by measuring the acceleration response to the impact of a light hammer tip at a few
beam points. A frequency analyzer and a modal extraction procedure is then used to
estimate x j , b and u j (x). Knowing these modal characteristics, Eqs. (1.65)–(1.67)
could be employed to predict the response of the beam to arbitrary loads q(x, t).
1.4.7 Complex-Valued Eigenvalues and Mode
Shapes
Typically, when solving unforced vibration problems with linear continuous systems, one inserts into the equation of motion a solution form u(x, t) = u(x)e
kt ,
where (u, k) 2 C
2 is an unknown mode shape and eigenvalue to be determined.
Generally, without assumptions on the character or presence of damping, both k
and u will complex-valued. Having determined these, the question may then arise
as how to come from (k, u) to the natural frequencies (damped and undamped), the
damping ratios, and the actual motions of the system.
Using nothing but trigonometric identities and algebra, one can show that the
system motions uðxÞe
kt can be written as uðx; tÞ ¼ AðxÞe
Àfxt cos ~
xt þ wðxÞ
ð
Þ ; where
x ¼ jkj ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ReðkÞ
2 þ ImðkÞ
2
q
is the undamped natural frequency, f = −Re(k)/x is
the damping ratio, ~
x ¼ ImðkÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À f
2
p
x is the damped natural frequency,
A(x) = |u(x)| is the spatial amplitude function, and wðxÞ ¼ arctan ImðuðxÞ=ReðuðxÞÞ
ð
Þ
is the spatial phase shift.
With mass- or stiffness-proportional damping the mode shapes will be real.
Then w = 0 for all x, implying motions of the form uðx; tÞ ¼ uðxÞe
Àfxt cos ~
xt
ð Þ; i.e.
with all points on the structure moving synchronously in either phase or antiphase
(depending on the sign change of u(x)); these are standing wave motions, with all
nodes (points with no motion) and antinodes (largest motion) fixed in space, and all
system points attaining maximum or minimum simultaneously.
Conversely, with non-proportional damping, the spatial phase w will generally
depend on x, meaning that different system points attain maximum and minimum at
different times, and that nodal points will move back and forth in time along the
structure; such motions are not standing waves but traveling waves.
1.4.8 Rayleigh’s Method
The Rayleigh quotient R[~ u] yields an upper-bound estimate for the lowest natural
frequency x 1 of any conservative elastic system:
R½~ u ¼
V max ð~ uÞ
x À2 T max ð~ uÞ
¼ ~
x
2
1 ! x
2
1 ;
ð1:68Þ
18
1 Vibration Basics
beam points. A frequency analyzer and a modal extraction procedure is then used to
estimate x j , b and u j (x). Knowing these modal characteristics, Eqs. (1.65)–(1.67)
could be employed to predict the response of the beam to arbitrary loads q(x, t).
1.4.7 Complex-Valued Eigenvalues and Mode
Shapes
Typically, when solving unforced vibration problems with linear continuous systems, one inserts into the equation of motion a solution form u(x, t) = u(x)e
kt ,
where (u, k) 2 C
2 is an unknown mode shape and eigenvalue to be determined.
Generally, without assumptions on the character or presence of damping, both k
and u will complex-valued. Having determined these, the question may then arise
as how to come from (k, u) to the natural frequencies (damped and undamped), the
damping ratios, and the actual motions of the system.
Using nothing but trigonometric identities and algebra, one can show that the
system motions uðxÞe
kt can be written as uðx; tÞ ¼ AðxÞe
Àfxt cos ~
xt þ wðxÞ
ð
Þ ; where
x ¼ jkj ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ReðkÞ
2 þ ImðkÞ
2
q
is the undamped natural frequency, f = −Re(k)/x is
the damping ratio, ~
x ¼ ImðkÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À f
2
p
x is the damped natural frequency,
A(x) = |u(x)| is the spatial amplitude function, and wðxÞ ¼ arctan ImðuðxÞ=ReðuðxÞÞ
ð
Þ
is the spatial phase shift.
With mass- or stiffness-proportional damping the mode shapes will be real.
Then w = 0 for all x, implying motions of the form uðx; tÞ ¼ uðxÞe
Àfxt cos ~
xt
ð Þ; i.e.
with all points on the structure moving synchronously in either phase or antiphase
(depending on the sign change of u(x)); these are standing wave motions, with all
nodes (points with no motion) and antinodes (largest motion) fixed in space, and all
system points attaining maximum or minimum simultaneously.
Conversely, with non-proportional damping, the spatial phase w will generally
depend on x, meaning that different system points attain maximum and minimum at
different times, and that nodal points will move back and forth in time along the
structure; such motions are not standing waves but traveling waves.
1.4.8 Rayleigh’s Method
The Rayleigh quotient R[~ u] yields an upper-bound estimate for the lowest natural
frequency x 1 of any conservative elastic system:
R½~ u ¼
V max ð~ uÞ
x À2 T max ð~ uÞ
¼ ~
x
2
1 ! x
2
1 ;
ð1:68Þ
18
1 Vibration Basics