translational mechanical systems - purdue university notes... · school of mechanical engineering...

School of Mechanical EngineeringPurdue University

ME375 Translation - 1

Translational Mechanical Systems

• Basic (Idealized) Modeling Elements• Interconnection Relationships -Physical Laws• Derive Equation of Motion (EOM) - SDOF• Energy Transfer• Series and Parallel Connections• Derive Equation of Motion (EOM) - MDOF



Variables

• x : displacement [m]• v : velocity [m/sec]• a : acceleration [m/sec2]• f : force [N]• p : power [Nm/sec]• w : work ( energy ) [Nm]

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

0

1

0

2

2

1 0

0

( ) ( ) ( )

( ) ( )

t

t

t

t

d x x vdtd d d dv v x x x adt dt dt dt

dp f v f x wdt

w t w t p t dt

w t f x dt



Idealized Modeling Elements

• Inertia (mass)• Stiffness (spring)• Dissipation (damper)



• Spring– Stiffness Element

– Idealization• Massless• No Damping• Linear

– Stores Energy

Basic (Idealized) Modeling Elements

– Reality• 1/3 of the spring mass may be

considered into the lumped model.

• In large displacement operation springs are nonlinear.

x2

K

x1

fS fS

2 1Sf K x x

(x2 x1)

fS



Practical Nonlinear Spring

Engine Mount:#T062 VERTICAL

-4000

-2000

0

2000

4000

-20 -15 -10 -5 0 5

DISP (mm)

LOAD

(N)

xxK 2force Restoring

Experimental Analytical

1

2

Kisolationfor

motions Small12

loads staticfor motions Large

xK 2



• Damper– Friction Element

– Dissipate Energy

Basic (Idealized) Modeling Elements• Mass

– Inertia Element

– Stores Kinetic Energy

2 1 2 1Df B x x B v v

x2x1

fDfD

fD

2 1x x

x

f1

f2

f3

1 2 3ii

M x f f f f

M



• Newton’s Second Law

• Newton’s Third Law– Action & Reaction Forces

• Displacement Law

Interconnection Laws

Linear

Momentum

EXTii

d M v M x fdt

x

KM

MK



Modeling Steps

• Understand System Function, Define Problem, and Identify Input/Output Variables

• Draw Simplified Schematics Using Basic Elements• Develop Mathematical Model (Diff. Eq.)

– Identify reference point and positive direction.– Draw Free-Body-Diagram (FBD) for each basic element.– Write Elemental Equations as well as Interconnecting

Equations by applying physical laws. (Check: # eq = # unk)– Combine Equations by eliminating intermediate variables.

• Validate Model by Comparing Simulation Results with Physical Measurements



In Class Exercise (Blood Sampler)

Schematic:



• EOM of a simple Mass-Spring-Damper System

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

• Multiply the above equation by the velocity term v : What have we done ?

• Integrate the second equation w.r.t. time: What are we doing now ?

Energy Distribution

TotalContribution Contribution Contribution

Applied Forceof Inertia of the Damper of the Spring

( )M x Bx K x f t

1 1 1 1

0 0 0 0

0 1

212 20 Total work done by the

applied force ( ) from time to

1 102 2

tt

t t t t

t t t t

f tt t

Bx dtKE M x PE K x E

M x x dt Bx x dt K x x dt f t v dt

xK

MB

f



• Suspension SystemMinimize the effect of the surface roughness of the road on the drivers’ comfort.

Example -- SDOF Suspension– Simplified Schematic (neglecting tire model)

Define the reference position for the displacement of the car as the position when the spring does not have any deflection (i.e., the neutral position)



SDOF Suspension

– Draw FBD – Apply Interconnection Laws

Q: Since gravity is always present, is there a way to represent the suspension system by subtracting the effect of gravity?



SDOF Suspension (II)• Relative Displacement Approach

Define the reference position as the position of the car when the system is at rest in the gravity field, i.e., the spring force balances the car’s weight.

– FBD



SDOF Suspension (II)– Interconnection Laws &

SimplificationQ: What are the differences between the two

models?

Q: Do the two models represent the same physical system? If they do, why are they different?



• Springs in Series

Series Connection

K1 K2

x1 x2

fSfSKEQ

x1 x2

fSfS



• Dampers in Series

Series Connection

x1 x2

fDfD

x2

fD

x1

fDB1 B2 BEQ



• Springs in Parallel

Parallel Connection

fS

KEQ

x1 x2

fS

x1 x2

fSfSK1

K2



• Dampers in Parallel

Parallel Connection

x2

fD

x1

fD

BEQ

x1 x2

fDfD B1

B2



• Suspension System

MDOF Suspension– Simplified Schematic (with tire model)



MDOF Suspension

– Draw FBD – Apply Interconnection Laws



MDOF Suspension

– Matrix Form

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