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Date: 24-5-2016
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Date: 29-10-2020
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Date: 25-10-2020
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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 − ω2o). The solutions to Equation (3) depend dramatically on the sign of α2. The α2 term is negative when γ 2/4 < ω2o 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>ω2o. 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 − ω2o)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.
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