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Me 163 Spring 1997 Directions for Project ME 163 SPRING 1997 DIRECTIONS FOR PROJECT GENERAL DESCRIPTION OF PROJECT In this project, you are asked to make a design decision for a vibration-reducing mount for a delicate object subjected to vibrations. The object is mounted on a spaceship floor which is vibrating. The mount for the object consists of a spring and damper. Your task will be to choose the spring constant of the mount so that both the relative displacement and absolute acceleration of the object fall within given limits. You will need to make full use of the material presented in class on sinusoidally forced vibrations. In the analysis part of this project, you will derive formulas for the amplitude of both the relative displacement and absolute acceleration of the object. In the design part of the project, you will use those formulas (and Mathematica) to make your design choice. PRACTICAL MATTERS You may work alone, or in groups of two. You may discuss the project freely with others. If you get significant help from someone else, that fact should appear as an acknowledgment or reference in your report. Any books or articles that you use should also be referenced. Your report should be brief but self-contained, so that a reader not having access to these directions will know exactly what you have done. A typical report organization might be something like the following: Introduction (with a general description of the problem); Formulation (the detailed quantitative formulation of the design problem); Analysis (derivation of the basic formulas to be used in the design calculations); Design Results (your final design choice, along with the calculations that support your choice); Summary (a summary of your design choices, along with a mention of any issues you think need to be considered in future work on systems of this type); References (books, articles or people consulted). Although typed reports are preferred, neatly handwritten reports are acceptable. Your project grade will be based on both the write-up (20%) and the technical content (80%). DETAILED DESCRIPTION OF PROJECT PROBLEM Captain’s Log, Stardate 7304.6 Background You are a second-year Starfleet Academy Cadet, Division of Mechanical Sciences, and you are aboard the U.S.S. Enterprise on your first deep-space cruise. The Enterprise is making its third visit to the Glia starsystem, a study which began on Stardate 5526.3 with an examination of an ecosystem on Glia-2, and which continued on Stardate 6872.1 with the launch of an unmanned ME 163 STARDATE 7304.6 PAGE 2 probe to the large airless planet Glia-4. One of the important events in the second expedition was the discovery of naturally occuring dilithium crystals (used in warp propulsion) on Glia-4. The focus of the present expedition is the development of a method to transport these crystals without damage from Glia-4 to the Enterprise. The basic problem is that the engine vibrations in the shuttlecraft Jade that goes between Glia-4 and the Enterprise are sufficiently great to damage the dilithium crystals. (The transporter cannot be used for dilithium because it converts dilithium into para-dilithium which does not work in warp propulsion). Your assignment is to design a spring- damper mount to protect these crystals while they are being transported. Dilithium Crystals and the Vibration Problem The dilithium crystals are very sensitive to vibrations. The Jade engine is a little out of synch, and produces unacceptably large vibrations which, much to the disgust of Mr. Scott, destroyed the first two batches of crystals ferried back to the orbiting Enterprise. To fix this problem, you have been asked to design a spring-damper mount to be attached to the floor of the Jade shuttlecraft. The purpose of this mount is to reduce the vibrations reaching the crystals. Vibration Analysis The situation is shown in Figure 1. We are given that, because of the engine vibrations, the displacement of the Jade floor from an inertial reference frame is =γ YtFF( ) A cos( t ) . (1) γ Thus the floor is vibrating with an amplitude AF and an angular frequency . The displacement of the dilithium container relative to the floor is denoted by YR(t). We let the equilibrium displacement (with no vibration) be the constant YE, and we let =− Wt() YRE () t Y (2) be the displacement from equilibrium. By finding the periodic response to the forced vibration, show that the amplitude of the periodic displacement of the container relative to the floor is given by γ 2 A = F WP 12/ . (3) k 2 γb 2 − γ 2 + m m Consider now the absolute acceleration of the dilithium container (the acceleration relative to the inertial frame). Show that the amplitude of this acceleration is given by 12/ k 2 bγ 2 aW= + , (4) PP m m where WP is given by equation (3). These formulas are the basis of the design calculations in the next section. ME 163 STARDATE 7304.6 PAGE 3 Figure 1. The dilithium container D has a total mass m, including the dilithium. The container is mounted to the floor F of the Jade shuttlecraft. The mount has a spring constant k and a damping constant b. The displacement of the container relative to the floor is YR. The displacement of the Jade floor from an inertial reference frame R is YF. System Parameters and Design Choice Science Officer Spock makes one trip in the shuttlecraft and takes measurements which show that the troublesome vibration has the following characteristics: ==γ -1 AF 0., 1 mm 500 s . (5) The mass of the dilithium plus container is given by m = 2 kg . (6) The dilithium crystal can be damaged by oscillatory inertial forces. In order to avoid this damage, we must insure that the amplitude of the absolute acceleration satisfies < 2 aP 22 m/ s . (7) ME 163 STARDATE 7304.6 PAGE 4 There is a second constraint on the displacement of the crystal. The field generated by the crystal will interfere with the shuttlecraft’s electronics if the crystal moves appreciably relative to the shuttlecraft. The maximum allowable displacement is only 90 µm, so we must require that < µ WP 90 m . (8) In principle, you can adjust both the spring constant k and the damping constant b in meeting these conditions. However, Captain Christopher Pike of the Enterprise, who fancies himself to be an engineering expert, tells you that when he was in school, he was taught that all vibration reducing mounts should have a damping parameter ζ = 0.8, where b ζ= . (9) 2 km Rather than offending the Captain and possibly ending your career almost before it has started, you accept the constraint (9). Now as you vary the spring constant k, you must also vary the damping constant b in such a way that ζ = 0.8 always, to keep the Captain happy. Your design task is to find a value or values of k which will satisfy all of the constraints. As a start, you might show that rigidly mounting the container to the floor of the Jade shuttlecraft will not work. When you have found your design value or values of k, use Mathematica to construct graphs illustrating both your design choice and the performance of the system. Be sure to verify that the constraints of equation (7) through (9) are satisfied. What effect do you think the Captain’s constraint (9) had on your solution? WORK SCHEDULE Intensive dilithium mining is scheduled to start on Stardate 7329.4. Therefore your final design report has to be submitted to Starfleet Senior Science Officer Clark by Stardate 7323.6 (Rochester time: Monday May 5 by 6 PM EDT). Do not fail to meet this deadline. The dilithium supply of the entire Federation depends on your successful work, as does your promotion to third- year Cadet status at Starfleet Academy..
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