Electric and Hybrid Drive Systems homework

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  = m a ) 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 physics, jounce is the fourth derivative of the position vector with respect to time, with the first, second, and third derivatives being velocity, acceleration, 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 [1]) 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” [1] [2] 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()

RwADgxTRr

wRwrollwgxT

wh

wTRgyTrollwh

ADdFwind

gxT

wgxTw

gyTw

roll

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dv

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dt

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b

b

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=

=

=+

=

(

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(

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(

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2

1

cosif 0

if 0 and cos

sgn()cosif 0 and cos

gyTo

RwRrgxTRwRrgxTogyT

RwRrgxTogyTRwRrgxTogyT

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b

b

bb

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vr

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350kg

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=

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00

22

10

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0.004,0.0040.2

0.00004 [s/]

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0.1[m]

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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

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displacement between wheel center and

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300Nm

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w

m

w

m

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+

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m

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m

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b

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= force the pushes the ball up or down t

he slope

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(

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This is the factor that

takes rolling inertia

into account as the

wheel accelerates.

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sin

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