Applied torque

Roark s formulas for torsional stress from a twisting motion can be expressed for beam loading conditions [4], e.g., for a power transmission application. Shear stress developed in a material subjected to a specified torque in a torsion test is estimated by [5] 

The angle of twist for a given applied torque can he estimated hy [6] 

When torque is applied, the modulus of elasticity in shear (shear modulus) can be estimated by using Poisson s ratio  

Shear modulus can he expressed with rheological variables [14]  

Sinusoidal time-varying (STV) flow demonstrates differences between elastic and viscous properties, and it demonstrates viscoelastic behavior in more complex time-varying flow [14]. 

---

Figure 5,16. It is assumed that by using an exactly symmetric cone a shear rate distribution, which is very nearly uniform, within the equilibrium (i.e. steady state) flow held can be generated (Tanner, 1985). Therefore in this type of viscometry the applied torque required for the steady rotation of the cone is related to the uniform shearing stress on its surface by a simplihed theoretical equation given as...

The radial pressure is not eonstant over the length of the hub, but in faet peaks at the projeeting portions of the shaft whieh resist eompression resulting in an inereased pressure at the ends of the hub, or stress eoneentration. For this reason, fretting fatigue failure may be antieipated when the applied torque is alternating. 

M = applied torque D = shaft diameter L = loading stress. 

We know that the eoeffieient of variation, C, of the applied torque is approximately 0.1, and that the final loading stress variable will have a similar level of variation beeause the dimensional variables have a very small varianee eontribution in eompar-ison. We also know the ultimate shear strength parameters of the weak link material, therefore substituting in equation 4.91 and rearranging to set the right-hand side to zero gives ... 

Finally, the interference between the applied torque and the pump shaft torque capacity can be analysed to determine if separation between them is too great leading... 

Figure 4.61 Weak link torque capacity shown relative to the applied torque and the torque capacity of the pump shaft...

As a guide, SM should be less than 10 for all eases of failure severity to avoid over-design. The distributions of the applied torque, weak link torque eapaeity and pump torque eapaeity are plotted to seale in Figure 4.61 for eomparison. 

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). 

Note that the direction of the applied torque will dictate the direction of the induced current in the elementary dynamo. 

Torsional stiffness is defined as the externally applied torque, T, in inch-pounds needed to turn the disk one radian (57.3°). Torque can be represented by the following equations ... 

Shear stress-strain data can be generated by twisting (applying torque) a material specimen at a specified rate while measuring the angle of twist between the ends of the specimen and the torque load exerted by the specimen on the testing machine. Maximum shear stress at the surface of the specimen can be computed from the measured torque that is the maximum shear strain from the measured angle of twist. 

The same apparatus was used, but quantities of paste were removed to give an air space in the vessel. On rapid agitation the volume increased, dependent on the air content required. Paste viscosities were measured, using a Stormer viscometer, which is a type of concentric cylinder viscometer. Although it is possible to obtain results in absolute terms, for comparative purposes the times for 100 revolutions of the rotor under a fixed applied torque were recorded. 

Solve the problem to determine a universal nondimensional rotation rate as a function of a nondimensional time. The solution can be accomplished numerically. The results must depend on the problem geometry and the applied torque. 

Section 3.2 for dynamic torsional tests. The factor K(180 - a ) is the applied torque. 

It is possible to write a general solution, in terms of the applied torque M, the relevant longitudinal shear modulus, G, the twist per unit length 6, and a form factor, F, thus... 

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). 

Torsional Stiffness (Torsion Constant). Torsional stiffness of a body is applied torque (Nm) divided by angle of twist (rad). Its SI unit is... 

In general, shear stress at one location (e.g., the bob surface in a concentric cylinder viscometer) is calculated from the dimensions of the sample gap and the measured or applied torque. Shear rate is calculated at the same location from sample gap dimensions and rotational speed. By making experimental measurements over a range of speeds or torques, the flow curve (shear stress versus shear rate) of the sample can be established. Suitable mathematical treatment of the flow curve data yields the sample s constitutive equation and rheological properties. 

Since the classical equation of motion equates the rate of change of angular momentum J to the applied torque Mj X H, the equation of motion for the magnetization of a solid is (damping terms neglected)... 

Force equilibrium considerations show that, in the absence of an applied torque, the strain and stress tensors are symmetric o-y = o, and Sy = s, . Consequently, there are really only six independent stresses for three directions that can be applied to strain a body. The mathematical representation of stress is thus ... 

The second moment of force about the same axis is the product of the force and the square of the distance between its line of action and the axis. This is the moment of inertia. The most direct example of its use is possibly connected with the motion of a rotating body, for which the rotational acceleration caused by an applied torque is calculated by dividing the torque by the moment of inertia of the body. The concept of a second moment has been extended to other less readily... 

The liquid phase of molecular matter is usually isotropic at equilibrium but becomes birefringent in response to an externally applied torque. The computer can be used to simulate (1) the development of this birefringence —the rise transient (2) the properties of the liquid at equilibrium under the influence of an arbitrarily strong torque and (3) the return to equilibrium when the torques are removed instantaneously—the fall transient. Evans initially considered the general case of the asymmetric top (C2 symmetry) diffusing in three-dimensional space and made no assumptions about the nature of the rotational and translational motion other than those inherent in the simulation technique itself. A sample of 108 such molecules was taken, each molecule s orientation described by three unit vectors, e, Cg, and parallel to its principal moment-of-inertia axes. 

The effect of the strong external field is to accentuate this difference— using the method first developed for achiral molecules described earlier in this review—that of aligning the molecules in the molecular dynamics cube with an externally applied torque.This may be used to simulate the effect of an electric field on an assembly of dipolar molecules using second-order... 

---