Dynamic System Modeling Presentation Part 1 - UBC Mechanical ...
Dynamic System Modeling Presentation Part 1 - UBC Mechanical ...
Dynamic System Modeling Presentation Part 1 - UBC Mechanical ...
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Lumped Model of a Distributed <strong>System</strong>: A Heavy Spring<br />
m s = mass of spring; k = stiffness of spring; l = length of spring<br />
One end fixed and the other end moving at velocity v<br />
Kinetic Energy Equivalence<br />
m<br />
Local speed of element δx = x s<br />
v. Element mass = δ x<br />
l<br />
<br />
Element kinetic energy KE =<br />
l<br />
l<br />
1 m x s x(<br />
v)<br />
2 l<br />
δ l<br />
2<br />
1 ms<br />
x 1 msv<br />
2 1 ms<br />
As δx → dx, Total KE = ∫ dx(<br />
v)<br />
= x dx<br />
3<br />
2 l l 2 l<br />
∫ =<br />
2 3<br />
0<br />
2<br />
l<br />
2<br />
2 v<br />
Equivalent lumped mass concentrated at free end = 1 × spring mass<br />
3<br />
Assumption: Conditions are uniform along the spring.<br />
0
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