Dynamics of Machines - Part II - IFS.pdf

1.6 Mechanical Systems with 1 D.O.F.
1.6.1 Physical System and Mechanical Model
(a)
(b)
(c)
Figure 4: (a) Real mechanical system composed of a turbine attached to an airplane flexible wing;
(b) Laboratory prototype built by a lumped mass attached to a flexible beam); (c) Equivalent
mechanical model with 1 D.O.F. for a lumped mass attached to a flexible beam.
1.6.2 Mathematical Model
It is important to point out, that the equations of motion in Dynamics of Machinery will frequently
have the form of second order differential equations: ¨y(t) = F(y(t), ˙y(t)). Such equations
can generally be linearized around an operational position of a physical system, leading to second
order linear differential equations. It means that the coefficients which are multiplying the
variables ¨y(t) , ˙y(t) , y(t) (co-ordinate chosen for describing the motion of the physical system)
do not depend on the variables themselves. In the mechanical model presented in figure 4 these
coefficients are constants: m1, d1 and k1. One of the aims of the course Dynamics of Machinery
is to help the students to properly find these coefficients so that the equations of motion can
really describe the movement of the physical system. The coefficients can be predicted using
theoretical or experimental approaches.
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