Natural frequency, of the system

This is an important equation that defines the behaviour of a vibrating body under different conditions of applied force or motion F y From this it can be inferred that the response or movement of object x will depend upon t) and 7 is termed the fraction of critical damping and w , the angular natural frequency of the system. With the help of these equations, the response characteristics of an object to a force can be determined. 

Critical speed is when the frequency of a periodic exciting frequency applied to the rotor-bearing support system corresponds to the natural frequency of the system. 

Critical speeds correspond to the natural frequencies of the gears and the rotor bearings support system. A determination of the critical speed is made by knowing the natural frequency of the system and the forcing function. Typical forcing functions are caused by rotor unbalance, oil filters, misalignment, and a synchronous whirl. 

It can be observed from the previous expression that when rigid support), then Lu t = lu or the natural frequency of the rigid system. For a system with a finite stiffness at the supports, or K/, = Kr, then lu is less than Lu t. Hence, flexibility causes the natural frequency of the system to be lowered. Plotting the natural frequency as a function of bearing stiffness on a log scale provides a graph as shown in Figure 5-15. 

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. 

So, at m2 mi, the natural frequencies of the system correspond to the independent vibrations of the mass M on the spring h and of the reduced mass m on the spring k 2. As the parameter B tends to unity for m2 mu the relative displacement of particles 1 and 2 is approximately described by the normal coordinate x2 ... 

By the very definition of the GF, the real parts of the poles of its frequency Fourier component correspond to natural frequencies of the system (see, for example, Eqs. (A1.23) or (A1.55)). Consequently, the spectrum of natural frequencies of the perturbed system, cop, should fit the equation... 

For a monoatomic lattice treated in the approximation of isotropic GFs, the equation is deduced for natural frequencies of the system ... 

The curve shows the amplitude of oscillation of an object or system as the frequency of the input oscillation is steadily increased. Start by drawing a normal sine wave whose wavelength decreases as the input frequency increases. Demonstrate a particular frequency at which the amplitude rises to a peak. By no means does this have to occur at a high frequency it depends on what the natural frequency of the system is. Label the peak amplitude frequency as the resonant frequency. Make sure that, after the peak, the amplitude dies away again towards the baseline. 

An alternative approach based on a mass spring model (MSM) has been proposed by Williams [ 1 ]. The test system is represented by a lumped mass model with the contact stifBiess between the striker and specimen being ki, this acts on an equivalent mass m which is 17/35 of the specimen mass, and the specimen stiffness is k2- A vital factor is the contact stiffness ki, which controls the dynamics of the system. In reality, this factor is not linear but for a case of the contact of a finite cylinder on plane used here, it can be approximated as linear. From this it is possible to calculate the natural frequency of the system o) where... 

A schematic diagram of the torsion pendulum is shown in Figure 1. Free oscillations are initiated by an angular step-displacement of the upper member of the pendulum. The response of the lower member is a damped wave at the natural frequency of the system, and therefore is related to the physico-mechanical properties of the specimen. 

A resonance of a system can be produced by excitation that oscillates at a frequency close to the natural frequency of the system, unlike a relaxation, which is the restoring action of a diffusive force of thermodynamic origin. Direct resonance or a one-photon process can occur within isolated intervals of the electromagnetic spectrum from ultraviolet to visible frequencies close to 10 Hz (electronic oscillator), in infrared with frequencies dose to 10 Hz (vibrational modes), and in the far infrared and microwave range with frequencies close to 10 Hz (rotational modes). 

Assuming small damping, this value is the natural frequency of the system. 

The lowest natural frequency of the system was earlier determined to be about 4 Hz. Since the frequency of driving excitation is 50% of the lowest natural frequency, a simple sensitivity an ysis was performed in order to determine its influence. It was found that as the driving frequency increased from 1.15 Hz to 2.44 Hz, the harmonic response increased only 5%. The results are listed in Table 42.4. 

The basic theory of seiche oscillations is similar to the theory of free and forced oscillations of mechanical, electrical, and acoustical systems. The systems respond to an external forcing by developing a restoring force that re-establishes equilibrium in the system. A pendulum is a typical example of such a system. Free oscillations occur at the natural frequency of the system if the system disturbed beyond its equilibrium. Without additional forcing, these free oscillations retain the same frequencies but their amplitudes decay exponentially due to friction, until the system eventually comes to rest. In the case of a periodic continuous forcing, forced oscillations are produced with amplitudes depending on friction and the proximity of the forcing frequency to the natural frequency of the S3 tem. ... 

The liquid-to-solid contact lines were pinned by a hydrophobic substrate this significantly reduced energy dissipation from movements of the contact lines and viscosity. The natural frequency of the system scaled with the radius of the lens as and the resulting very high frequency response was obtained with a modestly sized lens [41]. 

Next we consider an oscillatory system, such as (16.1), in which there is an oscillatory input of the concentration of the reactant A. In that case the response of the system is not only at the natural frequency of the system but also at other frequencies related to the frequency of the external perturbation of the reactant A. The variety of responses of a typical system are shown in Fig. 16.1. 

Chain vibration can cause very large inereases in chain tensile loading if the vibration occurs at or near the natural frequency of the chain. The added tension from vibration can sometimes be as large as the nominal tensile load. Here again, any varying tensile load from vibration will be added to the nominal tensile load. The natural frequency of the system should be calculated when necessary, as suggested in later chapters. Then, if a possible problem with vibration is found, the system should be redesigned to avoid the natural frequency. 

Equation (4.48) is useful in correcting difficulties due to vibration. If a motor rests upon a base that has some flexibility which may be expressed in terms of a spring constant (k) and the mass of the motor and base (m), then the natural frequency of the system will be given by Eq. (4.48). When the rotational frequency of the motor corresponds to the natural frequency of the mass-spring system, then the amplitude of vibration will quickly rise to a high level. Not only will the motor move violently, but also the vibration will be transmitted to the floor. There are several remedies for this situation ... 

The electric field of a laser is oscillating. Keldysh (Landau and Lifschitz, 1971) provided an estimate for the role of an oscillating field that is not resonant The Keldysh estimate for the onset of non-resonant ionization of an atom, for a field frequency lower than any natural frequency of the system, is consistent with our value of 10 W cm . Molecules are not like a hydrogen atom and, in particular, the gap between electronic states is very dependent on the intemuclear distances (Dietrich and Corkiim, 1992 Chelkowski and Bandrauk, 1995). This has interesting potential applications, e.g., S. Lochbrtmner et al.,... 

Two natural frequencies of the system can be found from the corresponding characteristic equation, derived by equating the denominator of Eq. 7 to zero ... 

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