Question 2.6: The six-element system shown in Fig. 2.12 is a simplified re...

The elemental equation for the spring k_1 in derivative form is

\frac{dF_{k1}}{dt}=k_1(v_1-v_2)                 (2.35)

Integration of Equation (2.35) with respect to time with x_1 and x_2, both zero in the relaxed state, yields

F_{k1} = k_1(x_1 − x_2).        (2.36)

For the mass m_1,

F_{k1} − F_{k2} − F_{b1} = m_1\frac{d^2x_2}{dt^2} .            (2.37)

For the damper b_1,

F_{b1} = b_1v_2.            (2.38)

For the spring k_2,

F_{k2} = k_2(x_2 − x_3).          (2.39)

For the mass m_2,

F_{k2}-F_{NLD}=m_2\frac{d^2x_3}{dt^2}             (2.40)

and for the nonlinear damper (NLD),

F_{NLD}=f_{NL}(v_3)=f_{NL}(\frac{dx_3}{dt})             (2.41)

Equations (2.36)–(2.39) may now be combined, yielding

m_1\frac{d^2x_2}{dt^2}+b_1\frac{dx_2}{dt}+(k_1+k_2)x_2=k_1x_1+k_2x_3,    (2.42)

and Eqs. (2.39)–(2.41) are combined to yield

m_2\frac{d^2x_3}{dt^2}+f_{NL}\left(\frac{dx_3}{dt}\right) +k_2x_3=k_2x_2.           (2.43)

It can be seen that two second-order differential equations are needed to model this fourth-order system (four independent energy-storage elements), one of which is nonlinear. The nonlinear damping term in Eq. (2.43) complicates the algebraic combination of Eqs. (2.42) and (2.43) into a single fourth-order differential equation model. In some cases the NLD characteristic may be linearized, making it possible to combine Eqs. (2.42) and (2.43) into a single fourth-order differential equation for x_2 or x_3. Because this system has four independent energy-storage elements, a set of four state variables is required
for describing the state of this system (e.g., x_2, v_2, x_3, and v_3, or F_{k1}, v_2, F_{k2}, and v_3). The exchange of energy among the input source and the two springs and two masses, together with the energy dissipated by the dampers, would require a very long and complicated verbal description. Thus the mathematical model is a very compact, concise description of the system. Further discussion of the manipulation and solution of this mathematical model is deferred to later chapters.