Page 88 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 3.9 Structural Damping 75

The amplitude of forced vibration can be found by substituting c into Eq. (3.1-3):

+(S)
Solving for X, we obtain
4 ^

1 -
ttF„
=
k — nib

1 - (i)
—
We note here that unlike the system with viscous damping, X/3^^ goes to oo when
0) = 0)^. For the numerator to remain real, the term must be less than 1.0.
3.9 STRUCTURAL DAMPING

When materials are cyclically stressed, energy is dissipated internally within the
material itself. Experiments by several investigators^ indicate that for most struc­
tural metals, such as steel or aluminum, the energy dissipated per cycle is
independent of the frequency over a wide frequency range and proportional to the
square of the amplitude of vibration. Internal damping fitting this classification is
called solid damping or structural damping. With the energy dissipation per cycle
proportional to the square of the vibration amplitude, the loss coefficient is a
constant and the shape of the hysteresis curve remains unchanged with amplitude
and independent of the strain rate.
Energy dissipated by structural damping can be written as
Wj = aX ^ (3.9-1)
where a is a constant with units of force/displacement. By using the concept of
equivalent viscous damping, Eq. (3.8-2) gives
iTC^^coX^ = aX^
or
(3.9-2)
7TCÜ
By substituting for c, the differential equation of motion for a system with
structural damping can be written as
mx + X F kx = FnSincuf (3.9-3)
(— )-*
\ 7TÙ) j
'A. L. Kimball, “Vibration Damping, Including the Case of Solid Damping,” Trans. ASME,
APM 51-52 (1929). Also B. J. Lazan, Damping of Materials and Members in Structural Mechanics
(Elmsford, NY: Pergamon Press, 1968).