In a system in void oscillates eigenfrequency

[mather]

hello!

in a system in void oscillates (eg a mass tied in a spring, whose other edge is fixed)

we say that if we exert a periodic force to it, with the same frequency that it oscillates (eigenfrequency), then its energy and oscillation width becomes infinite

if the periodic force has not exactly the same frequency, the system will gather energy from the force, but it will also lose energy, since the force won't be exerted at the correct moments, each time

however, which function calculates the evolution of the energy of the system, according to various frequencies of the exerted force?

thanks!

 

[HallsofIvy]

The equation is $d^2x/dt^2+ kx= f(x)$. If we let v= dx/dt, then $d^2x/dt^2= (dv/dt)= (dv/dx)(dx/dt)= v(dv/dx$. Then we have v(dv/dx)= f(x)- kx so we can write vdv= (f(x)- kx) dx and, integrating both sides, $(1/2)v^2= \int f(x)- (k/2)x^2+ C$. That is the same as $(1/2)v^2+ (k/2)x^2=\int f(x)dx+ C$
The left side is the total energy, kinetic plus potential, so that if there were no "f(x)", it would reflect "conservation of energy" in a closed system. With f(x), $\int f(x)dx$ is the external energy that is "pumped" into (when f(x) is positive) or out of (when f(x) is negative) the system.

 

[mather]

The equation is $d^2x/dt^2+ kx= f(x)$. If we let v= dx/dt, then [itex]d^2x/dt^2= (dv/dt)= (dv/dx)(dx/dt)= v(dv/dx). Then we have v(dv/dx)= f(x)- kx so we can write vdv= (f(x)- kx) dx and, integrating both sides, $(1/2)v^2= \int f(x)- (k/2)x^2+ C$. That is the same as $(1/2)v^2+ (k/2)x^2=\int f(x)dx+ C$<br /> The left side is the total energy, kinetic plus potential, so that if there were no "f(x)", it would reflect "conservation of energy" in a closed system. With f(x), $\int f(x)dx$ is the external energy that is "pumped" into (when f(x) is positive) or out of (when f(x) is negative) the system.[/itex]

$<br /> thanks but i didnt understand a thing<br /> <br /> my question basically is: how will a periodic force with frequency different than oscillating system's eigenfrequency, affect the energy of the system?<br /> <br /> would it possible to increase it, in a slower manner than if it had the same frequency as the eigenfrequency?<br /> <br /> also, a graph would help<br /> <br /> also, what about if the force's frequency varies? what would be the possible outcomes of that interaction?$

 

[mather]

anyone?

 

[mather]

?