Sinusoidal Case Research Articles

MODIFICATION OF FOOD CHARACTERISTICS WITH CELLULOSE HYDROCOLLOIDS II: The Modified Bingham Body ‐ A Useful Rheological Model*

AbstractRheological measurements were made with the Weissenberg Rheogoniometer on a series of unctuous materials including butter, margarine, an ointment, and gels prepared from sodium carboxymethylcellulose of a low degree of substitution. In steady shear, these substances show a sharply peaked stress overshoot. The stress response to an imposed sinusoidal strain is almost a square wave. These responses may be qualitatively described by a modified Bingham body consisting of a viscous element of viscosity, ν, a yield element having a yield stress of s̀0, r at rest and s̀0, s when moving (s̀0, r≥s̀0, s), and an elastic element of modulus, G, connected in series.The response of this model for imposed steady and sinusoidal shear has been calculated in terms of the model parameters. The boundary condition equations for the imposed sinusoidal shear case cannot be solved in a closed form. Numerical solutions were obtained by computer for the full range of the parameters using a successive approximation technique. The predictions of this model are in good agreement with experimental results, and enable such results to be interpreted in terms of meaningful physical parameters.

Optical Method for Measuring Amplitudes of Damped Spreading Ripples

An accurate method for measuring the amplitudes of sinusoidal ripples using the intersection of caustic surfaces has already been developed. By comparison with the sinusoidal case, we find that the same equations for the amplitudes apply to damped spreading ripples, for example, those produced by a small spherical source in a real liquid. The limitations on the ratio of the spatial absorption coefficient to the wave number and on the distance from the point source are discussed.

Distortion in Varicap FM Oscillators

The performance of FM systems, particularly FDMFM, is often limited by the distortion of key components. Voltagecontrolled oscillators (VCOs), used both as modulators and demodulators, may be a major source of distortion. This paper presents a method for calculating the distortion in varicap tuned oscillators, with the effects of both series and shunt fixed capacitors included. General distortion coefficients are derived, using a Taylor series expansion for the oscillator tuning curve, which are a function of the oscillator resonant circuit only. These coefficients are then used in the computation of harmonic distortion for the sinusoidal modulation case and in the computation of noise power ratio (NPR) with Gaussian modulation. The latter is particularly useful in the design of FDM-FM systems. NPR is plotted as a function of deviation for flat FM, phase modulation, and International Radio Consultative Committee (CCIR) preemphasis characteristics.

The uncorrelated output components of a nonlinearity

Using his characteristic-function approach, Rice (1945) obtained a double series for the autocorrelation function of a sinusoidal signal and Gaussian noise after passage through a memoryless nonlinearity. It is shown here that the output of the nonlinearity can be expressed as the sum of uncorrelated terms whose auto-correlation functions are the terms of Rice's double series. Such a decomposition of the output is shown to be generally possible if and only if the bivariate probability density functions of the input signal and the input noise can both be expressed in the diagonal form studied by Barrett and Lampard (1955), though not necessarily involving polynomials, as they can in the sinusoidal and Gaussian cases. In addition, a more direct and meaningful equation is found for the coefficients in Rice's double series.

Average Control for Sinusoidal- and Random-Vibration Testing

Time-division multiplexers (TDM) have found application in providing control from the average of several acceleration magnitudes for swept-sinusoidal vibration tests and, also, control to the average of several acceleration spectral densities for wide-band random tests. The vibration testing systems using the TDM are described. Pertinent system parameters are determined mathematically in order to limit signal harmonic distortion for the sinusoidal case, and to limit test-level modulation and spectral spreading for the random case. A special case for sinusoidal testing in which a narrow-band tracking filter is used on the output of the TDM is also discussed.

Average Control for Sinusoidal and Random-Vibration Testing

Time-division multiplexers (TDM) have found application in providing control from the average of several acceleration magnitudes for swept-sinusoidal vibration tests, and also control to the average of several acceleration spectral densities for wide-band random tests. The vibration testing systems using the TDM are described. Pertinent system parameters are determined mathematically in order to limit signal harmonic distortion for the sinusoidal case, and to limit test level modulation and spectral spreading for the random case. A special case for sinusoidal testing in which a narrow-band tracking filter is used on the output of the TDM is also discussed.

The scattering of plane waves from periodic surfaces

The reflection of scalar plane waves from periodic surfaces (in two dimensions) is studied. Particular attention is devoted to the sinusoidal surface upon which the wave function vanishes. The formulation of the problem rests upon Green's theorem, and it is shown that in the sinusoidal case one is led to an infinite set of coupled algebraic equations. These are essentially different from the equations derived by Rayleigh in his original formulation of the problem. Numerical results are obtained, and the more interesting properties of the results are discussed. As a sidelight, it is shown that the matrix of reflection coefficients for an arbitrary periodic nonabsorbing surface has interesting symmetry properties reminiscent of the quantum-mechanical S-matrix.

Inductance coefficients of rotating machines expressed in terms of winding space harmonic

The paper develops expressions for the self- and mutual-inductance coefficients of machine windings for the general case of nonsinusoidal winding distributions and nonuniform air gap. These expressions involve the harmonics of the winding distribution and the air-gap permeability. For some sample cases, numerical values are given and the departures from the ideal sinusoidal case assessed. The coefficients evaluated in the paper are those which form the starting point of the generalised mathematical theory of machines. The results are of significance in showing how, and to what extent, departures from the ideal machine modify the coefficients and introduce new terms not amenable to the usual mathematical treatment.

Membrane potential transients and membrane time constant of motoneurons

This paper is concerned with the interpretation of passive membrane potential transients produced in a neuron when intracellular microelectrodes are used to apply current across the soma membrane. It is also concerned with the specific problem of estimating the nerve membrane time constant from experimental transients in neurons having extensive dendritic trees. When this theory is applied to the most recent results published for cat motoneurons, the resulting membrane time constant estimates are significantly larger than the values estimated by Eccles and collaborators. The time course of soma membrane potential is solved for a variety of applied currents: current step, brief pulse of current, sinusoidal current, voltage clamping current, and a current of arbitrary time course. The sinusoidal case provides a theoretical basis for a purely electrophysiological method of estimating the fundamental ratio between combined dendritic input conductance and soma membrane conductance. Also included is the time course of passive decay to be expected to follow various somadendritic distributions of membrane depolarization or hyperpolarization. The discussion includes an assessment of the observations, hypotheses, and interpretations that have recently complicated our understanding of synaptic potentials in cat motoneurons. It appears that electrotonic spread between the dendrites and soma can account for the observations which led Eccles and collaborators to postulate a prolonged residual phase of synaptic current in cat motoneurons.

Finite Amplitude Sound Produced by a Piston in a Closed Tube

The problem stated in the title is solved for a gas with γ = − 1 by the method of a previous paper. First an arbitrary piston motion is considered. Then more detailed results are given for the sinusoidal case.

Transient conditions in electric machinery

The vector method is just as useful in solving problems involving transient conditions in electric circuits as it has proved to be when the currents and potentials are steady sinusoids. As far as the writer is aware, the vector method for determining transients in rotating electric machines was first used by L. Dreyfus. Previously the method had been applied to fixed combinations of resistances, inductances and capacitances by Kennelly and others. By making certain assumptions that are, however, quite reasonable in many cases, the transient currents in nearly all of the common types of electric machinery are damped sinusoids. Fortunately the damping is exponential and is thus readily accounted for. It is interesting to trace the development of the method. In the solution of all problems in direct currents the potentials, currents, and circuit constants are real numbers. In the corresponding problem in which the applied potentials are steady sinusoids, these quantities are all represented by complex numbers. In all other respects the working out of the solution is identical with that followed in the direct-current case. When the currents are damped sinusoids, they and the potentials and the circuit constants can still be represented by complex numbers. There is this difference, however; the vectors which represent the currents and potentials shrink exponentially as they rotate and the values of the circuit constants depend not only upon the frequency of the current, but also upon its rate of shrinking. Again the solution of any problem follows the same procedure that would the corresponding one in which the currents are steady sinusoids. In both the steady and damped sinusoidal cases the circuit constants depend upon the angular velocity of the vectors which represent the currents. In the former, the angular velocity is purely imaginary while in the latter it is complex, the real part being the rate at which the current vector shrinks and the imaginary portion, its angular velocity. In electric machinery in which rotating magnetic fields are produced, these fields shrink exponontially as they rotate when the currents are damped sinusoids. If these rotating magnetic fields are represented by vectors, the vectors will have a complex angular velocity just as do the currents. The e. m. f. which is produced by a steady sinusoidal variation of flux lags the flux by 90 degrees, whereas if the flux variation is a damped sinusoid, the angle of lag is less than 90 degrees, depending upon the damping. The mathematical relation, however, is the same, vis., the e. m. f. is proportional to the negative of the product of the flux and its angular velocity. It is then readily appreciated that the form of the solution for the transient state is the same as that which is used for the steady state. Before the method can be expected to give as accurate results as are obtained when predicting the steady operation, considerable experimental data must be obtained in order to determine the best methods of measuring the necessary constants, for these may be somewhat different during the transient period than during steady operation.

Transient Conditions in Electric Machinery

The vector method is just as useful in solving problems involving transient conditions in electric circuits as it has proved to be when the currents and potentials are steady sinusoids. As far as the writer is aware, the vector method for determining transients in rotating electric machines was first used by L. Dreyfus. Previously the method had been applied to fixed combinations of resistances, inductances and capacitances by Kennelly and others. By making certain assumptions that are, however, quite reasonable in many cases, the transient currents in nearly all of the common types of electric machinery are damped sinusoids. Fortunately the damping is exponential and is thus readily accounted for. It is interesting to trace the development of the method. In the solution of all problems in direct currents the potentials, currents, and circuit constants are real numbers. In the corresponding problem in which the applied potentials are steady sinusoids, these quantities are all represented by complex numbers. In all other respects the working out of the solution is identical with that followed in the direct-current case. When the currents are damped sinusoids, they and the potentials and the circuit constants can still be represented by complex numbers. There is this difference, however; the vectors which represent the currents and potentials shrink exponentially as they rotate and the values of the circuit constants depend not only upon the frequency of the current, but also upon its rate of shrinking. Again the solution of any problem follows the same procedure that would the corresponding one in which the currents are steady sinusoids. In both the steady and damped sinusoidal cases the circuit constants depend upon the angular velocity of the vectors which represent the currents. In the former, the angular velocity is purely imaginary while in the latter it is complex, the real part being the rate at which the current vector shrinks and the imaginary portion, its angular velocity. In electric machinery in which rotating magnetic fields are produced, these fields shrink exponontially as they rotate when the currents are damped sinusoids. If these rotating magnetic fields are represented by vectors, the vectors will have a complex angular velocity just as do the currents. The e. m. f. which is produced by a steady sinusoidal variation of flux lags the flux by 90 degrees, whereas if the flux variation is a damped sinusoid, the angle of lag is less than 90 degrees, depending upon the damping. The mathematical relation, however, is the same, vis., the e. m. f. is proportional to the negative of the product of the flux and its angular velocity. It is then readily appreciated that the form of the solution for the transient state is the same as that which is used for the steady state. Before the method can be expected to give as accurate results as are obtained when predicting the steady operation, considerable experimental data must be obtained in order to determine the best methods of measuring the necessary constants, for these may be somewhat different during the transient period than during steady operation.