Post on 26-Jun-2015
ME 330 Control Systems
SP 2011
Lecture 5
ScanYZTest(dsc,bl=0.1,rr=1.0,yscan=10,zscan=0):ScanYZTest(dsc,bl=0.1,rr=1.0,yscan=10,zscan=0):
Mechanical Systems
)()()()( tkxtxctxmtf f(t)
)()( 2 sXkcsmssF
From the Laplace transform
kcsms 21
F(s) X(s)Often seen in block-diagram representation
Newton’s 2nd Law governs mechanical systems, resulting equation of motion describes dynamical system.
Mechanical Systems Modeling Determining the equation of motion
)()()()( tkxtxctxmtf
f(t)
)()()()( 2 skXscsXsXmssF
Free-Body Diagram
)(tkx
m)(txc
)(txm
)(tf
)(skX
m)(scsX
)(2 sXms
)(sF
Summation of forces
Impedance
Summation of impedances
Mechanical Components
Table 2.4Force-velocity, force-displacement, and impedancefor springs, viscous dampers, and mass
Mechanical Components Table 2.5
Torque-angular velocity, torque-angular displacement, and rotational impedance for springs, viscous dampers, and inertia
Note: rotational mechanics are analogous to translational mechanics force => torque damper => damper mass => inertia
Multiple Degrees of Freedom Number of equations of motion required =
number of linear independent motions Linear independence: point of motion is allowed to
move even if all other points of motion are fixed. Also known as degrees of freedom.
Multiple Degrees of Freedom Analyze impedances for each degree of freedom.
1c
)(tf
2c
3c
)(11 sXk
m1)(11 ssXc
)(sF
)(12
1 sXsm
)(12 sXk
)(13 ssXc
)(22 sXk)(23 ssXc
)()()()(0
)()()()()(
232322
2123
223121312
1
sXkksccsmsXksc
sXkscsXkksccsmsF
)(22 sXk
m2
)(22 ssXc
)(22
2 sXsm
)(12 sXk)(13 ssXc
)(23 ssXc)(23 sXk
Transfer Function Laplace transform of equations of motion
Can solve for any transfer function
)()()(
)()()(
)()(
)(
)(
32322
223
2331312
1
32322
21
kksccsmksc
ksckksccsm
kksccsm
sF
sX
)()()(
)()()()(
)(
32322
223
2331312
1
232
kksccsmksc
ksckksccsm
ksc
sF
sX
Equations by Inspection For two degree of freedom system, the general
form is given bySum of
impedances connected to
the motion at x1
Sum of impedances
between x1 and x2
X1(s) X2(s)– = Sum of
applied forces at x1
Sum of impedances connected to
the motion at x2
Sum of impedances
between x1 and x2
X2(s)X1(s)– = Sum of
applied forces at x2
+
)()()()()()()()()(
)()()()()()()()()(
)()()()()()()()()(
332211
2233222121
1133222111
sFsXssXssXssXs
sFsXssXssXssXs
sFsXssXssXssXs
NNNNNN
NN
NN
Higher degree of freedom systems
Torsion Mechanical Systems Same rules apply as in translational mechanical systems
Sum of impedances connected to
the motion at 1
Sum of impedances
between 1 and 2
1(s) 2(s)– = Sum of
applied forces at 1
Sum of impedances
between 1 and 3
3(s)–
Sum of impedances connected to
the motion at 2
Sum of impedances
between 1 and 2
2(s)1(s)– = Sum of
applied forces at 2
Sum of impedances
between 2 and 3
3(s)– +
Sum of impedances connected to
the motion at 3
Sum of impedances
between 1 and 3
3(s)1(s)– = Sum of
applied forces at 3
Sum of impedances
between 2 and 3
2(s)– +
Torsion Mechanical Systems Summation of impedances
)()()(00
)()()()(0
0)()()()(
3232
322
33222
21
2112
1
ssDsDsJssD
ssDsKsDsJsK
sKsKsDsJsT
Next Lectures
Derivation of more mechanical system models Derivation of electrical system models