Dynamics of Machines - Part II - IFS.pdf
Dynamics of Machines - Part II - IFS.pdf
Dynamics of Machines - Part II - IFS.pdf
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equations <strong>of</strong> motion can really describe the movement <strong>of</strong> the physical system.<br />
After creating the mechanical model for the physical system, the next step is to derive the<br />
equation <strong>of</strong> motion based on the mechanical model. The mechanical model is built by lumped<br />
masses m1, m2, m1 (assumption !!!), springs with equivalent stiffness coefficient (calculated<br />
using Beam Theory) and dampers with equivalent viscous coefficient (obtained experimentally).<br />
While creating the mechanical model and assuming that the mass is a particle, the equation <strong>of</strong><br />
motion can be derived using Newton’s or Lagrange axioms. For the 3 D.O.F system one can<br />
write:<br />
M¨y(t) + D˙y(t) + Ky(t) = f(t) (75)<br />
or<br />
⎡<br />
⎣<br />
m11 m12 m13<br />
m21 m21 m23<br />
m31 m32 m33<br />
The mass coefficients<br />
⎫<br />
m11 = m1 + m2<br />
m12 = 0<br />
m13 = 0<br />
m21 = 0<br />
m22 = m3 + m4<br />
m23 = 0<br />
m31 = 0<br />
m32 = 0<br />
m33 = m5 + m6<br />
⎤⎧<br />
⎨ ¨y1<br />
⎦ ¨y2<br />
⎩<br />
¨y3<br />
⎪⎬<br />
⎪⎭<br />
⎫<br />
⎬<br />
⎭ +<br />
⎡<br />
⎣<br />
d11 d12 d13<br />
d21 d22 d23<br />
d31 d32 d33<br />
⎡<br />
+ ⎣<br />
⎤⎧<br />
⎨<br />
⎦<br />
⎩<br />
˙y1<br />
˙y2<br />
˙y3<br />
k11 k12 k13<br />
k21 k22 k23<br />
k31 k32 k33<br />
⎫<br />
⎬<br />
⎭ +<br />
⎤⎧<br />
⎨<br />
⎦<br />
⎩<br />
y1<br />
y2<br />
y3<br />
⎫<br />
⎬<br />
⎭ =<br />
⎧<br />
⎨<br />
⎩<br />
f1<br />
f2<br />
f3<br />
⎫<br />
⎬<br />
⎭ ejωt<br />
can easily be achieved either by measuring the masses or by having the material density and<br />
mass dimensions.<br />
The equivalent damping coefficients can be approximated by<br />
d11 = 2ξ k11m11<br />
d12 = 0<br />
d13 = 0<br />
d21 = 0<br />
d22 = 2ξ k22m22<br />
d23 = 0<br />
d31 = 0<br />
d32 = 0<br />
d33 = 2ξ ⎫<br />
⎪⎬<br />
(Approximation!!!) (78)<br />
⎪⎭<br />
k33m33<br />
or by assuming, for example, proportional damping D = αM + βK. The coefficients α and β<br />
can be chosen, so that the damping factor ξ <strong>of</strong> the first resonance is <strong>of</strong> the same order as the<br />
damping factor achieved in the previous section. Please, note that this is just an approximation<br />
50<br />
(76)<br />
(77)