Lecture4-ModellingMechanicalSystemsII

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Page : 1 EE406 Control Systems Lecture 4 : Modeling Mechanical Systems II

UCSI University Faculty of Engineering Kuala Lumpur, Malaysia Department of Mechatronics

Lecture 4

Modeling Mechanical Systems II

Mohd Sulhi bin Azman Lecturer

Department of Mechatronics UCSI University

[email protected]

1 August 2011

Contents

• Review of basic concepts:

– angular displacement, velocity and acceleration – torque, moment of inertia.

• Modeling simple rotational mechanical systems.

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Review : Torque

• Torque can be defined as

the tendency to produce

a change in rotational

motion. The defining

equation is:

• The distance d, is

measured perpendicular

to the line of action of

the force F.

( )

Force distance

T t

=

×

=

Fd

Review : Angular Displacement

• The angular displacement of a rigid body describes the amount of rotation. • Usually denoted by the

symbol Θ. A more applicable way of measuring the angular displacement is the radian. It is defined by the following equation:

arc length radius rad

S R

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Review : Angular Velocity

• Angular velocity is the time rate of change in angular

displacement.

• Usually denoted by the symbol ω (omega). It is defined by the following equation:

d dt

θ

ω

=

Review : Angular Acceleration

• Angular acceleration is the rate of the angular velocity. It is usually denoted by the symbol α (alpha).

• It is defined by the following equation:

2 2

d d

dt dt

ω θ

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Mass Moment of Inertia

• It is a measure of an object's

resistance to changes to its rotation. And it is also the inertia of a rotating body with respect to its axis of rotation.

• The symbol I or J are usually used to refer to the moment of inertia or polar moment of inertia. The defining equation is:

2

Torque Moment of inertia angular acceleration

d d

J J J

dt dt

ω θ

τ α

= ×

= = =

Introduction

• A rotational system can be analyzed similar to the translational system.

• A rotation will generally occur around a fixed axis.

• Torques will occur around the axis rotation and it

corresponds to forces.

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Torsional Spring

• A torsion spring is a spring that works by torsion or twisting; that is, a flexible elastic object that stores mechanical energy when it is twisted.

• The amount of force (actually torque) it exerts is proportional to the amount it is twisted.

Modeling Rotational Spring

• Consider the rotational spring, shown below:

• The angular displacement is:

• The angular velocity is:

[

1 2

]

( )

T t

=

K

θ

t

=

K

θ

t

−

θ

t

[

1 2

]

0 0

( )

t

( )

t

( )

T t

=

K

∫

ω τ τ

d

=

K

∫

ω τ ω τ

−

d

τ

1

( )

t

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Modeling Rotational Damper

• Consider the following viscous damper:

• The angular displacement is:

• The angular velocity is:

[

1 1

]

( )

T t

=

D

ω

t

=

D

ω

t

−

ω

t

1

( )

t

θ

2

( )

t

[

1 2

]

1 2

( )

d

( )

T t

D

t

D

t

dt

θ





=

−

=



ɺ

−

ɺ



lubricant

Modeling Rotational Mass

• A rotational mass is subjected to the mass moment of inertia – related to mass.

• The torque due to the angular displacement is:

• The torque due to the angular displacement is: 2

1 2

2

( )

d

t

( )

T t

J

t

dt

θ



θ



=



ɺɺ

−

ɺɺ



[

1 2

]

( )

d

t

( )

T t

J

t

dt

ω

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Newton’s Law for Rotational System

• The law is given as:

• Where:

– J = mass moment of inertia – Θ = angular displacement

2

Torques

J

d

dt

θ

=

∑

Example 1

• Consider the following rotational mechanical

system. Find the modeling equation for this

system.

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Solution to Example 1

• Sketch a suitable FBD for J1:

a. Torques on J1 due only to the motion of J1; b. Torques on J1 due only to the motion of J2; c. Final free-body diagram for J1

• Let’s write a suitable modeling equation for J

1

.

[

]

1

1 1 1 1 1 2

( )

in out

J D K

T t

T

t

T t

T

T t

J

t

D

t

K

t

T t

J

t

D

t

K

t

K

t

θ

=

+

=

+

−

=

+

−

ɺɺ

ɺ

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• Sketch a suitable FBD for J2:

a. Torques on J2 due only to the motion of J2; b. Torques on J2 due only to the motion of J1; c. Final free-body diagram for J2

Solution to Example 1

• Let’s write a suitable modeling equation for J

1

.

[

]

2

2 2 2 2 2 1

1 2 2 2 2 2

( )

0

( )

0

( )

0

( )

in out

J D K

T t

T

t

T

J

t

D

t

K

t

J

t

D

t

K

t

K

t

K

t

J

t

D

t

K

t

θ

=

+

=

+

−

=

+

−

= −

+

ɺɺ

ɺ

ɺɺ

ɺ

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• How can we find the transfer function? The procedure is the same. We take the forward Laplace transform and obtain a matrix. Then, use Cramer’s rule to find the desired transfer function.

• Taking forward Laplace transform and assume zero initial condition:

1 1 1 1 1 2

1 2 2 2 2 2

( )

0

J

t

D

t

K

t

K

t

T t

K

t

J

t

D

t

K

t

θ



+

−

=









−

+

=







ɺɺ

ɺ

ɺɺ

ɺ

L

2

1 1 1 2

2

1 2 2 2

( )

0

J s

D s

K

s

K

s

T s

K

s

J s

D s

K

s

θ



+



−

=









−

+



+



=

Solution to Example 1

• Next, we form a matrix:

• If we want to find the transfer function Θ1(s)/T(s),

then solve for T(s) by Cramer’s rule. 2 1 1 1 2 2 2 2

( )

0

s

T s

J s

D s

K

s

K

J s

D s

K

θ



+

−











=



 







−

+













2 2 2 1 ₂ 1 1 2 2 2

( )

0

( )

T s

K

J s

D s

K

s

J s

D s

K

J s

D s

K

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• On solving, we obtain:

• Re-arranging gives:

(

)

(

)(

)

2

2 2

1 ₂ ₂ ₂

1 1 2 2

( )

s

T s

J s

D s

K

J s

D s

K

J s

D s

K

θ

=

+

−

(

)(

)

2

1 2 2

2 2 2

1 1 2 2

( )

s

J s

D s

K

T s

J s

D s

K

J s

D s

K

θ

=

+

−

Systems With Gears

• Gears provide mechanical advantage to rotational systems.

• Some applications include in systems using a motor as its actuator.

• Consider the following gear:

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Gear Relationship

• As the gears turn, the distance travelled along each gear’s circumference is the same. Thus:

• The torque is:

• Thus, the torques are directly proportional to the ratio of the number of teeth.

1 1 2 2

2 1 1

1 2 2

r

N

r

N

θ

=

1 1 2 2

2 1 2

1 2 1

T

N

T

N

θ

=

Representing Gear-less FBD

• How can we do this? Easy. Consider the following system:

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Representing Gear-less FBD

• And we get:

• The modeling equation is:

(

)

1 2 1 1 2

1 2 2 2

1 2 2 1 2 1 ( ) ( ) ( ) ( ) in out

J D K

T T N

T t F F F

N N

T t J D K

N N

T s Js Ds K s

N

θ θ θ

θ = = + + = + + = + + ɺɺ ɺ

This is in-terms of Θ2

…(i)

• Let us represent the previous figure in terms of Θ1.

• Now, how to convert from Θ1 to Θ2?

1 2 2 1 1 2 1 2 N N N N θ θ θ θ =

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• Substituting (ii) into (i), we obtain:

• Simplifying and re-arranging gives:

• Now, we can think of the load as having been reflected from the output to the input.

(

2

)

2 1

1 1

1 2

( )N N

T s Js Ds K

N = + + θ N

(

2

)

1 2

1 1

2

2 2 2

2

1 1 1

1 1

2 2 2

( )

N T s Js Ds K

N

N N N

T s J s D s K

N N N

θ θ   = + +      _ _ _ _ _ _    =   +   +            

Generalizing the Result

• Rotational mechanical impedances (mass, spring

and damper) can be reflected through gear

trains by multiplying the mechanical impedance

ratio:

where the impedance to be reflected is

attached to the source shaft and is reflected in

the destination shaft.

2

Number of teeth of gear

of destination shaft Number of teeth of gear

on the source shaft

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Example 2

• Determine the transfer function Θ

₂

(s)/T

₁

(s) for

the following system:

• Note that there is one equation of motion for this system. Why? Because the inertias do not undergo linearly independent motion. They are tied by the gears. Thus:

Simplify

2 2

2

1

destination source

N N

 

= 

 

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Solution to Example 2

• The mass (inertia), spring and damper can now

be combined into equivalent systems. That is:

2 2 1 2 1 2 2 1 2 1 2 eq eq eq N

J J J

N N

D D D

N K K   =   +     =   +   =

• Let us now write a suitable modeling equation for this system:

• Taking the forward Laplace transform assuming zero-initial condition:

• The transfer function is now equals to:

2

1 2 2 2

1

( )

in out

eq eq eq

T T N

T t J D K

N θ θ θ

=

= ɺɺ + ɺ +

(

2

)

2

1 2

1

( )N _eq _eq _eq ( )

T s J s D s K s

N = + + θ

2 2 1

2 1

( ) ( )

( ) _eq _eq _eq

s N N

G s

T s J s D s K

θ

= =

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Gear Trains

• In order to eliminate gears with large radii, a gear train is used to implement large gear ratios by cascading smaller gear ratios.

Next Step

• Textbook reference : Chapter 2.

• Homework 4 has been posted on the course

website. Attempt them. You do not have to

submit Homework 4 as it will not be graded.

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Wise Word

"When the world says: ‘Give up’

Hope whispers: ‘Try it one more time’”