Forced Damped Oscillator

We shall first summarize the motion of the damped forced oscillator, which includes a force contribution driving the motion and supplying energy to the vibrating system. 

This expression shows diat if die detuning Acuj is negative (i.e. red detuned from resonance), dieii die cooling force will oppose die motion and be proportional to die atomic velocity. The one-diniensional motion of die atom, subject to an opposing force proportional to its velocity, is described by a damped haniionic oscillator. The Doppler damping or friction coefficient is die proportionality factor. 

Now suppose that the harmonic oscillator represented in Fig. 1 is immersed in a viscous medium. Equation (32) will then be modified to include a damping force which is usually assumed to be proportional to the velocity, -hx. Thus,... 

Ashkin and Dziedzic (1977) used the radiation pressure force of a laser beam to levitate microdroplets with the apparatus presented in Fig. 15. A polarized and electro-optically modulated laser beam illuminated the particle from below. The vertical position of the particle was detected using the lens and split photodiode system shown. When the particle moved up or down a difference signal was generated then a voltage proportional to the difference and its derivative were added, and the summed signal used to control an electro-optic modulator to alter the laser beam intensity. Derivative control serves to damp particle oscillations, while the proportional control maintains the particle at the null point. 

The examples we saw are for L C circuits supplied from a direct current source. What happens when an L C circuit is excited by an alternating current source Once again, oscillatory response will be present. The oscillatory waveform superimposes on the fundamental waveform until the damping forces sufficiently attenuate the oscillations. At this point, the system returns to normal operation. In a power system characterized by low resistance and high values of L and C, the effects would be more damaging than if the system were to have high resistance and low L and C because the natural frequencies are high when the values of L and C are low. The... 

Fig. 39. NO/CO reaction on Pt(IOO). (From Ref. 161.) (a) Damped oscillation of the rate if the temperature is held strictly constant, (b) Periodic modulation of the temperature by 2 K. causes forced oscillations with appreciable amplitude.

The interaction of a light wave and electrons in atoms in a solid was first analysed by H. A. Lorentz using a classical model of a damped harmonic oscillator subject to a force determined by the local electric field in the medium, see Equation (2.28). Since an atom is small compared with the wavelength of the radiation, the electric field can be regarded as constant across the atom, when the equation of motion becomes ... 

In Eq. (4-31), the first three terms describe a simple damped harmonic oscillator the first term is due to molecular accelerations, the second is due to viscous drag, and the third is due to the restoring force. Qq is the oscillator frequency, which is of order 10 sec", and p is a viscous damping coefficient. The crucial term producing the dynamic glass transition is, of course, the fourth term, which has the form of a memory integral, in which molecular motions produce a delayed response. The kernel m(t — t ) is determined self-consistently by the time-dependent structure. One simple choice relating m(s) to the structure is ... 

Typical examples of load-time traces of pre-cracked Charpy specimens at impact rates from 0.1 up to 3.3 m/s are shown in Fig. 4a for PVC. While the recorded non-damped signal up to about 0.5 m/s is of sufficient quality to directly determine the fracture force, Fp, significant force oscillations are visible on the signals for higher loading rates. 

For a weakly perturbed, harmonic damped driven oscillator, the resonance is shifted on the frequency scale depending on the sign of the interaction forces (Fig. 1.13). The availability of analytical expressions facilitates the applications of the weakly perturbed harmonic oscillator models for AM-AFM. Such harmonic models may be useful to illustrate the concepts used in AM-AFM well enough however, in most practical imaging cases, they do not describe the experiments [15]. 

Fig. 1.16 Average tip—sample force and oscillation frequency as a function of reduced amplitude (setpoint). The dashed resonance corresponds to a damped driven oscillator without sample—tip interactions. Reprinted from [16], copyright American Physical Society...

The torsional potential of mean force (Fig. 24) and the correlation function for the torsional motions of the Tyr-21 ring in BPTI suggest that the time dependence of A can be described by the Langevin equation for a damped harmonic oscillator (see Chapt. IV.C and D). 

To obtain the standard form of Onsager s theory [37,38], we next linearize the thermodynamic forces in eqs. (A. 15) and (A.28). This linearization reduces these equations to coupled damped harmonic oscillator equations of motion. 

Equation 6 describes a damped harmonic oscillator, suggesting that the coatings be studied in oscillation. In fact, this decision is forced on the experimenter by the inadequacy of gap-loading instruments. When applied to the situation represented by a free oscillation. Equation 6 is solved with 0 = 0. If forced oscillation provides the stress, the left-hand side is represented by Og sin 0)t, where u) is the circular frequency. Examples are given later of both types. 

Two unknowns require two measurements. For a free oscillator these measurements are the resonant frequency and the damping. For a forced oscillator the favored combination is the amplitude ratio and phase angle over a range of applied frequencies. This combination is not available for the evaluation of coatings because of the requirement that one surface be free. The two measurements described in the example below are damping and phase angle. 

We now discuss a damped harmonic oscillator, which is a harmonic oscillator that is subject to an additional force that is proportional to the velocity, such as a frictional force due to fairly slow motion of an object through a fluid. 

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