Response control of beam vibrations with a reciprocating dynamic damper

A method for damping vibrations in a beam traversed by a moving load, comprising:

• connecting a first flexible support having a linear spring constant, kl, beneath a first fixed end of the beam;
• connecting a second flexible support having a linear spring constant, kl, beneath a second fixed end of the beam;
• connecting a first torsional spring to the first end, wherein the first torsional spring has a torsional spring constant, kθ;
• connecting a second torsional spring to the second end, wherein the second torsional spring has the torsional spring constant, kθ;
• connecting a housing including a dynamic linear spring mass damper to the beam;
• generating, with a moving load, first vibrations in the beam;
• and
• actuating, with a motor, the dynamic linear spring mass damper to travel reciprocally between the first end and the second end, thereby generating second vibrations in the beam which are 180 degrees out of phase with the first vibrations.

A non-transitory computer readable medium having program instructions stored therein that, when executed by one or more processors, cause the one or more processors to perform a method for damping vibrations in a beam traversed by a moving load, comprising:

• receiving first vibration signals generated by a first accelerometer connected to the beam;
• connecting a dynamic linear spring mass damper to the beam;
• receiving second vibration signals generated by a second accelerometer of the dynamic linear spring mass damper;
• receiving position signals from a position sensor connected to a housing of the dynamic linear spring mass damper;
• Fourier transforming the first vibration signals, the second vibration signals and the position signals to a frequency space;
• calculating a force, F(x, t) of the moving load on the beam based on a weighted sum of a fourth partial derivative with respect to position of a frequency of the moving load, a first partial derivative with respect to position of the frequency of the moving load, a second partial derivative with respect to time of the frequency of the moving load, a displacement of a mass of the dynamic linear spring mass damper due to the second vibrations multiplied by a Fourier transform of a second force, (G(x, t)), on the beam generated by the second vibrations of the dynamic linear spring mass damper, and a first derivative of the displacement of the mass due to the second vibrations multiplied by the second force on the beam generated by the second vibrations of the a dynamic linear spring mass damper;
• and
• calculating the speed and travel distance of the dynamic linear spring mass damper needed to dampen the vibrations based on the force, F(x, t).