Angular displacements

In an early study of lysozyme ([McCammon et al. 1976]), the two domains of this protein were assumed to be rigid, and the hinge-bending motion in the presence of solvent was described by the Langevin equation for a damped harmonic oscillator. The angular displacement 0 from the equilibrium position is thus governed by... 

Figure H-7 is the schematic of a basic I/P transducer. The transducer shovvm is characterized bv (1) an input conversion that generates an angular displacement of the beam proportional to the input current, (2) a pneumatic amplifier stage that converts the resulting angii-...

The angular displacement can be computed using the following relationship ... 

Angular acceleration is the time rate of change of angular velocity which, in turn, is the time rate of change of angular displacement. 

A torsional spring of stiffness K, a mass of moment of inertia / and a fluid damper with damping coefficient C are connected together as shown in Figure 3.25. The angular displacement of the free end of the spring is 0 ( ) and the angular displacement of the mass and damper is 6a t). 

Fig. 4.14 DC servo-motor under armature control, e it) = Armature excitation voltage e it) = Backemf /a(t) = Armature current = Armature resistance = Armature inductance 6f = Constant field voltage if = Constant field current Tm = Torgue developed by motor 6 t) = Shaft angular displacement u] t) = Shaft angular velocity = dd/dt.

Rotation The angular displacement of a body about a specified axis in a specified direction in unit time. 

The bellows can, however, be incorporated into a properly designed expansion fitting as shown in Figure 22.23, which is capable of absorbing not only axial movement of the pipeline but some lateral and angular displacement as well. 

The form of Lz in Cartesian coordinates is Eq. (7-la), and it is clear that orbital angular momentum is related to angular displacement in the same way as the linear operators are related. 

The perturbation theory presented in Chapter 2 implies that orientational relaxation is slower than rotational relaxation and considers the angular displacement during a free rotation to be a small parameter. Considering J(t) as a random time-dependent perturbation, it describes the orientational relaxation as a molecular response to it. Frequent and small chaotic turns constitute the rotational diffusion which is shown to be an equivalent representation of the process. The turns may proceed via free paths or via sudden jumps from one orientation to another. The phenomenological picture of rotational diffusion is compatible with both... 

Position Feedback Encoders generate 512 pulses per revolution of the motor. One pulse correlates to the me2tsurement of 0.023 degree in angular displacement of the mandrel. 

Here /I is angular displacement of mass with respect to the point of equilibrium,... 

If D is the angular displacement of the spring with a spring constant of Cs, i.e. the torque required to give unit angular displacement, then the stress is... 

Figure 4.3. The rod (or disk) model for torsion and flexure of DNA. The DNA is modeled as a string of rods (or disks) connected by Hookean twisting and bending springs which oppose, respectively, torsional and flexural deformations. The instantaneous z and x axes of a subunit rod around which the mean squared angular displacements , j = x, z, take place are indicated. The filament is assumed to exhibit mean local cylindrical symmetry in the sense that for any pair of transverse x- and y-axes. Twisting = mean squared angular displacement about body-fixed x -axis = (/)y(t)2) (assumed).

The wavelength of the torsion normal mode with relaxation time r = 1 ns is A >50 bp for a >3.8x10 12 dyn-cm [from Eq. (4.34)]. Thus, the shortest torsion normal modes resolved in the FPA have wavelengths extending over about five full turns of the helix. The rms angular displacement of a base pair around its helix axis is about 18° at t= 1 ns and increases without bound as t goes to infinity. 

The chain of subunit symmetry axis vectors (bond vectors) is projected onto a plane containing the first vector. In this plane, the mean squared angular displacement of the (m + 1 )th vector with respect to the first is(109) <<5, 2> = m + l kBT/k = RJP. Thus, we set... 

Due to its large discrete jumps, rotation around the helix axis is not a continuous Gaussian random process, so its actual mean squared angular displacement cannot be used directly in C (t). However, <<5z(/)2> for the equivalent Gaussian random process that makes the same contribution to the FPA is obtained from the relation... 

Of the 3n coordinates needed to describe an n-atom molecule, three are used for center of mass motion, three describe angular displacement (rotation, hindered rotation, or libration) (two if the molecule is linear, 0 if monatomic), the remaining 3n—6 (3n—5, if linear, 3n—3 = 0, if monatomic) describe atom-atom displacements (vibrations). In some cases it may not be possible to separate translation cleanly from rotation and vibration, but when the separation can be made it is a convenience. Elementary treatments assume... 

In general, the 3D motion of the spherical pendulum is very complex, but for fixed initial angular displacements, values of the kinetic energy can be found (by trial and error) for which this motion is periodic. The approximation discussed above leads to the approximate description of the horizontal motion in terms of Mathieu functions, for which Flocquet analysis determines periodic solutions in terms of two integers k and n, which can be thought of as quantum numbers. 

The right-hand sided is a cubic polynomial in w[t] which is shown in Fig. 3, plotted as a function of w for W= —1/2 (corresponding to an initial angular displacement from the downward vertical of 60°) and three values of K. 

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