If we observe such a picture that some object makes movements from side to side and at the same time the deflection rate and the deflection distance are unchanged, then there is a term harmonic oscillations for such a movement.
Harmonic oscillations are oscillations that are symmetrical with respect to a certain center and are constant in time. Harmonic oscillations look like this:Harmonic oscillations
Harmonic oscillations are a theoretical model that does not occur in real life, but having assumed all responsibility for this assumption, it is possible to calculate real, more complex processes with an acceptable approximation.
To describe the graph seen above, we output the formula:
First of all, a sinusoid is suitable for the mathematical description of such oscillations, since a sinusoid has exactly the same view, its formula:
y = a + bsin(cx + d)
Since it is convenient to view the graph starting from zero, we will convert the sine wave to a cosine (the same +/pi;/2) and get rid of the shift a, because we are interested in the periodicity, not where it started from:
y = bcos(cx + d)
As for the distance from the axis to the point, y is usually written along the y axis, but in this case we will use x, because it is assumed that the movement occurs in one plane and the first plane that comes to mind is x (in general, the name does not matter, but someone got confused once ...). The coordinates of time, i.e. t, are postponed on the x axis , so we will write down the distance x and call the graph x(t).
x(t) = bcos(ct + d)
Such fluctuations are within a certain maximum possible value, such a value is called the amplitude, the symbol A is used for the amplitude:
x(t) = Acos(ct + d)
The constants c and d are the parameters of the sine wave, namely - c shows how fast the change occurs directions, d shows where the fluctuations begin. Common designations: ω – frequency of cyclic oscillations, φ – initial phase
Harmonic oscillation formula
The generally accepted formula looks like this:
x(t) = A cos (ωt + φ) (1)
Oscillation equations
The period T is related to the oscillation frequency ω, since ω is expressed in radians, then the period is expressed in terms of the oscillation frequency through the coefficient π:
T = 2π/ω
ω = 2π/T
The oscillation frequency f is the inverse of the period:
f = 1/T = ω/2π
Velocity and acceleration of position change are time derivatives:
v(t) = x'(t) = -A ω sin ( ωt + φ )
a(t) = x''(t) = -A ω2cos ( ωt + φ )
Based on equation (1): a(t) = - ω2x(t)
From where
d2x/dt2 + ω2x = 0
Since harmonic oscillations are periodic, in time T the pendulum will make a full cycle of movement, to the end in one direction and back, therefore, will be at the same point x(t)
x ( t + T ) = x (t)
Spring pendulum
We have considered the kinematics of oscillatory movements, now we will transfer it to the dynamics of body movement.
Based on the fact that the spring exerts resistance during tension and compression, it follows that there is a force, resisting movement. For a spring, the force of resistance to movement is calculated by the formula:
F = -kx
Where k is the spring stiffness coefficient, x is the displacement relative to the equilibrium point.
Based on the formulas:
F = -kx Spring stiffness equation
F = ma Newton's second law
a = d2x/dt2
Output:
m d2x/dt2 = -kx → m d2x/dt2 + kx/m = 0
Taking into account the equation d2x/dt2 + ω2x = 0:
ω2 = k/m
The oscillation equation of a spring pendulum
Based on the above formulas, the equation of harmonic oscillations of a spring pendulum is written as follows:
x(t) = A cos(√k/m · t + φ)
