the dotted line in Fig. 2-13. The displacement change with time is the sum of two decaying exponential functions, and system vibration is not maintained. Motion is aperiodic, and the body returns to rest without oscillation. It is also clear from this composite diagram that the overdamped system of equation (2-69) does not return to rest as rapidly as the previously discussed critically damped case.
Finally, consider the situation of a mechanical system with small damping (equivalent to a worn out shock absorber). This is generally referred to as an underdamped system where ξ<1, and the radical of equation. (2-65) is imaginary.
The constants S1 and S2 for this condition can be written as follows:
(2-70) Using equation (2-70) to recompute the S terms, and then including these new expressions into the general equation, the following solution for the equa-tion of moequa-tion for a under damped system was presented by W.T. Thomson as:
S1 ωc –ξ+ ξ2–1
This expression may be simplified to a more understandable format as:
(2-71) By inspection, equation (2-71) consists of the superposition of an oscillating sine wave plus an exponential term. In most cases, the amplitude of the sine wave is decreased by the exponential function with increasing time. The variable Y in this equation represents the peak intersection between the exponential function and zero time. The ϕ term is the timing lag between the oscillatory curve and a zero time starting point. For demonstration purposes, a response curve for an under damped system is plotted in Fig. 2-14. A displacement value of 10.0 was assigned to Y, the timing offset ϕ was set equal to zero, and a con-stant value of 5.0 was used for the undamped natural frequency ωc. This system exhibits an oscillatory motion with respect to time, and this is referred to as free vibration of the under damped mechanical system.
Another interesting point from examining (2-71) is that the term is multiplied by the time t to determine the number of radians. This suggests that the undamped natural frequency ωc is altered by the damping ratio ξ to produce a new frequency. In fact, this is commonly identified as the damped natural frequency or ωdamped critical for the mechanical system, and it is defined in the following manner:
(2-72)
Y 1=8.581 Y2=4.560 Y3=2.425 Y4=1.289
ωc 1–ξ2
ωdamped critical = ωc× 1–ζ2
Free Vibration with Damping 43
This is a very important equation because it directly influences the impact of damping upon a resonance. As shown in the forthcoming Fig. 2-18 for forced vibration — variations in damping produce major changes in amplitude and phase through a resonance. However, there is also a subtle shift in the resonant frequency as the damping is varied. Thus, the machinery diagnostician must be fully aware of the fact that changes in system damping will alter the behavior through a resonance in the following three ways:
❍ Significant changes in the peak amplitude at the resonance.
❍ Significant variations in the phase angle change across the resonance.
❍ Subtle change in the damped natural frequency (i.e., damped critical speed).
Fig. 2-14 for an under damped system shows that the oscillatory motion decays with time. Examination of a longer time record would reveal that the amplitude decrease is actually an exponential decay. The rate of this exponential decay may be quantified by the log decrement which is defined as follows:
(2-73) In equation (2-73), Y1 and Y2 represent any two successive amplitudes in the decaying dynamic signal. The natural logarithm of this ratio defines the damping as the log decrement δ. In some cases, particularly with lightly damped or short duration experimental data, it is necessary to examine multiple cycles of the decaying signal to determine the log decrement. For these situations, the right hand side of equation (2-73) may be used. Within this part of the expres-sion, the initial peak amplitude is still specified by Y1, and the amplitude follow-ing N number of cycles is identified as YN+1. The validity of this relationship may be checked by calculating the log decrement for different combinations. For instance, in Fig. 2-14, the first amplitude peak Y1 has a magnitude of 8.581, and the second peak Y2 is equal to 4.560. Using the first part of equation (2-73), the log decrement δ may be computed from these values in the following manner:
The same calculation may be performed using the first three cycles in Fig.
2-14. For this case the third peak Y4 has an amplitude of 1.289, and the log dec-rement δ may be computed with the right side of equation (2-73) as follows:
The same result of δ=0.632 has been reached using a single cycle and multi-Log Decrement δ Y1
ple cycles. Certainly this concept may be extended to the examination of various decaying dynamic data sets. It should also be mentioned that the log decrement δ may also be expressed in terms of the critical damping ratio ξ. It can be shown that the log decrement δ is accurately expressed as:
(2-74)
The decaying signal plotted in Fig. 2-14 was produced with a damping ratio of ξ=0.1. To check the validity of equation (2-74), this damping ratio may be used to calculate the log decrement as follows:
Once more the same value for the log decrement has been obtained. This provides confidence that equations (2-73) and (2-74) are compatible, and consis-tent. Depending on the available data, one expression may be easier to apply ver-sus the other. Another usable format for these expressions is obtained by solving equation (2-74) for the damping ratio ξ to produce the following:
(2-75)
Equation (2-75) is useful for determining the damping ratio based upon experimental or analytical values of the log decrement. Further examination of equations (2-74) and (2-75) reveals that the damping ratio ξ and the log decre-ment δ are closely related. For instance, the polarity of the log decrement and the damping ratio must be the same. If the damping ratio is positive, the log decre-ment must also be positive. Similarly, if the log decredecre-ment is negative, then the damping ratio must be negative. The physical significance of negative damping is depicted in Fig. 2-15. This diagram is based upon equation (2-71) where Y was assigned a value of 1.0, the timing offset ϕ was equated to zero, a value of 5.0 was
Fig. 2–15 Time Domain