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Constitutive Relationships for Basic
Mechanical Elements
Mass/Mass/InertiaInertiaSpringSpring DamperDamper
RotationalRotational
TranslationalTranslational
ElementElement
M
x
m
v
m
F
F
x
T
F
F
x
x
k
F
J
J
T
J
x
x
x
x
b
F
x
x
F
F
T
T )
K
T
T
T
B
T
Modeling Procedure1.
Draw a Free-Body-Diagram of each elementshowing all the forces/moments acting on it.
Write down the constitutive relationship between theexerted or transmitted force (or torque) and thecorresponding motion variables for each elementincluding power transforming elements.
Eliminate auxiliary variables to obtain a system ofdifferential equations for the system with the samenumber of equations and unknowns.
If desired, reduce the system of diff. eqs. in (3) to asingle differential eq. relating the output to the inputof the system.
Other Examples^
Mass-Mass
-Spring
Spring-
-Damper System
Damper System
^
2-^2
-Degree of Freedom Systems
Degree of Freedom Systems
^
TrnslationalTrnslational
^
RotationalRotational
^
Gear Train ExampleGear Train Example
School of Aerospace Engineering Georgia Institute of Technology
AE 3515 Modeling of Mechanical Systems
Mathematical Modeling of Mechanical Systems
Objective: Use applicable physical laws to derive a set of differential equations describing the behavior of a mechanical system.
Basic Elements: Mass or Inertia, Spring, Damper
Modeling Procedure:
- Draw the free-body-diagram of each element.
- Write down the constitutive relationship between the exerted or transmitted force (or torque) and the corresponding motion variables for each element including power transforming elements.
- Eliminate auxiliary variables to obtain a system of differential equations for the system with the same number of equations and unknowns.
- If desired, reduce the system of diff. eqs. in (3) to a single differential eq. relating the output to the input of the system.
