matplotlib can be installed via pip

HarmonicOscillator.py

import matplotlib.pyplot as plt
import math


def SmoothUpdate(objPos, objVel, finPos, deltaTime, dampingRatio, angularFrequency):
    """
    This function is a derivative of work done by:

    Copyright (c) 2008-2012 Ryan Juckett
    http://www.ryanjuckett.com/

    This software is provided 'as-is', without any express or implied
    warranty. In no event will the authors be held liable for any damages
    arising from the use of this software.

    Permission is granted to anyone to use this software for any purpose,
    including commercial applications, and to alter it and redistribute it
    freely, subject to the following restrictions:

        1. The origin of this software must not be misrepresented; you must not
         claim that you wrote the original software. If you use this software
         in a product, an acknowledgment in the product documentation would be
         appreciated but is not required.

        2. Altered source versions must be plainly marked as such, and must not be
         misrepresented as being the original software.

        3. This notice may not be removed or altered from any source
         distribution.
    """

    if dampingRatio < 0 or angularFrequency < 0:
        print("The damp ratio and angular frequency must be greater than 0.")
        return

    #initial position and velocity relative to the finish position
    x0 = objPos - finPos
    v0 = objVel

    if dampingRatio > 1:
        #over-damped equation
        za = -angularFrequency * dampingRatio
        zb = angularFrequency * math.sqrt(dampingRatio*dampingRatio - 1)
        z1 = za - zb
        z2 = za + zb
        exp1 = math.exp( z1 * deltaTime )
        exp2 = math.exp( z2 * deltaTime )

        #update motion
        c1 = (v0 - x0 * z2) / (-2 * zb) # z1 - z2 = -2*zb
        c2 = x0 - c1
        return finPos + c1*exp1 + c2*exp2 #objPos =
        #objVel = c1*z1*exp1 + c2*z2*exp2
    elif dampingRatio == 1:
        #critically damped equation
        exp1 = math.exp( -angularFrequency * deltaTime )

        #update motion
        c1 = v0 + angularFrequency * x0
        c2 = x0
        c3 = (c1*deltaTime + c2) * exp1
        return finPos + c3 #objPos =
        #objVel = (c1*exp1) - (c3*angularFrequency)
    else:
        #under-damped equation
        w = angularFrequency * dampingRatio #omegaZeta
        a = angularFrequency * math.sqrt(1 - dampingRatio*dampingRatio)
        exp1 = math.exp( -w * deltaTime )
        cos1 = math.cos( a * deltaTime )
        sin1 = math.sin( a * deltaTime )

        #update motion
        c1 = x0
        c2 = (v0 + w*x0) / a
        return finPos + exp1*(c1*cos1 + c2*sin1) #objPos =
        #objVel = -exp1*( (c1*w - c2*a)*cos1 + (c1*a + c2*w)*sin1)

plt.grid(True)

#x axis
xrange1 = [(2*math.pi*i)/32 for i in range(64)]

#y axis
yrange1 = [math.cos(i) for i in xrange1]
#yrange2 = [0 for i in xrange1]
#yrange2 = [math.cos(i)/math.pow(math.e, i) for i in xrange1]
#yrange3 = [-i/(2*math.pi) + 1 for i in xrange1]
#yrange4 = [math.cos(i)/math.pow(math.e, i) for i in xrange1]

yrange2 = [SmoothUpdate(1, 0, 0, i, 2, 1) for i in xrange1]
yrange3 = [SmoothUpdate(1, 0, 0, i, 1, 1) for i in xrange1]
yrange4 = [SmoothUpdate(1, 0, 0, i, .75, 1) for i in xrange1]
yrange5 = [SmoothUpdate(1, 0, 0, i, .5, 1) for i in xrange1]


plt.plot(xrange1, yrange1, "-", xrange1, yrange2, "-", xrange1, yrange3, "-", xrange1, yrange4, "-", xrange1, yrange5, "-")
plt.show()