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….

26