Problem 35 - An Introduction to the Mechanics of Solids in SI Units

Question

Illustrated is a schematic diagram of a cable-control system for the rudder of a subsonic jet trainer aircraft. The rudder lever arm is connected to the pilot's foot control by \(4.5\)-mm-diameter extra-flexible stainless-steel cable, a 25 -mm length of which has a spring constant of \(60 \mathrm{MN} / \mathrm{m}\) (including the effect of untwisting). The cables have an initial tension of \(1.4 \mathrm{kN}\). Cable length from rudder lever arm to the pilot's foot control is about \(6 \mathrm{~m}\). The pilot can push on his foot control with a force of about \(700 \mathrm{~N}\). In a static test of the rudder control system a force was exerted on the rudder and gradually increased until the pilot could no longer hold his foot control stationary. Through what angle had the rudder rotated when the force reached the level which just caused the pilot's foot control to move? Would this angle have been different if there had been no initial tension in the cables?

Step-by-Step Solution

Answer

Rudder rotates proportional to \(0.28m/r\), where r is rudder lever arm. Without initial tension, the rudder would rotate to a larger angle.

Step 1: Calculate cable displacement (stretching)

When pilot holds his foot control stationary it means that the force exerted by the pilot is equal to the force in the cable (spring force). We know that the spring force is equal to the spring constant times the spring displacement. The spring displacement is equal to the Force divided by the spring constant. So we have displacement = \(700N / 60MN/m\) = \(0.0117m\), given that the spring constant applies for a cable length of \(25mm\), the displacement for a cable length of \(6m\) is calculated by multiplying the displacement by the ratio of the lengths (i.e. \(6m/25mm\)): \(0.0117m * (6m/25mm) = 0.28m\)

Step 2: Calculate the angle of rotation

As for the angle, because we are looking at sufficiently small displacements, we may assume that the motion is approximately simple harmonic. Hence, the displacement is given by \(s = r * \Theta\), where \(s\) is the linear displacement, \(r\) is the radius (lever arm) and \(\Theta\) represents the angle in radians. Therefore, we rewrite the equation for \(\Theta\) (angle) where \(\Theta = s/r\). Now, from step 1 we know \(s = 0.28m\) and we can solve for \(\Theta\) = \(0.28m/r\). However, as the problem doesn't provide the value for the 'rudder lever arm', we can't calculate the exact angle and we could say the angle is proportional to \(0.28m/r\). If a value for lever arm is provided by the problem, it can be subtituted in the formula to calculate exact angle.

Step 3: Effect of no initial tension

If the cable had no initial tension, once the pilot exerted a force on his control, the tension in the system would need to build up from zero until the spring force in the cable was equal to the pilot's force, during which the pilot's control would have moved a certain distance. Therefore, the pilot would need to exert more force in order to hold his foot control stationary, resulting in greater stretching of the cable and hence the rudder would rotate to a larger angle.

Key Concepts

Cable-Control System

Understanding the mechanics of a cable-control system is fundamental when analyzing the movements and forces in an aircraft's rudder control, as shown in the given exercise. Such systems utilize cables to transmit control forces from the pilot’s commands to the aircraft's flight control surfaces.

In the scenario presented, the pilot exerts a force through a stainless-steel cable to maneuver the rudder. The key features of this system include the tension in the cable, the elasticity of the cable due to its spring constant, and the system’s ability to amplify or reduce the pilot's input through the lever arm.

The initial tension within the cable is significant as it ensures responsiveness during flight, reducing any slack that would delay the transfer of force. When the pilot's foot control is held stationary and force is applied, the cable undergoes displacement—or stretching—proportional to the force applied, governed by Hooke's Law, which states that the force in a spring is directly proportional to its extension, until the limit of elasticity is reached.

Force Displacement

The force displacement in solid mechanics refers to the change in position of an object when a force is applied. This concept is particularly important in the context of the exercise where the pilot’s foot control is linked to the rudder by a cable-control system.

The relationship between the applied force, the cable's spring constant, and the resulting displacement is represented by the formula \( \text{displacement} = \frac{\text{force}}{\text{spring constant}} \). Since materials follow Hooke's Law up to a certain point, we can determine how much the cable stretches when a specific force is applied. Understanding this fundamental relationship allows us to analyze the mechanics of the aircraft's control system and to predict the behavior of the rudder.

In the given problem, identifying the displacement caused by the pilot's force is crucial to determining how much the rudder would turn. The step by step solution calculates the displacement of the cable as if it were a spring, highlighting the central role of force displacement in analyzing mechanical systems.

Simple Harmonic Motion

The term simple harmonic motion (SHM) describes a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It's a fundamental concept in the study of oscillatory systems in mechanics.

In the provided exercise, we can approximate the movement of the rudder as a simple harmonic motion because the displacement is small and the system can be assumed to behave like a linear spring. The linear displacement from the center position is given by the equation \( s = r \times \Theta \), with \( s \) being the linear displacement, \( r \) the lever arm radius, and \( \Theta \) the angle in radians. If we were given the value of the rudder's lever arm, we could calculate the exact angle of rotation for the rudder using SHM principles.

However, without this information, we can still deduce that the movement is directly proportional to the linear displacement caused by the force applied by the pilot. This understanding is vital when designing control systems that require precise responses to pilot inputs.