Mechanical math

it Lecturr 5

Unit6 - NormalModes

This Unit is about vibrations and it continues from where MTH2L3 Units I wtd Mfi{215 Unit 5 left off. Its aim is to extend your ability to model systems in which vibration play an important role. It will use some of the mathematical techniques you have learnt, such as systems of differential equations, matrices and eigenvalues.

In our previous study of vibration, only one particle of mass m is involved, and only one displacement x is needed to completely describe its motion. However, in real mechanical systems, more than one particle with different masses are moving and thus more than one displacement needed to define the positions of the particles.

The number of degrees offreedom of a system is the smallest number of coordinates which are required to speciff the configuration of the system at any instant. The importance of the number of degrees of freedom is that it is equal to the number af equations af motion. (Refer to

For a one degree offreedora system without damplng and forcing, the homogeneous equation of motion has the form,

mti+kx=0 The solution to this equation of motion is,

x(t) : Acos(ott + 0) which is always sinusoidal, cortmonly called simple harmonie motion (or SHM in short).

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