Driven Oscillator Examples

If a damped oscillator is driven by an external force, the solution to the motion equation has two parts, a transient part and a steady-state part, which must be used together to fit the physical boundary conditions of the problem.

Constant applied force 恒定的施加力 If a constant force is applied to a damped oscillator, it will stretch out to a final position determined by its spring constant. However, depending upon the initial conditions, it may oscillate about that final position and then settle down to the final position. This is the practical implication of the transient solution to the motion equations. 如果对一个阻尼振子施加一个恒定的力，它会伸展到由其弹簧常数决定的最终位置。然而，根据初始条件的不同，它可能会围绕该最终位置 oscillate，然后最终稳定到该位置。这是运动方程暂态解的实用意义。

Resonant excitation 共振激发 If a sinusoidal driving force is applied at the resonant frequency of the oscillator, then its motion will build up in amplitude to the point where it is limited by the damping forces on the system. If the damping forces are small, a resonant system can build up to amplitudes large enough to be destructive to the system. Such was the famous case of the Tacoma Narrows Bridge, which was blown down by the wind when it responded to a component in the wind force which excited one of its resonant frequencies. 如果在振子的共振频率上施加一个正弦驱动力，其运动幅度将逐渐增大，直到被系统的阻尼力所限制。如果阻尼力较小，共振系统可以达到足够大的振幅，从而对系统造成破坏。这种现象在著名的塔科马海峡大桥事件中得到了体现，该桥因风力中包含一个激发其共振频率的成分，而被吹毁。

Changing initial conditions 改变初始条件 The effect of the transient solution on the behavior of a damped oscillator is strongly dependent upon the initial conditions. Although a driven system will ultimately settle down to a behavior determined by the driving force (steady-state solution), the early part of the motion can show a lot of variety. 力对阻尼振子行为的影响强烈依赖于初始条件。尽管驱动系统最终会趋于由驱动力（稳态解）决定的行为，但运动的早期部分可以表现出很多多样性。

Complex exciting forces 复杂激励力 It is the nature of a resonant system to respond strongly to influences which have frequencies close to its resonant frequency. If a complex exciting force is applied, i.e., one which will have many frequency components, then the system will tend to pick out the components which are close to its resonant frequency. The example given is that of a square-wave driving force 共振系统的特性是，它会强烈响应频率接近其共振频率的激励。如果施加一个复杂的激励力，即包含许多频率成分的力，那么系统倾向于选出接近其共振频率的那些成分。所举的例子是正弦波激励力。

Driven oscillator 驱动振子

Transient solution 暂态解

Steady-state solution 稳态解

Driven Oscillator Example: Constant Applied Force

If a constant force is applied to a damped oscillator, it will stretch out to a final position x = F/k determined by its spring constant. However, depending upon the initial conditions, it may oscillate about that final position and then settle down to the final position. This is the practical implication of the transient solution to the motion equations. This illustration shows transient behavior for damping coefficients which differ by a factor of 10.

Driven Oscillator Example

Since a resonant excitation will continue to contribute energy to the system at its natural frequency, the amplitude of oscillation will continue to grow until that energy is dissipated by the damping forces, or until something destructive happens to the system. There are many stories about mechanical resonances being produced by pieces of mechanical equipment which produce periodic vibrations, like air-compressors.

Driven Oscillator Example

Driven Oscillator Example

It is the nature of a resonant system to respond strongly to influences which have frequencies close to its resonant frequency. If a complex exciting force is applied, i.e., one which will have many frequency components, then the system will tend to pick out the components which are close to its resonant frequency. The example given is that of a square-wave driving force