rota-tional frequency ωc units of Radians per Second to Cycles per Second in accor-dance with equation (2-2) to yield the following:
(2-45) Clearly, the frequency of oscillation is a function of the spring constant, and the mass. This is the undamped natural frequency of the mechanical system. It is also commonly called the undamped critical frequency, and the subscript “c” has been added to identify frequencies ωc and Fc. In all cases, following an initial disturbance, the mass will oscillate (or vibrate) at this natural frequency, and the amplitude of the motion will gradually decay as a function of time. This reduc-tion in amplitude is due to energy dissipareduc-tion within a real mechanical system.
Although this result is simple in format, it does represent an extraordinar-ily important concept in the field of vibration analysis. That is, the natural fre-quency of a mechanical resonance will respond to an alteration of the stiffness and the mass. Often, the diagnostician has limited information on the effective stiffness, or equivalent mass of the mechanical system. However, changes in stiffness or mass will behave in the manner described by equation (2-44). In many instances, this knowledge of the proper relationship between parameters will allow a respectable solution to a mechanical problem.
Initially, the existence of a unique natural frequency that is a function of the mechanical system mass and stiffness may appear to be only of academic interest. In reality, there are field applications of this physical relationship that may be used to provide solutions for mechanical problems. For instance, if a mechanical system is excited by a periodic force at a frequency that approaches a natural resonant frequency of the mechanical system — the resultant vibratory
D ×ω2×eiωt
– K
M
--- ×D×eiωt
+ = 0
D×eiωt –ω2 K ---M
+
× = 0
ω2
– K
M
--- +
= 0
ωc K
---M
=
Fc 1 2π--- K
---M
×
=
Undamped Free Vibration 31
motion may be excessive, or even destructive. Three potential solutions to this type of problem were identified by J. P. Den Hartog5, in his text Mechanical Vibrations. Quoting from page 87 of this book:
“…In order to improve such a situation, we might first attempt to eliminate the force. Quite often this is not practical or even possible. Then we may change the mass or the spring constant of the system in an attempt to get away from the resonance condition, but in some cases this is also impractical. A third possibility lies in the application of the dynamic vibration absorber, invented by Frahm in 1909…The vibration absorber consists of a comparatively small vibratory system k, m attached to the main mass M. The natural frequency of the attached absorber is chosen to be equal to the frequency ω of the disturbing force. It will be shown that then the main mass M does not vibrate at all, and that the small sys-tem k, m vibrates in such a way that its spring force is at all instances equal and opposite to Po sin ω t. Thus there is no net force acting on M and therefore the mass does not vibrate…”
In his text book, Den Hartog proceeds to derive a detailed equation set that supports the above statements. He also examines torsional systems, and damped vibration absorbers. Thomson6 also discussed the utilization of both lateral and torsional vibration absorbers. However, for this discussion, the application of a simple lateral undamped spring mass vibration absorber will be reviewed. The fundamental engineering principles behind an absorber installation are illus-trated with the following case history.
Case History 1: Piping System Dynamic Absorber
The mechanical system under consideration consists of a pair of product transfer pumps that were subjected to a modification of the discharge piping to span across a new roadway. These essential pumps were motor driven at a con-stant speed of 1,780 RPM. The pumps had a successful eight year operating his-tory, with only minor seal problems, and one coupling failure. During a plant revision, the pump discharge piping was rerouted to a new pipe rack. Due to the design of the new rack, the discharge line was poorly supported, and problems began to appear on both pumps shortly after the piping modification.
Multiple seal failures were combined with repetitive bearing, and coupling failures. These two pumps that previously received maintenance attention only once or twice a year were now subjected to overhauls on a monthly basis. This increased maintenance passed unnoticed for a long time. Unfortunately, one night the main pump failed when the spare pump was out for repairs. This coin-cidence of mechanical failures forced a plant outage, and this event focused man-agement attention upon the reduced reliability of these pumps.
Vibration analysis of the pumps and the associated piping revealed a
domi-5 J.P. Den Hartog, Mechanical Vibrations, 4th edition, (New York: McGraw-Hill Book Company, 1956), p. 87.
6 William T. Thomson, Theory of Vibration with Applications, 4th Edition, (Englewood Cliffs, New Jersey: Prentice Hall, 1993), pp. 150-159.
k m⁄
nant motion at the pump running speed of 1,780 RPM. Comparison with histori-cal data revealed 1X vibration amplitudes on the pump and motor were ten to twenty times higher than previously measured. This machinery abnormality was coincident with vertical vibration levels in excess of 25 Mils,p-p at the middle of the unsupported discharge line (i.e., midspan of the road crossing).
A temporary brace was fabricated, and placed below the discharge line.
This support reduced the piping vibration, and also resulted in a drop in the pump synchronous motion. Considering the positive results of this test, and some preliminary calculations on the natural frequency of the piping span, it was concluded that the pump running speed was very close to a lateral natural frequency of the new discharge pipe.
Since a brace in the middle of the road was unacceptable as a long-term solution, other possibilities were examined and discarded. Finally, the applica-tion of a tuned spring mass vibraapplica-tion absorber was considered as a potential and practical solution. For this problem a simple horizontal cantilevered vibration absorber was designed to resemble the diagram in Fig. 2-9.
This device consists of a fabricated pipe saddle that is securely bolted to the outer diameter of the discharge pipe. It is physically located at the point of high-est vibration (i.e., center of the piping span). Since the pipe vibrates vertically, the absorber is positioned horizontally so that the cantilevered weight may also vibrate vertically. In this case, the spring consists of flat bar stock that has the most flexible axis placed in the direction of the desired motion. The overhung mass is bolted to the flat bar stock spring, and it may be moved back and forth to allow adjustment of the natural frequency.
By inspection of this damper assembly, it is apparent that the stiffness and mass of the spring, plus the overhung mass are equivalent to a simple spring mass system. The problem in designing an appropriate vibration absorber is now reduced to a reasonable selection of physical dimensions to obtain a natural fre-quency of 1,780 CPM for this installed assembly.
Several approaches may be used to determine an acceptable set of absorber Fig. 2–9 Typical Tuned
Spring Mass Vibration Absorber Assembly For Piping System
Fabricated Pipe Saddle Sliding Overhung Mass
Spring
Pipe I.D.
Undamped Free Vibration 33
dimensions. For example, a Finite Element Analysis (FEA) could be performed.
However, an FEA approach may become unnecessarily complicated and time consuming. Use of published beam natural frequency equations may also be con-sidered. However, one must be careful of published canned equations where the assumptions and boundary conditions may not be clearly explained or under-stood. Fortunately, a practical approach for performing these calculations was presented by John D. Raynesford7, in his Hydrocarbon Processing article on this subject. In this article, he considered the system as a simple spring mass assem-bly. The dimensions of the spring were combined with the overhung mass to pro-vide the basic elements for the absorber design. Specifically, Raynesford considered the total static deflection Ytotal of the vibration absorber to be associ-ated with the weight W, mass M, gravitational constant G, and the spring con-stant K of the assembly in the following manner:
(2-46)