Free vibration

Since the essential parts of the right-hand side of the differential equation system, given by Eqs. (9.27), have demonstrated their operability, the homogeneous solution will be examined in detail to complete the inspection of the left-hand side. As there is no anal dic approach available to capture the dynamic behavior, the subsequent comparison comprises the formulations with the developed beam finite elements and with the commercial shell finite elements. The resulting natural frequencies w for all modes up to the third torsional mode are given in Table 10.9 for the non-rotating system as well as in Table 10.10 for the rotating system. 

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The new instrument introduced for inspection of multi-layer structures from polymeric and composite metals and materials in air-space industry and this is acoustic flaw detector AD-64M. The principle of its operation based on impedance and free vibration methods with further spectral processing of the obtained signal. 

Dynamic techniques are used to determine storage and loss moduli, G and G respectively, and the loss tangent, tan 5. Some instmments are sensitive enough for the study of Hquids and can be used to measure the dynamic viscosity T 7 Measurements are made as a function of temperature, time, or frequency, and results can be used to determine transitions and chemical reactions as well as the properties noted above. Dynamic mechanical techniques for sohds can be grouped into three main areas free vibration, resonance-forced vibrations, and nonresonance-forced vibrations. Dynamic techniques have been described in detail (242,251,255,266,269—279). A number of instmments are Hsted in Table 8. Related ASTM standards are Hsted in Table 9. 

Free- Vibration Methods. Free-vibration instmments subject a specimen to a displacement and allow it to vibrate freely. The oscillations are monitored for frequency and damping characteristics as they disappear. The displacement is repeated again and again as the specimen is heated or cooled. The results are used to calculate storage and loss modulus data. The torsional pendulum and torsional braid analy2er (TBA) are examples of free-vibration instmments. 

The amplitudes of oscillations will depend upon weight, stiffness and configuration. The record of these oscillations is known as free vibration record. The rate of oscillations will determine the natural frequency of the object. Figure 14.20 shows one such free vibration record. 

Figure 14.20 A typical free vibration record (sine wave) illustrating natural frequency of vibration and level of damping of an object F f>...

Figure 5-6. Free vibration with viscous damping.

So far, the study of vibrating systems has been iimited to free vibrations where there is no externai input into the system. A free vibration system vibrates at its naturai resonant frequency untii the vibration dies down due to energy dissipation in the damping. 

The free vibration frequencies and mode shapes will be determined for plates with various laminations specially orthotropic, symmetric angle-ply, antisymmetric cross-ply, and antisymmetric angle-ply. The results for the different types of lamination will be compared to determine the influence of bend-twist coupling and bending-extension coupling on the vibration behavior. As with the deflection problems in Section 5.3 and the buckling problems in Section 5.4, different simply supported edge boundary conditions will be used in the several problems presented. 

The free vibration of an elastic continuum is harmonic in time, so Whitney chose a harmonic solution... 

The boundary conditions established by the machine design determine the freedom of movement permitted within the machine-train. A basic understanding of this concept is essential for vibration analysis. Free vibration refers to the vibration of a damped (as well as undamped) system of masses with motion entirely influenced by their potential energy. Forced vibration occurs when motion is sustained or driven by an applied periodic force in either damped or undamped systems. The following sections discuss free and forced vibration for both damped and undamped systems. 

To understand the interactions of mass and stiffness, consider the case of undamped free vibration of a single mass that only moves vertically, which is illustrated in Figure 43.12. In this figure, the mass, M, is supported by a spring that has a stiffness, K (also referred to as the spring constant), which is defined as the number of pounds tension necessary to extend the spring one inch. 

Note that, theoretically, undamped free vibration persists forever. However, this never occurs in nature and all free vibrations die down after time due to damping, which is discussed in the next section. 

Note that for undamped free vibration, the damping constant, c, is zero and, therefore, ji, is zero. 

The simple systems described in the preceding two sections on free vibration are alike in that they are not forced to vibrate by any exciting force or motion. Their major contribution to the discussion of vibration fundamentals is that they illustrate how a system s natural... 

In undamped forced vibration, the only difference in the equation for undamped free vibration is that instead of the equation being equal to zero, it is equal to Fo sin(ft)/) ... 

In the above equation, the first two terms are the undamped free vibration, while the third term is the undamped forced vibration. The solution, containing the sum of two sine waves of different frequencies, is itself not a harmonic motion. 

In a damped forced vibration system such as the one shown in Figure 43.14, the motion of the mass M has two parts (1) the damped free vibration at the damped natural frequency and (2) the steady-state harmonic motions at the forcing frequency. The damped natural frequency component decays quickly, but the steady state harmonic associated with the external force remains as long as the energy force is present. 

If the molecule moves without hindrance in a rigid-walled enclosure (the free enclosure ), as assumed in free volume theories, then rattling back and forth is a free vibration, which could be considered as coherent in such a cell. The transfer time between opposite sides of the cell t0 is roughly the inverse frequency of the vibration. The maximum in the free-path distribution was found theoretically in many cells of different shape [74]. In model distribution (1.121) it appears at a > 2 and shifts to t0 at a - oo (Fig. 1.18). At y — 1 coherent vibration in a cell turns into translational velocity oscillation as well as a molecular libration (Fig. 1.19). 

Easy availability of ultrafast high intensity lasers has fuelled the dream of their use as molecular scissors to cleave selected bonds (1-3). Theoretical approaches to laser assisted control of chemical reactions have kept pace and demonstrated remarkable success (4,5) with experimental results (6-9) buttressing the theoretical claims. The different tablished theoretical approaches to control have been reviewed recently (10). While the focus of these theoretical approaches has been on field design, the photodissociation yield has also been found to be extremely sensitive to the initial vibrational state from which photolysis is induced and results for (11), HI (12,13), HCl (14) and HOD (2,3,15,16) reveal a crucial role for the initial state of the system in product selectivity and enhancement. This critical dependence on initial vibrational state indicates that a suitably optimized linear superposition of the field free vibrational states may be another route to selective control of photodissociation. 

The parameter coq is called the frequency of free vibrations and it is related to the period of a motion of a particle by... 

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... 

In reality, however, there is always attenuation (kAO) and free vibrations may have a completely different behavior. In this light it is appropriate to distinguish three cases. 

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<function described by a sinusoid slightly decaying with time, and their frequency is approximately coq. 

With an increase of the parameter , the exponential decay becomes stronger but the frequency of free vibrations decreases. If = mo vibrations are described by a function... 

Now we consider a more general and practical case where free vibrations still play a certain role. Suppose that the vertical spring with mass is in a state of equilibrium until the moment t — 0, and then we apply some external force which acts during the time interval 0 < t < t. Correspondingly, Equation (3.103) can be written as... 

In order to study this case as well two others, let us consider free vibrations of the mass. 

Therefore, the mass vibrates around the point of equilibrium and in the presence of attenuation it returns to its original position /I = 0. It is clear that the period of free vibrations is... 

Assuming that at the initial instant the angular velocity dp/dt — 0, we conclude that the mass m, placed at any point around the point of equilibrium, remains at rest. Of course, it is only an approximation, because we preserved in the power series, (Equation (3.146)), only the linear term and discarded terms of higher orders. Formally, this case is characterized by infinitely large period of free vibrations... 

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