Sinusoidal function

The torsional strain is a sinusoidal function of the torsion angle. Torsional strain results from the barrier to rotation about single bonds as described for ethane on p. 56. For molecules with a threefold barrier such as ethane, the form of the torsional barrier is... 

A periodic motion or vibration that is a sinusoidal function of time, that is, motion along a line given by equation X = a cos(ft)/ + (f>), where t is time, a... 

In practice the assumption of the uniform heat release per unit length of the rod is not valid since the neutron flux, and hence the heat generation rate varies along its length. In the simplest case where the neutron flux may be taken as zero at the ends of the fuel element, the heat flux may be represented by a sinusoidal function, and the conditions become as shown in Figure 9.20. 

It is essential that the roots are pure imaginary and this indicates that the solution is expressed in terms of sinusoidal functions. The general solution of Equation (3.85) is... 

A and B are constants and they are independent of time. Bearing in mind that = X + iy, it is simple to find functions x t) and y t), which describe a motion of the pendulum on the earth s surface. In accordance with Equation (3.88) a solution is represented as a product of two functions. The first one characterizes a swinging of the pendulum with the angular velocity p, which depends only on the gravitational field and the length /, while the second is also a sinusoidal function and its period is defined by the frequency of the earth s rotation and the latitude of the point, (Foucault s pendulum). In order to understand the behavior of the pendulum at the beginning consider the simplest case when a rotation is absent, co — 0. Then, we have... 

Here A is the amplitude, cp the initial phase, and coo the frequency of free vibrations. Thus, in the absence of attenuation free vibrations are sinusoidal functions and this result can be easily predicted since mass is subjected to the action of the elastic force only. In other words, the sum of the kinetic and potential energy of the system remains the same at all times and the mass performs a periodic motion with respect to the origin that is accompanied by periodic expansion and compression of the spring. As follows from Equation (3.105) the period of free vibrations is... 

Thus, the function (t) is a product of two functions one of them is a decaying exponential, but the other is a sinusoidal function with a frequency p. For instance, if K<free vibrations are close to a function described by a sinusoid slightly decaying with time, and their frequency is approximately coq. 

Our analysis is based on the mathematical properly that given a stable process (or system) and a sinusoidal input, the response will eventually become a purely sinusoidal function. This output will have the same frequency as the input, but with different amplitude and phase angle. The two latter quantities can be derived from the transfer function. 

If the input is a sinusoidal function such that f(t) = Asincot, the output Y(s) is... 

Pb concentration in air and surface water (SW) data used as inputs are presented in the Figs. 12 and 13 and were created by applying the following sinusoidal functions ... 

It should be recognized that the discrete Fourier coefficients G(x, y, co) are represented by complex numbers. The real part Re(G(x, y, to)) of the complex number represents the amplitude of the cosine part of the sinusoidal function and the imaginary part Im(G(x, y, co)) represents the amplitude of the sine wave. 

Note that due to the sinusoidal function of the shear stress in Frenkel s model V Yb= p/(4dc)). The corresponding maximum value of the shear energy, W y is given by... 

At large radii the widths of the Bragg layers converges asymptotically to the conventional (Cartesian) quarter-wavelength condition. Mathematically, this can be explained by noting that for large radii the Bessel function can be approximated by a sinusoidal function divided by square root of the radius. From the physical... 

Physically, this amounts to considering emerging rays whose histories involve no more than one reflection from the 1-2 interface (figure 1). R is then a purely sinusoidal function of (and of film thickness d), oscillating about the value R0 - r23. 

Ii and I2 are represented by a sinusoidal function with time t, such that... 

The rheometer most often used to measure viscosity at low shear rates is the cone and plate viscometer. A schematic of a cone and plate rheometer is found in Fig. 3.24. The device is constructed with a moving cone on the top surface and a stationary plate for the lower surface. The polymer sample is positioned between the surfaces. Two types of experiments can be performed the cone can be rotated at a constant angular velocity, or it can be rotated in a sinusoidal function. The motion of the cone creates a stress on the polymer between the cone and the plate. The stress transferred to the plate provides a torque that is measured using a sensor. The torque is used to determine the stress. The constant angle of the cone to the plate provides an experimental regime such that the shear rate is a constant at all radii in the device. That is, the shear rate is independent of the radial position on the cone, and thus the shear stress is also independent of the position on the cone. 

Using the cone and plate rheometer the angle Q is forced in a sinusoidal manner, leading to linear strain being introduced in the polymer. The shear strain, y, is a sinusoidal function of time t with a shear rate amplitude of % as follows ... 

If at the Fermi level, the only surface Bloch wave of the material is a sinusoidal function with Bloch vector q. 

Hints and Help Assume first that the concentration in the river can be described by a sinusoidal function with a period of one week. Then remember that any deviation from the sinus mode can be accounted for by superposition of harmonic oscillations of higher frequencies (Fourier series). Use Fig. 25.7 to prove that the attenuation of... 

Positional Distribution Function and Order Parameter. In addition to orientational order, some liquid crystals possess positional order in that a snapshot at any time reveals that there arc parallel planes which possess a higher density of molecular centers than the spaces between these planes II the normal to these- planes is defined as the -axis, then a positional distribution function. g( ). can be defined, where gOd is proportional to the fraction of molecular centers between r and + Since yO is periodic, it can he represented as a Fourier scries (a sum uf a sinusoidal function with a periodicity equal to the distance between ihe planes and its harmonics). To represent the amount ol positional order, the coefficient in front of the fundamental term is used as the order parameter. The more Ihe molecules lend to form layers, the greater the coefficient in front of ihe fundaiucnlal sinusoidal lerm and [he greaicr the order parameter for positional order,... 

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