Vibrations forced

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 

We have obtained an inhomogeneous differential equation with constant coefficients. As follows from the theory of linear equations, its solution is a sum 

Here Te = mAg and. s(t) is displacement of mass with respect to an equilibrium at the first point. If we assume that this device starts to measure instantly, the function Agf is described by a step function, Fig. 3.7a. In order to find the motion of the mass due to the step-function force, forced vibrations, one can apply different approaches, for instance, the Fourier transform. We will solve this problem differently and suppose that the right hand side of Equation (3.118) behaves as 

In the case of the step function we have B — 0 but the coefficient A remains the 

Mechanical sensitivity and stability of vertical spring-mass system 

In this final chapter, we will attempt to review the findings attained during the various steps of theory derivation and subsequent validation as well as to present the prospect of potential future development. 

Adaptive fiber composites have been examined on the basis of a thorough, systematic treatment of the theory across a wide spectrum from piezoelectric material behavior to the dynamics of rotating structures. Making a point of consistency and continuity, the derived formulations are accompanied by several innovations and improvements, the most relevant of which will be recalled below  

Brockmann, Theory of Adaptive Fiber Composites, Solid Mechanics and Its Applications 161, 

To ensure the soundness of the various assumptions and simplifications made throughout the course of derivation as well as to exclude errors in the implementation, the obtained results have been successfully counterchecked using completely independent approaches. Therein, the following steps are regarded, 

Using the presented comprehensive formulation of the theoretical framework and the associated elementary examinations, the recognition and utilization of causal relationships, in view of the manipulation of structural behavior with adaptive means, is facilitated. With the resulting spatial beam finite elements, a versatile modeling tool can be provided as a basis for further investigations. 

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Sasaki N and Tsukada M 1999 Theory for the effect of the tip-surface interaction potentiai on atomic resoiution in forced vibration system of noncontact AFM Appl. Surf. Sc/. 140 339... 

The transmissibihty of an isolator varies with frequency and is a function of the natural frequency (J/) of the isolator and its internal damping. Figure 7 shows the transmissibihty for a family of simple isolators whose fundamental frequency can be represented as follows, where k is the stiffness of the isolator, N/m and m is the supported mass, kg. Figure 7 shows that an isolator acts as an amplifier at its natural frequency, with the output force being greater than the input force. Vibration isolation only occurs above a frequency of aboutv times the natural frequency of the isolator. 

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. 

Resonance-Forced Vibration. Resonance-forced vibration devices drive the vibration of the specimen. This can be over a range of frequencies that includes the resonant frequency, which is detected as a maximum in the ampHtude, or the instmment can be designed to detect the resonant frequency and drive the specimen at that frequency. An example of the resonance-forced vibration technique is the vibrating reed. A specimen in... 

Fig. 46. Schematic diagram of a dynamic mechanical analy2er based on the nonresonance-forced vibration principle (Rheovibron-type).

The Metravib Micromecanalyser is an inverted torsional pendulum, but unlike the torsional pendulums described eadier, it can be operated as a forced-vibration instmment. It is fully computerized and automatically determines G, and tan 5 as a function of temperature at low frequencies (10 1 Hz). Stress relaxation and creep measurements are also possible. The temperature range is —170 to 400°C. The Micromecanalyser probably has been used more for the characterization of glasses and metals than for polymers, but has proved useful for determining glassy-state relaxations and microstmctures of polymer blends (285) and latex films (286). 

Figure 5-17. Forced vibration with viscous damping.

Forced (resonant) vibration. In forced vibration the usual driving frequency in rotating machinery is the shaft speed or multiples of this speed. 

This speed becomes critical when the frequency of excitation is equal to one of the natural frequencies of the system. In forced vibration, the system is a function of the frequencies. These frequencies can also be multiples of rotor speed excited by frequencies other than the speed frequency such as blade passing frequencies, gear mesh frequencies, and other component frequencies. Figure 5-20 shows that for forced vibration, the critical frequency remains constant at any shaft speed. The critical speeds occur at one-half, one, and two times the rotor speed. The effect of damping in forced vibration reduces the amplitude, but it does not affect the frequency at which this phenomenon occurs. 

Figure 5-20. Characteristic of forced vibration or resonance in rotating machinery. (Ehrich, F.F., Identification and Avoidance of Instabiiities and Seif-Excited Vibrations in Rotating Machinery, Adopted from ASME Paper 72-DE-21, Generai Eiectric Co., Aircraft Engine Group, Group Engineering Division, May 11, 1972.)...

Resonant vibration. Any of the forced vibration loads, such as cyclic or misalignment loads, may have a frequency that coincides with a natural frequency of the rotating-shaft system, or any component of the complete power plant and its foundation, and may, thus, excite vibration resonance. 

For complex offshore structures or where foundations may be critical, finite-element analysis computer programs with dynamic simulation capability erm be used to evaluate foundation natural frequency and the forced vibration response. 

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. 

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. 

For damped forced vibrations, three different frequencies have to be distinguished the undamped natural frequency, = y KgJM the damped natural frequency, q = /KgJM — cgJ2M) and the frequency of maximum forced amplitude, sometimes referred to as the resonant frequency. 

The introduction of these somewhat mysterious functions allows certain differential equations to be converted into equivalent integral equations. Although the method is particularly useful in its application to partial differential equations, it will be illustrated here with the aid of a relatively simple example, the forced vibrations of a classical oscillator. 

An instrument for the measurement of heat buildup in vulcanised rubber by a forced vibration method. 

Forced-vibration instruments, 21 745 Force field calculations, 16 742 Force field energy, 16 742 Force field performance, 16 745 Force fields, 16 743-745 Force field simulations, 16 746-747 programs for, 16 746 Force modulation microscopy, 3 332 Forces, exponents of dimensions in absolute, gravitational, and engineering systems, 8 584t Forchlorfenuron, 13 43t, 53 Ford nuclear reactor, 17 594... 

The formula applies to results from both long-term creep and short-term forced vibration tests, but by itself provides a poor fit at longer durations. 

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