Abstract
In this chapter we introduce the topic of finite element analysis for beam vibration. In Chap. 8 we described continuous models based on EulerBernoulli beam theory. This is a reasonable approach for a simple geometry, such as the cylindrical shaft in a small DC electric motor, but is more difficult to implement for a more complex structure, such as a gas turbine fan. In this case, finite element analysis provides a common option for vibration studies. Rather than a continuous crosssection model, the geometry is discretized into small elements. The mass and stiffness matrices for the elements are then defined and assembled to provide a numerical solution for the behavior of the entire structure. In this chapter, we focus on axial and bending vibrations of beams that are discretized into elements along their lengths. We derive the associated stiffness and mass matrices and then show how the element matrices are combined to describe the entire beam. Finally, we arrange these matrices into the corresponding differential equations of motion and solve them to obtain the eigenvalues and frequency response functions, or FRFs. While our scope is limited to simple beam geometries, it provides a foundation that can be used for more complex structures.
Keywords
 Finite element analysis
 Axial beam element
 Mass matrix
 Stiffness matrix
 Kinetic energy
 EulerLagrange equation
 Dynamic matrix
 Transverse beam element
 Interpolation/shape function
 Frequency response function
Little things console us because little things afflict us.
—Blaise Pascal
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Notes
 1.
In other words, the element returns to its original length when the force is removed.
References
Logan DL (2011) A first course in the finite element method. Cengage Learning, Boston, Chapter 4.
Archer JS (1963) Consistent mass matrix for distributed mass systems. J Struct Div 89(4):161–178
Thomson WT, Dahleh MD (1998) Theory of vibration with applications, 5th edn. PrenticeHall, Inc., Upper Saddle River, Chapter 10.
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Exercises
Exercises

1.
Consider a cylindrical rod with a diameter of 10 mm and a length of 100 mm. One is end is fixed and the other is free. A 500 N axial load is applied at the free end. See Fig. P9.1. Determine the aluminum rod’s change in length (μm) if its elastic modulus is 70 GPa.

2.
Populate the 2 × 2 stiffness matrix for the axial element that represents a cylindrical aluminum rod with a diameter of 10 mm, length of 100 mm, and elastic modulus of 70 GPa. Substitute the numerical stiffness values (N/m).

3.
Populate the 2 × 2 mass matrix for the axial element that represents a cylindrical aluminum rod with a diameter of 10 mm, length of 100 mm, and density of 2700 kg/m^{3}. Substitute the numerical mass values (kg).

4.
Let a first cylindrical rod with a diameter of 10 mm and a length of 100 mm be joined endtoend to a second cylindrical rod with a diameter of 5 mm and a length of 50 mm. See Fig. P9.2. If the elastic modulus is 70 GPa and the density is 2700 kg/m^{3} for both aluminum rods, write the 3 × 3 stiffness matrix (N/m).

5.
Determine the first axial natural frequency for the fixedfree steel beam shown in Fig. P9.3. The elastic modulus is 205 GPa and the density is 7800 kg/m^{3}. Use 20 elements for the finite element model.

6.
Plot the four interpolation (or shape) functions from Eq. 9.80. For the horizontal axis, use \( \frac{x}{l} \) with limits from 0 to 1. For the vertical axis, use limits of −0.25 to 1.25.

7.
Populate the 4 × 4 stiffness matrix (N/m) for the transverse beam element that represents a cylindrical aluminum rod with a diameter of 10 mm, a length of 100 mm, and an elastic modulus of 70 GPa.
Is the stiffness generally higher or lower than the axial element for the same geometry and material from Problem 2?

8.
Populate the 4 × 4 mass matrix (kg) for the transverse beam element that represents a cylindrical aluminum rod with a diameter of 10 mm, a length of 100 mm, and a density of 2700 kg/m^{3}.

9.
Consider the freefree boundary condition steel beam displayed in Fig. P9.4, where the elastic modulus is 200 GPa and the density is 7800 kg/m^{3}. For a finite element model with 25 transverse elements, determine the length so that the first bending natural frequency is 1500 Hz.

10.
Plot the left end FRF for the 25 finite element model that represents a 10 mm diameter, 150 mm long cylindrical beam with freefree boundary conditions. Use an elastic modulus of 69 GPa, a density of 2750 kg/m^{3}, and solid damping factors of η = {0.005, 0.01, 0.02}. Plot the magnitudes for all three FRFs in a single figure.
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Schmitz, T.L., Smith, K.S. (2021). Finite Element Introduction. In: Mechanical Vibrations. Springer, Cham. https://doi.org/10.1007/9783030523442_9
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DOI: https://doi.org/10.1007/9783030523442_9
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