3.2.1 Geometrical Nonlinearities
Geometrical nonlinearities typically arise from large deflections or rotations, or
other purely kinematic characteristics. For example, the dynamics of the pendulum
in Fig. 3.1(a) is governed by
€ h þ x
2 sin h ¼ 0:
ð3:1Þ
The nonlinear sine-term expands to sinh = h –
1
6 h
3 + ⋯, showing that the familiar
linear approximation € h þ x
2 h ¼ 0 is valid only for small rotations h.
The beam in Fig. 3.1(b) is axially coupled with a linear spring. Here the longitudinal displacement w of the moving end is nonlinearly related to the transverse
deflection u. A single-mode, third-order approximation to the equations of dynamic
motion takes the form
€ a þ x
2
0 1 À P 0 =P c
ð
Þ a þ c
2 a
3
¼ 0;
ð3:2Þ
where a is the mid-span deflection, c
2 is a positive constant depending on the
spring stiffness, P 0 is the spring pre-load and P c the critical buckling load. Note that
for post-critical loads one has (1 – P 0 /P c ) < 0, so that there are three possible
equilibrium positions: a = 0 and a = ± (x 0 /c)(P 0 /P c – 1)
1/2 . Nonlinear systems
often have multiple states of equilibrium. Conversely, a system possessing more
than one equilibrium is certainly nonlinear.
For the clamped Euler column in Fig. 3.1(c) the internal moment is given by
M = EIj, where j is the curvature:
j ¼
u
00
ðsÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À u 0 ðsÞ
ð
Þ
2
r
¼ u
00
ðsÞ 1 þ
1
2
u
0
ðsÞ
ð
Þ
2 þ Á Á Á
:
ð3:3Þ
Thus, for rotations u′(s) that are finite (but not very large), a third order nonlinearity will appear in the equation of motion. A single-mode approximation to the
equation of motion takes the form of (3.2), with a positive linear stiffness coefficient. For small rotations, u′(s) ( 1, one has s % x and j % u′′(s), and thus the
well-known linear expression for the bending moment, M = EIu′′(x).
Fig. 3.1(d) shows a system causing nonlinear inertial terms to appear in the
equations of motion. Transverse oscillations of the beam are accompanied by small
horizontal displacements w of the moveable end-mass m. hence the mass exerts an
axial force –mẅ) on the beam. For finite rotations, w is nonlinearly related to the
transverse deflection u. A single-mode, third-order approximation to the equation
describing transverse motions takes the form.
€ a þ x
2
0 a þ g a€ a þ _
a
2
À
Á
a ¼ 0;
ð3:4Þ
3.2 Sources of Nonlinearity
97
Geometrical nonlinearities typically arise from large deflections or rotations, or
other purely kinematic characteristics. For example, the dynamics of the pendulum
in Fig. 3.1(a) is governed by
€ h þ x
2 sin h ¼ 0:
ð3:1Þ
The nonlinear sine-term expands to sinh = h –
1
6 h
3 + ⋯, showing that the familiar
linear approximation € h þ x
2 h ¼ 0 is valid only for small rotations h.
The beam in Fig. 3.1(b) is axially coupled with a linear spring. Here the longitudinal displacement w of the moving end is nonlinearly related to the transverse
deflection u. A single-mode, third-order approximation to the equations of dynamic
motion takes the form
€ a þ x
2
0 1 À P 0 =P c
ð
Þ a þ c
2 a
3
¼ 0;
ð3:2Þ
where a is the mid-span deflection, c
2 is a positive constant depending on the
spring stiffness, P 0 is the spring pre-load and P c the critical buckling load. Note that
for post-critical loads one has (1 – P 0 /P c ) < 0, so that there are three possible
equilibrium positions: a = 0 and a = ± (x 0 /c)(P 0 /P c – 1)
1/2 . Nonlinear systems
often have multiple states of equilibrium. Conversely, a system possessing more
than one equilibrium is certainly nonlinear.
For the clamped Euler column in Fig. 3.1(c) the internal moment is given by
M = EIj, where j is the curvature:
j ¼
u
00
ðsÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À u 0 ðsÞ
ð
Þ
2
r
¼ u
00
ðsÞ 1 þ
1
2
u
0
ðsÞ
ð
Þ
2 þ Á Á Á
:
ð3:3Þ
Thus, for rotations u′(s) that are finite (but not very large), a third order nonlinearity will appear in the equation of motion. A single-mode approximation to the
equation of motion takes the form of (3.2), with a positive linear stiffness coefficient. For small rotations, u′(s) ( 1, one has s % x and j % u′′(s), and thus the
well-known linear expression for the bending moment, M = EIu′′(x).
Fig. 3.1(d) shows a system causing nonlinear inertial terms to appear in the
equations of motion. Transverse oscillations of the beam are accompanied by small
horizontal displacements w of the moveable end-mass m. hence the mass exerts an
axial force –mẅ) on the beam. For finite rotations, w is nonlinearly related to the
transverse deflection u. A single-mode, third-order approximation to the equation
describing transverse motions takes the form.
€ a þ x
2
0 a þ g a€ a þ _
a
2
À
Á
a ¼ 0;
ð3:4Þ
3.2 Sources of Nonlinearity
97