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Sunday, October 5, 2014

Acceleration Warning

From the post "Gravity Wave and Schumann Resonance",

gw=geiwt=goei(xre+wt)

ix  is a circular wave front parallel to the surface of Earth.

gwt=iwgeiwt

which makes gwt  also parallel to the surface of Earth.  From which we formulate a gravitation potential term  GPE, the rate of change of gravitational potential,

dGPEdt=mgwth=iwmgheiwt

where  h is the reference height along  g, and  m  the mass of a body.

Then we consider the same body in motion on the surface of Earth with speed  v,

KE=12mv2

dKEdt=12mdv2dt=mv.a

Since both KE and GPE, is in the same direction, because of the term  i,  we can have resonance when,

dKEdt=dGPEdt

12mdv2dt=iwmgheiwt

Since,  g=d2hdt2,

12dv2dt=iwhd2hdt2eiwt

Multiplying  i  to both sides,

i.12dv2dt=whd2hdt2eiwt ---(1)

i.vdvdt=i.dxdtd2xdt2=whd2hdt2eiwt

Because of the factor  i  that rotates  dv2dt  to the direction of  h,   dv2dt acts like a driving force along  h.  Changing the rate of change of  v2  can drive the system (a car travelling perpendicular changing gravitational wave) to resonance.

We know that  x  is not oscillating; not a sinusoidal,  Therefore,  h(t)  must have a sinusoidal time component to cancels the  eiwt  factor in the differential equation (1).  Consider,

h=ho(t)ebt  then,

h=ho(t)ebt+bho(t)ebt

h=ho(t)ebt+2bho(t)ebt+b2ho(t)ebt={ho(t)+2bho(t)+b2ho(t)}ebt

then,

whd2hdt2eiwt=who(t){ho(t)+2bho(t)+b2ho(t)}e2bteiwt

when,

e2bt=eiwt

the sinusoidal time component in the differential equation cancels, ie,

h=ho(t)ei(w/2)t

We can simplify further,

i.12dv2dt=whd2hdt2eiwt

i.12dv2dt=who(t){ho(t)+2(iw2)ho(t)+(iw2)2ho(t)}

i.12dv2dt=who(t){ho(t)iwho(t)w24ho(t)} ---(*)

i.12dv2dt  is along real  h  and  iwho(t)  is along  x.

Equating the rotated parts,

Since v  anf  h  are not parallel to start with, we cannot simply equate imaginary to imaginary part nor real to real.  Parallel components are equated after rotation by  i.

From  (*),

12dv2dt=who(t)ho(t)w34h2o(t) ---(2)

and

w2ho(t)ho(t)=0

12w2ddt{ho(t)}2=0=d(A2)dt

where  A2  is a contant

h2o(t)=2A2w2

ho(t)=i2Aw

From  (2),  since,      ho(t)=0,    ho(t)=0

12dv2dt=vdvdt=v.a=0w342A2w2

a=+wA22v    this is the condition for resonance

And,

h=i2Awei(w/2)t

Aw  has the unit of  m when  A  has the unit of ms-1, a velocity,  which is why it is multiplied by the factor  i.

h=2Awei{(w/2)tπ/2}

When t=0,  at the on set of resonance, considering magnitudes only,

h(t=0)=2Aw

12mv2=mgh(0)=mg2Aw

A=w2g2v2,   h=v22gei{(w/2)tπ/2}

The condition for a car flip becomes,

a=wA22v=w{w2g2v2}22v=w316g2v3

So, when the acceleration  a  is,

a=(2*pi*7.489)^3/(16*9.087^2)*v3=78.86v3  in ms-2

we will experience oscillation perpendicular to the direction of travel at a frequency of 7.489/2 = 3.745 Hz.  Since, this is an exchange of KE and GPE,  the greater the velocity the greater the amplitude of this resonance will be.  This often happens at low speed, when the acceleration is pressed on hard.

More importantly and more frequently, this can also happen when,

a=wA22v

ie.  in sudden deceleration, the car can also flip when,

a=-78.86v^3

Note;  With the initial condition of h=re  when  t=0  at the onset of resonance, the result is a huge number that do not make sense.  This is because  h  is the change in height not the absolute height.