Tuned-Mass Damper on a Cantilever Beam

In this article, we compare the performance of a tuned-mass damper mounted at the end of a cantilever beam to the Lanchester damper which was shown in the previous article.  The classic single-degree-of-freedom (SDOF) tuned-mass damper is sketched in the figure below.  The design approach is to find the equivalent SDOF system for the cantilever beam’s mode of interest and then use the design formulas for an optimal SDOF TMD to determine the stiffness and damping of the absorber.

SDOF tuned-mass damper
SDOF tuned-mass damper

The equivalent SDOF primary system  of the cantilever beam near natural frequency of interest can be determined from the results of an FEA analysis, or analytically for a system as simple as this cantilever beam.  The relevant parameters are the effective modal mass and the natural frequency.

cantilever beam
Cantilever beam (steel) with dimensions 5 mm thick x 20 mm wide x 100 mm long having a total mass of 0.0785 kg

The tuning rules for an SDOF tuned-mass damper can be found in a many different references including Mechanical Vibrations by J.P. Den Hartog (google books link).

The tuning ratio of the absorber is given by

    \begin{equation*} \frac{\omega_n}{\Omega_n}=\frac{\sqrt{k/m}}{\sqrt{K/M_{\mbox{effective}}}} =\frac{1}{1+\mu} \end{equation*}

where \mu=m/M_{\mbox{effective}} is the mass ratio.  The damping ratio of the absorber is given by

    \begin{equation*} \frac{c}{c_{\mbox{crit}}}=\sqrt{ \frac{3\mu}{8 \left( 1+\mu \right) ^3}} \end{equation*}

where c_{\mbox{crit}}=2m \Omega_n is the critical damping ratio

The absorber properties are

    \begin{equation*} k=m \Omega_n^2 \left( \frac{1}{1+\mu} \right)^2 \end{equation*}

    \begin{equation*} c=2m \Omega_n \sqrt{ \frac{3\mu}{8 \left( 1+\mu \right) ^3}} \end{equation*}

The figure below shows the transfer function for a tuned-mass damper of 5% added mass.  We performed two designs.  In the first case, the TMD is optimized for the first bending mode and in the second case, it is optimized for the second mode.  Because of the tuned-mass damper adding additional degrees of freedom and being relatively lightly damped, the original mode splits into two modes (one with the beam and damper moving in phase and one with the beam and damper moving out of phase).

Collocated transfer function X(s)/F(s) at the end of the cantilevered beam with tuned-mass damper optimized for first and second modes (5% added mass)
Collocated transfer function X(s)/F(s) at the end of the cantilevered beam with tuned-mass damper optimized for first and second modes (5% added mass)

Damping with tuned-mass damper optimized for first mode
Mode 1 – 13.5 and 13.3% damping
Mode 2 – 0.7% damping
Mode 3 – 0.24% damping
Mode 4 – 0.13% damping

Damping with tuend-mass damper optimized for second mode
Mode 1 – 0.7% damping
Mode 2 – 11 and 16% damping
Mode 3 – 1.7% damping
Mode 4 – 0.8% damping

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