of gimbals. Each tunnel contains a small test object, which is held on the tunnel's axis by means of a suspension system, perhaps electrostatic, and held at rest relative to the sphere by slowly rotating the latter by means of inertia reaction wheels, governed by a servomechanism. Fundamentally, one balances the gravitational forces on the test objects by centrifugal force, determines the latter by measuring the components of angular velocity, and calculatesG from the resulting balance. It is better to use three tunnels than one because their use minimizes the effects of the Earth's gravity-gradient. Many other measurements and corrections are required. The latter arise from Earth gravity-gradient, aerodynamic drag (with the tethered sphere), gravitational forces produced by the spacecraft itself, and the force reductions produced by the empty space in all three tunnels. After the consideration of these effects there is a presentation and discussion of the equations required to reduce the observations to obtainG. There then follow the extra equations, not needed in the reduction, that are required for a computer simulation to investigate the possible extraction of a test object and to aid in designing the servomechanisms. In Appendix B, I have devised another version of the experiment, in which the sphere is kept intact, but has short thin hollow ‘vestigial tunnels’ attached to the outside of the sphere, along perpendicular diameters. These external tunnels would contain the test objects and the suspension systems. The servomechanisms would then have to prevent collision of a test object with the sphere, as well as extraction. This second method could allow for some inhomogeneities in the sphere, would require no accurate drilling, and would make the suspension systems more accessible for construction and adjustment.

Methods for estimating vehicle transfer functions requiring only the geometric and inertia characteristics

Abstract: Progress toward the development of generally applicable methods for calculating vehicle transfer functions is presented. Such methods would allow control engineers with little knowleged in hydrodynamics to estimate the vehicle dynamics during the preliminary design phase.

Inertia sensor switch

Abstract: Inertia switch with mass mounted between inclined structure and, when at rest, contacting spring switch extension means.

Inertia switch with oscillating controller restrained against return to normal position

Abstract: The disclosure is of a circuit-breaker suitable for use in a vehicle and adapted to function automatically to interrupt the circuit from the battery of the vehicle if the latter suffers a shock or impact, the circuit-breaker comprising a sealed insulating housing in which is mounted an inertia weight which can be displaced by a horizontal shock force from any direction and thereby releases a conducting contact plate which is moved by springs so as to break the circuit.

About the first integrals of the generalized problem of translatory-rotary motion of rigid bodies

Abstract: The existence of ten first integrals for the classical problem of the motion of a system of material points, mutually attracting according to Newtonian law, is well known. The existence of the analogous ten first integrals for the more complicated problem of the motion of a system of absolutely rigid bodies, whose elementary particles mutually attract according to the Newtonian law, was established by the author (Duboshin, 1958, 1963, 1968). In his later papers (Duboshin, 1969, 1970), the problem of the motion of a system of material points, attracting each other according to a more general law, was considered and, in particular, it was shown under what conditions the ten first integrals, analogous to the classical integrals, may exist for this problem. In the present paper, the generalized problem of translatory-rotatory motion of rigid bodies, whose elementary particles acting upon each other according to arbitrary laws of forces along the straight line joining them, is discussed. The author has shown that the first integrals for this general problem, analogous to the integrals of the problem of the translatory-rotatory motion of rigid bodies, whose elementary particles acting according to the Newtonian law, exist under certain well known conditions. That is, it has been established that if the third axiom of dynamics (action = reaction) is satisfied, then the integrals of the motion of centre of inertia and the integrals of the moment of momentum exist for this generalized problem. If the third axiom is not satisfied, then the above mentioned integrals do not exist. The third axiom is a necessary but not a sufficient condition for the existence of the tenth integral-the energy integral. The tenth integral always exists if the elementary particles of the bodies acting with a force, depend only on the mutual distances between them. In this case the force function exists for the problem and the energy integral can be expressed in a well known form. The tenth integral may exist for some more general case, without expressing the principle of conservation of energy, but permitting calculation of the kinetic energy, if the configuration of a system is given. The problem, in which the elementary particles acting according to the generalized Veber's law (Tisserand, 1896) has been cited as an example of this more general case.