Find the steady-state solution of the differential equation.

Match the quantity with the appropriate units (units may used more than once, some units may not be used).

(a) Steady-state amplitude, \(X\)

(i) \(\mathrm{m}\)

(b) Steady-state amplitude of torsional oscillations, \(\Theta\)

(ii) none

(c) Magnification factor, \(M(r, \zeta)\)

(iii) \(\mathrm{N}\)

(d) Transmissibility ratio, \(T(r, \zeta)\)

(iv) \(\mathrm{N} / \mathrm{m}^{2}\)

(e) Acceleration amplitude, \(\omega^{2} X\)

(v) \(\mathrm{rad}\)

(f) Relative displacement amplitude, Z

(vi) \(\mathrm{N} \cdot \mathrm{s} / \mathrm{m}\)

(g) Frequency ratio, \(r\)

(viii) \(\mathrm{N} \cdot \mathrm{s} \cdot \mathrm{m} / \mathrm{rad}\)

(h) Equivalent viscous-damping coefficient for Coulomb damping, \(c_{\text {eq }}\)

(ix) \(\mathrm{N} \cdot \mathrm{s}\)

(i) Ratio of friction force to excitation force, \(\iota\)

(x) \(\mathrm{N} \cdot \mathrm{m}\)

(j) Hysteretic-damping coefficient, \(h\)

(xi) \(\mathrm{m} / \mathrm{s}^{2}\)

(k) Energy captured by energy harvester, \(E\)

(xii) \(\mathrm{W} /\) cycle 

(l) Average power captured by energy harvester, \(\bar{P}\)

(xiii) \(\mathrm{N} / \mathrm{m}\)