A mass m, supported only by a bearing having a pressurized film of viscous fluid, undergoes translation (i.e., motion along a straight line in the x direction) as the result of having a time-varying input force Fi applied to it, as shown in Fig. 2.10. The object is to develop a mathematical model

Question

A mass m, supported only by a bearing having a pressurized film of viscous fluid, undergoes translation (i.e., motion along a straight line in the x direction) as the result of having a time-varying input force [latex]F_i[/latex] applied to it, as shown in Fig. 2.10. The object is to develop a mathematical model that relates the velocity [latex]v_{1}[/latex] of the mass to the input force [latex]F_i[/latex].Expressed verbally, the system responds to the input force as follows. Initially the input force accelerates the mass so that its velocity increases, accompanied by an increase in its kinetic energy; however, as the velocity increases, the damper force increases, opposing the action of the input force and dissipating energy at an increasing rate. Thus the action of the damper is to reduce the acceleration of the mass resulting from the input force. If the input force is then removed, the damper force continues to oppose the motion of the mass until it comes to rest, having lost all of its kinetic energy by dissipation in the damper.

Accepted Answer

Newton’s second law applied to the mass m yields

[latex]F_i-F_b=m\frac{dv_1}{dt}.[/latex] (2.25)

The elemental equation for the damper is

[latex]F_b=b(v_1-v_2).[/latex] (2.26)

Equations (2.25) and (2.26) constitute a necessary and sufficient set of two equations containing the two unknowns [latex]F_b[/latex] and [latex]v_1(v_2[/latex] is zero here).

Combining Eqs. (2.25) and (2.26) to eliminate [latex]F_b[/latex] gives

[latex]m\frac{dv_1}{dt}+b(v_1-v_2)=F_i[/latex]. (2.27)

Because the bearing block is rigidly connected to ground,[latex]v_2=0[/latex] , leaving

[latex]m\frac{dv_1}{dt}+bv_1=F_i [/latex]. (2.28)

Again, this is a simple first-order system, requiring only one state variable, [latex]v_1[/latex], for describing its state as a function of time.

Question 2.4: A mass m, supported only by a bearing having a pressurized f...

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