Engineering g Problem Solving Process

Engineering Problem Solving Process g g g

• GIVEN – State briefly and concisely (in your own words) the information given.

• FIND – State the information that you have to find.

• DIAGRAM – A drawing showing the physical situation with all quantities involved should be included.

• BASIC LAWS – Give appropriate mathematical formulation of the basic laws that

id l h bl

you consider necessary to solve the problem.

• ASSUMPTIONS – List the simplifying assumptions that you feel are appropriate in the problem.

ANALYSIS Carry through the analysis to the point where it is appropriate to

• ANALYSIS – Carry through the analysis to the point where it is appropriate to substitute numerical values.

• NUMBERS – Substitute numerical values (using a consistent set of units) to obtain a numerical answer. The significant figures in the answer should be obtain a numerical answer. The significant figures in the answer should be consistent with the given data.

• CHECK – Check the answer and the assumptions made in the solution to make sure they are reasonable. Estimate the answer. Check the units, if appropriate.

• IMPORTANT:

– This solution procedure is required.

– Include name and section number on each page.

– Indicate problem title / number at the top of each page.p p p g – Staple pages together.

– Always use engineering computation paper.

– Always start a problem solution on a new page – Always start a problem solution on a new page.

– Always use pencil.

– Always use a straight edge and a compass.

– Never write on the back of a page.

– Handwriting and diagrams must be legible and work should not be crowded.

– Keep all assigned work in a binder for reference when studying for exams.

– Professional Presentation! Take Pride in Your Work!

The Three Main Challenges

for an

Automotive Engineer Automotive Engineer

are:

Economy, Emissions,

and Performance

and Performance

Electromagnet

( )

f x,i C x

⎛ ⎞

= ⎜ ⎟

⎝ ⎠ Infrared LED

Phototransistor

⎝ x ⎠

Levitated Ball

Magnetic Levitation System

Electromagnetic Valve Actuator

F C l A t ti E i

For a Camless Automotive Engine

Engineering Problem

A steel ball of mass m is

connected to ground by a linear spring with a spring constant K and an unstretched position given and an unstretched position given by x₀. An electromagnet exerts a force f_e on the ball. This attractive force is proportional to the square

f

( )

e 2

f x,i C i x

⎛ ⎞

= ⎜ ⎟

⎝ ⎠

of the current in the electromagnet and inversely proportional to the square of the distance between the ball and the electromagnet.

⎝ x ⎠

the ball and the electromagnet.

For a constant current in the

electromagnet, the ball will come

t ilib i iti Fi d th

to an equilibrium position. Find the equilibrium position or positions, and assess the stability of each one.

Problem Solution Problem Solution

• Given

– Ball has mass m.

– Ball is connected to ground by a linear spring.

Spring has constant K and an unstretched position given by x – Spring has constant K and an unstretched position given by x₀. – An electromagnet exerts a force f_e on the ball. This force is

proportional to the square of the current in the electromagnet and inversely proportional to the square of the distance between and inversely proportional to the square of the distance between the ball and the electromagnet.

• Find:

– For the ball in static equilibrium, find the position(s) of the ball, i.e., the distance(s) x from the electromagnet, and comment on the stability of the system(s) in static equilibrium.

Diagram:

( )

e 2

f x,i C ⎛ i ⎞

= ⎜ ⎟

e

( ) ,

⎜ x ⎟

⎝ ⎠

• Basic Laws:

N t ’ 2

L f M ti ∑ F ^G = ma ^G – Newton’s 2

Law of Motion

• Assumptions:

x x

F = ma

∑ ∑

• Assumptions:

– Ball is magnetic.

– Gravity constant g = 9.81 m/s² and is directed downwards.

– Spring is linear with a spring constant K (N/m).

– The electromagnetic force is proportional to the square of the current in the electromagnet and inversely proportional to the square of the distance between the ball and the electromagnet.

– Electromagnetic force does not change with time or temperature.

– Current in the electromagnet is constant ig ₀₀.

– Electromagnetic force proportionality constant C is known.

– There are no forces other than the forces in the x direction.

Analysis:

F

= 0 static equilibrium

∑

_{( )} ⎛ i

⎞

e 0

f mg K(x x) 0 f K(x x) mg

− + + − =

= − +

∑ _{( )}

e 2

f x,i C i x

⎛ ⎞

= ⎜ ⎟

⎝ ⎠ f = mg

( )

e 0

2

e 2

f K(x x) mg f x,i C i

= +

⎛ ⎞

= ⎜ ⎟

⎝ ⎠ ^{f = K x} ( ^x )

f

= mg

e

( )

⎜ x ⎟

⎝ ⎠ ^f

^{= K x} (

^{− x} )

Free Body Diagram

Numbers:

Ci

= 1 (assume for simplicity) m 3.06

K 225

=

=

0 2

x 0.267 i

=

⎛ ⎞ SI Units

2 0

C i K(x x) mg

x 1

⎛ ⎞

= − +

⎜ ⎟

⎝ ⎠

2

1 225x 90

x = − +

Unstable Equilibrium

Stable Equilibrium

f

K(x

− x) + mg

Stability

q

f

= K(x

x) + mg f = K(x − x) + mg

MARGINALLY