The present trend in rotating equipment is toward increasing design speeds, which increases operational problems from vibration; hence the importance of vibration analysis. A thorough appreciation of vibration analysis will aid in the diagnoses of rotor dynamics problems.
This chapter is devoted to vibration theory fundamentals concerning undamped and damped freely oscillating systems. Application of vibration theory to solv- ing rotor dynamics problems is then discussed. Next, critical speed analysis and balancing techniques are examined. The latter part of the chapter discusses impor- tant design criteria for rotating machinery, specifically bearing driver types, and design and selection procedures.
Mathematical Analysis
The study of vibrations was confined to musicians until classical mechanics had advanced sufficiently to allow an analysis of this complex phenomenon.
Newtonian mechanics provides an approach which, conceptually, is easy to understand. Lagrangian mechanics provides a more sophisticated approach, but it is intuitively more difficult to conceive. Since this book uses some basic concepts, we will approach the subject using Newtonian mechanics.
Vibration systems fall into two major categories: forced and free. A free system vibrates under forces inherent in the system. This type of system will vibrate at one or more of its natural frequencies, which are properties of the elastic system. Forced vibration is vibration caused by external force being impressed on the system. This type of vibration takes place at the frequency of the exciting force, which is an arbitrary quantity independent of the natural frequencies of the system. When the frequency of the exciting force and the natural frequency coincide, a resonance condition is reached, and dangerously large amplitudes may result. All vibrating systems are subject to some form of damping due to energy dissipated by friction or other resistances.
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The number of independent coordinates, which describe the system motion, are called the degrees of freedom of the system. A single degree of freedom system is one that requires a single independent coordinate to completely describe its vibration configuration. The classical spring mass system shown in Figure 5-1 is a single degree of freedom system.
Systems with two or more degrees of freedom vibrate in a complex manner where frequency and amplitude have no definite relationship. Among the multi- tudes of unorderly motion, there are some very special types of orderly motion called principal modes of vibration.
During these principal modes of vibration, each point in the system follows a definite pattern of common frequency. A typical system with two or more degrees of vibration is shown in Figure 5-2. This system can be a string stretched between two points or a shaft between two supports. The dotted lines in Figure 5-2 show the various principal vibration modes.
Figure 5-1.System with single degree of freedom.
Figure 5-2.System with infinite number of degrees of freedom.
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Figure 5-3.Periodic motion with harmonic components.
Most types of motion due to vibration occur in periodic motion. Periodic motion repeats itself at equal time intervals. A typical periodic motion is shown in Figure 5-3. The simplest form of periodic motion is harmonic motion, which can be represented by sine or cosine functions. It is important to remember that harmonic motion is always periodic; however, periodic motion is not always harmonic. Harmonic motion of a system can be represented by the following relationship:
x=Asinωt (5-1)
Thus, one can determine the velocity and acceleration of that system by differentiating the equation with respect tot
Velocity= dx
dt =Aωcosωt=Aωsin(ωt+π
2) (5-2)
Acceleration= d2x
dt2 = −Aω2sinωt =Aω2sin(ωt+π ) (5-3) The previous equations indicate that the velocity and acceleration are also har- monic and can be represented by vectors, which are 90◦and 180◦ahead of the displacement vectors. Figure 5-4 shows the various harmonic motions of dis- placement, velocity, and acceleration. The angles between the vectors are called phase angles; therefore, one can say that the velocity leads displacement by 90◦ and that the acceleration acts in a direction opposite to displacement, or that it leads displacement by 180◦.
Figure 5-4.Harmonic motion of displacement, velocity, and acceleration.
Undamped Free System
This system is the simplest of all vibration systems and consists of a mass suspended on a spring of negligible mass. Figure 5-5 shows this simple, single degree of freedom system. If the mass is displaced from its original equilibrium position and released, the unbalanced force, the restoring (−Kx) of the spring, and acceleration are related through Newton’s second law. The resulting equation
Figure 5-5.Single degree of freedom system (spring mass system).
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can be written as follows:
mx¨ = −Kx (5-4)
This equation is called the motion equation for the system, and it can be rewritten as follows:
¨ x+K
mx=0 (5-5)
Assuming that a harmonic function will satisfy the equation, let the solution be in the form
x=C1sinωt+C2cosωt (5-6)
Substituting Equation (5-6) into Equation (5-5), the following relationship is obtained:
−ω2+ K M
x =0
which can be satisfied for any value ofxif ω=
%K
M (5-7)
Thus, the system has a single natural frequency given by the relationship in Equation (5-7).
Damped System
Damping is the dissipation of energy. There are several types of damping—
viscous damping, friction or coulomb damping, and solid damping. Viscous damping is encountered by bodies moving through a fluid. Friction damping usually arises from sliding on dry surfaces. Solid damping, often called structural damping, is due to internal friction within the material itself. An example of a free vibrating system with viscous damping is given here.
As shown in Figure 5-6, viscous damping force is proportional to velocity and is expressed by the following relationship:
Fdamp= −cx˙
wherecis the coefficient of viscous damping.
Figure 5-6.Free vibration with viscous damping.
The Newtonian approach gives the equation of motion as follows:
mx¨= −kx−cx˙ (5-8)
or it can be written as mx¨+cx˙+kx=0
The solution to this equation is found by using the trial solution
x =c(ert) (5-9)
which when substituted in Equation (5-8) yields the following characteristic equation:
r2+ c
mr+ k m
ert =0 (5-10)
This equation is satisfied for all values oftwhen
r1,2= −c 2m±
&
c2 4m2 − k
m (5-11)
from which the general solution is obtained as follows:
x =e−c2mt
C1e
%
c2 4m2−mk(t )
+C2e−
%
c2 4m2−mk(t )
(5-12)
The nature of the solution given by Equation (5-9) depends upon the nature of the roots,r1andr2. The behavior of this damped system depends upon whether
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the root is real, imaginary, or zero. The critical damping coefficientcc can now be defined as that which makes the radical zero. Thus,
c2 4m2 = k
m
which can be written as c
2m =
%k
m=ωn (5-13)
One can therefore specify the amount of damping in any system by the damping factor
ζ = c cc
(5-14) Overdamped system. If c2/4m2 > k/m, then the expression under the radical sign is positive and the roots are real. If the motion is plotted as a function of time, the curve in Figure 5-7 is obtained. This type of nonvibratory motion is referred to as aperiodic motion.
Critically damped system. Ifc2/4m2 =k/m, then the expression under the radical sign is zero, and the rootsr1 andr2 are equal. When the radical is zero and the roots are equal, the displacement decays the fastest from its initial value as seen in Figure 5-8. The motion in this case also is aperiodic.
This very special case is known as critical damping. The value ofc for this case is given by:
c2cr 4m2 = k
m
Figure 5-7.Overdamped decay.
Figure 5-8.Critcal damping decay.
c2cr=4m2k m=4mk Thus,
ccr=√
4mk=2m
%k
m=2mωn
Underdamped system. If c2/4m2 < k/m, then the roots r1 and r2 are imaginary, and the solution is an oscillating motion as shown in Figure 5-9. All the previous cases of motion are characteristic of different oscillating systems, although a specific case will depend upon the application. The underdamped system exhibits its own natural frequency of vibration. When c2/4m < k/m,
Figure 5-9.Underdamped decay.
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the rootsr1andr2are imaginary and are given by
r1,2= ±i
&
k m− c2
4m2 (5-15)
Then the response becomes
x=e−(c/2m)t
C1ei
% mk−c2
4m2
+C2e−i
% mk− c2
4m2
which can be written as follows:
x=e−(c/2m)t[Acosωdt+Bsinωdt] (5-16)
Forced Vibrations
So far, the study of vibrating systems has been limited to free vibrations where there is no external input into the system. A free vibration system vibrates at its natural resonant frequency until the vibration dies down due to energy dissipation in the damping.
Now the influence of external excitation will be considered. In practice, dynamic systems are excited by external forces, which are themselves periodic in nature. Consider the system shown in Figure 5-10.
The externally applied periodic force has a frequencyω, which can vary inde- pendently of the system parameters. The motion equation for this system may be obtained by any of the previously stated methods. The Newtonian approach will be used here because of its conceptual simplicity. The freebody diagram of the massmis shown in Figure 5-11.
Figure 5-10.Forced vibration system.