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_______________________________________ Design of a Bi-Metallic Strip for a Thermal Switch Statics and Mechanics of Materials II 4/13/15 William Slaughter Arielle Iacocca, Sean McCarthy, Hunter Christman, Matthew Nitkiewicz _______________________________________ Abstract This report covers the requirements needed in order to select a bi-metallic strip for a thermal switch to close an electrical circuit. The Introduction section covers a broad description of the project itself as well as any needed constraints. The Analysis and Design section covers the needed calculations, figures, etc. in a step by step process in order to cover the selection of materials. The Results section displays our final results with the addition of a discussion concerning any shortcomings of the switch. Introduction In this particular project the objective was to design a bi-metallic strip for a thermal switch. The switch will be used to open or close an electrical contact at a predetermined temperature The constraints of the project is that the strip must be 3 in. long, with one end fixed, similar to a cantilevered beam. Strip 1 is to have a height of h1and strip 2 is to have a height of h2. The width of the strips will be the same and set to the variable b. The free-end of the beam is to have a deflection of .05 in. when there is a 50∘ F change in the temperature, relative to its reference temperature. After all variables are set accordingly, the two materials used must be specified along with the respective dimensions for the bi-metallic strip. The project’s success is measured by the fact that it is able to close an electrical circuit by deflecting .05 inches without fracturing. In addition, the strip should be relatively easy to manufacture and should be of suitable size for use in a thermostat. Analysis & Design In order to achieve the desired actuation under the change in temperature, an equation was needed for the end deflection of the beam in terms of material properties and dimensions. Finding this relationship was the primary aim of the analysis. To start, the strains due to thermal expansion and the stress due to the welding constraint was considered. The thermal and stress related deformations were considered by assuming that the first strip has the higher thermal expansion coefficient, as shown in Figure 1. Figure 1 Since the final amount of deformation of the two strips must be equal, Equation 1 can be written. Equation 1 The only variable in equation 1 that is not clearly defined is the resulting tension and compression in the strips. Note that to maintain force equilibrium, the two forces must be equal in magnitude. The magnitude of these forces was then solved for. To make expressions more compact, the misfit strain was defined as shown in Equation 2. Equation 2 Equation 3 Figure 2 As is evident in Figure 2, the forces create a moment about the center of the strips whose magnitude is given by Equation 4. The two forces act along the neutral axes of each strip. Note that throughout this analysis, the actual heights of each strip are undetermined. To maintain moment equilibrium about point O, a clockwise moment needs to be created that is equal in magnitude to the moment generated by the force couple. Equation 4 Considering the two strips as a single cantilever composite beam, the deflection can be related to the bending moment at point O. Using a table of calculated beam deflections and noting the clockwise direction of the moment that resulted from the force couple, Equation 5 can be written. It is important to realize that the equation given by the table only applies to beams made of a single material. Equation 5 To use Equation 5, the composite beam needs to be transformed into a beam with an equivalent flexural stiffness which only consists of a single material. The top strip material was chosen for the transformed section and a variable n was introduced as a ratio of the Young’s Moduli of the two materials. Equation 6 Figure 3 Figure 3 shows the geometries of the transformed cross section of the beam that represents the two strips. By assuming that the two heights of the strips would be somewhat similar, the neutral axis for the transformed section was drawn somewhere in the lower half of the beam. This neutral axis intersects the centroid of the area which can, after some simplification, be calculated by Equation 7. Equation 7 As is shown in Figure 3, Equation 7 was calculated with the zero point for y occurring at the base of the area. From here, the second moment of area can be calculated with respect to the neutral axis, making sure to keep in mind the geometric assumptions as made in Figure 3. Equation 8 Note that in Equation 8, yc still needs to be substituted in from Equation 7. Once that is done, the moment of area will be defined in terms of dimensions and material properties as desired. Furthermore, all variables that need to be substituted into Equation 5 are also in the proper format. Equation 3 can be substituted into Equation 4 to obtain the bending moment, which can be inserted into equation 5. The flexural stiffness, EI, given by Equation 6 and Equation 8 can be inserted into the denominator of Equation 5. After doing some lengthy algebra, the equation simplifies as shown in Equation 9. Equation 9 Equation 2, which defines the misfit strain, should also be substituted in within Equation 9. With the maximum deflection of the two strips given in terms of strip dimensions, material properties, and the change in temperature of the strips, analysis was able to be done by setting different values for these parameters and considering the actuating conditions for the deflection and change in temperature. Results After plugging in different types of common gauges of metals with a set of extra calculations to help adjust how much we are off, we finally came up with steel and zinc being perfect for the design. The zinc is strip will be .02 inches thick and the steel portion is going to be .0134 inches thick. This corresponds to 10 gauge sheet zinc and 29 gauge stainless steel. The width of the strip will be one tenth of an inch and the length will be three inches. The width was chosen as to make it small enough to fit inside a standard thermometer. Strip: Zinc: 3 by .02 by .1 inches Steel: 3 by .0134 by .1 inches Some potential shortcomings with the design is that the strip is very thin, so the switch may break if an outside force is applied. There can be defects in the metals or at the joinings causing the strips to slide over one another instead of bending, or they may form cracks and break. At very high temperature the metals will start to have plastic deformation and at lower temperatures the strip is more prone to fracture.