Design Project 1 Design of a Cheap Thermal Switch

Transcription

1 Design Project 1 Design of a Cheap Thermal Switch ENGR 0135 Statics and Mechanics of Materials 1 October 20 th, 2015 Professor: Dr. Guofeng Wang Group Members: Chad Foster Thomas Hinds Kyungchul Yoon John Schoellkopff Sean Varley

2 ~ Abstract ~ Contained in this report is a new design specification of a cheap thermal switch where data, calculations, theoretical analysis, and processes used to derive this new design will be detailed thoroughly. For the original design, the switch is deemed closed at 180 degrees Fahrenheit, but for this project, modifications must be made to this original design. More specifically, the cross sectional area of the aluminum strip must be adjusted to allow the switch to be activated at 115 degrees Fahrenheit. Through statics and mechanics analysis such as the effect of temperature on deformations and analyzing a statically indeterminate problem, an equation was derived detailing the ideal solution for the dimensions of the parts to be included in our switch. Throughout this report, those dimensions will be revealed through mathematical deduction with the assistance of free body and multicomponent assembly analysis. ~ Introduction ~ In this project, it is required to design a thermal switch [1] for incorporation into a low cost product. The switch consists of three metal strips clamped rigidly together in a vertical manner with sufficient space between each strip. In the design, the only loads that are induced are caused by temperature changes; hence, there are no initial stresses. When the actuating temperature is achieved, the central aluminum strip snaps aside making contact with one of the outer steel strips. These steel outer strips are electrically conducting where contact with the inner strip closes a circuit. Currently, the switch is designed to close when the temperature increases to approximately 180 degrees fahrenheit; however, it is the goal of this project to design a new switch by only varying the dimensions of the central strip so that the closure of the switch occurs when the temperature increases to approximately 115 degrees fahrenheit. With this switch, as the temperature increases, the lengths of both the aluminum and steel strips must remain equal because of the rigid end pieces. Hence, the compressive force in the aluminum strip increases as the result of a temperature increase. Finally, when the critical temperature is met, the aluminum strip buckles and contacts the rigid steel strip. Therefore, this problem will be approached in such a way that an equation of the critical change in temperature Δ T cr will be derived and checked with the original design for correctness. This formula will then be used to study different designs that satisfy the modified design specifications where in the end the best solution will be provided. It has also been hypothesized that when the cross sectional area of the middle aluminum strip is decreased, the temperature required to buckle the middle strip will also decrease. Additionally, both our hypotheses and future mathematical calculations were generated making several assumptions. Those assumptions include: the plates on the bottom and top remain rigid, the buckling of the aluminum strip does not change its length, the cross sectional area of the strips do not change under temperature change, the thickness of the aluminum strip is less than its width, and we are only varying the thickness and width of aluminum.

3 Figure [1] Pictorial Representation of Thermal Switch ~ Analysis & Design ~ Below in figure [2] is a free body analysis of the thermal switch that will be used in conjunction with the mathematical derivation to determine the dimensions of the new switch. Figure [2] Initial Free Body Analysis

4 Variable Declarations: E = Young s Modulus P = Force w = Width T = Temperature (degrees Fahrenheit) A = Area L = Length t = Thickness α = Thermal Expansion Coefficient ***Note: The subscripts of s and a denote steel and aluminum respectively To begin this segment of the project, an equation specifying the critical change in temperature ΔT will be established in terms of E, E, α, α, t, t, w, w, L. cr a s a s a s a s We first start by making assumptions about the scenario. We assume that the net force applied to the rigid plate must be zero, otherwise the plate would move. This results in our first equation which defines a relationship between the force applied by each of the strips. Another assumption that must be made is that the plate is fully rigid, therefore the change in length of the aluminum strip must equal the change in length of each of the steel strips. Change in length for either strip is equal to the sum of the change due to mechanical stress and the change due to thermal stress. Combining these equations, and eliminating length from each side gives our second equation, which relates the force on each beam to the change in temperature. Our next equation comes from the area of each of the strips. The cross sectional area of each strip is given by multiplying its width and thickness. These results give our third and fourth equation. Equation (5) was derived from the substitution of the minimum second moment of

5 inertia of the cross sectional area into the critical axial compressive load. Finally, the critical axial compressive load was determined to be the opposite of the compressive load of aluminum. Our next step in solving for ΔT cr is to solve for ΔT in equation (2). This gives us a result for Δ T in terms of our forces and material properties. This equation is then combined with (1) in order to give us a result, equation (6) that is only in terms of the load on the aluminum column and the material properties. Equation (7) is created by combining equation (6) with equation (5). This gives a result that is only based on material properties and sizes. This is what we were looking for.

6 Further algebraic work is shown which simplifies equation (7) to its final state, which makes solving for Δ T easier and this is also our final equation which will determine the dimensions outlined in the discussion. In order to ensure that our equation make sense, we will substitute the original specifications of the design into the equation and solve. Δ T =(( * 1/16 2 )/ 3*4 2 )*((10,000*¼*1/16)/(2*30,000*⅛*1/16) + 1 )*(1/(12.5x x10 6 ) Δ T degrees F ahrenheit ~ Discussion ~ The thermal switch operates by expanding three separate metal bars that are bounded by two rigid plates. Due to the unique thermal properties of the system, the steel bars expand less quickly than the aluminum bar as temperature is increased. Since the aluminum bar expands but is restricted by the expansion of the steel bars and the rigid plate boundaries, the aluminum bar buckles. During buckling, the aluminum bar comes in contact with one of the steel bars, completing the circuit and powering the switch. Given the equations above, the exact cross sectional dimensions of the aluminum bar were found in order for the device to activate at 115 degrees Fahrenheit. The equation that was found to relate the critical change in temperature to the properties of the bars of the device was extremely accurate, since the predicted critical change in temperature of the initial dimensions was 181 degrees Fahrenheit. The change in temperature that was given was 180 degrees Fahrenheit (1 degree difference). This proves that the equation is valid and can be applied to determine the dimensions of the aluminum based on a new change in critical temperature. The hypothesis mentioned earlier regarding the decrease in the cross sectional area was correct. The initial area of the aluminum bar was inches by inches. The resulted area was 0.05 inches by 0.3 inches. Since both the thickness and width decreased, the area decreased as well. The solution was reached through physical concepts of statics and mechanics of materials. Specifically, using the definition of a rigid body, the solution proved that the deformations of the steel and aluminum bars were equal. Also, force equilibrium was used to

7 prove that the force of the steel bar was equal to negative one half of the force of the aluminum bar. Finally, the force of the aluminum bar was proved to be equal to the critical force at the aluminum bar when it is buckling. This is correct because the aluminum bar buckles at a far lower temperature than steel. Therefore, the force by which the aluminum bar buckles is the force on the aluminum bar due to temperature change. In order to calculate the final dimensions of the aluminum cross section, the final equation must be solved for w s. Using algebra, the equation simplifies to In order for this equation to be positive, Solving the inequality for t a, After substituting the known values for ΔT, α, and α, t cr a s a must be less than.0544 inches. From this finding, we arbitrarily set the thickness of the rod to.05 inches, and subsequently solved the equation for w a, which produced a width of.3 inches. These results are consistent with the constraints we assumed, including w a >t a. We then substituted these dimensions into the simplified equation to find the change in critical temperature, which resulted in 115 degrees Fahrenheit, which confirms our finding. ~ Conclusion ~ Through the various calculations and data analysis of a cheap thermal switch, a mathematical model was produced resulting in several modifications that would lower the switch's activation temperature to 115 degrees Fahrenheit. Through analysis, it was determined that as the dimensions of the middle aluminum section are decreased, the force required to buckle the aluminum beam lowers accordingly. As our results in previous sections indicate, it was determined that dimensions of 0.05 inches by 0.3 inches will produce the 115 degree Fahrenheit activation requirement established in the introduction.