Compute the natural frequency and plot the solution of a...

Compute the natural frequency and plot the solution of a spring-mass system with mass of $2 \mathrm{~kg}$ and stiffness of $4 \mathrm{~N} / \mathrm{m}$, and initial conditions of $x_{0}=1 \mathrm{~mm}$ and $v_{0}=0 \mathrm{~mm} / \mathrm{s}$, for at least two periods.

Key Concepts

Initial Conditions

In the context of oscillatory systems, initial conditions—specifically the initial displacement and velocity—are used to determine the amplitude and phase of the simple harmonic motion. They are critical for deriving the specific solution to the differential equation governing the system.

Period of Oscillation

The period of an oscillatory system is the time taken to complete one full cycle and is inversely related to the natural frequency by the relation T = 2?/?. Understanding the period is essential when plotting the motion for a given number of cycles or time intervals.

Natural Frequency

This concept refers to the inherent oscillation rate of a system, calculated for a spring-mass system using the formula ? = ?(k/m), where k is the stiffness and m is the mass. It is a fundamental property that determines the period and behavior of the oscillatory motion in the absence of any damping forces.

Simple Harmonic Motion

This is the type of motion exhibited by an ideal spring-mass system, where the restoring force is directly proportional to the displacement. The resulting differential equation leads to a solution expressed as a combination of sine and cosine functions, which describes the periodic oscillations of the system.

Transcript

00:01 Okay, from the example, we know that our general solution is going to take the form y of t is equal to c1, cosine omega, not t, plus c2 sine omega, not t.

00:18 So then now we need to solve for c1 and c2.

00:23 So y prime of t this is going to be equal to so c1 or sorry omega not c1 and then negative that so negative omega 0 c1 cozyne omega not t and then plus omega not c2 sign or sorry not sign this should be sign c1, sine, and this should be cosine omega -not t.

00:57 So remember, we're plugging in y of 0.

01:01 So y of 0, it's just going to be equal to c1, which is equal to y -not, according to this initial condition.

01:09 Then we also need to plug in y -prime of 0.

01:11 So this goes to 0, and then this is just 1 here.

01:16 So we get omega -0 -c2.

01:18 So this is equal to omega -0, c2, is equal to v -0 because of this initial condition here.

01:28 So then c -2 is going to be equal to v -0 over omega -0 like so.

01:34 So our general solution, y of t, is going to be equal to y -not, cosine omega -0 -t, plus c2, or sorry, plus v -0 divided by omega -0, sine omega -0 t and then now the circular frequency is just going to be omega -not right omega -o -not is our circular frequency so omega -not equals well omega -not right then next we're going to find the amplitude so the amplitude has the formula okay a knot is equal to the square root of c1 squared plus c2 squared.

02:30 So plugging in our c1 and c2, remember c1, c1 is equal to, sorry, c1 is equal to y0...

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