LA PLANEACIÓN DE CENTROS COMERCIALES DESDE EL PROMOTOR

reduced to functions of the damping ratio ξ, and the critical speed ratio Ω. This is an expected result since it is common knowledge that vibration amplitudes in the vicinity of a resonance are directly related to the frequency offset from the center frequency of the resonance. Furthermore, the damping ratio should also participate in determining the response characteristics since it is the fundamen-tal indicator of energy dissipation.

This relationship is easier to understand in the graphic format of Fig. 2-18.

Within this diagram, a family of five curves are plotted over a critical speed fre-quency ratio Ω between 0 and 3.0, and a damping ratio ranging from an under-damped system at ξ=0.1 to an overdamped condition of ξ=2.0. This general type of data display is referred to as a response plot or a Bode plot. In most cases, the synchronous 1X vibration amplitude and phase are plotted against rotative speed instead of the non-dimensional values used for Fig. 2-18. However, the con-cepts of tracking synchronous amplitude and phase as a function of speed are the same for a dimensional or a non-dimensional system.

In either case, an amplitude increase occurs at the natural or critical fre-quency (Ω=1) of the system. This amplitude response is coincident with a sub-stantial phase shift at the critical speed. It is also apparent that the magnitude of the amplitude response, and the amount of the phase shift are both directly related to the system damping.

Y Y_o

Forced Vibration 49

Specifically, an under damped mechanical system (e.g., ξ=0.1) exhibits a large amplitude change through the resonant frequency, combined with a phase shift that approaches 180°. Conversely, a heavily damped system (e.g., ξ=2.0) dis-plays no amplitude change through the critical speed region, and only a minor variation in phase angle to reveal the presence of the natural frequency.

It is meaningful to recall that the forcing function for equation (2-88) was based upon a constant driving force. That is, the same force is applied to the sys-tem at every frequency. In actuality, forced vibration is often excited by a force that varies with machine speed. For instance, centrifugal force due to unbalance varies as the square of the speed. Hence, it is desirable to consider revising the amplitude ratio of equation (2-88) to include a forcing function that incorporates an Ω² term. This would simulate the relationship between the amplitude response, and the mass unbalance forcing function. In the simplest format, it can be shown that the amplitude ratio for an unbalance response is described by the next equation that includes the speed squared Ω² term in the numerator:

(2-90)

For this forced unbalance response, the phase relationship remains identi-Fig. 2–18 Calculated

Bode Plot Of Forced Response For A Simple Mechanical System

cal to the previously presented equation (2-89). If the amplitude ratio and phase for a forced unbalance condition are plotted as a function of the critical speed ratio, Fig. 2-19 emerges. Note that the damping relationship remains consistent with the previous discussion, and the amplitude exhibits the most significant change. In this forced unbalance case, the amplitude ratio at low speeds approaches zero due to a small driving force (i.e., low unbalance force). At the critical speed of Ω=1, the magnitude of the peak is governed by the damping. At frequencies above the critical speed, the amplitude and phase remain fairly con-stant for each value of damping ratio. Hence, above a resonance, it is normal to encounter a plateau region where synchronous amplitude and phase remain rea-sonably constant with increasing machine speed. This behavior will be demon-strated with actual examples of machinery vibration data throughout this text.

A calculated or a measured Bode plot is excellent for observing rotational speed vibration amplitude and phase as a function of speed (frequency ratio).

However, there are conditions when a detailed examination of phase changes is required. Fortunately, the same vector information that is used to construct a Bode plot may also be viewed in terms of polar coordinates. In this type of data presentation, the vectors (amplitudes and angles) at each speed (or speed ratio) are plotted in a polar format. The heads of the vectors are then connected to form a continuous line. This type of diagram is generally referred to as a polar plot,

Fig. 2–19 Calculated Bode Plot Of Unbalance Response For A Simple Mechanical System

180 150 120 90 60 30 0

Phase Lag (Degrees)

ξ=0.2

Low Damping,ξ=0.1

ξ=0.5 ξ=2.0

ξ=1.0 High Damping,

ξ=2.0

ξ=0.1

0 1 2 3 4 5

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Amplitude Ratio

Critical Speed Frequency Ratio Low Damping,ξ=0.1 ξ=0.2

ξ=0.5

ξ=2.0 ξ=1.0

(Ω)

Forced Vibration 51

and it has many applications within the domain of machinery malfunction anal-ysis. A typical example of this type of data is shown in Fig. 2-20. In this diagram, the conditions of low damping (i.e., ξ=0.1 and 0.2) were extracted from Fig. 2-19, and these curves were replotted in a polar coordinate format. The peak of the resonance occurs at Ω=1, which is coincident with a 90° phase angle shift. As in the Bode plot, the high speed condition is identified as Ω=3.0.

Vector angles on the polar plot are always plotted against rotation. This phase lag logic is directly associated with the vibration measurement systems used to analyze the machinery behavior. In virtually all cases, it is highly desir-able (if not mandatory) to generate data plots that are physically representative of the machinery geometry. This topic will be discussed in much greater detail in the subsequent chapters 3, 5, 6, 7 and 11.

Before leaving the calculated Bode and polar plots for this forced response of a spring mass damper system, the magnitude of the vibration at the natural frequency should be examined in greater detail. From the previous definitions, it is clear that the critical speed frequency ratio Ω is equal to unity (ω/ωc=1) at the natural frequency. Substituting this value of Ω=1 into either equations (2-88) or (2-90) yields identical results. Specifically, when a value of Ω=1 is placed into equation (2-88), the following result is obtained:

Fig. 2–20 Calculated Polar Plot Of Unbalance Response For A Simple Mechanical System

Thus, at the natural resonance of Ω=1, the Amplitude Ratio is reduced to:

(2-91)