AE352_F23_HW1

AE352: Aerospace Dynamical Systems (Fall 2023) Homework 1 Due Date: 6 th Sep 2023, 11:59 pm Instructions 1. You are allowed TWO two-day extensions in the semester. You can take an extension for any two homework assignments of your choice, no questions asked. We will keep track of how many extensions you have taken. 2. This homework has 3 problems. Only one problem of the 3 will be graded for correctness, and the remaining problems will be graded for completion (0%/50%/100%). Problem 1 Consider a UAV flying such that its center C is at altitude a ( t ) and at a distance l ( t ) down-range from a payload as shown in Figure 1. Attached to the UAV is a manipulator with a hook at the end (shown at point H ). The manipulator can be extended or retracted to pick up payloads and has length h ( t ) at time t . Its angle with the vertical at any instant t is θ ( t ) . The payload has an attachment point P where the UAV’s manipulator can hook into to it. The point P is at a fixed height p from the ground. We would like to find a set of suitable reference frames to model the scenario where the UAV is trying to pick up the payload with its hook. Figure 1: Payload pickup by UAV. 1. Define intermediate frames such that (a) You start with the frame F = ( O, ⃗ f 1 , ⃗ f 2 , ⃗ f 3 ) as your fixed frame. (b) Your final frame (name it Q ) has one axis aligned (parallel or anti-parallel) with the vector − → CH . (c) You follow the “bridging” technique from lecture notes to define frames between these two frames. That is, each new frame you define must be either (1) a rotation about one axis of the old frame or (2) a translation of the old frame. Tip : When you rotate a frame about an axis, that axis remains unchanged in the new frame! Use the following table format to list out your frames in a structured way. We give an example below. 1

Frame Origin Basis Vectors Rotation or Translation Basis conversion F O ( ⃗ f 1 , ⃗ f 2 , ⃗ f 3 ) N/A N/A G O ′ ( ⃗ f ′ 1 , ⃗ f ′ 2 , ⃗ f ′ 3 ) Translation by −− → OO ′ ( ⃗ f ′ 1 , ⃗ f ′ 2 , ⃗ f ′ 3 ) = ( ⃗ f 1 , ⃗ f 2 , ⃗ f 3 ) H O ′ ( ⃗ f ′′ 1 , ⃗ f ′′ 2 , ⃗ f ′′ 3 ) Rotation ϕ about ⃗ f ′ 3 ⃗ f ′′ 1 = cos ϕ ⃗ f ′ 1 + sin ϕ ⃗ f ′ 2 , ⃗ f ′′ 2 = − sin ϕ ⃗ f ′ 1 + cos ϕ ⃗ f ′ 2 , ⃗ f ′′ 3 = ⃗ f ′ 3 Q ... ... ... ... 2. Express the following vectors in the specified coordinates: (a) { − → OP } F = (b) { − → CH } Q = (c) { −− → OH } F = (d) Write out a simple criterion (an equation) for when the hook H coincides with the payload attachment point at P . 3. In the process of trying to affix itself to the payload, we want the hook to be at zero velocity relative to the payload (otherwise we’re possibly swinging and crashing the hook into the payload at high speed..!). First, find the following velocities, and note that θ ( t ) , h ( t ) , a ( t ) and l ( t ) are all time varying quantities in our problem. (a) ⃗ v Q ( H ) (b) ω Q / F (c) ⃗ v F ( H ) (d) Write a criterion for the velocity of the hook to be zero relative to the attachment point P . 2

Problem 3 Consider a helicopter in a banked turn with fixed bank angle θ . The center of mass H of the helicopter traces a circle at an angular rotation rate of ˙ ϕ rad/s while maintaining a fixed distance R from the point O . The propeller of the helicopter has radius r and rotates about axis ⃗ g 3 (at rotation rate ˙ ψ ). The center P of the propeller is at a distance h from the center of mass H i.e | − → HP | = h . We are interested in calculating the speed of the tip T of the propeller. To do this, we have defined some frames in Figure 3. F = ( O, ⃗ e 1 , ⃗ e 2 , ⃗ e 3 ) is the fixed frame and G = ( H, ⃗ g 1 , ⃗ g 2 , ⃗ g 3 ) 1 is the helicopter fixed frame (it does NOT rotate with the propeller). Figure 3: Helicopter in banked turn. 1. Define additional frames of interest according to the bridging technique discussed in class. One of these frames, call it the propeller frame H , must have its origin at P and one axis matching ⃗ g 3 . The other two axes lie in the plane of the propeller. This frame must also be rotating with the propeller . 2. Find − → ω H / F . 3. Apply Bour’s formula to compute the velocity ⃗ v F ( T ) of the tip T of the propeller. 4. You may have seen in other classes that the velocity of the tip of a propeller must always be subsonic for a variety of reasons. The propeller diameter of a Boeing AH-64 Apache Helicopter is 14.6 m. Assume its rotor speed is 290 RPM during the banked turn. Assume θ = π/6, R = 1200 m, ˙ ϕ = 0.1 rad/s, and for simplicity, h = 0 m . Is the tip speed subsonic 2 ? 1 ⃗ g 2 is not indicated in the figure but its direction should be apparent to you. Why? 2 Assume flight at sea level at standard temperature. 4