The data in Table D.3 assumes there is no axial loading. Generally the presence
of axial tension increases the effective transverse stiffness, and thus also increases
the natural frequencies. Conversely, compressive loading reduces natural frequencies (the lowest becoming zero with the approach to buckling loading). The effect of
axial loading on natural frequencies increases with beam slenderness, i.e. as a beam
become more ‘string-like’ the effect of tension increases (because relatively more of
the transverse stiffness comes from tension). Bokaian (1988, 1990) provides formulas for natural frequencies and mode shapes of axially tensioned or compressed
beams for different standard boundary conditions (see also Brøns et al. 2021, Sah
et al. 2018, Hermansen and Thomsen 2018).
Table D.3 Natural frequencies and mode shapes for flexurally vibrating uniform Bernoulli-Euler
beams
i,ii
Boundary supports
Frequency
equation
Roots k j of the
frequency equation
iii
Mode shape
iv u j (x)
a) Clamped-clamped
cos k cosh k ¼ 1
4.7300
7.8532
10.9956
14.1372
!(2j + 1)p/2
J k j
x
l
À Á À
JðkjÞ
HðkjÞ H k j
x
l
À Á
b) Clamped-hinged
tan k ¼ tanh k
3.9266
7.0686
10.2102
13.3518
!(4j + 1)p/4
J k j
x
l
À Á À
JðkjÞ
HðkjÞ H k j
x
l
À Á
c) Clamped-free
(= cantilever)
cos k cosh k ¼ À1
1.8751
4.6941
7.8548
10.9955
!(2j–1)p/2
J k j
x
l
À Á À
GðkjÞ
FðkjÞ H k j
x
l
À Á
d) Clamped-guided
tan k ¼ À tanh k
2.3650
5.4978
8.6394
11.7810
!(4j–1)p/4
J k j
x
l
À Á À
FðkjÞ
JðkjÞ H k j
x
l
À Á
e) Hinged-hinged
(= simply
supported)
sin k ¼ 0
jp
sin jp
x
l
À Á
f) Hinged-guided
cos k ¼ 0
( 2 j–1)p/2
sin ðj À
1
2 Þp
x
l
À
Á
g) Guided-guided
sin k ¼ 0
jp
cos jp
x
l
À Á
h) Free-free
cos k cosh k ¼ 1
Same as for (a)
G k j
x
l
À Á À
JðkjÞ
HðkjÞ F k j
x
l
À Á
i) Free-hinged
tan k ¼ tanh k
Same as for (b)
G k j
x
l
À Á À
GðkjÞ
FðkjÞ F k j
x
l
À Á
j) Free-guided
tan k ¼ À tanh k
Same as for (d)
G k j
x
l
À Á À
HðkjÞ
FðkjÞ F k j
x
l
À Á
i
The natural frequencies are x j = k j
2
(EI/qA)
1/2 /l
2
, where k j is given in the table
ii Adapted from Flügge (1962) with permission from the publisher
iii Lowest roots numerically, or all roots analytically, or limits for large j (indicated by !)
iv
F(u) = sinhu + sinu; G(u) = coshu + cosu; H(u) = sinhu − sinu; J(u) = coshu − cosu
488
Appendix D: Natural Frequencies and Mode Shapes for Structural Elements
of axial tension increases the effective transverse stiffness, and thus also increases
the natural frequencies. Conversely, compressive loading reduces natural frequencies (the lowest becoming zero with the approach to buckling loading). The effect of
axial loading on natural frequencies increases with beam slenderness, i.e. as a beam
become more ‘string-like’ the effect of tension increases (because relatively more of
the transverse stiffness comes from tension). Bokaian (1988, 1990) provides formulas for natural frequencies and mode shapes of axially tensioned or compressed
beams for different standard boundary conditions (see also Brøns et al. 2021, Sah
et al. 2018, Hermansen and Thomsen 2018).
Table D.3 Natural frequencies and mode shapes for flexurally vibrating uniform Bernoulli-Euler
beams
i,ii
Boundary supports
Frequency
equation
Roots k j of the
frequency equation
iii
Mode shape
iv u j (x)
a) Clamped-clamped
cos k cosh k ¼ 1
4.7300
7.8532
10.9956
14.1372
!(2j + 1)p/2
J k j
x
l
À Á À
JðkjÞ
HðkjÞ H k j
x
l
À Á
b) Clamped-hinged
tan k ¼ tanh k
3.9266
7.0686
10.2102
13.3518
!(4j + 1)p/4
J k j
x
l
À Á À
JðkjÞ
HðkjÞ H k j
x
l
À Á
c) Clamped-free
(= cantilever)
cos k cosh k ¼ À1
1.8751
4.6941
7.8548
10.9955
!(2j–1)p/2
J k j
x
l
À Á À
GðkjÞ
FðkjÞ H k j
x
l
À Á
d) Clamped-guided
tan k ¼ À tanh k
2.3650
5.4978
8.6394
11.7810
!(4j–1)p/4
J k j
x
l
À Á À
FðkjÞ
JðkjÞ H k j
x
l
À Á
e) Hinged-hinged
(= simply
supported)
sin k ¼ 0
jp
sin jp
x
l
À Á
f) Hinged-guided
cos k ¼ 0
( 2 j–1)p/2
sin ðj À
1
2 Þp
x
l
À
Á
g) Guided-guided
sin k ¼ 0
jp
cos jp
x
l
À Á
h) Free-free
cos k cosh k ¼ 1
Same as for (a)
G k j
x
l
À Á À
JðkjÞ
HðkjÞ F k j
x
l
À Á
i) Free-hinged
tan k ¼ tanh k
Same as for (b)
G k j
x
l
À Á À
GðkjÞ
FðkjÞ F k j
x
l
À Á
j) Free-guided
tan k ¼ À tanh k
Same as for (d)
G k j
x
l
À Á À
HðkjÞ
FðkjÞ F k j
x
l
À Á
i
The natural frequencies are x j = k j
2
(EI/qA)
1/2 /l
2
, where k j is given in the table
ii Adapted from Flügge (1962) with permission from the publisher
iii Lowest roots numerically, or all roots analytically, or limits for large j (indicated by !)
iv
F(u) = sinhu + sinu; G(u) = coshu + cosu; H(u) = sinhu − sinu; J(u) = coshu − cosu
488
Appendix D: Natural Frequencies and Mode Shapes for Structural Elements