Moment of Torque

The integral with respect to the cross-sectional area in the moment of torque can alternatively be expressed in vector notation as 

If the cross section is singly connected, the first integral taken along the boundary surface where the condition given by Eq. (17.222b) holds, becomes null. Moreover, the first integral is taken along the boundary where F = 0, so that 

The value of the resulting integral on the right-hand side of Eq. (17,240) can 

The function f could be tabulated for different values of d/b. Thus ioi d = b (square section), we obtain 

Finally, Eqs. (17.235) and (17.242) lead to the following expression for the moment of torque of a viscoelastic rod  

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The moment of torque can be calculated in the usual way as the difference between the torque acting on the solid cylinder and the torque corresponding to the part removed to produce the hollow cylinder. For the solid cylinder, and according to the prescriptions made in the main text [Eqs. (17.235) and (17.245),... 

Gmin- minimal moment of torque, AG- the increment of torque in rubber compound during vulcanization, to2- scorch time, t9o- optimal vulcanization time, k- constant of the vulcanization rate... 

The Joule is equal to a 1 N m, but is reserved for a unit of energy and can have more than one application, as discussed later. When we get into thermal stresses and heat transfer, it is confusing to nse Jonle as a bending moment of torque and as a thermal unit. We will spend more time later on the proper use of metric units. 

If the metal specimen does not break after impact, then it absorbed the energy of impact, which defines its toughness. We wiU discnss tonghness later. The comparable impact energy nsed in the U.S. Customary system is the ft-lbf. The unit ft-lbf can be nsed as an energy unit or as a bending moment of torque. The thermal unit in the U.S. Customary system is the British Thermal Unit, or BTU. For reference, 1 BTU approximately equals 1055 Joules, or 1.055kJ (kilojoules). 

In the case of a polyatomic molecule, rotation can occur in three dimensions about the molecular center of mass. Any possible mode of rotation can be expressed as projections on the three mutually perpendicular axes, x, y, and z hence, three moments of inertia are necessar y to give the resistance to angular acceleration by any torque (twisting force) in a , y, and z space. In the MM3 output file, they are denoted IX, lY, and IZ and are given in the nonstandard units of grams square centimeters. 

Moment of Force (Torque or Bending Moment). Moment of force is force times moment arm (lever arm). Its SI unit is N-m. 

A centrifugal compressor driven through V-belts at a speed of 4500 r.p.m. having the torque curve as shown in Figure 2.18 and a moment of inertia /WK of 2.50 kgm employs a squirrel cage motor with the following parameters ... 

AT = average accelerating torque over the speed interval (difference between motor and load torque) g = gravitational constant WR -- torsional moment of inertia... 

A flywheel of moment of inertia / sits in bearings that produee a frietional moment of C times the angular veloeity uj t) of the shaft as shown in Figure 2.7. Find the differential equation relating the applied torque T t) and the angular veloeity uj t). 

Figure 2.8 shows a reduetion gearbox being driven by a motor that develops a torque T tn(t). It has a gear reduetion ratio of and the moments of inertia on the motor and output shafts are and /q, and the respeetive damping eoeffieients Cm and Cq. Find the differential equation relating the motor torque CmfO and the output angular position 6a t). 

A field eontrolled d.e. motor develops a torque Tm t) proportional to the field eurrent k t). The rotating parts have a moment of inertia / of 1.5 kg m and a viseous damping eoeffieient C of 0.5 Nm s/rad. 

The calculation for torque is a primary example of a single degree of freedom in a mechanical system. Figure 43.15 represents a disk with a moment of inertia, /, that is attached to a shaft of torsional stiffness, k. 

Moment of inertia It is the ratio of torque applied to a rigid body free to rotate about a given axis to the angular acceleration thus produced about that axis. 

This is an equation of rotation of an elementary mass around the y-axis. Here r can be treated as the moment of inertia of the unit mass and dco/dt is the angular acceleration. The product gx characterizes the torque with respect to the point 0. Multiplying Equation (3.49) by dm and performing integration over the pendulum mass, we obtain... 

Equations 12 through 14 may be derived from thermodynamic and mechanical expressions for the transverse force and torque acting on a surface which intersects the interfacial region ( ). Moments of an isotropic pressure force and a surface tension acting at R are... 

A kinetic argument shows that ay - oy always. Any imbalance between these two would lead to an angular acceleration of a volume element. If this volume element were shrunk, then the torque would reduce in proportion to the linear dimension cubed, but the moment of inertia would reduce in proportion to the fifth power of the linear dimension, so that the angular acceleration would increase as the reciprocal of the square of the size of the volume element, becoming infinite in the limit. Thus reductio ad absur-dum, ay = cry. Hence there are only six independent components of the stress tensor. 

Few details of the apparatus are given in ISO 4663, it is simply stated that means shall be provided to measure frequency to 1% ( 5% in a transition region), amplitude to 1% and, for method C, the supplied energy to 2%. It is suggested that a moment of inertia of about 0.03 gm is suitable for the inertia member which may be a disc or rod. For methods B and C the torsion wire should be of such dimensions that its restoring torque is not more than 25% of the total restoring torque due to the test piece and suspension. BS 903 (equivalent to method B of ISO 4663) suggests that moments of inertia between 50 and 500 g cm are suitable and states that the tensile strain on the test piece should be between 0 and 5%. The British... 

The important point to note here is that the 2nd moment of Ky(t) depends on the 2nd and 4th moments of y(t). The 2nd moments of each of the three previously mentioned autocorrelation functions may be calculated from ensemble averages of appropriate functions of the positions, velocities, and accelerations created in the dynamics calculations. Likewise, the 4th moment of the dipolar autocorrelation function may also be calculated in this manner. However the 4th moments of the velocity and angular momentum correlation functions depend on the derivative with respect to time of the force and torque acting on a molecule and, hence, cannot be evaluated directly from the primary dynamics information. Therefore, these moments must be calculated in another manner before Eq. (B.3) may be used. 

First, if the specimen is subjected to axial stress a as well as the twist then as Biot215 has shown, Eq. (2.1) needs to be modified so that M = M — al, where M is the applied torque and I the second moment of area of the cross section with respect to the twist axis. Second, the St. Venant principle needs to be modified, as shown above, if the material is highly anisotropic, and very high length to diameter ratios may be necessary (Folkes and Arridge125). 

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