0.10 Simultaneous equations

Simultaneous equations

Introduction

The applications of simultaneous equations in the modeling, analysis and design of engineering systems are numerous. This module presents several applications drawn from engineering that illustrate uses of simultaneous equations.

Unmanned air vehicle (robotics)

Many applications in engineering involve the quantities speed, distance and time. In some applications, one is presented with two situations. In the two situations, the speed of an object differs. The following is an example involving an Unmanned Air Vehicle (UAV) whose speed varies depending upon whether the UAV travels in the same direction of the wind as opposed to the situation where the UAV travels in the opposite direction in opposition to the wind. Simultaneous equations can be used to solve problems relating to situations such as this.

Example 1: In a closed course surveillance flight, the downwind leg of length18.6 miles is completed in 2.0 hours. The upwind leg which is also 18.6 miles in length is completed in 3.5 hours. Find both the speed of the UAV and the speed of the wind.

To address such a problem it is critical to begin with the definition of variables. We will let V be the airspeed of the UAV measured in miles/hr. The wind speed expressed in miles/hr will be represented by the variable W .

We know that the distance that an object travels is equal the product of its speed and time. When the UAV travels downwind, its speed will be equal to the sum of its airspeed and the windspeed. When the UAV travels upwind, its total speed will equal to the difference of its airspeed minus the windspeed. The distance that the UAV travels is the same (18.6 miles) whether it travels downwind or upwind.

We can use this information to establish two equations

From equation (1) we obtain

which can can yield an expression for the variable V

Next, we can substitute this expression for V into equation (2). By doing so, we will be able to solve for W .

Dividing each side of the equation by (3.5) yields

This equation can be readily solved for the variable W as follows

Next, we substitute this value for W into equation (4) to solve for V .

Thus we conclude that the UAV airspeed is 7.305 miles/hr and the windspeed is 1.995 miles/hr.

Analysis of a torsion beam

Torque is a term that is used to describe the tendency of a force to rotate an object about an axis, a fulcrum or a pivot. Just as a force is a push or a pull, a torque can be thought of as a twist. Loosely speaking, torque is a measure of the turning force on an object such as a bolt. For example, pushing or pulling the handle of a wrench connected to a nut or bolt produces a torque (turning force) that loosens or tightens the nut or bolt.

Consider the torsion beam shown in Figure 1.