The field of vibration control considers the following fundamental principles:
Fig 1. Single Degree of Freedom System
Starting from the equilibrium of the acting forces in the system
FI + FD + FE = F(t)
FI = inertia force [N]
FD = damping force [N]
FE = elastic force [N]
F(t) = excitation force [N]
the differential equation describing the motion of the mass is given by:
mẍ + dẋ + kx = F(t)
In the case of no external excitation, but an initial displacement of the mass leading to an elastic force of the spring, the solution of the equation describes a free, undamped oscillation – the mass oscillates with its natural angular frequency ωo, i.e., its natural frequency fo=1/2π√(k/m).
In the case of a sinusoidal (harmonic) excitation force with a frequency f, the change in maximum displacement of the system (amplitude amplification) is given by:
η = f/fo
D = Damping
Vibration transmissibility is defined by the ratio of transmitted forces divided by the exciting forces. Fig. 2 shows the transmissibility factor VT = (FB(t) /F(t)) over the tuning factor n. The tuning factor is defined as the ratio of the excitation frequency f divided by the natural frequency f0. It clearly shows that the attenuation of vibration transmission takes place only if the factor is beyond the value of
There is no reduction of dynamic forces possible when the natural frequency (= tuning frequency) of a single degree of freedom system (SDOF) is close to or above the relevant excitation frequencies. The dynamical amplification V being the ratio between transmitted and excited force equals:
In the case of base isolation, the tuning frequency of the elastically supported system has to be fo << f x 1/ √2 to avoid resonances with the excitation frequencies f.
Fig 2: transmissibility curve depending on the damping
Fig 3. Amplitude amplification versus excitation frequency
The basic characteristics of a vibratory system are defined by the oscillating mass, the damping as well as the elastic properties of the system. In case of an excitation around the natural frequency of the system,
i.e. around resonance, the resulting amplitude of the oscillation is specifically high (Fig. 2). When changing the size of the oscillating mass or the elastic properties of the system the natural frequency of the system will change as well. By that change it can be achieved that the natural frequency of the system and the excitation frequency are no longer in the same range i.e. the system is no longer in resonance for the given excitation frequency and the amplitude of the oscillation is reduced. In fact the amplitude amplification function will shift depending on the natural frequency (Fig 4). De-tuning can be done by changing the mass or – the typical approach by stiffening the system i.e. increasing the stiffness k. Obviously, with the new natural frequency resonance will occur at another excitation frequency. Therefore it has to be made sure that the new natural frequency is not matching any other possible excitation frequency. Like damping, de-tuning can be applied as a retrofit measure. However, significant de-tuning is often only possible ina narrow frequency range as f0 ~ k
Fig 4. Amplitude amplification versus excitation frequency
A standard method of introducing damping or balancing forces is the application of a Tuned Mass Damper (TMD). Excitation forces are compensated by mass forces so that parts of the structure (main construction) remain almost at rest in a specific frequency range. In vibration damping, an additional oscillator consisting of mass and stiffness (undamped absorber) or of mass, stiffness, and damping (damped absorber) is attached to the main structure for calming.
The undamped absorber is usually used for stationary harmonic oscillation problems, where the absorber natural frequency fT is tuned to the excitation frequency. Theoretically, the vibration amplitudes of the main system can be reduced to zero if no damping is present.
For stationary broadband excitation, which is essential in the range of a natural system frequency, the use of an optimally damped absorber is useful. The natural frequency ω_(T,opt) and the damping ratio Dopt of the optimally damped absorber can be designed according to the Den Hartog formula:
Fig 5. Amplitude amplification versus excitation frequency