Damping forced vibration

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. 

With damped forced vibration, the only difference in its equation and the equation for damped free vibration is that it is equal to Fo sin(ft)/) as shown below instead of being equal to zero. 

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. 

Nonclassically Damped Forced Vibrations of MDOF Structures Under Earthquake Excitation... 

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. 

Figure 5-17. Forced vibration with viscous damping.

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. 

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. 

The intensity of the energy dissipation can be estimated by the damping force acting on a particle i vibrating in a velocity Vi, which follows a linear relation [21]. 

Before considering particular test methods, it is useful to survey the principles and terms used in dynamic testing. There are basically two classes of dynamic motion, free vibration in which the test piece is set into oscillation and the amplitude allowed to decay due to damping in the system, and forced vibration in which the oscillation is maintained by external means. These are illustrated in Figure 9.1 together with a subdivision of forced vibration in which the test piece is subjected to a series of half-cycles. The two classes could be sub-divided in a number of ways, for example forced vibration machines may operate at resonance or away from resonance. Wave propagation (e.g. ultrasonics) is a form of forced vibration method and rebound resilience is a simple unforced method consisting of one half-cycle. The most common type of free vibration apparatus is the torsion pendulum. 

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. 

For s = 0 and for s - as, the interferogram given by Eq. (A 1.2) exhibits the properties quoted in Section 3.2. The functions I (v) and I (s) are a Fourier transform pair well known from the theory of forced vibrations and resonance of damped oscillators. One reason why they are very useful to demonstrate problems of Fourier transform spectroscopy is that the different contributions to I v) and to I (s) can be studied separately (cf. Fig. 12). 

We have already studied the complementary function of this equation in connection with damped oscillations. Any particular integral represents the forced vibration, but there is one particular integral which is more convenient than any other. Let... 

In a forced vibration ejqreriment, the damping peak for polycarbonate occurred at 150°C at a fiequency of 1 Hz. What worild be the location of this peak if the fiequency were 1000 Hz Polycarbonate has a... 

It is important to mention that due to the complex part, the hysteretic damping model is only valid in case of steady-state force-excited vibrations. If a damped free vibration is to be modeled, a viscous damping model has to be used. 

The plot of these functions, called the frequency response of the system, presents one of the most important features in forced vibration. In a lightly damped system, when the forcing frequency is close to the natural frequency (r 1), the amplitude of the vibration can get extremely large. This phenomenon is called resonance (consequently, the natural... 

A various number of procedures are available for performing DMA experiments. They differ on the type of vibrational excitation (forced vibrations, free damped vibrations and resonant vibrations). 

Another important concern for piezoelectric resonator applications is the amount of energy lost due to internal damping forces. A system capable of resonant oscillations will have, associated with it, a figure of merit called the material quality factor Q. The quality factor relates the energy stored to the energy dissipated per cycle. Mechanical losses can arise from intrinsic mechanisms such as thermal phonons (lattice vibrations) or defects in crystals. As the temperature increases, the thermal phonon population increases and the intrinsic phonon loss will also increase. The presence of this loss causes the stress and strain to differ in phase angle. A measure of this deviation is the relationship... 

Shear force The force that damps horizontally vibrating sharp tips within about 5 nm of a scanned surface used for AIM and highly resolved SNOM, particularly on rough surfaces. 

---