Forcing functions

The force function /(r) differs from that in ref. [65] by a factor r, yielding simpler expressions. Some errors in that reference have been corrected. 

The linear speed of sound in the Hquid is yi, B, and n are constants that should be set to the appropriate values for water. Any acoustic forcing function is included in the pressure at infinity term, (0- The pressure at the bubble wall, P(R), is given by... 

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. 

In its simplest form, damping is neglected and no external forcing function is applied, resulting in the equation... 

For a complex, multimass system like that shown in Figure 9-15, the equations of motion become quite complex, especially if a forcing function exists and internal damping is included. Inertial damping (damping to ground) is neglected. The equations of motion for this system would take ihe form ... 

It, therefore, appears that the equilibrium approximation is a special case of the steady-state approximation, namely, the case i > 2- This may be, but it is possible for the equilibrium approximation to be valid when the steady-state approximation is not. Consider the extreme but real example of an acid-base preequilibrium, which on the time scale of the following slow step is practically instantaneous. Suppose some kind of forcing function were to be applied to c, causing it to undergo large and sudden variations then Cb would follow Ca almost immediately, according to Eq. (3-153). The equilibrium description would be veiy accurate, but the wide variations in Cb would vitiate the steady-state description. There appear to be three classes of practical behavior, as defined by these conditions ... 

To initiate a chemical relaxation it is necessary to perturb the system from its initial equilibrium position. This is done by applying a forcing function, which is an appropriate experimental stress to which the system responds with a shift in equilibrium configuration. Forcing functions can be transient (a sudden, essentially discontinuous Jolt ) or periodic (a cyclic stress of constant frequency). 

The sensitivity of the equilibrium constant to temperature, therefore, depends upon the enthalpy change AH . This is usually not a serious limitation, because most reaction enthalpies are sufficiently large and because we commonly require that the perturbation be a small one so that the linearization condition is valid. If AH is so small that the T-jump is ineffective, it may be possible to make use of an auxiliary reaction in the following way Suppose the reaction under study is an acid-base reaction with a small AH . We can add a buffer system having a large AH and apply the T-jump to the combined system. The T-jump will alter the Ka of the buffer reaction, resulting in a pH jump. The pH jump then acts as the forcing function on the reaction of interest. 

Ultrasonic absorption is a so-called stationary method in which a periodic forcing function is used. The forcing function in this case is a sound wave of known frequency. Such a wave propagating through a medium creates a periodically varying pressure difference. (It may also produce a periodic temperature difference.) Now suppose that the system contains a chemical equilibrium that can respond to pressure differences [as a consequence of Eq. (4-28)]. If the sound wave frequency is much lower than I/t, the characteristic frequency of the chemical relaxation (t is the... 

Equation (4-30) shows that when w is very small, alf- = A -I- B absorption occurs by both the chemical and classical modes. In terms of Fig. 4-2, there is no phase-lag between the pressure and concentration curves. When to is very large, a// = B the chemical system is unable to absorb energy from the high-frequency forcing function. When w is comparable with 1/t, a.lf - is a function of frequency, and alf- passes through an inflection point when w = t thus t is found by measuring absorption as a function of frequency. 

There is a cause, referred to as a forcing function, for every frequency component in a machine-train s vibration signature. 

Vibration does not just happen. There is a physical cause, referred to as a forcing function, and each component of a vibration signature has its own forcing function. The components that make up a signature are reflected as discrete peaks in the TFT or frequency-domain plot. 

Frequency is defined as the number of repetitions of a specific forcing function or vibration component over a specific unit of time. Take for example a four-spoke wheel with an accelerometer attached. Every time the shaft completes one rotation, each of the four spokes passes the accelerometer once, which is referred to as four cycles per revolution. Therefore, if the shaft rotates at 100 rpm, the frequency of the spokes passing the accelerometer is 400 cycles per minute (cpm). In addition to cpm, frequency is... 

Electric motors are susceptible to a variety of forcing functions that cause instability or imbalance. The narrow-bands established to monitor the fundamental and other harmonics of actual running speed are useful in identifying mechanical imbalance, but other indices also should be used. 

Gear couplings When properly installed and maintained, gear-type couplings do not generate a unique forcing function or vibration profile. However, excessive wear, variations in speed or torque, or over-lubrication results in a forcing function. 

Universal joints There are a variety of universal joints used to transmit torsional power. In most cases, this type of intermediate drive is used where some misalignment between the drive and driven unit is necessary. Because of the misalignment, the universal s pivot points generate a unique forcing function that influences both the dynamics and vibration profile generated by a machine-train. 

V-belts are common intermediate drives for fans, blowers, and other types of machinery. Unlike some other power-transmission mechanisms, V-belts generate unique forcing functions that must be understood and evaluated as part of a vibration analysis. The key monitoring parameters for V-belt-driven machinery are fault frequency and mn-ning speed. 

Most of the forcing functions generated by V-belt drives can be attributed to the elastic or mbber band effect of the belt material. This elasticity is needed to provide the traction required transmitting power from the drive sheave (i.e., pulley) to the driven sheave. Elasticity causes belts to act like springs, increasing vibration in the direction of belt wrap, but damping it in the opposite direction. As a result, belt elasticity tends to accelerate wear and the failure rate of both the driver and driven unit. 

Both of these abnormal forcing functions (i.e., turbulent flow and operation outside of the effective control range) increase the level of vibration. However, when the instability is relatively minor, the resultant vibration occurs at the vane-pass frequency. As it become more severe, there also is a marked increase in the broadband energy. 

In most cases, running speed is the forcing function that excites the natural frequency of the dynamic component. As a result, rotating equipment is designed to operate at primary rotor speeds that do not coincide with the rotor assembly s natural frequencies. Most low- to moderate-speed machines are designed to operate below the first critical speed of the rotor assembly. 

Operational model forcing function for variable slope (3.13.4)... 

This model also can accommodate dose-response curve having Hill coefficients different from unity. This can occur if the stimulus-response coupling mechanism has inherent cooperativity. A general procedure can be used to change any receptor model into a variable slope operational function. This is done by passing the receptor stimulus through a forcing function. 

Operational Model Forcing Function for Variable Slope... 

Therefore, the operational model for agonism can be rewritten for variable slope by passing the stimulus equation through the forcing function (Equation 3.51) to yield... 

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