Page 4 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

Equivalent Masses, Springs and Dampers
Equivalent masses

m M Mass (M) attached at end of m eq = M + m
spring of mass m 3

M Cantilever beam of mass m 33
m carrying an end mass M m eq = M + 140 m

M Simply supported beam of m eq = M + 0.5 m
mass m carrying a mass
m M at the middle

Coupled translational and J 0
rotational masses m eq = m + 2
J 0 R
2
R J eq = J 0 + mR
m

m 1 m 2 m 3 Masses on a hinged bar l 2 2 l 3 2
m eq 1 = m 1 + a b m 2 + a b m 3
l 1 l 1
l 1
l 2
l 3

Equivalent springs
Rod under axial load k eq = EA
1l = length, A = cross sectional area2 l

Tapered rod under axial load k eq = pEDd
1D, d = end diameters2 4l

Helical spring under axial load Gd 4
(d = wire diameter, k eq = 8nD 3
D = mean coil diameter,
n = number of active turns)

Fixed-fixed beam with 192EI
load at the middle k eq = l 3