Torque pendulum

FIGURE 5.21 The torque pendulum. The description of the stressed state in the interfacial film. 

The principal scheme of the torque pendulum is shown in Figure 5.21. A disk of radius R is placed at the surface of the liquid or at the interface between the two liquids (a polar aqueous surfactant solution phase and a nonpolar phase, that is, hydrocarbon or fluorocarbon). The disk is suspended on a thin wire that serves as a dynamometer. Turning this wire by an angle f produces a torque M, while the turn of a cuvette by an angle ( ) with respect to the disk leads to the total shear deformation of the adsorption film. The principle of operation of this instrument is similar to that of a rotation viscometer, with one principal difference in the torque pendulum it is not possible to utilize a thin gap between a disk and a cuvette. Because of this the stresses and the deformations in the film are not uniform. 

The rheological behavior of the lAL formed between a nonpolar phase (HL, FL) and an aqueous solution of a surfactant (HS, FS) was studied by the rotation suspension ( torque pendulum ) method [13-15]. Its principle is illustrated in Figure 3.1. As a rule, the conditions of constant rate of revolution Q of the cylindrical vessel were used, that is, the constant shear strain rate. Both the revolution rate and the surfactant concentrations were varied over a broad range. 

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... 

Exercise. For a pendulum in a potential 1/(0) and subject to a constant torque x this equation is... 

Solution. The restoring torque for a torsion pendulum is — kd, where 6 is the angle of rotation (see Fig. 8.18) and k is the torsion constant. The equation of motion [19] is then... 

One can determine the shear modulus of a fiber from a torque per unit area versus twist curve. In practice, a simple apparatus called a torsion penduliun is used more commonly. An experimental setup to measure the shear modulus of small fibers is shown in Fig. 9.8 (Mehta, 1996). The torsional pendulum, placed in a vacuum oven, allows the measurement of shear modulus as a func-... 

Various methods (1-1) have used to determine the dynamic mechanical properties of polymers. Many of the instruments described are well known and are widely used (torsional pendulum, rheovibron, vibrating reed, and Oberst beam ASTM D4065-82). Newer instruments like the torqued cylinder apparatus (4), resonant bar apparatus (5) and Polymer Laboratories Dynamic Mechanical Thermal Analyzer (6) are becoming more popular in recent times. 

Recently, Douarche et al verified the transient ES FR and steady state ES FR for a harmonic oscillator (a brass pendulum in a water-glycerol solution, that is driven out of equilibrium by an applied torque). They also developed a steady state relation for a system with a sinusoidal forcing, and showed that the convergence time was considerably longer in this case. 

Measurements of creep in torsion can be made very accurately. The reason is that deformation can be measured by measuring the large deflections of a light beam. A convenient way to simultaneously obtain shear dynamic and transient data is to combine both types of measurements in the same equipment (4). Usually this requires only small modifications of the experimental device. For example, the cross bar in a torsion pendulum can be removed and replaced by weights and pulleys to apply a constant torque to the upper clamp. In this way, a torsion creep apparatus is obtained (Fig. 7.8). The... 

A circular rod of a viscoelastic material of length h and radius R located between the two clamps of a torsion pendulum is rotated slightly from its equilibrium position by a deflecting torque. The torque is released, and the system begins to oscillate. Calculate the resonance frequency of the system. 

Condensed-matter physics (Josephson junction, charge-density waves) Mechanics (Overdamped pendulum driven by a constant torque)... 

We now consider a simple mechanical example of a nonuniform oscillator an overdamped pendulum driven by a constant torque. Let 0 denote the angle between the pendulum and the downward vertical, and suppose that d increases counterclockwise (Figure 4.4.1). 

To think about this problem physically, you should imagine that the pendulum is immersed in molasses. The torque F enables the pendulum to plow through its vis-... 

The dimensionless group y is the ratio of the applied torque to the maximum gravitational torque. If / > 1 then the applied torque can never be balanced by the gravitational torque and the pendulum will overturn continually. The rotation rate is nonuniform, since gravity helps the applied torque on one side and opposes it on the other (Figure 4.4.2). 

This mechanical analog has often proved useful in visualizing the dynamics of Josephson junctions. Sullivan and Zimmerman (1971) actually constructed such a mechanical analog, and measured the average rotation rate of the pendulum as a function of the applied torque this is the analog of the physically important I-V curve (current-voltage curve) for the Josephson junction. 

Pendulum driven by constant torque) The equation 0-t-sin0 = y describes the dynamics of an undamped pendulum driven by a constant torque, or an undamped Josephson junction driven by a constant bias current. 

This section deals with a physical problem in which both homoclinic and infinite-period bifurcations arise. The problem was introduced back in Sections 4.4 and 4.6. At that time we were studying the dynamics of a damped pendulum driven by a constant torque, or equivalently, its high-tech analog, a superconducting Josephson junction driven by a constant current. Because we weren t ready for two-dimensional systems, we reduced both problems to vector fields on the circle by looking at the heavily overdamped limit of negligible mass (for the pendulum) or negligible capacitance (for the Josephson junction). 

Now we re ready to tackle the full two-dimensional problem. As we claimed at the end of Section 4.6, for sufficiently weak damping the pendulum and the Josephson junction can exhibit intriguing hysteresis effects, thanks to the coexistence of a stable limit cycle and a stable fixed point. In physical terms, the pendulum can settle into either a rotating solution where it whirls over the top, or a stable rest state where gravity balances the applied torque. The final state depends on the initial conditions. Our goal now is to understand how this bistability comes about. 

Suppose we slowly decrease 7, starting from some value / > 1. What happens to the rotating solution Think about the pendulum as the driving torque is reduced, the pendulum struggles more and more to make it over the top. At some critical value 4 < 1, the torque is insufficient to overcome gravity and damping, and the pendulum can no longer whirl. Then the rotation disappears and all solutions damp out to the rest state. 

The gravitational torque is like that of an inverted pendulum, since water is pumped in at the top of wheel (Figure 9.1.6). 

To check that the sign is correct, observe that when sin 6 > 0 the torque tends to increase co, just as in an inverted pendulum. Here g is the effective gravitational constant, given by g = gg sin a where gg is the usual gravitational constant and a is the tilt of the wheel from horizontal (Figure 9.1.1). 

The period of oscillation of a physical pendulum (the leg) moving in response to a torque is given by the moment of inertia of the leg and the maximum torque ... 

Dynamic mechanical measurements can mostly be divided into two groups. The reaction of a sample to a once applied light torque can be measured with the torsion pendulum. The sample oscillates freely, whereby the amplitude decreases steadily with each cycle for viscoelastic materials. The ratio of two successive amplitudes is constant for ideal viscoelastic materials. This procedure yields shear moduli. The torsion pendulum allows measurements to be relatively easily made the disadvantage is that the frequency is not an independent variable with this method. 

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