In the past two weeks, I have finished Unit 1 and moved on to Unit 2. Starting here, I will be working with Second Order Differential Equations with constant coefficients. In this blog, I will briefly introduce a method to solve these equations and move on to the Spring Model.
Introduction to Second Order Differential Equations with Constant Coefficient:
The Second Order Differential Equations means the equation now has the second-order derivative of the independent variable. Here is the standard form of a Second Order Differential Equation:
In this standard form, t is the independent variable, y is a function of t, and A and B are constants. In this blog, we are only working on a homogeneous equation, so the right-hand side is 0 instead of another constant. If you remember integrating factor from First Order DE, note that the same method can no longer lead us to a solution, as we can no longer transform the left-hand side into a form of . To solve this new system, we have to use another tool, Characteristic Equations.
The idea of characteristic equations is first to find specific solutions for y. Then based on the given initial conditions or identity of the equation to derive the general solution. More than often, we would first try .
Apply this to all terms involving y, we have:
The standard form then transforms into
If you are familiar with the graph of y=e^x, you probably remember that the curve never intersects with the x-axis, which means that it never equals to 0. As a result, must equals 0. In fact, is called the characteristic equation of a Second Order DE with constant coefficients. In a scenario when A and B are given, we can solve for r. Then, we can plug our answer(s) back into the original equation to find solutions for y. If you are interested in knowing more details about his method or solving an example problem, please visit here.
The Spring Model:
The spring model is the first place where we need to use the Second Order DE. Imagine there is spring with one side attached to a wall and the other side attached to a block. Apparent, when we pull/push the spring to either direction and release, the block will oscillate back and forth. In fact, if there is no friction or no internal energy loss, the block will oscillate forever! But why is it so? I will introduce the model using the Second Order DE today and analyze its movement in my next blog.
If you remember the materials from your Physics class, the restoring force of a spring is:
In this equation, k represents the spring constant, and x represents the distance of the block from its equilibrium position.
From Newton’s Second Law, F=m*a, in which . By plugging a(t), we would have . This is also known as Hooke’s Law.
In this blog, however, we are considering a more complex scenario, in which there is a dashpot on the other side of the block. Therefore, we need to introduce the damping constant, b.
As we only have linear damping, we can easily find
Therefore, we have the following Second Order DE with constant coefficients:
If we divide all terms by the factor of m, we will have the following standard form
When the block is not speeding up or slowing down (in other words, at rest or moving at a constant speed), we will have solvable homogeneous form.
Now, we have found the DE for spring and dashpot. When k, b, and m, are given, we can solve this equation to find the position of block at any specific time.
In the next few weeks, I will keep working on Second Order DE. In specific, I will work on Harmonic Oscillators and Superposition. Once I build up my knowledge, I will come back and further analyze the Spring Model and its characteristics, including how and why it oscillates in an ideal scenario.
Arthur Mattuck, Haynes Miller, Jeremy Orloff, and John Lewis. 18.03SC Differential Equations. Fall 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.
Tseng, Z. S. (2008). Notes-2nd order ODE pt1. Retrieved October 21, 2018, from Penn State University Department of Mathematics website: http://www.math.psu.edu/tseng/class/Math251/Notes-2nd%20order%20ODE%20pt1.pdf
Second order physical systems [Photograph]. (n.d.). Retrieved from https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-ii-second-order-constant-coefficient-linear-equations/modes-and-the-characteristic-equation/MIT18_03SCF11_s12_1text.pdf
10 ways to play and learn with springs [Photograph]. (2015, March 10). Retrieved from https://leftbraincraftbrain.com/10-ways-to-play-and-learn-with-springs/