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# A Damping Problem

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A Damping Problem

? 2008 Waterloo Maple Inc

Introduction

The system consists of a machine containing a large rotating mass. This is to be placed on an elastic material. The damping, , is to be determined by visualising the damped amplitude as a function of frequency and damping.

Parameters of the System

The machine has a mass = 1000 , and the rotating mass is = 500. The elastic material has spring rate = 100000   . Let be the radius of the crankshaft and ω  its angular velocity.

Solution

The oscillation of the machine is described by the following differential equation, where c describes the damping coefficient:

 (3.1)

Solving this gives:

 (3.2)

Since we are only interested in the stationary state, the exponential functions (which describe the transient effect) can be ignored.

 (3.3)

The force exerted on the base is:

 (3.4)

To determine the amplitude, we set the oscillating part to its maximal value, using the following addition theorem:

 (3.5)

 (3.6)

 (3.7)

From this, the damped amplitude is given by :

 (3.8)

The unbalancing force (without damping) is:

 (3.9)

Therefore the unbalancing amplitude is:

 (3.10)

The quality of the damping is determined by the ratio of the amplitudes,

 (3.11)

We determine the maximum amplitude with respect to :

 (3.12)

Choosing the positive root, which corresponds to the physical reality:

 (3.13)

 (3.14)

 (3.15)

Graphical Display of the Solution

We insert the parameters into the formula for the amplitude and display it.

#Parameter Insertion & Plot

Conclusion

Maple enables us to solve the differental equation exactly. With this symbolic solution we could calculate and visualise the ratio of the amplitude of the oscillation to the amplitude delivered to the base. This ratio is important for the quality of the damping.

Reference

Heinz Waller and Reinhard Schmidt, Schwingungslehre f?r Ingenieure, Wissenschaftsverlag.