Newton’s 2nd Law: Translational Motion

• Newton’s 2nd law governs the relation between acceleration and force

• Acceleration is proportional to force, and inversely proportional to mass

F=ma

where, • F = the vector sum of all forces applied to each body in

a system, newton (N)• a = the vector acceleration of each body w.r.t. an

inertial reference frame (m/sec2)• m = mass of the body (kg)

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Newton’s 2nd Law: Rotational Motion

• Newton’s 2nd law governs the relation between angular acceleration and moment (torque)

• Angular acceleration is proportional to moment, and inversely proportional to moment of inertia

M=I

where, • M = the sum of all external moments about the center

of mass of a body in a system, (N-m)• = the angular acceleration of the body w.r.t. an

inertial reference frame (rad/sec2)• I = body’s moment of inertia about its center of mass

(kg-m2)

I

M

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Mass Spring Dashpot System• Applying Newton’s 2nd law,

• Taking the Laplace transform

• Transfer function

𝑚�̈�=−𝑏 �̇�−𝑘𝑦+ 𝑓f

(𝑚𝑠2+𝑏𝑠+𝑘 )𝑌 (𝑠 )=𝐹 (𝑠)

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MATLAB SimulationMass Spring Dashpot System

• Transfer function

• m=1, k=1

• Case study– b=1 (underdamped <1)– b=2 (critically damped =1)– b=3 (over damped >1)

f

num = 1den = [1 b 1]sys = tf(num, den)step(sys)

40 5 10 15

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (seconds)

Am

plitu

de

underdamped

critically damped

overdamped

Cruise Control Model

• Example 2.1– Write the equations of motion– Find the transfer function

• Input: force u• Output: velocity

Cruise control model Free-body diagram5

Cruise Control Model

• Example 2.1– Applying Newton’s 2nd law

– Since

- Transfer function

Free-body diagram

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Cruise Control Model

• MATLAB Simulation

– Transfer function– num = 1/m– den = [1 b/m]– Sys = tf(num*u, den)– Step(sys)

Parameter values: u=500, m=1000kg, b=50Ns/m

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0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8

9

10Step Response

Time (seconds)

Am

plitu

de

A Two-Mass System: Automobile Suspension

• Two masses: Car (m2) and Tire (m1)• Problem

– Write equations of motion for the automobile and wheel motion

– Find the transfer function

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A Two-Mass System: Automobile Suspension

• Free body diagram of each body

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Rotational Motion: Pendulum

• Example– Derive equation of motion

• Nonlinear equation• Linear approximation

– Find transfer function

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Moment

• Moment

Pendulum

– Applying Newton’s 2nd law for rotational motion, M=I

– Equation of motion

– Linear equation of motion

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Is it reasonable assumption?

SIMULINK of Pendulum

Linear model

Nonlinear model

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SIMULINKMass Spring Dashpot System

• Dynamic equation of motion

• Draw the block diagram

f

14

𝑚�̈�=−𝑏 �̇�−𝑘𝑦+ 𝑓

SIMULINKMass Spring Dashpot System

• Dynamic equation of motion• m=1, k=1• Case study

– b=1 (underdamped <1)– b=2 (critically damped =1)– b=3 (over damped >1)

f

150 5 10 15

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (seconds)

Am

plitu

de

underdamped

critically damped

overdamped

𝑚�̈�=−𝑏 �̇�−𝑘𝑦+ 𝑓

y

1

m

1

k

f

1

b

1s

Integrator1

1s

Integrator

�̈� �̇�

Rotational Motion: Satellite Attitude Control Model

• Example– Derive the equation of motion– Find transfer function

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Rotational Motion: Satellite Attitude Control Model

• Example– Applying Newton’s 2nd law for

rotational motion, M=I

• Find transfer function

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Combined Motion:Rotational and Translational Motion

• Inverted pendulum mounted car– Input: force u– Output:

– Derive equations of motion

Unstable system

Combined Motion:Rotational and Translational Motion

– Position of the center of gravity of the pendulum rod

– Rotational motion of pendulum

Free body diagram

Combined Motion:Rotational and Translational Motion

– Horizontal motion of the center of pendulum

– Vertical motion of the center of gravity of pendulum

– Horizontal motion of cart

Free body diagram

Combined Motion:Rotational and Translational Motion

– For a small angle

Free body diagram

Flexible Read/Write for a Disk Drive

• Examples– Find the equations of motion– Find transfer function

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Flexible Read/Write for a Disk Drive

• Equations of motion

• Transfer function

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