7.7 Exercises
201
When the program is executed, the original equation should be solved both
with Newton’s method and via fixed point iterations (as just described). Compare
the output from the two methods.
Filename: fixed_point_iteration.py.
Exercise 7.8: Solve Nonlinear Equation for a Vibrating Beam
An important engineering problem that arises in a lot of applications is the vibrations
of a clamped beam where the other end is free. This problem can be analyzed
analytically, but the calculations boil down to solving the following nonlinear
algebraic equation:
cosh β cos β = −1,
where β is related to important beam parameters through
β
4
= ω
2
EI
,
where is the density of the beam, A is the area of the cross section, E is Young’s
modulus, and I is the moment of the inertia of the cross section. The most important
parameter of interest is ω, which is the frequency of the beam. We want to compute
the frequencies of a vibrating steel beam with a rectangular cross section having
width b = 25 mm and height h = 8 mm. The density of steel is 7850 kg/m
3 , and
E = 2 × 10 11 Pa. The moment of inertia of a rectangular cross section is I =
bh 3 /12.
a) Plot the equation to be solved so that one can inspect where the zero crossings
occur.
Hint When writing the equation as f (β) = 0, the f function increases its
amplitude dramatically with β. It is therefore wise to look at an equation with
damped amplitude, g(β) = e −β f (β) = 0. Plot g instead.
b) Compute the first three frequencies.
Filename: beam_vib.py.
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing,
adaptation, distribution and reproduction in any medium or format, as long as you give appropriate
credit to the original author(s) and the source, provide a link to the Creative Commons licence and
indicate if changes were made.
The images or other third party material in this chapter are included in the chapter’s Creative
Commons licence, unless indicated otherwise in a credit line to the material. If material is not
included in the chapter’s Creative Commons licence and your intended use is not permitted by
statutory regulation or exceeds the permitted use, you will need to obtain permission directly from
the copyright holder.
201
When the program is executed, the original equation should be solved both
with Newton’s method and via fixed point iterations (as just described). Compare
the output from the two methods.
Filename: fixed_point_iteration.py.
Exercise 7.8: Solve Nonlinear Equation for a Vibrating Beam
An important engineering problem that arises in a lot of applications is the vibrations
of a clamped beam where the other end is free. This problem can be analyzed
analytically, but the calculations boil down to solving the following nonlinear
algebraic equation:
cosh β cos β = −1,
where β is related to important beam parameters through
β
4
= ω
2
EI
,
where is the density of the beam, A is the area of the cross section, E is Young’s
modulus, and I is the moment of the inertia of the cross section. The most important
parameter of interest is ω, which is the frequency of the beam. We want to compute
the frequencies of a vibrating steel beam with a rectangular cross section having
width b = 25 mm and height h = 8 mm. The density of steel is 7850 kg/m
3 , and
E = 2 × 10 11 Pa. The moment of inertia of a rectangular cross section is I =
bh 3 /12.
a) Plot the equation to be solved so that one can inspect where the zero crossings
occur.
Hint When writing the equation as f (β) = 0, the f function increases its
amplitude dramatically with β. It is therefore wise to look at an equation with
damped amplitude, g(β) = e −β f (β) = 0. Plot g instead.
b) Compute the first three frequencies.
Filename: beam_vib.py.
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing,
adaptation, distribution and reproduction in any medium or format, as long as you give appropriate
credit to the original author(s) and the source, provide a link to the Creative Commons licence and
indicate if changes were made.
The images or other third party material in this chapter are included in the chapter’s Creative
Commons licence, unless indicated otherwise in a credit line to the material. If material is not
included in the chapter’s Creative Commons licence and your intended use is not permitted by
statutory regulation or exceeds the permitted use, you will need to obtain permission directly from
the copyright holder.