SECOND ORDER LINEAR Des WITH CONSTANT COEFFICIENTS

Second order homogenous linear differential equation with constant coefficients The general formula for such equation is To solve this equation we assume the solution in the form of exponential function: If then and the equation will change into after dividing by the e λx we obtain We obtained a quadratic characteristic equation. The roots are and

There exist three types of solutions according to the discriminant D 1) If D>0, the roots λ 1, λ 2 are real and distinct 2) If D=0, the roots are real and identical λ 12 =λ 3) If D<0, the roots are complex conjugate λ 1, λ 2 where α and ω are real and imaginary parts of the root

If we substitute we obtain Further substitution is sometimes used and then considering formula we finally obtain where amplitude A and phase φ are constants which can be obtained from initial conditions and ω is angular frequency. This example leads to an oscillatory motion. This is general solution in some cases, but …

Example of the second order LDE – a simple harmonic oscillator Evaluate the displacement x(t) of a body of mass m on a horizontal spring with spring constant k. There are no passive resistances. If the body is displaced from its equilibrium position (x=0), it experiences a restoring force F, proportional to the displacement x: From the second Newtons law of motion we know Characteristic equation is We have two complex conjugate roots with no real part

The general solution for our symbols is No real part of λ means α=0, and omega in our case The final general solution of this example is Answer: the body performs simple harmonic motion with amplitude A and phase φ. We need two initial conditions for determination of these constants. These conditions can be for example The particular solution is From the first condition From the second condition

Example 2 of the second order LDE – a damped harmonic oscillator The basic theory is the same like in case of the simple harmonic oscillator, but this time we take into account also damping. The damping is represented by the frictional force F f, which is proportional to the velocity v. The total force acting on the body is The following substitutions are commonly used Characteristic equation is

Solution of the characteristic equation where δ is damping constant and ω is angular frequency There are three basic solutions according to the δ and ω. 1) δ>ω.Overdamped oscillator. The roots are real and distinct 2) δ=ω.Critical damping. The roots are real and identical. 3) δ<ω.Underdamped oscillator. The roots are complex conjugate.

Damped harmonic oscillator in the Mathematica All three basic solutions together for ω=10 s -1 Overdamped oscillator, δ=20 s -1 Critically damped oscillator, δ=10 s -1 Underdamped oscillator, δ=1 s -1