Heavy damping

Heavy damping occurs when the degree of damping is sufficiently large that the system returns sluggishly to its equilibrium position without making any oscillations at all. This corresponds to the mass in Figure 1.1 being immersed in a fluid of large viscosity like syrup. For this case the oscillatory part of our solution, cos ωt in Equation (2.1), is no longer appropriate. Instead we replace it with the general function f (t), i.e.

x = exp(−βt)f (t ). .........(1)

Substituting x and its derivatives into letting β = γ/2 gives

.........(2)

or

............(3)

Figure 1.1 An example of a damped mechanical oscillator showing an oscillating mass immersed in a viscous fluid.

where α² = (γ ²/4 − ω²_o). The solutions to Equation (3) depend dramatically on the sign of α². The α² term is negative when γ ²/4 < ω²_o and this leads to an oscillatory solution with the complex form f (t) = Aexp i(αt + φ). This solution is not appropriate for the case of heavy damping where there is no oscillation. In fact it corresponds to the case of light damping. The α² term is positive when γ ²/4>ω²_o. In this case Equation (3) has the general solution

f (t) = Aexp(αt) + B exp(−αt),

giving
........(4)
This is the non-oscillatory solution that we require. The term (γ ²/4 − ω²_o)^1/2 is clearly less than γ/2 and so the exponents of both exponential terms are negative in sign. Hence the displacement reduces to zero with time and there is no oscillation.