A model for the vertical part of a vehicle passenger seat is shown in Figure Q4a. This comprises a uniform beam, built-in at the base, of height L and bending stiffness El,and mass distribution m kg/m. A large headrest is modelled as a concentrated mass M positioned at the top of the beam. Assuming a 1-DOF Generalised Displacementy(x, t) = (x)z(t) with displacement pattern y(x) = (x/L)', use the Principle of Virtual Work, to obtain an equation of motion for free vibration of the seat in terms of variable z(t), making use of the expression for single coordinate virtual bending work w given \delta \mathrm{v}_{\mathrm{B}}=\left(\left(\int_{0}^{t} \operatorname{EI}\left(\psi^{*}(\mathrm{x})\right)^{2} \mathrm{dx}\right) \delta \boldsymbol{z}\right) \mathrm{z} .

Fig: 1

Fig: 2

Fig: 3

Fig: 4