W4 System Modeling1
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Transcript of W4 System Modeling1
BMFA 3313CONTROL SYSTEMS
TOPIC : Modeling in the Frequency Domain
•Introduction•Translational Mechanical System•Rotational Mechanical System
•Systems with Gears•Electromechanical System
1
Learning OutcomesAt the end of this topic, students should be able to:
• Find a mathematical model, called a transfer function for linear, time invariant (LTI) mechanical translational & rotational systems and electromechanical systems.
2
Mathematical Modeling for Translational Mechanical System
• A mathematical modeling for an element or system is an equation or set of equation that define the relationship between input and output of the system
• Mechanical systems have 3 passive, linear components. Two of them, the spring and the mass, are energy-storage elements; and one of them, the viscous damper, dissipates energy.
• These mechanical elements are shown in next table.
• In the table, K, fv and M are called spring constant, coefficient of viscous friction and mass, respectively.
Translational Mechanical System
Force-velocity, force-displacement, and impedance translational relationshipsfor springs, viscous dampers, and mass
Translational Mechanical System
Steps:• The mechanical system requires one differential equation, called the
equation of motion.
– Assume +ve direction of motion ~ to the right- Assume –ve direction of motion to the left
• Draw a free body diagram, placing on the body all the forces that act on the body
• Use Newton’s Law to form differential equation (setting the sum equal to zero)
• Assume zero initial condition• Take Laplace Transform• Find Transfer Function
Example (One equation of motion)
• Find the transfer function, X(s)/F(s), for the system of Figure below:
• Step 1 ~ Draw the free-body diagram (using differential equation)– Place on the mass all forces felt by the mass
– Mass is traveling toward the right (force points to the right)
– All other forces impede the motion and act oppose it.
• Step 2 ~ Write the differential equation using Newton’s Law (sum to zero all of the forces)
)()()()(
2
2
tftKxdt
tdxf
dt
txdM v
• Step 3 ~ Taking the Laplace Transform assuming zero initial condition
• Step 4 ~ Solving the transfer function & block diagram
KsfMssF
sX
v
)(
1
)(
)(2
)()()(
or
)()()()(
2
2
sFsXKsfMs
sFsKXssXfsXMs
v
v
Take the Laplace transform of force displacement column in previous Table,
we obtain:
• for the spring, F(s) = KX(s)
• for the viscous damper, F(s) = fvsX(s)
• for the mass, F(s) = Ms2X(s)
This approach is more simple rather than to write the differentiate equation
Mechanical Translation Components
Example (2 degrees of freedom)
Find the transfer function, X2(s)/F(s), for the system of figure as shown below.
*The system is 2 degree of freedom since each mass can be moved in horizontal direction while the other is held still ~ 2 equations
• Step 1 ~ (If we hold M2 and move M1 to the right)
a) Draw forces on M1 due to only motion of M1
b) Draw forces on M1 due only motion of M2
c) Draw all the forces on M1
Example (2 degrees of freedom)
• Step 2 ~ (If we hold M1 and move M2 to the right)
a) Draw forces on M2 due to only motion of M2
b) Draw forces on M2 due only motion of M1
c) Draw all the forces
Example (2 degrees of freedom)
• Step 3 ~ Laplace transform of the equation of motion from
Step 1c and Step 2c
• Step 4 ~ The transfer function X2(s)/F(s) is
0)()()(:2
)()()(][:1
232322
2123
223121312
1
sXKKsffsMsXKsfcStep
sFsXKsfsXKKsffsMcStep
vvv
vvv
)(2 sX
)(
)()(
)( 232 KsfsG
sF
sX v
3232
2223
2321312
1
KKsffsMKsf
KsfKKsffsM
vvv
vvv
Example (2 degrees of freedom)
Solve by Cramer’s Rule
Transfer Function of Translational Mechanical Systems
X1(s) X2(s) X3(s) F(s)
+ (Sum of Impedance related to X1)
- (Sum of
Impedance btw X1
and X2)
Applied force
- (Sum of
Impedance btw X1
and X2)
+ (Sum of Impedance related to X2)
- (Sum of
Impedance btw X2
and X3)
Applied force
- (Sum of
Impedance btw X2
and X3)
+ (Sum of Impedance related to X3)
Applied force
To produce the Equation of Motions
Eqn 1
Eqn 2
Eqn 3
Exercise
Write but do not solve, the Laplace transform of the equations of motion for
the system shown in Figure below.
Mathematical Modeling for Rotational Mechanical System
• Rotational mechanical systems are handled the same way as translational mechanical systems, except that torque replaces force and angular replaces translational displacement.
• Table next shows the components along with the relationships between torque and angular velocity, as well as angular displacement.
• Notice that the symbols for the components look the same as translational symbols, but they are undergoing rotation and no translation.
• The values of K, D and J are called spring constant, coefficient of viscous friction and moment of inertia, respectively.
Rotational Mechanical SystemsRotational Mechanical System
Torque-angular velocity, torque-angular displacement, and impedancerotational relationships for springs, viscous dampers, and inertia
Example (Two equation of motion) • Find the transfer function, for the rotational system shown in the figure
below. The rod is supported by bearings at either end and is undergoing torsion. A torque is applied at the left, and the displacement is measured at the right.
a. Physical system;b. Schematic;c. Block diagram
*2 degree of freedom since each inertia can be rotated while the other is held still ~ two equations
Solve by Cramer’s Rule
Transfer Function of Rotational Mechanical Systems
Θ1(s) Θ2(s) Θ3(s) T(s)
+ (Sum of Impedance related to Θ1)
- (Sum of
Impedance btw Θ1
and Θ2)
Applied torque
- (Sum of
Impedance btw Θ1
and Θ2)
+ (Sum of Impedance related to Θ2)
- (Sum of
Impedance btw Θ2
and Θ3)
Applied torque
- (Sum of
Impedance btw Θ2
and Θ3)
+ (Sum of Impedance related to Θ3)
Applied torque
To produce the Equation of Motions
Eqn 1
Eqn 2
Eqn 3
Transfer Functions for Systems with Gears
• Rotational Systems associated gear train driving the load
• Gears provide mechanical advantage to rotational systems
2
1
2
1
1
2
2211
N
N
r
r
rr
Transfer Functions for Systems with Gears
1
2
2
1
1
2
2211
N
N
T
T
TT
Transfer functions for (a) angular displacement in lossless gears and (b) torque in lossless gears
Transfer Functions for Systems with Gears - Example
Transfer Functions for Systems with Gears - Example
Rotational mechanical impedance can be reflected through gear trains by multiplying the mechanical impedance by the ratio;
where the impedance to be reflected is attached to the source shaft and is being reflected to the destination shaft.
2
shaftongearofteethofNumber
shaftongearofteethofNumber
source
ndestinatio
Transfer Functions for Systems with Gears - Example
Transfer Functions for Systems with Gears
Study Example 2.22 on page 74
To be continued….
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