Consider the differential equation of motion for the free vibration of a damped single-degreeof-freedom system given by

Show that Eq. (E.1) remains the same irrespective of the units used by considering the following data and systems of units:

a. SI units: \(m=2 \mathrm{~kg}, c=800 \mathrm{~N}-\mathrm{s} / \mathrm{m}, k=4000 \mathrm{~N} / \mathrm{m}\).

b. Metric engineering units: (mass \(-\mathrm{kg}_{\mathrm{f}}-\mathrm{s}^{2} / \mathrm{m}\), force \(-\mathrm{kg}_{\mathrm{f}}\), length \(-\mathrm{m}\), time \(-\mathrm{s}\) ); \(1 \mathrm{~kg}_{\mathrm{f}}-\mathrm{s}^{2} / \mathrm{m}=9.807 \mathrm{~kg}, 1 \mathrm{~kg}_{\mathrm{f}}=9.807 \mathrm{~N}, 1 \mathrm{~kg}\).

c. Metric absolute units (cgs system): (mass - gram, force - dyne \(\left(\right.\) gram \(\left.-\mathrm{cm} / \mathrm{s}^{2}\right)\), length \(\mathrm{cm}\), time \(-\mathrm{s}), 1 \mathrm{~g}=0.001 \mathrm{~kg}, 1\) dyne \(=10^{-5} \mathrm{~N}, 1 \mathrm{~cm}=0.01 \mathrm{~m}\).

Transcribed Image Text:

mx + cxkx = 0 (E.1)