Help for the pattakon "gear" program.
A MOTORCYCLE IS RUNNING ON THE STREET
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You can see the reciprocating piston. It is connected to the crankshaft
through the connecting rod. On crankshaft is connected a gear that rotates
making the next big gear (of clutch) to rotate with the proper speed. In
our case, the gear fixed on the crankshaft has 14 teeth and the big gear
connected to the clutch, has 30 teeth, so primary transmission is 30/14.
On the circumference of these gears you can see bright dots which move as
if a spot of chalk was on the real gear. The axis of this big gear is also
the axis of primary axle of the gearbox. On primary axle there are three
gears. They have 20, 26 and 32 teeth. On the next axle to the left, there
are three more gears with 40, 34 and 28 teeth (Why so many ? Why not ?..)
The 40 teeth gear cooperates with the gear of primary with the 20 teeth so
this gear ratio is 2:1 or 2.0. The gear with the 34 teeth cooperates with
that with 26 teeth of primary axle, so this gear ratio is 34:26 or 1.31.
The third gear of the secondary axle, of the gearbox, has 28 teeth and
cooperates with the gear with 32 teeth of primary axle, so this third gear
ratio is 28:32 or .875. That is we have three available pairs of gears to
transmit the motion of the big gear (and clutch) to the sprocket of the
gearbox. This gearbox sprocket is secured to the secondary axle of gear-
box, and has 20 teeth. The other sprocket secured to the rear wheel (wheel
sprocket) has 50 teeth and is connected to the gearbox sprocket by chain.
The bright dots on the sprockets give their speed. There is also a bright
dot fixed on the circumference of the tyres giving their speed.
To change gear in the gearbox, just press key 1, 2, 3 or `. For example,
pressing once key 2, the bright circles are moved from last used pair of
gearbox to the second pair. The rotation speed of the engine increases or
decreases to match with motorcycle speed, etc.
The bright dots fixed on the road give the speed of motorcycle.
The graph on the screen is the graph of the maximum available force that
the rear tyre can apply to the road (and also the road can apply to the
motorcycle) as function of the velocity. There are three curves, one per
gearbox ratio. You can see just bellow this graph the accelerator position
(or throttle position), the selected gear and motorcycle velocity, the
motor speed (in rounds per minute) and the past time. There are also on
the screen the distance covered by the motorcycle since last reset and the
braking distance (in case of use of brakes).
The bright dot on the force/velocity graph, is moving as the throttle is
opened, and always gives the velocity of the motorcycle and the present
available force, according to selected gear. As you easily can see, the
acceleration of the bike/rider depends heavily on the selected gear in the
gearbox and the moment that this gear was selected.
The necessary time to reach some velocity is the measure of right or
wrong use of gearbox. The less time, the best use. To suspend running
of program, use key pause.
The force/velocity graph is easily taken from a graph of power or a
graph of torque of the engine. The form of curves in force / velocity
diagram is exactly the form of the curve : torque / speed of engine.
Notice that the time passes as if the bike was real: when the acceleration
force is high, the speed increases rapidly.
On the screen you can also see the steering of the bike (not in its proper
position). It has the clutch control, the brakes control, the throttle or
accelerator control, tachometer and speedometer. The curve besides the
steering indicates the throttle opening.
CLUTCH OPERATION
To find the clutch slipping compare the rotary motion of clutch big gear
(indicated by bright dot on it) with the rotary motion of primary axle.
With 5, 6 and 9 keys we control the minimum used rotation speed of engine
This is the r.p.m we keep the engine rotating the time we engage the
clutch. Why this minimum r.p.m. is so important ? What really makes the
clutch being so indispensable for starting ?...
The electric motor and the steam engine can have at low rotation speed
and even at zero rotation speed enough torque. These two kinds of motors
can use the clutch only to alter gear in gearbox. With engine immovable
we select proper gear ratio and then accelerator is pressed. Since there
is torque from zero r.p.m there is also accelerating force on the load.
In the case of the internal combustion engines, the torque equals zero
when the engine is not rotating. Also if r.p.m falls below some minimum
value, the engine not only stops offering power or torque but also turns
off. That is : internal combustion engines cannot rotate below a minimum
speed. Now imagine two shafts, a rotating driving shaft and another driven
shaft, interconnected to each other trough a clutch. The clutch makes the
disconnection of the two shafts possible, and also can transfer all or
part of the torque of the driving shaft to the driven shaft. This part
depends on the position of the clutch control lever. The clutch is also
characterized by the maximum torque it can pass : this depends on clutch
diameter, on number of disks, on friction coefficient of disk plates, on
strenght of used springs. If the torque offered by the driving shaft is
more than maximum then the clutch disks start slipping and the torque
passed to the driven shaft is only the maximum characteristic of clutch
(As clutch is ruined this maximum torque decreases). The passing of the
torque through the clutch to the driven shaft takes place no matter what
the driven shaft rotation speed is. Even with driven shaft immovable,
torque passes to it. Half disks of the clutch rotate with driving shaft
speed and the rest disks rotate with driven shaft speed. When the driven
shaft rotates with different speed than driving, then there is slipping
between the two sets of disks.
So, in the case of bike, even with the primary gearbox shaft immovable,
(and also the bike immovable when no tyre slipping and no neutral in the
gearbox) the torque produced by the engine multiplied with the primary
transmission ratio is transferred - at most - to the primary shaft of
the gearbox (the percentage depends on clutch control lever position).
In this way, even with the motorcycle stopped, the engine can rotate and
can offer - with the slipping of clutch disks - torque to the primary
shaft of gearbox. Here note that instead the clutch can pass all torque
from one shaft to the other (even in case of disks slipping) it does not
pass all the power from driving shaft to driven while slipping. The
percentage of power passed from driving to driven shaft is proportional to
rotation speed of driven shaft: the power passed is zero when the driven
shaft is immovable and is the whole power offered by driving shaft when
the clutch is completely engaged (that is when no disks slipping).
Note that the power a shaft transfers is proportional to both torque and
speed. What is going on with the power not passed to the driven shaft ?
It is lost as friction on the surfaces of the clutch disks, increasing
their temperature and decreasing their life (due to surface ruin). The
relevant speeds and the amounts of power which pass and those which are
lost into clutch are all shown on screen (use key J).
Now, why to alter minimum r.p.m. ? The time of clutch engaging, as was
said, torque is transferred to the primary shaft of gearbox. This torque
according to gear ratio selected, generates a force on the rear tyre
contact surface with the road, which force accelerates the mass of bike
and rider ( after the subtraction of resistance due to friction and rest
surroundings influence ). This force is analogous to the torque on primary
shaft and to the gear ratio selected. The torque on the primary shaft of
the gearbox depends on the throttle opening, the clutch lever position
and the present r.p.m. (rotation speed). Supposing correct clutch lever
use, completely opened throttle and also clutch strong enough (strong
enough to pass at least the maximum torque of engine multiplied with the
ratio of the primary transmission), then the force that pushes the bike,
according to selected gear in gearbox, depends on used r.p.m. The curves
of the FORCE - VELOCITY diagram have exactly the form of the curve in
the diagram TORQUE - R.P.M. So as we see in the force velocity diagram
of our bike on screen, if the minimum r.p.m. becomes 3000 instead 1250
then the available maximum torque (and so the available maximum force to
push the bike) becomes significantly higher. When r.p.m. minimum is at
1250 r.p.m. then the resulting force may not be enough to start quickly
(or accelerate quickly) the bike. In case of resistance due to uphill
road or of strong opposite wind or of extra big mass (so increased rolling
friction) with the 1250 r.p.m the bike may not be able even to start.
In this last case, the motor produces power, the clutch transfers torque
to the primary gearbox shaft and a force (but not adequate) is pushing
the bike forward. All the power produced is lost on the clutch disks. If
now the minimum used r.p.m. is increased to 3000, then the force that
pushes forward the bike to start moving is more than resistance and the
bike start moving forward. The maximum forward pushing force is when the
r.p.m. is at 5000. Although in this rotation speed is not produced the
maximum power (see diagram of power distribution on screen as r.p.m is
increased) here is where the maximum force on bike - due to engine - is
achieved. Exactly here (at 5000 r.p.m) is the speed where engine gives
its maximum torque. If the minimum used r.p.m. is selected to 7500 r.p.m
then the available force to accelerate the bike is much lower.
Try to understand that power and torque are two different things. If the
torque that a shaft transfers is known and the rotation speed is known
too, to find the power transferred by the shaft it is enough to multiply
the rotation speed with the torque. The result is a quantity analogous to
the power (analogy factor depends on used units).
We said that the 5000 r.p.m is the r.p.m. we can have the best starting.
The bad news now: when minimum r.p.m. is set at 5000 then initially all
and then a percentage of the power produced by the engine (depending on
clutch slipping) is lost into clutch. And at 5000 r.p.m. is produced,
with completely opened throttle, almost all the power that the motor can
produce. Try to understand that bike's starting ability is better when as
minimum used r.p.m. is taken the speed where motor produces maximum torque
and not where motor produces maximum power. Check this selecting so strong
uphill that: with 5000 minimum used r.p.m. bike can hardly start and with
6000 minimum used r.p.m, where maximum power is produced, bike cannot move
(until r.p.m lower to values where more accelerating force is available).
TO SUMMARIZE FOR CLUTCH
The clutch can transfer torque from a rotating shaft (driving shaft) to
another shaft, no matter what the speed of the second shaft is. Using the
clutch we can have on the driven shaft the torque the driving shaft offers
at the same or at any higher speed (even when driven shaft is immovable).
POWER DISTRIBUTION DIAGRAM
A: Power produced by the motorcycle engine and measured on crankshaft.
C: Power lost on clutch due to clutch disks slipping. In the force
velocity diagram this power is proportional to the area of a rectangular
with two sides passing by bright dot (and parallel to axes), third side
on the velocity axis and last side parallel to force axis at the speed
where the clutch is fully engaged (according selected gear as well as
minimum used r.p.m.). This power is darker to show power loss.
T: Power measured on rear tyre with selected throttle position and
clutch completely engaged (this is a percentage - almost constant - of
the A power). This T power in force velocity diagram is proportional to
the area of the rectangular with one corner on the (0,0) point, sides
parallel to axes , above side passing by bright dot and right side
passing by the bright dot, if the clutch is completely engaged, or by the
speed where clutch is completely engaged (depending on gear selected
and minimum used r.p.m.).
L: L is the T power minus the power C lost in clutch. So L power is the
net power available on the contact surface between rear tyre and road.
F: The F power is the resistance power of surroundings to the motion of
motorcycle. It is the sum of the power loss due to rolling friction (which
is proportional to the force that motorcycle applies normally to the road,
to the velocity and to the rolling friction coefficient), the power loss
due to aerodynamic friction (this is proportional to cubic power of the
velocity, when no wind, to air drag factor and to frontal area), the power
transformed into dynamic energy due to the elevation of total weight (this
is proportional to weight and to elevation velocity) and the braking power
when brakes are on (this is proportional to the force applied normally to
the road, to the velocity and to friction coefficient of tyres-road). From
the above four parts of F power, that due to weight elevation is the only
one that is not lost: This power is absorbed and stored as dynamic energy
of bike/rider and can be transformed again -at least partially - on next
downhill back to kinetic energy. Here notice that when a force is moving
along its direction with some velocity, the produced or absorbed power is
this force times its velocity.
R: The R power is what is left to accelerate the motorcycle. It is the
power L produced by the engine and arrived to the rear tyre plus the power
produced due to weight lowering when downhill (dynamic energy changes back
to kinetic energy), plus the power due to wind help (due to aerodynamic
friction when the wind is faster than bike), minus F power.
WEIGHT (e.g M++++++++++M for 50 Kg more total mass.)
The total mass of the engine can be altered using M key. Increasing the
total mass, the rolling friction increases. Also the necessary minimum
throttle to start moving the motorcycle increases too, especially in case
of uphill road. Also braking distance is affected. Study the influence of
increased weight on performances. Also study the ability of the motorcycle
to start moving on an uphill road as the weight is increased. To alter the
total mass without interrupting the program, you can use N or B or . keys.
UPHILL/DOWNHILL ROADS (e.g. G---G for 3 more degrees downhill)
Use G key to select uphill or downhill road, and gradient of it. The
gradient of the road significantly affects the starting ability and the
performances of motorcycle. Also the braking distance. When downhill road,
there is, even at zero velocity, a force trying to accelerate the bike
When the gradient is very small this force cannot surpass the rolling
friction force. As the gradient becomes more negative (you can use 7,
4 or 8 keys to do this without interrupting program with g selection) then
there is a gradient after which motorcycle starts accelerating without
engine help. Study the influence of downhill on performances. See the case
of maximum available speed and accelerations. The influence of a downhill
road is similar to that due to a more powerful engine.
PERFORMANCES...
If the maximum speed that the engine reaches is not in the position where
is available the maximum power on rear tyre (power L on the power diagram
selected with J key) then there is better selection of gear ratio which
gives even higher maximum velocity. But the gear ratios also influence
the acceleration of the bike. Using just two of the three available gear
ratios (e.g. 2nd and 3rd instead of 1st and 3rd) you can try to achieve
the best acceleration from 0 to 60 velocity (no units referred). Also the
right and the wrong gear change can be shown. If the first gear is too
early changed then we have poor performances but this also happens when
first gear ratio is changed to second gear, too late. Where is the best
point to take place this change? If best performances is the point, then
it is enough - looking at power distribution diagram - to alter gear in
such a way that the R power (which accelerates the bike) to have always
its maximum available value. It is easier to look at Force-Velocity
diagram and to alter gear using always the maximum available force.
ECONOMY...
The typical case - when the point is not the best performances - is when
the bike moves with a constant speed (much lower than maximum available).
Then - to overcome the Air / Road influence - it is necessary to keep the
throttle opened more or less depending on gear selected. The point is that
to keep the velocity constant we need some power to overcome surroundings
resistance. Maybe in this velocity to be available more than one ratios in
the gearbox. The characteristic of typical engine is that its efficiency
is almost proportional to its load (or its throttle opening). With the
gear ratios available we can select that one which leads to heavier load.
This is the rule for best economy: to select the longer available ratio.
The motor rotates slower and the throttle is more opened.
The important is to know what is that we try to achieve : Best economy ?
Best performances ? Longer clutch life ? Longer engine life ? etc.
KEYS
With key F aerodynamic friction and rolling friction are also included.
With the J key the power distribution is on or off. See that at high
rotation speeds the T power is a smaller percentage of the power produced
on engine crankshaft.
Pressing once , key the motorcycle starts decelerating until standstill
or until another key is pressed.
Here we suppose that friction coefficient between rear tyre and road is
constant. Also we take under mind not only the force from the brakes, but
also (in case where Air / Road friction included) the influence of Air and
of Rolling friction. The result is that without Air/Road friction the
braking distance (that is the distance from the point the brakes were
pressed to the point the bike stopped moving) are longer, especially at
high speeds. Study the influence of uphill or downhill roads on braking.
With P [ ] \ keys you select 1/4, 1/2, 3/4 and 1/1 throttle opened. If
you want also the intermediate values of throttle opening, then use the
QAZX or C keys. Keep pressed X key and the throttle is opened (how much
is written at Throttle: ... on the screen) or keep pressed A key to go
to less throttle opening. Here see that with half throttle the engine
does not produce half power but more. This is not problem if the relation
between opened throttle and torque produced is known. This relation in our
case was selected to be close to experimental data found into books.
Notice here that it was supposed (in an effort to simplify things) that
the inertia resistance of the engine was negligible and that the internal
frictions of the engine were taken as negligible too. Last assumptions are
an answer on how engine rotates with throttle=0.00 at 5000 r.p.m. without
load. Also these assumptions are to prevent questions like : How quickly
the engine alters rotation speed when from 3rd gear in gearbox and high
rotation speed is selected 1st gear in gearbox and why the engine doesn't
decelerates motorcycle as in reality.
If we want to know the influence of internal engine friction and of motor
inertia on total behaviour (to explain why, selecting 1st from 3rd gear
in gearbox when high rotation speed, the rear tyre or the clutch must
start slipping) the number of assumptions and the difficulty in computing
would be in disproportion to the gain in accuracy of results. That is why
we made the above assumptions: to simplify computing and to omit things
that, after all, affect behaviour of motorcycle no more than a very small
percentage of the whole time.
With key ; you can decrease time step dt. The accuracy of results is then
increased (into limits, since accuracy of computer arithmetic calculations
is limited too), but also running of program get slower. With keys ' and /
you can increase or reset to initial value the time step dt.
With keys { and } and | you can control the frequency that the procedure
which controls the random changes in wind speed and in road slope is
activated. If these changes happen rarely press { as necessary. Use }
for more changes in time unit. Use | to reset frequency to initial value.
With the $ key you can lower the speed of the graph: how quickly the
piston reciprocates and the gears rotate. With % key the speed of moving
parts becomes higher and the key # resets this speed.
With the 6 key the minimum used r.p.m is increased. With 5 decreased
and with 9 is reset to initial value. From this minimum rotation speed of
engine is greatly dependent the acceleration but also the life of clutch.
You can finally control wind speed with < and > and ? keys.
AN EXAMPLE...
Let take the case that we want to keep our speed at 100. We start the
program and we press keys FJ66666668888xxxxxxxxxxxxxxxx The motorcycle
starts moving. When the velocity becomes more than e.g. 40 we press 2 to
select 2nd gear ratio in gearbox. After speed 70 with key 3 is selected
third gear in gearbox. If the throttle is not enough opened, key X is used
When the motorcycle get close to 100 speed then with A key the throttle is
closed enough to keep the speed around 100. There is a downhill ahead
(just press 777777777) and the engine accelerates (not forget to look at
power distribution diagram). Now to keep speed 100 it is necessary the key
A. We select throttle position (with A and X keys) so that speed is near
100. Now we close completely the throttle (this is done pressing once Z).
The motorcycle decelerates (look at power distribution). The engine does
not produce any power but the downhill road surpasses the Air and Rolling
friction and keeps the motorcycle in motion. If the gradient changes (use
keys 888) the bike decelerates till a new lower speed. Any moment we can
use the brakes to stop engine and see (under present conditions) how much
distance is necessary to stop... Things are not much different in reality.
With the keys < > and ? you can alter or reset wind velocity. Things are
similar to the case of aerodynamic friction loss. If the wind velocity is
equal to the motorcycle speed, then it is like the case of no air at all.
Especially in the case of wind velocity greater than motorcycle speed, the
air offers energy to the motorcycle (look at F and R power). We took here
that the coefficient of air friction, when wind speed is greater than bike
speed, is 1.5 times this coefficient in the typical case (that is when the
wind speed is lower than bike's speed) : to understand this look the shape
of motorcycle according to watching point (front / back). In case of extra
strong wind, even the brakes could be unable to keep the bike stopped.
Even with the brakes on the bike could accelerate to a velocity where the
relevant velocity (of wind as regard motorcycle) becomes low enough so
that braking force can keep bike NOT accelerating. Now combining the speed
of the wind with uphill and downhill roads, with clutch use, with throttle
opening, with total mass etc the game never ends.
RANDOMISED SURROUNDINGS
Pressing the key ! once the program changes randomly the wind velocity
as well as the road uphill or downhill. The reality is not far away. The
road and the wind is - more or less - undefined. If somebody wants to keep
let say constant speed, then he has to select proper gear in gearbox as
well as properly opened the throttle. And as time passes and surroundings
alter, also throttle position and brakes and gear selected must be
adjusted... To see surroundings random influence just press ! key. Wait
looking at screen.
EVEN MORE HELP FOR CLUTCH
If the clutch is still a mystery for you, just imagine a table with
wheels on its legs and a tablecloth on it. When this tablecloth is pulled
horizontally and slowly, the table stays immovable although a horizontal
force is applied to it, due to cloth weight and friction of tablecloth
with table surface. The friction force between table and tablecloth cannot
surpass the friction force between table and floor. Now we put a book on
the tablecloth and the cloth is pulled again slowly. Again the table stays
immovable. The friction force between tablecloth and table cannot surpass
the friction force between table and floor. Although the force applied on
the cloth passes to the table surface, it cannot move the table. Although
energy is absorbed - to move the tablecloth with the book on it - all this
energy is lost as friction between table and tablecloth. As we add books
on the table-cover and pull it horizontally, we reach a point where the
table starts moving. That moment there is slipping between cloth and table
and the friction force between cloth and table surpasses the friction
between floor and table. In case the cloth is slipping on the table and
the table moves on the floor, the tablecloth passes all the force applied
to it (by our hands) to the table but only a percentage of the given
to cloth energy passes to the table. If we add more books on the cloth
on the table then comes a moment when the tablecloth does not slip on the
table surface. The table is moved as if the force applied to the cloth
was immediately applied to the table. Now all the power (or energy)
offered to tablecloth by the hands is passed to the table. No power loss
on contact surface of table and tablecloth. Suppose now that the cloth
on the table is kept by its four corners by four men, and a fifth man adds
books on it as the rest men walk around the table. The friction torque
that the cloth transfers to the table initially is not enough to move the
table (friction torque between floor and table is not surpassed). Also all
power given to the cloth is lost as friction on cloth with table contact
surface. As more books are located to the tablecloth, comes a moment the
table starts rotating. If there is slipping of the cloth on the table then
a percentage of the power offered to the cloth (by the men who keep it) is
lost as friction and another is passed to the table. If the books are
enough to stop slipping then not only all the torque - offered to the
tablecloth - is transferred to the table, but also there is no power loss
on contact surface, so all power is passed to the table.
The clutch operation is similar. Like the tablecloth on the table,
there are in the clutch the slipping disks, and like the books added on
tablecloth (to increase friction force between cloth and table) there are
springs in the clutch that push disks in contact (and this spring force
is controlled by clutch lever position). Think the rest by yourself...
HOW THIS PROGRAM WORKS...
If there was a way to compute the velocity of the bike, after a small
time interval, from its present velocity and present conditions, then the
problem would be solved. In our case:
All the data we have is the geometry of the system (gears, sprockets and
rear tyre dimensions) the friction coefficients, the total weight or mass
and the FORCE - VELOCITY diagram taken with a dynamometer (better measured
at rear tyre). If force - velocity diagram is not available, then a simple
TORQUE - ROTATION SPEED diagram can be used.
- According to selected gear in gearbox, the minimum used r.p.m., the
throttle opening and the velocity the program specifies the force - due to
the engine action - that arrives on the rear tyre / road contact surface.
- Now according to the total weight and to the road slope the program
specifies the force - opposite to the direction of the motion - which is
the rolling friction. It is : W * cos(g) * f0, where W is the total
weight, g is the angle between horizontal plane and road line (that is g
is the gradient or the slope of the road) and f0 is the coefficient of the
rolling friction. Notice that rolling friction is proportional to the
force normally applied to the road, that is when road slope is high then
rolling friction get lower.
- Now according again to road slope, the program specifies the road slope
influence : - W * sin(g) , where W is the total weight and g as above.
The minus sign is to say that at uphill roads (where g is taken positive)
this force is opposite to the motion and at downhill roads (where g is
negative) this force is at the direction of the motion.
- Now according to wind speed the program specifies the force that the
wind applies to the bike. It is :
-A * ( Vm - Vw ) ^ 2 when Vb >= Vw
or B * ( Vm - Vw ) ^ 2 when Vb < Vw,
where Vm is the motorcycle speed, Vw is the wind speed (signed quantities)
and A and B are two constants proportional to frontal area and to the air
drag coefficient (A is that measured normally and B is with the bike/rider
looking behind). Note that X ^ 2 means X squared. In the program is taken
that B=1.5*A. Think why. Notice that when the wind speed is exactly the
speed of the bike, then the bike moves like there was no air drag at all.
- Now according to brake's lever position (on or off) program specifies
the force : - W * cos(g) * f1 , if brakes are on
0 , if brakes are off
which is the braking force (due to brakes action) always opposite to the
direction of the motion. Here W is as above, g the road slope angle as
above and f1 is the friction coefficient between road and tyres. Notice
that braking force is proportional to the normal force that bike applies
to the road and not just to the total weight. This means that the maximum
available braking force (due to brakes) is taken when level road (with the
tyre / road friction coefficient f1 unchanged).
Finally the force that is really applied on bike/rider is the sum of all
the above (five different parts). Since the total mass of bike/rider is
known and also known is the force which is applied on this mass (we have
already mention the computation way) the acceleration or deceleration of
the bike is computed too ( Newton's low : F = m * a , that is when a force
F is applied on a mass m then the resulting acceleration is a as above).
Knowing the present velocity of the bike and the present acceleration of
it, the velocity of the motorcycle, after a small (or infinitesimal) time
interval dt, will be : v(t+dt) = v(t) + a(t) * dt , where t is the time,
a(t) is the acceleration (or deceleration) of the bike at the time
t, v(t) is the velocity of bike at the time t and v(t+dt) is the computed
velocity of the bike the time moment t+dt. The above relation is right
only if the dt time interval is infinitesimal. If - as happens in this
program - the dt interval is small enough, the relation is very close to
right. The action of long time interval dt can be shown too (use keys ; '
and / to control dt and measure e.g. time necessary to reach speed 100).
THAT IS ALL : We start with that bike's velocity at time 0 is V0 (the
initial conditions). We compute the total force applied to the bike this
time moment 0 (it consists of five parts described above) and dividing
this force with the total mass of the bike, the acceleration a(0) (or
deceleration) at time 0 is computed. According to selected time step (it
must be very small if we want accuracy of results) the velocity at the
time moment t1 = 0+dt will be: V0+a(0)*dt. So we find the velocity at time
t1=0+dt. It is v(0+dt). Now the program searches if something is changed
(e.g. if it is altered the wind speed or the slope of the road or the
throttle position, or the gear ratio, or the rotation speed of the engine
etc) and according to new conditions computes the force applied on the
bike the time moment 0+dt. Then the acceleration a(0+dt) is computed as
said before, and finally the velocity v(0+dt+dt) is computed as:
v(0+dt+dt) = v(0+dt) + a(0+dt)*dt. So we know the velocity of the bike the
time moment 2*dt. Again the program computes the force applied on the bike
the moment 0+dt+dt (or 2*dt) and also the acceleration at the time 2*dt.
Then v(0+dt+dt+dt) = v(0+dt+dt)+a(0+dt+dt)*dt. The above cycle is again
and again repeated and so we can find what happens at any moment. We have
to repeat the above until dt+dt+...+dt becomes equal or very close to the
selected time moment. Notice that the covered distance is computed with a
similar way from the velocity of the bike and the time step dt.
A few words for the coefficients of friction. To get the f0 in a real
bike it is enough, with rider on, to put a quintal (steelyard) between
bike and our hand and to pull with a constant low speed (where aerodynamic
friction is negligible) the bike on a level road. What the quintal says is
the total rolling friction resistance. Now, with the f0 measured, air
friction coefficient A can be measured too. We select a long road with
constant slope and leave the bike with the rider to accelerate on this
slope without engine help. The bike accelerates till a final speed where
it stops accelerating. There, the air friction on the bike equals to the
accelerating force due to road slope ( -W * sin(g) remember ) minus the
rolling friction force (just before we said the way f0 to be measured). So
coefficient A can be computed too, with simple and palpable way. Of course
accurate measure of these coefficients is another story, but do we really
need such accuracy ? ...
All the above are just an approximation of what really happens when
A MOTORCYCLE IS RUNNING ON THE STREET.... We tried to omit the unimportant
and focus on important things. We made a number of assumptions trying
to simplify the problem but also trying to stay close to reality...
Other approaching methods, more accurate, could also be used, but since
undefined/unknown factors are so many, it would be just waste of time: If
the total weight of bike/rider is given as 196.289345 Kp, something is
going wrong. The value 196 Kp is more correct for total weight. It is (or
better it seems to be) not as accurate as the 196.289345 Kp value, but it
is more correct. Think why...
Try the program. Search the details. Think. There are much to discover...
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