Harmonic oscillation

Dear Ladies and Gentlemen,

Im still and desperately searching for a function of either velocity or
accelaration ( d(t) = f(v)
or/and d(t) = f(a)), which expresses the damped oscillation distance d(t)
caused by
varying velocity or accelaration.

I found this formula, which express a mass m in [kg] hanging in a spring k
in [N/m],

d(t) = C*[e^(-t/tau)]*cos(o*t - phi)

where:
d(t) = distance in [m] to the time t in [s],
C = distance different from neutral position in [m],
o = SQRT(kg/m), resonance frequency in [rad/s]
tau = time constant for damping oscillation
phi = initial phase

The oscillation is caused by a non specific up/down motion of the of the
spring, performed by a person.

I can measure the up/down distance, speed and acceleration and I know all
parametres in the formula but C and phi.

I gess that both C and phi in some way must be derived from the
acceleration.

My idea was to measure the distance, speed and acceleration in steps of
delta t = t[n]-t[n-1] and then step by step calculate C and phi to be
inserted in :

d(t) = C*[e^(-t/tau)]*cos(o*t - phi) where

where: d(t)=f(t)/k, tau=2*m/b, o=SQRT(k/m)

and as far as I know it should be the solution to the differential equation
m*d'' + b*d' +k*d = f(t) also mentioned by Anselm Proschniewski in an
earlier thread.

where: m=mass, b=damping, k=stiffness, f(t)=force, d=displacement, d'=speed,
d"=acceleration.

I'm not sharp in differential-, integral equations, but have some basic
knowledge about this and Laplace Transformation. I also know a part of the
laws in physics as

f=m*a, E=1/2*m*v^2, Hooks law etc, etc... but I can't solve my problem
Further more I use a math tool from Texas Instrument called DERIVE 5.06.

What do I miss ?

Best regards
Torben W. Hansen
Denmark