-x(t) wo 1 m F(t) C In the figure the variables and parameters have the following meaning: • m = mass (kg) • c = damping

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-x(t) wo 1 m F(t) C In the figure the variables and parameters have the following meaning: • m = mass (kg) • c = damping constant (kg/s) - proportional to the speed of m k = spring constant (N/s/s) - proportional to the distance m is from it's • X(t) = position of m as a function of time • F(t) = A force applied to mas a function of time . In the background notebook, we determined that: * + 260,* +62x F(1) = m

1. Write python code that uses RK 4th order to solve the ODE above. For the first part of the project, you will assume F(t) is zero. Later you will be using F(t), so plan accordingly.

m Background - Unforced Systems (F(t) = 0) Applying Newton's Second Law to the system in the figure + 4x + kx = F) Introducting a dot notation for the time derivatives and dividing through by me gives *+*+ ** = F One thing of interest is to solve for x() the homencis case that is when FO) = 0. The easiest way to do this is to propose that the solution is of the form()cell where and are unknown With this proposed solution & C# and x = Che Substituting gives cicek + Ce derde + hende = 0 (1? + A + Cel = 0 Sinice Cen) would be a trivial solution, we conclude that 12 + 2+ = 0 Before proceeding it's good to introduce a quantity that is really important for oscillating/ibrating systems, the natural frequency. This is the angular frequency that an undamped system will tend to vibrate at (see other rets for examples of this);

From the quadratic formula 2 -1934 V-o. The value of the damping constant c that would cause the square root to go away in the equation above would be c = 2mm. This is known as the critical damping constant, ce critical dormping is when a system as shown in the figure is perfectly damped so that an oscillation can only begin to happer, but then the dampet just barely stops the oscillations So let's define the ratio of the actual damping constant c to the critical damping constant to be be the dimensionless damping coefficient So that ce i=- + a) + V 2. Finally i = -50, a.VE-T Note that and are based on positive constants, chand and so we can assume they will also be positive