fig-1: displacement V/s time curve for under damped system
Logarithmic decrement is defined as the natural logarithm of the ratio of successive amplitude on the same side of mean position.
The rate of decay in the amplitudes of under-damped system is measured by the parameter known as logarithmic decrement.
Rate of decay in amplitudes depends on the amount of damping present in the system. So if the damping is more, then the rate of decay will also be more.
Let A and B are the two points on the successive cycles which shows maximum deflection as shown in figure.
The periodic time:
The amplitude at time t1 and t2 are:
Taking ratio, we get;
The logarithmic decrement is given by;
The logarithmic decrement can also be determined as follows;
δAdding upto ’n’ termsnδ∴ nδOrnδ∴δ=loge(x1x0)=loge(x2x1)=loge(x3x2)=…=loge(xnxn−1)=loge(x1x0)+loge(x2x1)+loge(x3x2)+…+loge(xnxn−1)=loge(x1x0 .x2x1 .x3x2 .… .xnxn−1)=loge(xnx0)=n1loge(xnx0)
x0 = amplitude at the starting position
xn = amplitude after ‘n’ cycles
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