*Notes to a video lecture on http://www.unizor.com*

__Forced Oscillation 1__

Oscillation is

**forced**, when some periodic force is applied to an object that can potentially oscillate.

A subject of our analysis will be an object on a spring with no other forces acting on it except one external force that we assume is periodical.

As we know, in the absence of external force the differential equation describing the movement of an object on a spring follows from the Hooke's Law and the Newton's Second Law

**m·x"(t) = −k·x(t)**or

**m·x"(t) + k·x(t) = 0**with a general solution

**x(t) = A·cos(ω**_{0}·t) + B·sin(ω_{0}·t)where

*A*and

*B*are any constants;

*ω*is an inherent

_{0}= √k/m*natural angular frequency*of oscillation of this particular object on this particular spring.

The equation above is

*homogeneous*because, together with any its solution

*x(t)*, the function

*C·x(t)*will be a solution as well.

Consider a model of

**forced**oscillation of an object on a spring with a periodic external force acting on it with the following characteristics:

*elasticity*of a spring

*k*,

*mass*of an object is

*m*,

*force*applied to an object is

*F(t)=F*.

_{0}·cos(ω·t)In this lecture we consider a case of angular frequency

*ω*of the periodic external function

*F(t)=F*to be

_{0}·cos(ω·t)__different from the inherent__

*natural angular frequency**ω*of oscillations without external forces:

_{0}=√k/m

**ω ≠ ω**_{0}= √k/mIf external force

*F(t)*, described above, is present, the differential equation that describes the oscillation of an object looks like

**m·x"(t) = −k·x(t) + F(t)**or, assuming the external force is a periodic function

*F(t)=F*,

_{0}·cos(ω·t)

**m·x"(t)+ k·x(t) = F**_{0}·cos(ω·t)The function on the right of this equation makes this equation

*non-homogeneous*.

Our task is to analyze the movement of an object on a spring in the presence of a periodic external function, as described by the above differential equations and initial conditions.

Notice that if some

*non-homogeneous*linear differential equation has two partial solutions

*x*and

_{1}(t)*x*, their difference is a partial solution to a corresponding

_{2}(t)*homogeneous*linear differential equation.

Indeed, if

**m·x**_{1}"(t)+ k·x_{1}(t) = F_{0}·cos(ω·t)and

**m·x**_{2}"(t)+ k·x_{2}(t) = F_{0}·cos(ω·t)then for

*x*

_{3}(t)=x_{1}(t)−x_{2}(t)

**m·x**_{3}"(t)+ k·x_{3}(t) = 0From the above observation follows that,

**in order to find a general solution to a**.

*non-homogeneous*linear differential equation, it is sufficient to find its one partial solution and add to it a general solution of a corresponding*homogeneous*equationIn our case we already know the general solution to a corresponding

*homogeneous*equation

**m·x"(t)+ k·x(t) = 0**So, all we need is to find a single partial solution to

**m·x"(t)+ k·x(t) = F**_{0}·cos(ω·t)and add to it the general solution to the above

*homogeneous*equation.

Considering our differential equation, that describes the forced oscillation with a periodic external function, contains function

*x(t)*, its second derivative and function

*cos(ω·t)*on its right side, it's reasonable to look for a partial solution

*x*in a form of

_{p}(t)*x*and find such a coefficient

_{p}(t)=C·cos(ω·t)*C*that the equation is satisfied.

Then

**x**_{p}(t) = C·cos(ω·t)

**x'**_{p}(t) = −C·ω·sin(ω·t)

**x"**_{p}(t) = −C·ω²·cos(ω·t)Substituting these into our differential equation, we get

**−m·C·ω²·cos(ω·t) +**

+ k·C·cos(ω·t) = F+ k·C·cos(ω·t) = F

_{0}·cos(ω·t)Since this is supposed to be true for any time value

*t*, the coefficients at

*cos(ω·t)*must satisfy the equation:

**−m·C·ω² + k·C = F**_{0}from which follows

*[*

**C = F**

= F

= F_{0}/(k−m·ω²) == F

_{0}/m·((k/m)−ω²) == F

_{0}/*]*

**m·(ω**_{0}²−ω²)We have found a partial solution to our

*non-homogeneous*equation:

*[*

**x**_{p}(t)=F_{0}·cos(ω·t)/*]*

**m·(ω**_{0}²−ω²)Now we can express the general solution to a differential equation that describes the

**forced**oscillation by a periodic external function as

*[*

**x(t)=F**_{0}·cos(ω·t)/*]*

**m(ω**_{0}²−ω²)

**+**

+ A·cos(ω+ A·cos(ω

_{0}·t) + B·sin(ω_{0}·t)where

*A*and

*B*are any constants determined by initial conditions

*x(0)*and

*x'(0)*;

*ω*is an inherent

_{0}= √k/m*natural angular frequency*of oscillation of this particular object on this particular spring;

*F*is an external periodic force acting on an object.

_{0}·cos(ω·t)It's beneficial to represent

*A·cos(ω*through a single trigonometric function as follows.

_{0}·t)+B·sin(ω_{0}·t)Let

**D = √A²+B²**Angle

*φ*is defined by

**cos(φ) = A/D**

**sin(φ) = B/D**(on a Cartesian coordinate plane this angle is the one from the X-axis to a vector with coordinates

*A,B*})

Then

*[*

**A·cos(ω**

= D·_{0}·t)+B·sin(ω_{0}·t) == D·

*]*

**cos(φ)·cos(ω**

+ sin(φ)·sin(ω_{0}·t) ++ sin(φ)·sin(ω

_{0}·t)

**=**

= D·cos(ω= D·cos(ω

_{0}·t−φ)Now the general solution to our differential equation that describes forced oscillation looks like

*[*

**x(t)=F**_{0}·cos(ω·t)/*]*

**m(ω**_{0}²−ω²)

**+**

+ D·cos(ω+ D·cos(ω

_{0}·t−φ)where

*F*is an amplitude of a periodic external force;

_{0}*ω*- angular frequency of a periodic external force;

*m*- mass of an object on a spring;

*k*- elasticity of a spring;

*ω*is an inherent natural angular frequency of oscillation of this particular object on this particular spring;

_{0}=√k/m*t*is time;

*D*and

*φ*parameters are determined by initial conditions for

*x(0)*and

*x'(0)*.

As you see from the formula for a general solution, it's essential that

*ω*, that is an angular frequency of the external force is not equal to an inherent natural angular frequency of the object and a spring in the absence of external force.

_{0}≠ωTheir equality renders zero in the denominator of a formula, invalidating the whole approach.

What happens if these frequencies are equal is a subject of the next lecture.

The solution to a forced oscillation in a case of

*ω*represents a superposition of two sinusoidal functions with different frequencies.

_{0}≠ωLet's represent it graphically for some specific case of initial conditions and parameters.

Initial conditions:

**x(0) = 0**

**x'(0) = 0**From these initial conditions follows

*[*

**0 = F**_{0}/*]*

**m(ω**_{0}²−ω²)

**+ D·cos(φ)**

**0 = D·ω**_{0}·sin(φ)The second equation results in

*, from which the first equation gives*

**φ=0***[*

**D = −F**_{0}/*]*

**m(ω**_{0}²−ω²)Parameters:

**m=1**

**k=4**

**ω**_{0}=√k/m=2

**F**_{0}=3

**ω=5**The parameters above define the value of

*D*:

**D=−3/(4−25)=1/7**With the above initial conditions and parameters the oscillation of the object is described by function

*[*

**x(t)=−3·cos(5t)/21+cos(2t)/7 =**

==

*]*

**cos(2t)−cos(5t)**

**/7**The graph of this function looks like this