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Continuous Beam Modeling

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Abstract

In Chaps. 15 we discussed the solution of discrete, lumped parameter models. For multiple degree of freedom systems, we employed modal analysis to enable us to transform the coupled equations of motion in local (model) coordinates into modal coordinates. In this coordinate frame, the equations of motion were uncoupled and we could apply single degree of freedom solution techniques. In Chap. 6 we shifted our attention to the “backwards problem”, which is representative of a common task for vibration engineers. In this problem, we begin with measurements of an existing structure and use this information to develop a model. We again used discrete models to describe the system behavior.

Keywords

  • Euler-Bernoulli beam theory
  • Boundary condition
  • Transverse vibration
  • Frequency response function
  • Solid damping
  • Anti-resonant frequency
  • Rotation frequency response function
  • Rigid body mode
  • Timoshenko beam model

Continuity in everything is unpleasant.

—Blaise Pascal

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Fig. 8.1
Fig. 8.2
Fig. 8.3
Fig. 8.4
Fig. 8.5
Fig. 8.6
Fig. 8.7
Fig. 8.8
Fig. 8.9
Fig. 8.10
Fig. 8.11
Fig. 8.12
Fig. 8.13
Fig. 8.14
Fig. 8.15
Fig. 8.16
Fig. 8.17
Fig. 8.18
Fig. 8.19
Fig. 8.20
Fig. 8.21

Notes

  1. 1.

    A beam can be described as a structure where one dimension is much larger than the other two dimensions.

  2. 2.

    A simply supported beam is pinned at one end and has a rolling support at the other.

  3. 3.

    We consider the resonant case because this is where damping has the most significant effect. Its influence is less at frequencies far from resonance.

  4. 4.

    For a continuous beam there are an infinite number of modes for an infinite bandwidth.

  5. 5.

    The authors credit Dr. W.T. Estler (retired, National Institute of Standards and Technology) with this figure.

  6. 6.

    The beam’s cross-section is not required to be circular as in the torsion vibration analysis in Sect. 8.7.

  7. 7.

    This means that the beam’s expansion and contraction in the directions normal to the oscillating axial deflection are ignored.

References

  1. Bishop R, Johnson D (1960) The mechanics of vibration. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  2. Chapra S, Canale R (1985) Numerical methods for engineers with personal computer applications. McGraw-Hill, New York, NY, Section 7.1

    Google Scholar 

  3. Blevins RD (2001) Formulas for natural frequency and mode shape. Krieger Publishing, Malabar, FL, Table 8.1

    Google Scholar 

  4. Weaver W Jr, Timoshenko S, Young D (1990) Vibration problems in engineering, 5th edn. Wiley, New York, Section 5.12

    Google Scholar 

  5. Hutchinson J (2001) Shear coefficients for Timoshenko beam theory. J Appl Mech 68:87–92

    CrossRef  Google Scholar 

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Exercises

Exercises

  1. 1.

    Consider a uniform cross-section fixed-free (i.e., clamped-free or cantilever) beam.

    1. (a)

      Sketch the first bending mode shape (lowest natural frequency).

    2. (b)

      Sketch the second mode shape (next lowest natural frequency).

    3. (c)

      On your sketches in parts (a) and (b), identify any node location(s).

  2. 2.

    In describing beam vibrations using Euler-Bernoulli beam theory, we derived the equation of motion \( \frac{\partial^4Y}{\partial {x}^4}-{\lambda}^4Y=0. \)

    1. (a)

      In the equation of motion, what does x represent physically?

    2. (b)

      In the equation of motion, what does Y represent physically?

    3. (c)

      Write the equation for λ (it replaces several other variables) and describe what each variable represents (include the SI units).

  3. 3.

    Consider a fixed-free beam. The general solution to the equation of motion can be written as Y(x) = Acos(λx) + Bsin(λx) + Ccosh(λx) + Dsinh(λx). To determine the four coefficients, A through D, four boundary conditions are required. Write the four boundary conditions (in the table) as a function of x and y for the beam shown in Fig. P8.3.

    at x = 0 at x = L
    1. 3.
    2. 4.
  4. 4.

    Consider the free-sliding beam shown in Fig. P8.4a. Direct and cross FRFs were measured at six locations and the imaginary parts are provided for the frequency interval near its second bending natural frequency of 350 Hz in Figs. P8.4b, P8.4c, P8.4d, P8.4e, P8.4f, and P8.4g. Given the FRF data, sketch the mode shape corresponding to the second natural frequency. Normalize the mode shape to a value of 1 at the free end.

  5. 5.

    Complete the following for the transverse deflection of a free-free cylindrical beam. The beam’s diameter is 15 mm diameter steel and it is 480 mm long. The beam material 6061-T6 aluminum with ρ = 2700 kg/m3, E = 70 GPa, ν = 0.35, \( G=\frac{E}{2\left(1+\nu \right)} \), and η = 0.002.

    1. (a)

      Plot the transverse deflection FRF over a frequency range of 10,000 Hz. Use a semi-logarithmic scale.

    2. (b)

      How many modes are captured in this bandwidth (excluding the rigid body modes)?

    3. (c)

      What is the natural frequency of the first (non-rigid) bending mode?

  6. 6.

    Complete the following for the torsion vibration of a free-free cylindrical beam. The beam’s diameter is 15 mm diameter steel and it is 480 mm long. The beam material 6061-T6 aluminum with ρ = 2700 kg/m3, E = 70 GPa, ν = 0.35, \( G=\frac{E}{2\left(1+\nu \right)} \), and η = 0.002.

    1. (a)

      Plot the torsion FRF over a frequency range of 10,000 Hz. Use a semi-logarithmic scale.

    2. (b)

      How many modes are captured in this bandwidth (excluding the rigid body mode)?

    3. (c)

      What is the natural frequency of the first (non-rigid) torsion mode?

  7. 7.

    Complete the following for the axial vibration of a free-free cylindrical beam. The beam’s diameter is 15 mm diameter steel and it is 480 mm long. The beam material 6061-T6 aluminum with ρ = 2700 kg/m3, E = 70 GPa, ν = 0.35, \( G=\frac{E}{2\left(1+\nu \right)} \), and η = 0.002.

    1. (a)

      Plot the axial FRF over a frequency range of 10,000 Hz. Use a semi-logarithmic scale.

    2. (b)

      How many modes are captured in this bandwidth (excluding the rigid body mode)?

    3. (c)

      What is the natural frequency of the first (non-rigid) torsion mode?

  8. 8.

    Consider the transverse vibration of a free-free cylindrical beam. If the diameter of a solid beam is d, determine the outer diameter, do, of a hollow beam with the same length and material properties to give the same natural frequencies as the solid beam if the inner diameter, di, is one-half of the outer diameter, di = 0.5do.

  9. 9.

    For a 25 mm diameter 6061-T6 aluminum rod (ρ = 2700 kg/m3 and E = 70 GPa,) with a nominal length of 190 mm and an associated uncertainty of 0.2 mm, determine the uncertainty in the second bending natural frequency, u(fn, 2) (in Hz) if free-free boundary conditions are imposed. You may neglect the uncertainty in E, ρ, and d.

  10. 10.

    The Timoshenko beam model is more accurate than the Euler-Bernoulli beam model because it includes the effects of ______________ and ______________.

Fig. P8.3
figure 22

Fixed-free beam model

Fig. P8.4a
figure 23

Free-sliding beam model

Fig. P8.4b
figure 24

Direct FRF \( \frac{X_1}{F_1} \) for the free-sliding beam

Fig. P8.4c
figure 25

Cross FRF \( \frac{X_2}{F_1} \) for the free-sliding beam

Fig. P8.4d
figure 26

Cross FRF \( \frac{X_3}{F_1} \) for the free-sliding beam

Fig. P8.4e
figure 27

Cross FRF \( \frac{X_4}{F_1} \) for the free-sliding beam

Fig. P8.4f
figure 28

Cross FRF \( \frac{X_5}{F_1} \) for the free-sliding beam

Fig. P8.4g
figure 29

Cross FRF \( \frac{X_6}{F_1} \) for the free-sliding beam

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Schmitz, T.L., Smith, K.S. (2021). Continuous Beam Modeling. In: Mechanical Vibrations. Springer, Cham. https://doi.org/10.1007/978-3-030-52344-2_8

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  • DOI: https://doi.org/10.1007/978-3-030-52344-2_8

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