# Selection Methodology of an Electric Actuator for
## 3 Analysis  
### Analytical Approach  
For the purpose of analytical solution, planer kinematics have been used to design and evaluate the parts used to extend and retract the gear with the attention to geometry of landing gear in the retracted and extended position while planer kinetics has been used for the evaluation of power requirement by the landing gear [16]. Basic guidelines for theoretical approach suggests that starting from the geometrical layout, the designer must replace the system with a mathematical model, while doing so satisfactory moment arms must be ensured for actuation along with the simplest kinematics possible for actuation. The designer must make sure to approximate the proper actuator \"dead length\" in the design layout. Following are suggested values. With no internal lock, the dead length should be 6 to 7 inch. With one internal lock, the dead length should be 8-11 inch and, for two internal locks, dead length should be in range of 12 to 15 inch [17]. While deciding the time for retraction of gear, it must be kept in mind that gear need to be retracted before the airplane reaches 75 percent of its speed with maximum forward acceleration [18]. Figure 3 shows the required dimensions of landing gear for analytical solution of retraction. The landing gear was treated as rigid body while performing the theoretical calculations for the retraction system. Time dependent general equations for the landing system were developed that can be used for retraction of any landing gear with same mechanism. When fully extended, the landing gear is at an angle of 107.94 degree as shown in Figure 3. During retraction, the angle starts to decrease as a function of time, so angle \(\theta\) at any position during retraction cycle is given as,  
\[\theta=\theta_{1}-\Delta\theta_{t} \tag{1}\]  
where \(\theta_{1}\) is the initial angle of landing gear when fully extended and \(\Delta\theta_{t}\) is the change in angle of landing gear at any given time \(t\). The retraction sequence of nose landing gear when retracted slowly without jerks can be modeled as a quadratic polynomial instead of step function, so that the velocity increases slowly to its maximum value and then decreases. If \(\dot{\theta}_{t}\) is the angular velocity of landing gear, then  
\[\dot{\theta}_{t}=c(t^{2}-t\times t_{max}) \tag{2}\]  
Here, \(c\) is the constant, whose value depends upon maximum retraction angle and time of nose landing gear and \(t_{max}\) is the maximum time of landing gear retraction or extension sequence. Figure 3: Nose Landing Gear dimensions for retraction mechanism (all dimensions are in cm). By integrating angular velocity, a change in angle of retraction or extension can be obtained, as shown in Equation (3). \[\Delta\theta_{t}=c(\frac{t^{3}}{3}-\frac{t^{2}t_{max}}{2}) \tag{3}\]  
For the 1600 Kg aircraft considered in this research, nose landing gear retracts by 88 deg in maximum time of 15 s; therefore, if \(t=t_{max}=15\sec\) and \(\Delta\theta_{t}=88\deg\), then by putting the values in (3), the value of c comes out to be \(-0.1565\), so  
\[\Delta\theta_{t}=-0.1564(\frac{t^{3}}{3}-\frac{t^{2}t_{max}}{2}) \tag{4}\]  
\[\dot{\theta}_{t}=-0.1564(t^{2}-t^{2}t_{max}) \tag{5}\]  
From the above equations, the desired angle and angular velocity of landing gear can be found at any time \(t\). Applying law of cosine to the Figure 3 gives  
\[A_{t}^{2}=D^{2}+M^{2}-2 DMcos(\theta) \tag{6}\]  
The length of actuator at any time \(t\) is given by  
\[A_{t}=\sqrt{D^{2}+M^{2}-2 DMcos(\theta_{1}-c(\frac{t^{3}}{3}-\frac{t^{2}t_{max} }{2}))} \tag{7}\]  
where, \(A_{t}\) is the length of actuator, \(D\) is the distance between actuator mount and landing gear strut mount, \(M\) is the moment arm, and \(\theta_{1}\) is the initial angle of landing gear when gears are in fully extended position. The length of an actuator (\(A_{0}\)) at fully extended position is calculated by setting \(t=0\) in Equation (7) and, at retracted position, the actuator length (\(A_{t}\)) is calculated by setting \(t=t_{max}\).