Page 400 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 12.4 Holzer Method 387

Setting the determinant of the preceding equation to zero, we obtain the frequency
equation
to“ - 36.3676ao)2 + 177.0377«^ = 0 (d)
where
EA,
« = — ^ (e)
inul
The two roots of this equation are
0)^ = 5.7898a and = 30.5778a
Using these results in Eq. (c), we obtain
C2 = 0.03689Cj for mode 1
Cl = -0.63819C2 for mode 2
The two natural frequencies and mode shapes are then
£-4,, /' \ .r ^ ^ • 3 X
0)^ = 2.4062 u^(x) = 1.0sm -I- 0.03689sm

EA, . 2>ttx
io, = 5.5297 Ui{x) = -0.63819sin + 1-0 n n ^

12.4 HOLZER METHOD

When an undamped system is vibrating freely at any one of its natural frequencies,
no external force, torque, or moment is necessary to maintain the vibration. Also,
the amplitude of the mode shape is immaterial to the vibration. Recognizing these
facts, Holzer^ proposed a method of calculation for the natural frequencies and
mode shapes of torsional systems by assuming a frequency and starting with a unit
amplitude at one end of the system and progressively calculating the torque and
angular displacement to the other end. The frequencies that result in zero external
torque or compatible boundary conditions at the other end are the natural
frequencies of the system. The method can be applied to any lumped-mass system,
linear spring-mass systems, beams modeled by discrete masses and beam springs,
and so on.
Holzer’s procedure for torsional systems. Figure 12.4-1 shows a tor­
sional system represented by a series of disks connected by shafts. By assuming a
frequency co and amplitude = 1, the inertia torque of the first disk is

Holzer, Die Berechnung der Drehschwingungen (Berlin: Springer-Verlag, 1921).