What is dampening power

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A model that only consists of a spring and a mass reflects many properties and behaviors of a real system (e.g. when the spring constant is doubled). However, the model does not describe the behavior to a frictional force that occurs always and everywhere.

In the case of a damped oscillation, the amplitude decreases over time. The lost energy is converted into heat. The total vibration of a system is made up of different parts, the most important of which are listed here:

  • Material damping: Attenuation based on the properties of the material. For example, wood has a different material damping than metal.
  • System damping: The system damping depends on the construction of the vibrating system. Solid structures dampen differently than trusses. Riveted constructions have greater damping than welded ones.
  • Bearing damping: Mainly friction damping, depending on the nature and quality of the bearings used. The bearing damping is either proportional to the speed or proportional to the square of the speed.
  • Ambient attenuation: When the system does not vibrate in a vacuum, ambient damping occurs. If the surrounding medium is in motion, one speaks of hydrodynamic damping. If the medium is at rest, hydrostatic friction occurs.
  • Damping by vibration damper: In many systems, damping is desired, so special vibration dampers are installed (e.g. the "shock absorber" in the car)

Vibration proportional to the speed

In hydraulic dampers, the damping force F is a good approximationd proportional to the speed of the piston, one speaks of a “linear damping behavior”. The damping force that occurs is the product of the speed v and a constant d:

The symbol for such a damper looks like this:

The proportionality factor d is the Damping coefficient with the unit kg / s. The damping force is always opposite to the speed.

In the following system we assume that the static equilibrium position is at x = 0:

At any point in time t, let m be deflected by x and have a velocity greater than 0 in this position. Then the following two forces act on the mass:

Spring return force:

Damping force:

Focal point:

converted to a homogeneous differential equation:

Here we divide by m and introduce two damping parameters, namely:

  • Natural angular frequency of the undamped system:
  • Decay constant

and get:

Solution of the differential equation

We now use the approach

and insert into the differential equation:

After dividing by the common factors, we get the characteristic equation:

We use the PQ formula to determine the two Eigenvalues (Solutions for λ):

There are three possibilities how the ratio of the decay constant to the natural angular frequency can be. The dimensionless quantity is used to differentiate between the three cases and to be able to evaluate the influence of the damping

practically. They are known as the Degree of damping. In the obsolete standard DIN 1311-2 from 1974, vibrations are included

referred to as “weakly damped”. In the new version of August 2002, a system is only called for

weakly attenuated, otherwise “strongly attenuated”.

A system with

means “very strongly damped”.

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