Middle East Technical University Department of Aerospace Engineering
AEE 582 Supplementary Notes Fall 2009 Dr. Ali Tรผrker Kutay
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Spring-Mass-Damper System Example
Consider the following spring-mass system:
Motion of the mass under the applied control, spring, and damping forces is governed by the following
second order linear ordinary differential equation (ODE):
๐๐ฆ +๐ต๐ฆ +๐พ๐ฆ =๐ข
(1)
Taking the Laplace transform of (2) yields the following transfer function from the input ๐ข to the output ๐ฆ:
๐บ ๐ =๐ฆ ๐
๐ข ๐ =1
๐๐ 2+๐ต๐ +๐พ
(2)
Setting the denominator of the transfer function to zero yields the characteristic equation of the above
system:
๐๐ 2+๐ต๐ +๐พ= 0
(3)
Roots of the characteristic equation are called the poles of the system and give important information
about the dynamic characteristics of the system. Poles of a system can be
real (imaginary part equal to zero): ๐1=๐1
๐2=๐2
or complex (nonzero imaginary part): ๐1=๐+๐๐
๐2=๐ โ ๐๐
Response to Initial Conditions
Response of a second order system to nonzero initial conditions with no control input can be obtained by
solving (2) with ๐ข= 0.
For real poles the solution is
๐ฆ ๐ก =๐1๐๐1๐ก+๐2๐๐2๐ก with ๐1=1
๐2โ๐1 ๐2๐ฆ 0 โ ๐ฆโฒ 0
๐2=1
๐2โ๐1 โ๐1๐ฆ 0 +๐ฆโฒ 0
where ๐ฆ 0 and ๐ฆโฒ 0 are the initial position and velocity.
For complex poles the solution is
๐ฆ ๐ก =๐๐๐ก ๐1cos ๐๐ก +๐2sin๐๐ก with ๐1=๐ฆ 0
๐2=โ๐
๐๐ฆ 0 +1
๐๐ฆโฒ 0
For the mass-spring system poles are obtained from (3) as
๐
๐ข
๐ฆ
๐
๐ข
๐ฆ
๐พ๐ฆ
๐ต๐ฆ
๐: mass in kg
๐ฆ: position in m
๐ข: control input in N
๐พ: linear spring constant in N/m
๐ต: damping coefficient in Ns/m
