Acoustics and Vibration Animations
Daniel A. Russell, Ph.D.Graduate Program in Acoustics The Pennsylvania State University All text and images on this page are ©2004-2011 by Daniel A. Russell and may not used in other web pages or reports without permission. |

A sinusoidal force F_{0}sin *w t* acts on an undamped main mass-spring system (without the absorber mass attached). When the forcing frequency equals the natural frequency of the main mass the response is infinite. This is called resonance, and it can cause severe problems for vibrating systems.

When an absorbing mass-spring system is attached to the main mass and
the resonance of the absorber is tuned to match that of the main mass, the motion
of the main mass is reduced to zero at its resonance frequency. Thus, the energy of the main mass is apparently "absorbed" by the tuned dynamic absorber.
It is interesting to note that the motion of the absorber is ** finite** at this resonance frequency, even though there is

If no damping is present, the response of the 2-DOF system is infinite at these new frequencies. While this may not be a problem when the machine is running at its natural frequency, an infinite response can cause problems during startup and shutdown. A finite amount of damping for both masses will prevent the motion of either mass from becoming infinite at either of the
new resonance frequencies. **However** if damping is present in **either** mass-spring element, the response of the main mass will **no longer be zero** at the target frequency.

## Animation of the main mass and dynamic absorber at three frequencies.
The 2-DOF system has two natural frequencies, corresponding to the two natural modes of vibration for the system. In the lower frequency mode both masses move in the same direction, in-phase with each other. In the higher frequency mode the two masses move in opposite direction, 180° out of phase with each other. The animation below shows the motion of the 2-DOF system at normalized forcing frequencies of (undamped classical tuned dynamic absorber), and f_{middle}=1 (opposite-phase mode). The arrows in the movie represent the magnitude and phase of the force applied to the main mass.f_{right}=1.3 |

The plots below show the displacements as a function of normalized frequency (driving frequency divided by natural frequency of main mass). The red dashed curve shows the displacement response of the undamped main mass alone. Notice that when the driving frequency matches the natural frequency, the response is inifinite.

An absorber mass (20% of the main mass) is tuned to the resonance frequency of the main mass and attached. The blue curve represents the displacement of the main mass after the classical undamped tuned dynamic absorber has been attached. Notice that the main mass has zero displacement at the original problem frequency. Notice, also, that there are now two new resonance frequencies. The green curve represents the displacement of the absorber mass. Notice that the displacement is infinite at the same two resonance frequencies, but that the response at the target frequency is finite (approximately 4.8).