Page 151 - Intro Predictive Maintenance

142 An Introduction to Predictive Maintenance

For damped forced vibrations, three different frequencies have to be distinguished:
the undamped natural frequency, Kg M ; the damped natural frequency,
c
2
cg c ˆ
q = Kg c - Ê ; and the frequency of maximum forced amplitude, sometimes
M Ë 2 M ¯
referred to as the resonant frequency.

7.5.2 Degrees of Freedom
In a mechanical system, the degrees of freedom indicate how many numbers are
required to express its geometrical position at any instant. In machine-trains, the rela-
tionship of mass, stiffness, and damping is not the same in all directions. As a result,
the rotating or dynamic elements within the machine move more in one direction than
in another. A clear understanding of the degrees of freedom is important because it
has a direct impact on the vibration amplitudes generated by a machine or process
system.

One Degree of Freedom
If the geometrical position of a mechanical system can be defined or expressed as a
single value, the machine is said to have one degree of freedom. For example, the
position of a piston moving in a cylinder can be specified at any point in time by mea-
suring the distance from the cylinder end.

A single degree of freedom is not limited to simple mechanical systems such as the
cylinder. For example, a 12-cylinder gasoline engine with a rigid crankshaft and a
rigidly mounted cylinder block has only one degree of freedom. The position of all
its moving parts (i.e., pistons, rods, valves, cam shafts) can be expressed by a single
value. In this instance, the value would be the angle of the crankshaft; however, when
mounted on flexible springs, this engine has multiple degrees of freedom. In addition
to the movement of its internal parts in relationship to the crank, the entire engine can
now move in any direction. As a result, the position of the engine and any of its inter-
nal parts requires more than one value to plot its actual position in space.

The definitions and relationships of mass, stiffness, and damping in the preceding
section assumed a single degree of freedom. In other words, movement was limited
to a single plane. Therefore, the formulas are applicable for all single-degree-of-
freedom mechanical systems.

The calculation for torque is a primary example of a single degree of freedom in a
mechanical system. Figure 7–15 represents a disk with a moment of inertia, I, that is
attached to a shaft of torsional stiffness, k.

Torsional stiffness is defined as the externally applied torque, T, in inch-pounds needed
to turn the disk one radian (57.3 degrees). Torque can be represented by the follow-
ing equations: