Damped Oscillation, Forced Oscillation, and Resonance
A periodic external force with an amplitude of 0.20 N acts on a 1.0 kg object attached to a spring with a spring constant of 4.0 N/m.
Adjust conditions:
The sliders adjust the values
十分時間がたつと、振幅は一定の力F₀による伸びF₀/kの何倍か？
抵抗係数だけを10倍にすると、定常振幅はF₀/kの何倍か？
The equations explain the phenomenon
1A damping force and a periodic external force are included.m d²xdt²=−k x− b dxdt+F₀ cos ωt……
The motion of the system is the sum x = xfree + xforced, where xfree solves the equation without the external force and xforced is the steady-state oscillation at the driving angular frequency. Step 10 explains why the solutions can be added.The component xfree accounts for the difference between the initial position and velocity and those of the steady-state oscillation. The system's initial position of 0 and initial velocity of 0 are not applied directly to xfree. The end of Step 4 explains how its coefficients are determined.
2An exponential function is substituted into the equation without the external force.x=eλt⇒m λ²+b λ+k=0……
3The solutions fall into 3 cases according to the damping strength.λ=−γ ± √(γ² − ω₀²)……
4A solution at the driving angular frequency is found.xforced=Re(A eiωt)⇒(k − mω² + i b ω) A=F₀……
5The amplitude is determined.|A|=F₀√((k − mω²)² + (bω)²)……
6The system is driven at its natural angular frequency.|A|=F₀b ω₀……
7The phase lag of the displacement relative to the external force is determined.tan δ=2γωω₀² − ω²……
8Numerical calculations show the transition to steady-state oscillation.x(t)→|A| cos(ωt − δ)……
9Q measures how weak the damping is.Q=ω₀2γ=kb ω₀……
Further study: motion under multiple external forces
10The solution without the external force and the steady-state solution are added.x=xfree+xforced……
What this chapter has shown
When the system is driven at its natural angular frequency, the steady-state displacement lags the external force by 1/4 of a period, and the velocity and external force point in the same direction. The amplitude grows when the work done by the external force, averaged over each period, exceeds the energy lost through damping. In steady-state oscillation, the external force and damping force cancel, and the amplitude is F₀/(bω₀).
5.0 timesAmplitude ÷ F₀/k
With all other conditions held fixed, increasing the damping coefficient by a factor of 10 makes the steady-state amplitude 0.5 times F₀/k.
In everyday lifeA tall building can sway strongly when the period of earthquake motion is close to its natural period.
Detailed summary
Even when the form of a differential equation's solution is unknown, substituting an exponential function turns it into an algebra problem. With resistance, the amplitude of oscillation decays exponentially. When a periodic external force is applied, the amplitude rises sharply near the natural frequency. When the driving angular frequency equals the natural angular frequency, the spring force provides the net force that produces acceleration, while the external force balances the resistive force, so resistance determines the amplitude. The same 1 equation can explain swings, tuning forks, and radios.