The first step in the analysis of the system is to consider how the input will affect the system. The input is a step function of 0.1 meter applied to the input variable, x2. When x2 is 0.1 meter, the displacement x3 can be directly determined using the gear ratio.
The gear ratio between two gears is directly related to the size of the gears. Because the number of teeth on a gear is directly related to the size of the gear, the gear ratio can be defined as shown in (1).
The gear ratio also relates the angular displacement of each gear. With a gear ratio of three, one revolution of the larger gear (gear 2) corresponds to three revolutions of the smaller gear (gear 1). In other words, the angular displacement of gear 1 is three times larger than the angular displacement of gear 2 as shown in (2).
The angular displacement of each gear is directly related to the displacements
x2 and x3. These relationships are found in (3) and
(4). These relationships are only valid for q
1 and q 2 in radians.
By combining (3) and (4) with (2), a relationship between x2
and x3 can be found and a value of x3 can be obtained.
This relationship and value can be found in (5).
The next step in the analysis is to find an equation in terms of the output variable, the force in spring 1. This will be done by means of a free body diagram of the mass. Free body diagrams of the gears do not have to be taken into consideration because the motion of the gears is already known from the input. The free body diagram of the mass can be found in Figure 2.
Figure 2. Free body diagram of the mass.
By summing the forces acting on the mass (positive being up), (6) can
be found. This equation relates the forces on the system with the inertia
of the mass.
The force exerted by the damper is the proportional to the velocity of the mass. The force exerted by the spring is proportional to the position of the mass. The acceleration of the mass is the second derivative of the position of the mass. These three facts can be used to find (7), which relates the displacement, x3, to known parameters in the form of a differential equation.
The value of x3 is known from (5) to be 0.05 meter. Because this value is a constant, the derivative of x3 is zero. Using these facts, (7) can be written in its canonical form as shown in (8).
Using Mathcad to evaluate the second order differential equation in (8), a graph of the output (the force in spring 1) as a function of time can be found (see Appendix A). Figure 3 shows a plot of the output versus time. Mathcad requires the differential equations to be in state variable form. These state variable equations are shown below.
