The following metrics are demonstrated by example throughout this document. The viscous damping ratio is defined in terms of the damping coefficients as

The critical damping coefficient is

The amplification factor is

The amplification factor can be calculated from measured frequency response function data via the half-power method.

where is the peak center frequency and is the difference between the two -3 dB points on either side of the peak.

The loss factor is

The logarithmic decrement is

The four damping types are summarized in the following table. The description is for the free vibration of a single-degree-of-freedom system due to initial displacement or velocity.

Type | Value | Response Description |
---|---|---|

Undamped | Simple harmonic motion, sinusoidal response | |

Underdamped | Damped sine response with exponential decay | |

Critically damped | Border between the overdamped and underdamped cases | |

Overdamped | Sum of two decaying exponentials with no oscillation |

The first three of these types are shown by examples in Figure 2.1. The underdamped type emphasized throughout this document.

Damping occurs as vibration energy is convert to heat, sound or some other loss mechanism. Damping is needed to limit the structural resonant response. Common sources are:

- viscous effects
- Coulomb damping, dry friction
- aerodynamic drag
- acoustic radiation
- air pumping at joints
- boundary damping

The dominant source for assembled structures is usually joint friction. This damping mechanism may be nonlinear due to joint microslip effects. The damping value tends to increase at higher excitation levels.

Damping may also decrease as the natural frequency increases, such that the amplification factor increases with natural frequency. The equation per Reference [10] for approximating Q for an electronic system subjected to a sine base input is

where

A | = 1.0 for beam-type structures = 0.5 for plug-in PCBs or perimeter supported PCBs = 0.25 for small electronic chassis or electronic boxes |

Gin | Sine Base Input (G) |

**Beam structures:** several electronic components with some interconnecting wires or cables.

**PCB:** printed circuit board well-populated with an assortment of electronic components.

**Small electronic chassis:** 8-30 inches in its longest dimension, with a bolted cover to provide access to various types of electronic components such as PCBs, harnesses, cables, and connectors.

The bridge in Figure 4.1 is near the University of Alabama in Huntsville. The author walked to the center of the bridge and mounted the triaxial accelerometer in Figure 4.2 on the deck floor using double-sided tape. He was the only person on the bridge. The ambient vibration response of the bridge was negligibly low in each of the axes, while he remained standing. The wind was very light on this day. The author then jumped up-and-down on the bridge to excite its vibration modes. The response in the vertical axis was significant. The response in each of the lateral axes remained negligibly low.

The jumps were performed near the 80 second mark. The accelerometer data was band-passed filtered from 1 to 20 Hz. The response after 82 seconds represents the free vibration decay. A damped sine curve-fit synthesis was performed on the vertical acceleration time history as shown in Figure 4.3, using trial-and-error with convergence. The bridge’s fundamental frequency is 2.2 Hz with 0.16% damping. The damping ratio is very low. But note that the fundamental frequency and damping may be nonlinear. A pedestrian’s vertical forces correspond to each footfall, and typically occur at 2.0 Hz. This is very close to the bridge’s 2.2 Hz natural frequency.

Also note that there is a potential for pedestrians to synchronizing their steps with the bridge motion and with one another. This behavior is instinctive rather than deliberate. Pedestrians find that walking in synchronization with the motion of a bridge is more comfortable, even if the oscillation amplitude is initially very small. This cadence makes their interaction with the movement of the bridge more predictable and helps them maintain their balance. But the synchronization also causes the pedestrians’ gait to reinforce the bridge’s oscillation in a resonance-like manner. These sorts of problems occurred after the opening of the London Millennium Bridge in 2000. Both passive and tuned mass dampers were added to the bridge for vibration control. Retrofitting the University Drive Bridge with dampers is unnecessary due to its low pedestrian traffic volume. But this could be an interesting project for the nearby engineering students.

A modified L-1011 aircraft carries the Pegasus vehicle up to an altitude of nearly 40,000 ft and a speed of Mach 0.8, as shown in Figure 5.1. Pegasus is suspended underneath the aircraft by hooks, where it develops an initial displacement due to gravity. The strain energy is suddenly released at the onset of the drop transient, causing Pegasus to oscillate nearly as a free-free beam. This is a significant “coupled loads” event for the payload which is enclosed in the fairing at the front end of the vehicle. The Pegasus first stage then ignites. The payload is eventually delivered into a low earth orbit. A certain Pegasus/payload configuration was analyzed via a finite element modal analysis. The resulting wire mesh model of the fundamental bending mode is shown in Figure 5.2. Note that this was a different Pegasus configuration than that represented by the flight data in Figure 5.3.

The data was measured in the transverse axis at the payload interface. The response is nearly a textbook quality damped sinusoid, with an exponential decay. The natural frequency and damping ratio are identified via a synthesized damped sine, curve-fit.

The logarithmic decrement for a starting peak and a peak n cycles later is

The logarithmic decrement for the two peaks shown in Figure 5.4 is

The logarithmic decrement value is equivalent to 1.2% damping in agreement with the previous damped sine curve-fit. The logarithmic decrement method has been included for historical reasons. The damped sine curve-fit method is more robust.

The Transamerica Pyramid is built from a steel frame, with a truss system at the base. The height is 850 ft (260 m). Reference [11] gives natural frequency and damping as obtained in the 1989 Loma Prieta earthquake and from ambient vibration. The ambient vibration was presumably due to wind, low level micro-tremors, mechanical equipment, outside street traffic, etc.

Direction | Loma Prieta Earthquake | Ambient Vibration | ||
---|---|---|---|---|

fn (Hz) | Damping | fn (Hz) | Damping | |

North-South | 0.28 | 4.9% | 0.34 | 0.8% |

East-West | 0.28 | 2.2% | 0.32 | 1.4% |

The results show non-linear behavior with an increase in damping during the severe earthquake relative to the benign ambient vibration.