4.9: Damped Resonance
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We now solve the dimensionless equations given by (4.8.5), (4.8.6) and (4.8.7), written here as ..x+a.x+x=cosβt, where the physical constraints of our three applications requires that α>0. The homogeneous equation has characteristic equation r2+αr+1=0, so that the solutions are r±=−α2±12√α2−4.
When α2−4<0, the motion of the unforced oscillator is said to be underdamped; when α2−4>0, overdamped; and when α2−4=0, critically damped. For all three types of damping, the roots of the characteristic equation satisfy Re(r±)<0. Therefore, both linearly independent homogeneous solutions decay exponentially to zero, and the long-time asymptotic solution of (???) reduces to the non-decaying particular solution. Since the initial conditions are satisfied by the free constants multiplying the decaying homogeneous solutions, the long-time asymptotic solution is independent of the initial conditions.
If we are only interested in the long-time asymptotic solution of (???), we can proceed directly to the determination of a particular solution. As before, we consider the complex ode ..z+α.z+z=eiβt, with xp=Re(zp). With the ansatz zp=Aeiβt, we have −β2A+iαβA+A=1, or A=1(1−β2)+iαβ=(1(1−β2)2+α2β2)((1−β2)−iαβ).
To determine xp, we utilize the polar form of a complex number. The complex number z=x+iy can be written in polar form as z=reiϕ, where r=√x2+y2 and tanϕ=y/x. We therefore write (1−β2)−iαβ=reiϕ, with r=√(1−β2)2+α2β2,tanϕ=−αβ1−β2.
Using the polar form, A in (4.9.6) becomes A=(1√(1−β2)2+α2β2)eiϕ, and xp=Re(Aeiβt) becomes xp=(1√(1−β2)2+α2β2) Re{ei(βt+ϕ)}=(1√(1−β2)2+α2β2)cos(βt+ϕ).
We conclude with a couple of observations about (4.9.11). First, if the forcing frequency ω is equal to the natural frequency ω0 of the undamped oscillator, then β=1 and A=1/iα, and xp=(1/α)sint. The oscillator position is observed to be π/2 out of phase with the external force, or in other words, the velocity of the oscillator, not the position, is in phase with the force. Second, the value of β that maximizes the amplitude of oscillation is the value of β that minimizes the denominator of (4.9.11). To determine βm we thus minimize the function g(β2) with respect to β2, where g(β2)=(1−β2)2+α2β2.
Taking the derivative of g with respect to β2 and setting this to zero to determine βm yields −2(1−β2m)+α2=0, or βm=√1−α22≈1−α24, the last approximation valid if α<<1 and the dimensionless damping coefficient is small. We can interpret this result by saying that small damping slightly lowers the “resonance” frequency of the undamped oscillator.