me450 2 translational unfilled

8/9/2019 ME450 2 Translational Unfilled

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Translational MechanicalSystems

System Variables

Ideal System Modeling Elements

Series and Parallel Connections of Elements

Interconnection (Physical) Laws

Derive Equation of Motion (EOM) - SDOF

Energy Transfer

Derive Equation of Motion (EOM) - MDOF


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Mechanical and Nuclear Engineering

pennState

System Variables

2

f : force [N] (T variable)

v : velocity [m/sec] (A variable)

x : displacement [m]

a : acceleration [m/sec2]

p : power [Nm/sec]

w : work ( energy ) [Nm]

1 [Nm] = 1 [J] (Joule)

ME450-Translational


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pennState

Real Spring Distributed system

Spring mass (may be considered

into the lumped model).

In large displacement operation

springs are nonlinear.

SpringEnergy Storage (Stiffness) element

Idealization

Massless

No Damping

LinearElemental equation

Energy stored

System Modeling Elements (Idealized)

3

K

(x2 -x1)

fS

ME450-Translational


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pennState

System Modeling Elements (Idealized)

DamperEnergy Dissipation (Friction) element

Elemental equation

4

MassEnergy Storage (Inertia) element

Elemental equation

Energy stored

x2x1

fDfD

x

f1

f2

f3M

http://en.wikipedia.org/wiki/Dashpot

http://en.wikipedia.org/wiki/Door_closer

Energy Dissipation Rate

ME450-Translational


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pennState

Series Connection

Spring in Series

5

x1 x2

fSfSKEQ

x1 x2

fS

ME450-Translational


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pennState

Series Connection

Damper in Series

6

x1 x2

fDfD

x2

fD

x1

fDB1 B2 BEQ

ME450-Translational


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Mechanical and Nuclear EngineeringpennState

Parallel Connection

Spring in Parallel

7

KEQ

x1 x2

fS

x1 x2

fSfSK1

K2

ME450-Translational


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Parallel Connection

Damper in Parallel

8

x2

fD

x1

fDBEQ

fDfDB1

B2

ME450-Translational


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Interconnection Laws

Newtons Third Law

- Action & Reaction Force

Ensure the continuity of the force in the system

9

K

M

x x

sfsfsf

0

ME450-Translational


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Modeling Steps

Identify reference point and positive direction.

Draw Free-Body-Diagram (FBD) for each basic element.

Write Elemental Equations as well as InterconnectingEquations by applying Newtons third laws.

Obtain Equations of Motion (EOM): Combine Equationsby eliminating intermediate variables. (Check: # eq = #unknown = #DOF)

10ME450-Translational


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Lumped Model of a Flexible Beam

11

x

K

M

K,M

x

B,K,Mxxs

f

K

M

B

g

f

x

Vertical Beam, horizontal deflection Horizontal Beam, vertical deflection

ME450-Translational


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Vertical Single Degree of Freedom(SDoF) System

12

B,K,Mx

xs

f

K

M

B

g

f

x

Draw Free-Body-Diagram (FBD)

M

x

sx

dx

Write Elemental Equations

ME450-Translational


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Suspension System (without tiremodel)

Suspension System Minimize the effect of the surface

roughness of the road on thedrivers comfort.

14

http://www.phelpsandwilliams.co.uk/suspension

K

M

B

x

xpx

ME450-Translational


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Horizontal Two Degree of Freedom(TDoF) System

16

DOF = 2

2f

K

1K

1B

1M

K2

M

2B

2K

1x 2x

1f

Absolute coordinates

FBD

1x 2x

2f

K

1 1K x

1 1B x

1M

K2

M

2 2 1

B x x-

2 2 1

K x x-

1f

Newtons law

1 1 1 1 1 1 2 2 1 2 2 1 1

2 2 2 2 1 2 2 1 2

M x B x K x B x x K x x f t

M x B x x K x x f t

- - - -

- - - -

ME450-Translational


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Horizontal Two Degree of Freedom(TDoF) System

17

2f

K

1K

1B

1M

K2

M

2B

2K

1f


1 1 1 2 1 1 2 1 2 2 2 2 1M x B B x K K x B x K x f t - - 2 2 2 2 2 2 2 1 2 1 2M x B x K x B x K x f t - -

Relative coordinates

1 1 1 1 1 1 2 21 2 21 1M x B x K x B x K x f t - - 2 1 2 21 2 21 2 21 2M x M x B x K x f t

ME450-Translational


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Two DoF System, Matrix Form of EoM

18


1 1 1 2 2 1 1 2 2 1 1

2 2 2 2 2 2 2 2 2

0

0

M x B B B x K K K x f t

M x B B x K K x f t

- - - -

Relative coordinates

1 1 1 2 1 1 2 1 1

2 2 21 2 21 2 21 2

0

0 0

M x B B x K K x f t

M M x B x K x f t

- -

2f

K

1K

1B

1M K

2M

2B

2K

1f

ME450-Translational


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Suspension System (with tire model)Vertical Two DoF System (cont.)

20

M2

M1

ME450-Translational


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Suspension System (with tire model)Vertical Two DoF System (cont.)


Matrix Form


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Energy Distribution of a System

22

EOM of a simple Mass-Spring-Damper System

We want to look at the energy distribution of the system. How should we start ?

TotalContribution Contribution ContributionApplied Forceof Inertia of the Damper of the Spring

( )M x Bx K x f t

x

K

M

Bf

ME450-Translational

me450 2 translational unfilled

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