Simple Harmonic Motion
The differential equation of a damped Simple Harmonic Motion is given by m*d2x/dt2 + r*dx/dt +sx =0 or d2x/dt2 + (r/m)dx/dt +(s/m)x =0 or
d2x/dt2 + 2k*dx/dt +w20 x =0 where r is damping coefficient , s is stiffness constant having dimension of surface tension, 2k=r/m and w20 = s/m, x is displacement from mean position.
Solution is x=Aeαt where A is amplitude and is a constant and α is a constant. dx/dt=αx, d2x/dt2 =α2x. Hence x* (α2 +2kα +w20)=0 or (α2 +2kα +w20)=0
α =-k ±√(k2 -w20)=-k ±w1 where √(k2 -w20)=w1 and general solution is
x=A1*e[-k +√(k2 -w20)]t + A2*e[-k -√(k2 -w20)]t
=e-kt [A1ew1t + A2*e-w1t ]
dx/dt =w1e-kt [A1ew1t - A2*e-w1t] -ke-kt [A1ew1t + A2*e-w1t]
If we take x=Aeiwt , the differential equn becomes x* (-w2 +2kiw +w20)=0 or w=ik ±i√(k2 -w20) and x=A1*e[-k +√(k2 -w20)]t + A2*e[-k -√(k2 -w20)]t
=e-kt *[A1e[√(k2 -w20)]t + A2*e[-√(k2 -w20)]t ]=e-kt *[A1ew1t + A2*e-w1t ] So the solution is same as the previous one.
The value of A1, A2 depend on Boundary conditions.
Boundary condition 1:
x=0 at t=0 i.e. the mass is at the mean position at t=0. Putting this condition, We get A1+A2=0
So x =e-kt [A1ew1t - A1*e-w1t ] =A1e-kt [ew1t - e-w1t ]
or x=A1e-kt [ew1t - e-w1t ]
dx/dt=w1A1e-kt [ew1t + e-w1t] - kA1e-kt [ew1t - e-w1t ]
Velocity imparted at mean position =dx/dt (0)=2w1A1
To find out the time t1, the mass takes to reach amplitude A1, we put dx/dt =0 at A1
0=w1A1e-kt1 [ew1t1 + e-w1t1] - kA1e-kt1 [ew1t1 - e-w1t1 ]
or w1[e2w1t1+1] =k[e2w1t1-1] or
t1=logC1/2w1 where C1=(k+w1)/(k-w1)
* Let A1 + A2=A, A1-A2=B, then A1=(A+B)/2, A2=(A-B)/2. Putting the values and rearranging,
x=e-kt *[A (ew1t + e-w1t)/2 + B (ew1t - e-w1t)/2 ] = e-kt *[Acoshw1t +Bsinhw1t]
x=e-kt *[Acoshw1t +Bsinhw1t] Here we assume that w1 > 0.......... case of over damping.
dx/dt=(-Ak+Bw1)e-kt *coshw1t +(Aw1-Bk)e-kt *sinhw1t=e-kt*[(-Ak+Bw1)*coshw1t +(Aw1-Bk)*sinhw1t]
Value of A,B determined by Boundary conditions. If we put the condition ,
At t=0,x=0, then A=0 and x=Be-kt *sinhw1t=(ew1t - e-w1t)/2*Be-kt
dx/dt=(-k)Be-kt[(ew1t - e-w1t)/2] +Be-kt[(ew1t + e-w1t)/2]*w1 which at t=0 equals Bw1 which means velocity v(0) at t=0 is equal to Bw1 which is the initial velocity imparted to the mass to enable it execute SHM. Either you fix initial velocity and find out B or fix B and find out initial velocity.
If we put t=0, and try to find
value of x, x=A at t=0. and dx/dt=(-Ak+Bw1) at t=0
* in The displacement in heavy damping, the amplitude decays exponentially with time and the hyperbolic terms increase. The net effect at first is increase but soon the exponential term dominates and displacement slowly falls off to zero.What is the maximum displacement at time t1 ?
at t=t1, dx/dt=0 or[(A-B)/(A+B)] *(w1+k)/(w1-k) = e2w1t ; or t1=logC/2w1 where C=[(A-B)/(A+B)] *(w1+k)/(w1-k) . When boundary conditions are
such that A=0, C=(k+w1)/(k-w1)
|Critical damping : w1=0
x=e-kt *Acoshw1t=Ae-kt Here we assume that w1 = 0.......... case of critical damping.
When w1 is nearly equal to zero, w1=δ and x=A1eδt+A2e-δt=A1+A1δt+A2-A2δt=(A1+A2)+(A1-A2)δt=A+Bt
Decay is purely exponential snd faster than over damping.
|The amplitude can , in general , decay exponentially with respect to time or space. In case of former, we call it relaxation time and for later, skin depth as in case of em waves in a conducting media. The criteria is that amplitude to fall to 1/e of its fixed value. For mechanical oscillator, relaxation time=1/k=2m/r. For skin depth=√(2/ωσμ) and has dimension of length.|