![]() In addition, an approximate sine-shaped curve shape was obtained, based on a 4th order approximate differential equation applicable for small angles. The exact differential equation for a bent ruler was derived and solved numerically. From the figure, it is clear that the numerical approximate solution overestimates the midpoint deflection and that the relative error is about 10% for an axial displacement of 90 cm (i.e. Here is the numerical sine-shaped approximate solution and is the numerical solution of the exact differential equation. In figure 13, the relative midpoint deflection, i.e. This can be compared to equation ( 28) which for L 0 = 2 m can be rewritten to, using SI-units. Axial displacement, a (cm)Ī simple curve fit using the least square method to a function of the type A( a) = Ca D gave the values C = 0.85 m 0.5 and D = 0.48 for the experimental data presented in figure 12. Table 1. Experimental data for wooden folding ruler. Thus, the high school students may in the recent future hopefully be provided by numerical tools and knowledge to form a foundation for solution of more complex problems in physics and mathematics. In Sweden, programming and use of symbollical and numerical mathematical tools will be included more in the high school courses in mathematics, according to a new regulation from Skolverket (The Swedish National Agency for Education). This simple expression could be used as a rule of thumb for carpenters to determine the midpoint deflection when the axial displacement is known or vice versa. A result of the investigations described in this report is that the midpoint deflection of the folding ruler can be obtained using a simple analytical expression, if the axial end displacement is known. Ī problem which sometimes occur for carpenters is that the folding ruler is bent. the length of the elastica is not assumed constant) is presented by Magnusson et al. ![]() For a historical review of the elastica, its solution and applications, see. In 1859 Kirchoff discovered that the same differential equation can be used to describe the motion of a simple pendulum. The mathematical problem of describing the curve shape of a bent elastica (e.g. In the present paper, the physics behind the phenomenom is studied, using the exact differential equation and some more approximate methods, together with experiments.Įlastica is a latin expression, which can be translated as 'thin strip of elastic material'. Gere and Timoshenko for a general introduction to solid mechanics and Timoshenko and Gere or Nguyen for more detailed studies of stability problems in solid mechanics. This type of stability problem is denoted buckling in solid mechanics. However, if the force is increased above a critical value, the ruler is suddenly bent and then a small increase in compressive force yields a large increase in deflection. If the compressive forces are small enough, there is zero transverse displacement (deflection). This curve shape is denoted catenary and is described mathematically by a hyperbolic cosine function.Īnother curve shape is formed by a ruler subjected to axial compressive forces at both ends. One of them is the curve shape of a rope or a metal chain fixed at both ends and hanging freely, only influenced by gravity due to its own mass. There are several different curve shapes occurring in nature. The experimental investigations, the algebraical expressions and the numerical simulations can be useful in high school teaching and at undergraduate university level. It is found that the algebraic sine-shaped solution gives reasonable results for moderate axial displacements. A comparison between measured data for a bent ruler, the numerical solution of the exact differential equation, the numerical solution of the approximate equation and the algebraic expression is presented. In addition a simple algebraic relationship between the midpoint deflection and the axial displacement is derived. An expression for the length of the bent ruler is derived and using the constraint that this length is constant, an equation for determination of a relationship between the midpoint transverse displacement (deflection) and the known axial displacement is obtained. Using this equation together with appropriate boundary conditions, an approximate curve shape for the ruler is determined, which turned out to be a sine function with no dependence on material parameters or cross-sectional data. This exact differential equation is then simplified to a fourth order differential equation using a small angle approximation. ![]() The exact nonlinear differential equation for the bent ruler is derived and solved numerically. A ruler, where one of its ends is axially displaced a known distance, is studied using analytical, numerical and experimental methods. ![]()
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