Continuous Beam Modeling

Abstract

In Chaps. 1 through 5 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.

Exercises

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

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

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

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

• 2. In describing beam vibrations using Euler–Bernoulli beam theory, we derived the equation of motion $\left( {{{{{\partial^{\,4}}Y}} \left/ {{\partial {x^4}}} \right.}} \right) - {\lambda^4}Y = 0$ _.

  • (a) In the equation of motion, what does x represent physically?

  • (b) In the equation of motion, what does Y represent physically?

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

• 3. Consider a fixed-free beam. The general solution to the equation of motion can be written as $Y(x) = A\cos \left( {\lambda x} \right) + B\sin \left( {\lambda x} \right) + C\cosh \left( {\lambda x} \right) + D\sinh \left( {\lambda x} \right)$. 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. 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–P8.4g. 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. Complete the following for the transverse deflection of a free-free cylindrical beam. The beam’s diameter is 15 mm and it is 480 mm long. The beam material is 6061-T6 aluminum with $\rho = 2,\!700\,{\hbox{kg}}/{{\hbox{m}}^{{3}}}$, $E = 70\,{\hbox{GPa}}$, $\nu = 0.35$, $G = {{E} \left/ {{2\left( {1 + \nu } \right)}} \right.}$, and $\eta = 0.002$.

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

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

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

• 6. Complete the following for the torsion vibration of a free-free cylindrical beam. The beam’s diameter is 15 mm and it is 480 mm long. The beam material is 6061-T6 aluminum with $\rho = 2,\!700\,{\hbox{kg}}/{{\hbox{m}}^{{3}}}$, $E = 70\,{\hbox{GPa}}$, $\nu = 0.35$, $G = {{E} \left/ {{2\left( {1 + \nu } \right)}} \right.}$, and $\eta = 0.002$.

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

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

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

• 7. Complete the following for the axial vibration of a free-free cylindrical beam. The beam’s diameter is 15 mm and it is 480 mm long. The beam material is 6061-T6 aluminum with $\rho = 2,\!700\,{\hbox{kg}}/{{\hbox{m}}^{{3}}}$, $E = 70\,{\hbox{GPa}}$, $\nu = 0.35$, $G = {{E} \left/ {{2\left( {1 + \nu } \right)}} \right.}$, and $\eta = 0.002$.

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

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

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

• 8. Consider the transverse vibration of a free-free cylindrical beam. If the diameter of a solid beam is d, determine the outer diameter, d _o , 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, d _i , is one-half of the outer diameter, ${d_i} = 0.5{d_o}$.

• 9. For a 25 mm diameter 6061-T6 aluminum rod ($\rho = 2,\!700\,{\hbox{kg}}/{{\hbox{m}}^{{3}}}$ and $E = 70\,{\hbox{GPa}}$) with a nominal length of 190 mm and an associated uncertainty of 0.2 mm, determine the uncertainty in the second bending natural frequency, $u\left( {{f_{{n,2}}}} \right)$ (in Hz), if free-free boundary conditions are imposed. You may neglect the uncertainty in E, ρ, and d.

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