Applied MechanicsApplied MechanicsRigid Body MechanicsRigid Body Mechanics

StaticsStaticsDynamicsDynamics

KinematicsKinematicsKineticsKinetics

Deformable Body MechamicsDeformable Body MechamicsElasticityElasticityPlasticityPlasticityViscoelasticityViscoelasticity

Fluid MechanmicsFluid MechanmicsLiquidsLiquidsGasesGases

Statics Vs. Deformable Body Statics Vs. Deformable Body MechanicsMechanics

Statically Determinant Statically Indeterminant

How is the problem different if the beam is rock? Wood? Bone? Tooth Enamel?

ElasticityElasticity

The beam can bend to conform to the constraints of the supports.

(You have done these types of problems in ENGR 220).

Elasticity vs. PlasticityElasticity vs. Plasticity

Elastic: You can bend it, but it will come right back when you remove the external force (like a spring).

Plastic: You can bend it, and it will stay where it left off (like soldering wire).

Some materials tend to behave elastically, some plastically, but….

Almost all materials will have both behaviors.

Small deflection – elastic.

Large deflection – plastic.

Stress-Strain CurveStress-Strain Curve(or Force-Deflection)(or Force-Deflection)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5 3 3.5

Strain

Stress

Elasticity vs. ViscoelasticityElasticity vs. Viscoelasticity

Elastic: Force is proportional to deflection (e.g. a spring).

Viscoelastic: Force is proportional deflection AND to rate of deflection (e.g. silly putty).

Viscous: Force is proportional to deflection only (e.g. a shock absorber or a fluid).

Mechanical AnalogyMechanical Analogy

F = kx spring (Think of k as Young’s Modulus)

(Circuits: q = CV or V = q/C)

F = C dx/dt shock absorber (dashpot)

(Circuits: V=iR, or V = R dq/dt)

What about F = m d2x/dt2 ?

Same as F = ma – Newton’s law of motion.

(Circuits: V = L di/dt = L d2q/d2t)

Mass is inductance

If we know something about how If we know something about how circuits behave, we know circuits behave, we know

something about how mechanical something about how mechanical systems behave.systems behave.

One major difference: Circuits tend to be discrete. Mechanical systems tend to be distributed. We must deal with spatial relationships.

Styrofoam CupStyrofoam Cup

Is a styrofoam cup elastic, plastic or Is a styrofoam cup elastic, plastic or viscoelastic?viscoelastic?

Place your answer here:Place your answer here:

YES

Important Stuff from Chapter 1Important Stuff from Chapter 1

Units must matchUnits must match

Hierarchy of Mechanical MeasuresHierarchy of Mechanical Measures

)(

)(

2

22

sec

cm/sec

)cm(

cm

dt

xd

dt

dva

dt

dxv

x

s

m

gm

m/N

N

Nseccm 2

dt

dEP

xdFdFE

dt

vdmamF

)(

Important Stuff from Chapter 1Important Stuff from Chapter 1

Energy and Torque (Same units, but …Energy and Torque (Same units, but …

mN

xdFdFE

FrT

Torque is a vector, Energy is a scalar.

Torque is instantaneous, Energy happens over time.

Important Stuff from Chapter 1Important Stuff from Chapter 1

Know your greek letters:Know your greek letters:

muis

nu is

etais

gamma isepsilonis

upsilon isxiis

zetais

Modeling and ApproximationModeling and Approximation

““In general, it is always best to begin with a In general, it is always best to begin with a simple basic model that represents the simple basic model that represents the system. Gradually, the model can be system. Gradually, the model can be expanded on the basis of experience …”expanded on the basis of experience …”

But …But …

Do not throw out the baby with the bath Do not throw out the baby with the bath water.water.

Generalized ProcedureGeneralized Procedure

1.1. Select the systemSelect the system2.2. Postulate characteristicsPostulate characteristics3.3. Simplify with approximationsSimplify with approximations4.4. Relate body parts to mechanical elementsRelate body parts to mechanical elements5.5. Construct the mechanical modelConstruct the mechanical model6.6. Apply principles of mechanicsApply principles of mechanics7.7. Solve for unknownsSolve for unknowns8.8. Compare to experimentsCompare to experiments9.9. Repeat from step 3Repeat from step 3

From Chapter 2From Chapter 2

You need to know how to You need to know how to manipulate vectors.manipulate vectors.

Vectors are the only way to Vectors are the only way to represent systems in represent systems in space.space.

Review: Dot product, cross Review: Dot product, cross product, how to calculate product, how to calculate them, and what they them, and what they meanmean..

For distributed systems: For distributed systems: 2,,,, vfv

Vector ArithmeticVector Arithmetic

Add Head to tailAdd Head to tail

Subtract – reverse direction and then addSubtract – reverse direction and then add

A + B

A - B

A

B