Longitudinal Vibrations of Finite Beams

Abstract

In Sect. 1.2, the free vibration of a simple mass–spring system was discussed. It was shown that the time dependence of the free vibrations was determined by the natural frequency or eigenfrequency of the system plus the losses.

Problems

If nothing else is stated, assume that the area of the cross section of the beam is S. Material parameters are $E = E_0 (1 + i\eta )$ and $\rho$.

6.1

A beam is clamped at both ends at $x = 0$ and $x = L$. Determine the eigenfunctions and corresponding eigenfrequencies for longitudinal vibrations of the beam.

6.2

A beam is clamped at $x = 0$ and free at the other end at $x = L$. Determine the eigenfunctions and corresponding eigenfrequencies for longitudinal vibrations of the beam.

6.3

Determine the eigenfunctions and eigenfrequencies for a beam with resiliently mounted ends as shown in Fig. 6.1c. L-waves only.

6.4

Determine the eigenfunctions and eigenfrequencies for a beam with periodic boundary conditions. L-waves only.

6.5

Determine Green’s function for a clamped beam. Consider only L-waves traveling along the axis of the beam.

6.6

A force ${F}'(x,t) = F / L \cdot \exp (i\omega t)$ per unit length excites a beam along its axis. The length of the beam is L. The beam is clamped at both ends. Determine the response of the beam by using the appropriate Green’s function.

6.7

A force ${F}'(x,t) = F / L \cdot \sin ({\uppi }x / L) \cdot \exp (i\omega t)$ per unit length is exciting a beam along its axis. The length of the beam is L. The beam is clamped at both ends. Determine the response of the beam by using the mode summation technique.

6.8

A beam is clamped at one end. A static force F is stretching the beam at the other end in the direction of its axis. At time $t = 0$ the beam is released. The beam is for $t > 0$ vibrating freely with one end clamped and the other free. Determine the displacement of the beam as function of time.

6.9

A beam is clamped at both ends and excited at midpoint by a harmonic force $F_0 \cdot \exp (i\omega t)$ along the axis of the beam. The frequency of the driving force is well below the first natural frequency of the beam. Determine the response of the beam. Use the mode summation technique.

6.10

Two straight beams, each with a length of L, are joined together along their axes. One beam has a cross-sectional area of S, the corresponding area for the other beam is 4S. The thicker beam is clamped at one end. The other end of the construction is free. Determine the first eigenfrequency of the beam construction.

6.11

A beam, length L, is mounted in between two identical structures. The point mobility of the adjoining structures is Y at the mounting positions with respect to longitudinal vibrations of the beam. Determine the dispersion equation which gives the natural frequencies for the system. In particular consider the case when each of the adjoining structures is a rigid mass m.

6.12

A mass m is mounted on a rod, geometrical parameters S and L, material parameters E and $\rho$. The rod is in turn mounted to a plate with the point mobility Y. The mass is excited by a force $F_0 \cdot \exp (i\omega t)$. Determine the power transmitted to the plate.

6.13

Two beams are coupled. The axes of the beams coincide. Beam 1 $(L_1 ,S_1 ,E_1 ,\rho _1 )$ is clamped at one end and firmly coupled to beam 2 $(L_2 ,S_2 ,E_2 ,\rho _2 )$ at the other end. A force $F_0 \cdot \exp (i\omega t)$ is exciting the free end of beam 2 along its axis. Determine the velocity of the junction.

6.14

A beam, length L, is clamped at both ends. The beam is excited by a force $F_ \cdot \exp (i\omega t)$ at midpoint along the axis of the beam. Determine the response of the beam by using the matrix method.

6.15

Show that the eigenfunctions for a beam with both ends resiliently mounted are orthogonal. Longitudinal waves only.

6.16

Two homogeneous beams are coupled at the ends along a straight and vertical line. The cross-sectional area of beam 1, the upper beam, is S. The cross-sectional area of beam 2 is 2S. The beams have the same length L and the same material parameters, Young’s modulus E and density $\rho$. The upper end of the coupled beams is denoted 1 and the bottom end by 2. Determine the point and transfer mobilities $Y_{11}$, $Y_{12}$, $Y_{21}$ and $Y_{22}$ of the structure consisting of the two coupled beams. Consider only longitudinal waves propagating along the axes of the beams.