Question 2.3: Consider the spring–damper system shown in Fig. 2.9. This co...

As an introductory aid in visualizing the action of each member of the system and defining variables, Fig. 2.9 shows diagrams of three different kinds for this system: a cross-sectioned mechanical drawing showing the system in its initial relaxed state with F_i=0[Fig. 2.9(a)], a stylized diagram showing the system in an active, displaced state when the force F_i is acting [Fig. 2.9(b)], and a free-body diagram of the system “broken open” to show the free-body diagram for each member of the system [Fig. 2.9(c)]. Applying Newton’s third law at point (1) yields

F_i=F_k+F_b       (2.18)

For the elemental equations,

F_k=k(x_1-x_2),     (2.19)

F_b=b(v_1-v_2).     (2.20)

Definitions:

V_1\equiv\frac{dx_1}{dt}.        (2.21)

V_2\equiv\frac{dx_2}{dt}.       (2.22)

The system is now described completely by a necessary and sufficient set of five equations containing the five unknown variables x_1,F_k,F_b,v_1 and  v_2 Note: The number of independent describing equations must equal the number of unknown variables before one can proceed to eliminate the unwanted unknown variables.

Combining Eqs. (2.18)–(2.22) to eliminate variables F_k,F_b,v_1and  v_2 yields

b(\frac{dx_1}{dt}-\frac{dx_2}{dt})+k(x_1-x_2)=F_i.        (2.23)

Note that x_2 and dx_2/dt have been left in Eq. (2.23) for the sake of generality to cover the situation in which the right-hand side of the system might be in motion, as it could be in some systems. Because the right-hand side here is rigidly connected to the frame of reference g, these variables are zero, leaving

b\frac{dx_1}{dt}+kx_1=F_i         (2.24)

This first-order differential equation is the desired mathematical model, describing in a very concise way the events described earlier in verbal form. It may be noted, in the context of state variables to be discussed in Chap. 3, that a first-order system such as this requires only one state variable, in this case x_1, for describing its state from instant to instant as time passes.
Given the initial state of the system in Example 2.3 and the nature of F_i as a function of time, it is possible to solve for the response of the system as a function of time. The procedure for doing this is discussed in later chapters.
The steps involved in producing mathematical models of simple mechanical systems are also illustrated in the following additional examples.