Lumped dynamics

depends on what the Qatar is pulling which we don’t know at this time. We will set it to zero i.e. = 0 .

Problem 1: (Definition) Review the notes and chapter, and complete the missing definition below:

Problem 2 : (Lumped dynamics) Show that an equivalent lumped dynamics model for the quarter car with wheel can be expressed as

where is a unitless factor to account for equivalent mass related to the translational mass and all rotating inertias. See Appendix A for further explanation.

Problem 3: (Equivalent mass) The quantity is also called the equivalent mass of the quarter car. Use the given parameter values to find an approximate value of factor and equivalent mass .

Problem 4: (Initial acceleration) Suppose the quarter car is initially at rest ; there is no wind velocity, i.e., ; the car is on a horizontal flat elevation, i.e., . If a constant tractive torque is applied at the wheel axle, what is the initial acceleration ?

Problem 5: (Required torque/thrust) Under the same initial condition of Problem 4, what necessary tractive torque would be required to produce an acceleration of half gravitational acceleration, i.e.,?

Problem 6: (Final velocity). A constant tractive torque is applied to the wheel, and there is no wind velocity, i.e., . After a while, the car reaches a steady state velocity. What is the final velocity as ? Hint: .

 Problem Max score 1. Definition 15% 2. Lumped dynamics 20% 3. Equivalent mass 15% 4. Initial acceleration 15% 5. Required torque 15% 6. Final velocity 20%

Appendix A: Rotational Inertia Coefficient

Example 1. A box sliding up or down a slope

Example 2: A wheel rolling up or down a slope

You are to perform the same analysis in Prob 2 of HW1, for the quarter car. Appendix B: Notes from Wikipedia on derivatives of position, for your reading pleasure.

Rate of change in Position/Location is Velocity – first order difference

Rate of change in Velocity is Acceleration – second order difference

Rate of change in Acceleration is Jerk/Jolt – third order difference

Rate of change in Jerk/Jolt is Snap – fourth order difference

Rate of change in Snap is Crackle – fifth order difference

Rate of change in Crackle is Pop – sixth order difference

Momentum equals mass times velocity! Force equals mass times acceleration! Yank equals mass times jerk! Tug equals mass times snap! Snatch equals mass times crackle! Shake equals mass times pop!!

It is well known that the first derivative of position (symbol  x ) with respect to time is velocity (symbol  v ) and the second is acceleration (symbol  a ).  It is a little less well known that the third derivative, i.e. the rate of change of acceleration, is technically known as jerk (symbol  j ).  Jerk is a vector but may also be used loosely as a scalar quantity because there is not a separate term for the magnitude of jerk analogous to speed for magnitude of velocity. In the UK jolt has sometimes been used instead of jerk and may be equally acceptable.

Many other terms have appeared in individual cases for the third derivative, including pulse, impulse, bounce, surge, shock and super acceleration.  These are generally less appropriate than jerk and jolt, either because they are used in engineering to mean other things or because the common English use of the word does not fit the meaning so well.  For example impulse is more commonly used in physics to mean a change of momentum imparted by a force of limited duration [Belanger 1847] and surge is used by electricians to mean something like rate of change of current or voltage.  The terms jerk and jolt are therefore preferred for rate of change of acceleration.  Jerk appears to be the more common of the two.  It is also recognised in international standards:

In ISO 2041 (1990), Vibration and shock – Vocabulary, page 2: “1.5 jerk: A vector that specifies the time-derivative of acceleration.” Note that the symbol  j  for jerk is not in the standard and is probably only one of many symbols used.

As its name suggests, jerk is important when evaluating the destructive effect of motion on a mechanism or the discomfort caused to passengers in a vehicle.  The movement of delicate instruments needs to be kept within specified limits of jerk as well as acceleration to avoid damage.  When designing a train the engineers will typically be required to keep the jerk less than 2 metres per second cubed for passenger comfort.  In the aerospace industry they even have such a thing as a jerkmeter; an instrument for measuring jerk.

In the case of the Hubble space telescope, the engineers are said to have even gone as far as specifying limits on the magnitude of the fourth derivative.  There is no universally accepted name for the fourth derivative, i.e. the rate of change of jerk, The term jounce has been used but it has the drawback of using the same initial letter as jerk so it is not clear which symbol to use.  Another less serious suggestion is snap (symbol  s ), crackle (symbol  c ) and pop (symbol  p ) for the 4th, 5th and 6th derivatives respectively.  Higher derivatives do not yet have names because they do not come up very often.

Since force ( F  = ma ) is rate of change of momentum ( p , symbol clashes with pop) it seems necessary to find terms for higher derivatives of force too.  So far yank (symbol  Y ) has been suggested for rate of change of force, tug (symbol  T ) for rate of change of yank, snatch (symbol  S ) for rate of change of tug and shake (symbol  Sh ) for rate of change of snatch.  Needless to say, none of these are in any kind of standards, yet.  We just made them up on usenet.

In physicsjounce is the fourth derivative of the position vector with respect to time, with the first, second, and third derivatives being velocityacceleration, and jerk, respectively; in other words, the jounce is the rate of change of the jerk with respect to time. Jounce is defined by any of the following equivalent expressions: where is jerk, is acceleration, is velocity, is position, is time.

The notation (used in ) is not to be confused with the displacement vector commonly denoted similarly. Currently, there are no well-accepted designations for the derivatives of jounce. The fourth, fifth and sixth derivatives of position as a function of time are “sometimes somewhat facetiously”   referred to as “Snap,” “Crackle” and “Pop”.

The dimensions of jounce are distance per (time to the power of 4). In SI units, this is “metres per quartic second”, “metres per second per second per second per second”, m/s4, m · s-4.

HW1P Longitudinal Dyn QuaterCar.docx 1 May 8, 2019

(

)

(

)

(

)

2

(eqn (2.26) textbook)

(eqn (2.27) modified)

(eqn (2.26) )

sgn()0.5

sin

sin

sgn()

wRwrollwgxT

wh

wTRgyTrollwh

gxT

wgxTw

gyTw

roll

dv

mFFFF

dt

dv

mFFF

dt

d

JTFbFr

dt

FvCAvv

Fmg

Fmg

Fmmg

vF

F

w

r

b

b

=—

=—

=-+

=+

=

=

=+

=

(

)

(

)

(

)

2

1

cosif 0

if 0 and cos

sgn()cosif 0 and cos

gyTo

RwRrgxTRwRrgxTogyT

RwRrgxTogyTRwRrgxTogyT

CCvv

FFFvFFFCF

FFFCFvFFFCF

b

b

bb

ì

ï

ï

–=–<

í

ï

–=–>

ï

î

whwh

vr

w

=

22

350kg

15kg

0.2 m

0.5kg m

w

wh

wwwh

m

m

r

Jmr

=

=

=

»

00

22

10

3

2

0.004,0.0040.2

0.00004 [s/]

1.2754kg/m

0.2,0.20.4

0.1[m]

DD

F

CC

CmC

CC

A

r

=££

=

éù

=

ëû

=££

=

=

2

0.01 [m]

9.81[m/s]

b

g

=

=

Rr

F

22

mass of quarter car [kg]

mass of wheel [kg] radius of wheel [m]

moment of inertia of wheel [kg m]0.5 app

roximating the wheel as a short cylinder

.

??

?

??

?

wwh

wwwh

RwRr

gxTwgxT

gyT

r

m

mr

Jmr

FF

F

FF

F

F

=

==

==

=

==

=

1

?

??

?

?

displacement between wheel center and

point of normal tire reaction [m]

?

??

??

??

??

?

oll

TR

TRTR

wh

dF

o

wh

wh

wind

T

TF

r

g

b

CA

CC

dv

v

dt

d

dt

v

b

r

w

w

=

===

=

=

=

=

==

==

==

==

=

wh

dvb

kmFFFFFF

dtr

=—–

2

1

ww

m

wh

mJ

k

m

mr

=++

eqm

mkm

=

m

k

eq

m

(0)0

v

=

()0

wind

vt

=

0

b

=

300Nm

TR

T

=

(0)

dv

dt

TR

T

(0)

0.5

dv

g

dt

=

()

vt

t

®¥

()

0

dvt

ast

dt

=®¥

m

k

2

2

,,

dxdx

xxx

dt

dt

==

&&&

1

m

k

=

x

f

sin

mg

bx

b

&

b

m

x

sin

x

mxmgbxf

b

=–+

&&&

2

1

w

w

m

w

m

J

k

r

=

+

w

m

sin

Roll

m

F

g

b

sin

wRoll

mxFmg

b

=-

&&

= force the pushes the ball up or down t

he slope

Roll

F

w

xr

q

=

&&

&&

wwRollwgyTw

JFrFbr

qt

=–

&&

(

)

(

)

This is the factor that

takes rolling inertia

into account as the

wheel accelerates.

2

sin

sin

sin

wwwgy

w

Tw

w

wg

w

ww

yT

ww

ww

ww

w

w

w

w

w

m

k

m

x

JrFbr

JFb

mxmg

mxmg

r

x

rr

J

mmg

r

x

r

r

b

b

t

t

t

æö

ç÷

èø

æö

ç

+

+

æö

+

=

÷

ç÷

è

èø

=

=–

ø

&&

&&

&&

&&

142

&

43

&

gyT

Fb

b

,,

xxx

&&&

sin

mg

b

,,

qqq

&&&

b

gyT

F

w

w

w

m

J

t

Roll

F

ECE 4/595 Electric

& Hybrid

Drive Systems

Prof Ka C Cheok

HW1P Longitudinal Dyn QuaterCar.docx

1

May 8, 2019

HOMEWORK

ASSIGNMENT 1

I

ssued:

8

May ’1

9

Due:

15

May ‘1

9

Submit

an

electronic copy of the assignment on our Moodle Course Page.

Referring to Section 2.7.4 of Chap 2, class notes and lecture, t

he

m

otion

of a quarter

car and

its

wheel

in

the

direction

tangential

can be

modeled by

:

(

)

(

)

(

)

2

(eqn (2.26) textbook)

(eqn (2.27) modified)

(eqn (2.26) )

sgn()0.5

sin

sin

sgn()

wRwrollwgxT

wh

wTRgyTrollwh

gxT

wgxTw

gyTw

roll

dv

mFFFF

dt

dv

mFFF

dt

d

JTFbFr

dt

FvCAvv

Fmg

Fmg

Fmmg

vF

F

w

r

b

b

=—

=—

=-+

=+

=

=

=+

=

(

)

(