Lanchester Damper on a Cantilever Beam

In this article, we show the robust and broadband performance  of a Lanchester damper applied to a cantilever beam and how it achieves good performance without tuning and good performance over a number of modes, not just the primary mode.

lanchester damper
Lanchester damper

The Lanchester Damper is a reaction mass damper with robust performance because it is insensitive to tuning and and provides good performance over a wide frequency range whereas a traditional tuned-mass damper is quite sensitive to tuning and only provides damping to the primary system over a relatively narrow frequency range.  In the ideal case, the Lanchester damper consists of a mass and a linear dashpot as shown in the figure below.  The Lanchester Damper has been known as an effective vibration suppression device since before 1910 as it was credited to Frederick W. Lanchester (wikipedia link) to alleviate rotary vibrations in an early combustion engine.  The history of the Lanchester damper is quite interesting and involves a significant patent dispute between Rolls Royce and Lanchester (see the text Royce and the Vibration Damper by Tom C. Clarke, ISBN 978-1872922188 amazon.com link).  The design formulas for the Lanchester damper can be found in a number of different references including the classic text Mechanical Vibrations by J.P. Den Hartog (amazon.com link).

The tuning rules for an SDOF Lanchester damper are presented in the text by Den Hartog.  The tuning rules were originally derived by looking at the frequency response function of the primary mass while varying the damping of the absorber and recognizing that there are two fixed points through which all curves pass.  One of those fixed points is at zero frequency and by choosing the damping such that the other point is the maximum of the transfer function, then the optimal design is obtained.

Consider a cantilever steel beam with dimensions 5 mm thick x 20 mm wide x 100 mm long having a total mass of 0.0785 kg.  Its frequency response with a collocated force and measurement at the tip is shown in the figure below (note: there are also additional modes which are unobservable such as torsion and transverse bending which are not shown).

cantilever beam
Cantilever beam
Transfer function X(s)/F(s) at end of beam without Lanchester damper (0.1% damping in all modes).
Transfer function X(s)/F(s) at end of beam without Lanchester damper (0.1% damping in all modes).

We choose to place the Lanchester damper at the tip because it has the largest motion during the first mode.  Intuitively, this is the best place to place a damper for the first mode.  We use an absorber mass of 5% of the beam’s total mass.

To design the Lanchester damper, we first calculate the effective modal mass for the original undamped system for the vibration mode of interest.  This is result is easily obtained during the modal analysis using a finite element solver such as Ansys.  For the first vibration mode, the effective modal mass is 0.0196 kg, of 25% of the beam’s mass.  When applying the SDOF Lanchester design formulas, one must use the effective mass.  The damper value is given by

    \begin{equation*} c=2 m_{\mbox{\tiny{damper}}} \Omega_n \sqrt{ \frac{1}{2 (2+\mu) (1+\mu)}} \end{equation*}

where \mu=\frac{m}{M_{\mbox{\tiny{effective}}}} is the effective mass ratio, m_{\mbox{\tiny{damper}}} is the mass of the absorber, and \Omega_n is the undamped natural frequency of the primary system.

Because it is impractical to have the ideal Lanchester damper of a dashpot without a spring, we set the natural frequency of the absorber system to be 50 Hz.  By placing the absorber’s natural frequency well below the first resonance of interest, we achieve a practical design, but the design formulas and performance still hold.  The Lanchester damper uses relatively large damper values so the absorber system itself is significantly overdamped (infinitely overdamped in the ideal case).

Then, we construct a state-space representation of the undamped cantilever beam using the results of the finite element modal analysis using the command SPMWRITE within Ansys.  Using the method described by Andrew Wilson, we can obtain the dynamic model for the beam with the absorber.

The frequency response of the undamped system and system with the Lanchester damper optimized for the first vibration mode is shown below.

Transfer function X(s)/F(s) at end of cantilever beam with and without Lanchester damper (5% added mass)

We see from the damped transfer function that we also obtain significant damping in the higher vibration modes

Damping with Lanchester damper optimized for first mode
Mode 1 – 5% damping
Mode 2 – 1.4% damping
Mode 3 – 0.5% damping
Mode 4 – 0.25% damping

If instead of optimizing the damper value for the first mode, we instead optimize for the second mode, we obtain the the transfer function shown below.  In this case the damping ratios

Damping with Lanchester damper optimized for second mode
Mode 1 – 1.4% damping
Mode 2 – 4.1% damping
Mode 3 – 2.7% damping
Mode 4 – 1.5% damping

Collocated transfer function X(s)/F(s) for cantilever beam without damper, with Lanchester damper optimized for first mode, and damper optimized for second mode (5% added mass in both cases).

A recent journal article out of the Netherlands (“Broadband damping of non-rigid-body resonances of planar positioning stages by tuned mass dampers” by Verbaan, Rosielle, and Steinbuch in Mechatronics Vol 24, Issue 6, Sept 2016, pp 712-723) demonstrates the use of multiple Lanchester dampers applied to bending vibrations of a plate indicative of a motion stage.  Although the authors do not refer to them as Lanchester dampers, they are using the same damper concept developed in the early 1900’s.

 

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