Develop a function that will simulate and animate the unsteady behavior of a pin-pin beam. Then use an AIAA format report to answer the following five questions. Use comments for any notes or distinguishes within the code and follow the rubric.
Microsoft Word - AERO 300 Laboratory 8.docx AERO 300 Laboratory Unsteady behavior of a pin-pin beam The lab (Section 5) will be due before your next lab section (.zip file submitted to PolyLearn). 1 Objectives This lab introduces students to the concept of structural vibration and the AIAA meeting papers format. Using the programming skills from previous labs, students will create a function that simulates deflection over time for a vibrating beam. Conclusions will be made in an AIAA format report. 2 Introduction You are the structural designer of a new long-range communication spacecraft. The spacecraft has a main body where guidance, control, and propulsion systems are designed. The spacecraft is very large due to customer-required performance, and cannot fit in the launch vehicle! As a result, the spacecraft is split into a main body and a deployable antenna and electronics systems that are stored on the end of a telescoping boom approximately 3.14 meters (it is exactly pi meters) in length. Because the antenna/communication module is so large, it weighs nearly the same as the main body, so the beam supports equal weights at both ends. Attachments to the beam end links are via pin connections. You are designing the beam to withstand micrometeorite attacks. There are two things you must consider: deflection and vibration. If the beam deflects too much, it will fracture. However, if the beam's frequency is too high, it will delaminate and shatter because it is a composite beam. You identify the worst case for deflection as a single micrometeorite impact at the center of the beam. As the meteorite hits, its energy is dissipated into the beam, resulting in the following deflection: ?(?) = sin(?) (1) The length of the beam is pi and that ensures that the two modules remain stationary with respect to each other. Because the beam has one full arc, or half of a sin wave, this case is the first mode. You identify another case, where two meteorites traveling in opposite directions impact the boom, causing a deflection: ?(?) = sin(2?) (2) Because the beam has two full arcs, or one full sin wave, this case is the second mode. The deflection for both cases is determined to be less than the maximum allowable. However, the vibrational behavior is still unknown. If we wanted to express the initial deflection for any mode m, we would write: ?(?) = sin(??) (3) 3 Procedure Finite differencing is a way to express derivatives and find approximations to PDEs or ODEs at discrete points. For our unsteady beam problem, we will start with an initial deflection and velocity and allow the beam to vibrate over time. We will break up the beam into points and use finite differences along the axis of the beam to describe it in space, and do the same with time to describe the beam over time. Finite differences are typically broken down into forward, backward, and central differencing. While forward and backward differencing use points in front and behind the current point to estimate the derivative, central differencing uses points behind and in front simultaneously. We will use central differencing in space and backward differencing in time. Thus, we will need boundary conditions on both sides of our beam along with an initial condition. Because our beam is of pin-pin type, we have zero moment and zero deflection at the ends. Our initial condition is given by the sine wave of Eq. (3) and an initial velocity. Since we have our initial position and the derivative at that point (the initial velocity), we can do a first order projection to a new time step. For high accuracy, we will use 5 points from a previous time step to define a new time step. Let x(i,j+1) represent the current position, but for a new time step. We can express it in terms of several positions from the current time step as shown in the stencil below. In equation form, this becomes: (4) where Q is a constant that determines the size of the step that is taken. In Eq. (4), the represents data at the current time step and the left-hand side is the updated value after a timestep. Note that Q must be small and negative for stability. Equation (5) displays the relationship between Q and the spatial and temporal step sizes: (5) where C is a constant = -9.3979e-6. If Q becomes positive or too large, the solution will be unstable. The spacing, , is determined by the length of the beam divided by the number of points minus one. Once the spatial step size ∆x is known, the temporal step size can be determined as ∆t = .Q?Δx3 (6) A good value for Q is -0.2 and using this you can get an initial Δt. If your time step is not constant, then you can recalculate Q remembering that a negative, small value for Q insures stability. Equation (4) gives us first order accuracy in time and only requires 2 initial conditions (position and velocity). After the first new time is created however, we now have displacements at two previous times. For the third time step onward, we can write the new time step as a function of two previous time steps, giving us second order accuracy in time. The figure above then becomes: And the equation becomes: (7) The last condition to satisfy is along the spatial boundaries. Let's look at i-2, the column just next to the boundary condition. For the i-2 term we look for W(0,j), which does not exist and is outside our beam. This is also the case on the opposite side of the beam with the i+2 term. However, because we have a pin-pin beam, the moment at the end is zero and the curvature is therefore zero. So, we can write the following relations; one for the left side of the beam and one for the right: (8) (9) 4 Pre-lab Assignment There is no pre-lab for this assignment. 5 Lab Assignment 1. Develop a function that will simulate and animate the unsteady behavior of a pin-pin beam: function [freq] = beamvibe(mode, n_pts, plots) where l freq is the frequency of oscillation, in Hz (cycles per second) l mode is the number of sin half cycles in the initiation condition l n_pts is the desired number of points l plots is a switch to turn the plots off. 'yes' means plot, 'no' means don't plot (hint: you may find the function strcmp useful) Use an AIAA format report to answer the following questions. 2. For the first mode (y = sin(x)), animate the beam's behavior over 0.1 seconds using an adequate number of points. What is the frequency of vibration? Repeat this process for the two impact (y = sin(2x)) case. What is the frequency of vibration now? 3. Now manually force your Q to be positive. What happens? 4. Repeat the second mode calculation for a longer period of 1 second and animate. Does the structure appear to have any internal damping? Is this realistic or accurate? 5. The maximum allowable frequency is 180 Hz (cycles per second). Will the structure survive the two-meteorite impact? Why/why not? If not, what could be done (either to the structure or your analysis) to correct this? 6. In the introduction, we noted that the deflection due to the two-impact collision would not break the structure. What does your answer to #5 tell you? Are deflections and steady state stresses the only thing you should consider in the design of a structure? 7. Use your function to plot frequency vs. mode for modes 1 through 20, with frequency on a log scale. Make sure you use enough points to capture 10 full sine waves. Title your plot/axes as well. Give thought to this problem. Put thought into the relationship. What can you say about the beam as the mode of vibration goes up? Think in terms of speeds, accelerations, forces, and stresses. 6 Material to be submitted Your code must be submitted on PolyLearn before the next lab. However, in addition to the code, you must also submit an AIAA report answering the questions in section 5. The AIAA technical paper format and template can be found here: https://www.aiaa.org/publications/journals/Journal-Author Your report should have the following: Abstract: The abstract should summarize your entire report. Give what you are doing, why, and a summary of your important results. All of this should be done briefly. Your abstract is what people will read quickly to determine if the rest of your paper is worth diving into. Introduction: The introduction should give the background on your subject. Why are you doing what you are doing? What knowledge is gained? Methods: Summarize how finite differencing works and what orders of accuracy you used. How did you implement it? You do not need to go through the details of all the equations, etc. Assume you are presenting at a structural dynamics conference where the audience is familiar with finite difference methods. Similarly, you do not need to explain every