Damping natural frequency

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. 

NOTE These serve as useful reference points for subsequent analysis of the damped natural frequencies. 

The eigenvalues and eigenvectors A come out to be complex. The eigenvalues have the shape 0 + jo, where a is the damping of the mode (negative value), and o is the damped natural frequency. As the matrices have real coefficients, the solutions come in pairs of complex conjugates. 

Damped natural frequency = (Dn Jl-Input shock at time t Duration of shock pulse Forced frequency = /T Maximum relative displacement. 

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. 

Damping reduces the transmissibihty at the natural frequency, but increases the transmissibihty at higher frequencies. The natural frequency of isolators made from most materials also can be expressed as a function of the static deflection of the isolator due to the load imposed by the supported equipment that is, / = 5/v where 5 is the static deflection of the isolator, cm (5). 

Bellows can vibrate, both from internal fluid flow and externally imposed mechanical vibrations. Internal flow liner sleeves prevent flow-induced resonance, which produces bellows fatigue failure in minutes at high flow velocities. Mechanically induced resonant vibration is avoided by a bellows with a natural frequency far away from the forcing frequency, if known. Multiple-ply bellows are less susceptible to vibration failure because of the damping effect of interply friction. 

The effect of pulsating flow on pitot-tube accuracy is treated by Ower et al., op. cit., pp. 310-312. For sinusoidal velocity fluctuations, the ratio of indicated velocity to actual mean velocity is given by the factor /l + AV2, where X is the velocity excursion as a fraction of the mean velocity. Thus, the indicated velocity would be about 6 percent high for velocity fluctuations of 50 percent, and pulsations greater than 20 percent should be damped to avoid errors greater than 1 percent. Tne error increases as the frequency of flow oscillations approaches the natural frequency of the pitot tube and the density of the measuring fluid approaches the density of the process fluid [see Horlock and Daneshyar, y. Mech. Eng. Sci, 15, 144-152 (1973)]. 

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

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. 

Natural frequency. This parameter for a single degree of freedom is given by lu = yjk/m. Inereasing the mass reduees lu , and inereasing the spring eonstant k inereases it. From a study of the damped system, the damped natural frequeney loj = ujn J — C is lower than... 

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 natural frequencies of a damped system are essentially the same as the undamped systems for all realistic values of damping. 

Equations (3.42) and (3.43) are the standard forms of transfer functions for a second-order system, where K = steady-state gain constant, Wn = undamped natural frequency (rad/s) and ( = damping ratio. The meaning of the parameters Wn and ( are explained in sections 3.6.4 and 3.6.3. 

Consider a second-order system whose steady-state gain is K, undamped natural frequency is Wn and whose damping ratio is (, where C < 1 For a unit step input, the block diagram is as shown in Figure 3.18. From Figure 3.18... 

Equation (3.71) can only be used if the damping is light and there is more than one overshoot. Equation (3.67) can now be employed to calculate the undamped natural frequency... 

Since the Bulk Modulus of hydraulic oil is in the order of 1.4 GPa, if m and F[ are small, a large hydraulic natural frequency is possible, resulting in a rapid response. Note that the hydraulic damping ratio is governed by Cp and A c. To control the level of damping, it is sometimes necessary to drill small holes through the piston. 

Thus, for a settling time of 0.1 seconds for a system that is critically damped, the undamped natural frequency is... 

Note that r locus and r locf ind works for both continuous and discrete systems. The statement squar e provides square axes and so provides a round unit circle. The command zgr id creates a unit circle together with contours of constant natural frequency and damping, within the unit circle. When examp76.m has been run, using r locf ind at the MATLAB prompt allows points on the loci to be selected and values of K identified (see Figure 7.20)... 

The natural frequency, co associated with the mode shape that exhibits a large displacement of the pump is compared with the fundamental frequency, of the wall. If co is much less than ru, then the dynamic interaction between the wall and the loop may be neglected, but the kinematic constraint on the pump imposed by the lateral bracing is retained. If nearly equals nr , the wall and steam supply systems are dynamically coupled. In which case it may be sufficient to model the wall as a one-mass system such that the fundamental frequency, Wo is retained. The mathematical model of the piping systems should be capable of revealing the response to the anticipated ground motion (dominantly translational). The mathematics necessary to analyze the damped spring mass. system become quite formidable, and the reader is referred to Berkowitz (1969),... 

Show that when = 0 (natural period of oscillation, no damping), the process (or system) oscillates with a constant amplitude at the natural frequency (O,. (The poles are at [Pg.61]

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