Edexcel A Level Further Maths: Core Pure:复习笔记8.3.2 Damped or Forced Harmonic Motion

What is damped harmonic motion?

• If we add a term representing a resistive force to the simple harmonic motion equation, the new equation describes a particle undergoing damped harmonic motion
  • Depending on the situation being modelled, this resistive force may represent such phenomena as friction or air resistance that resist the motion of the particle
• The standard damped harmonic motion equation is of the form

• • Note that that is the same as the simple harmonic motion equation, except for the addition of the damping term
  • x is the displacement of the particle from a fixed point O at time t
  • k is a positive constant representing the strength of the damping force
  • is a positive constant representing the strength of the restoring force that accelerates the particle back towards point O
• The damped harmonic motion equation is a second order homogeneous differential equation, and may be solved using the standard methods for such equations
  • This will involve using the auxiliary equation to find the complementary function for the equation
• You should, however, be familiar with the three main cases:
  • CASE 1:
    • The auxiliary equation has two distinct real roots, both of which are negative
    • This is known as heavy damping (sometimes also referred to as overdamping)
    • The general solution will be of the form  where α and β are the roots  of the auxiliary equation
    • Because α and β are both negative, the two exponentials will decay to zero as t increases
    • Therefore the particle’s displacement will also decay to zero, without any oscillations occurring
    • However the decay to zero will not happen as quickly as in Case 2 (critical damping) below

• • CASE 2:
    • The auxiliary equation has a single repeated root, which is negative
    • This is known as critical damping
    • The general solution will be of the form  where  is the repeated root of the auxiliary equation
    • Because α is negative, the exponential will decay to zero as t increases
    • Therefore the particle’s displacement will also decay to zero, without any oscillations occurring (although depending on the values of A and B it is possible that the particle will change direction once as the decay to zero occurs
    • For a given value of , the displacement of the critical damping case will decay to zero faster than any instance of the heavy damping case
  • CASE 3:
    • The auxiliary equation has complex roots which form a complex conjugate pair
    • This known as light damping (sometimes also referred to as underdamping)
    • The general solution will be of the form , where  and
    • Because p is negative, the exponential will decay to zero as t increases
    • Therefore the particle’s displacement will also decay to zero
    • However the cosine and sine terms mean that the particle will continue to oscillate with decreasing amplitude as the decay to zero occurs
  • In all three cases, initial or boundary conditions given in a question may allow you to work out the precise values of the arbitrary constants A and B
• The following displacement-time graph illustrates the behaviour displayed by a particle for each of the three cases

What is forced harmonic motion?

• If we add a term representing an external ‘driving’ force to the damped harmonic motion equation, the new equation describes a particle undergoing forced harmonic motion
• The standard forced harmonic motion equation is of the form

• • Note that it is the same as the damped harmonic motion equation, except for the addition of the driving term f(t) on the right-hand side of the equation
  • x is the displacement of the particle from a fixed point O at time t
  • k is a non-negative constant representing the strength of the damping force
    • If k = 0 then there is no damping force
  • is a positive constant representing the strength of the restoring force that accelerates the particle back towards point O
• The forced harmonic motion equation is a second order non-homogeneous differential equation, and may be solved using the standard methods for such equations
  • This will involve using the auxiliary equation to find the complementary function for the equation
  • It will also involve finding the particular integral for the equation, based on the form of f(t)
  • Initial or boundary conditions given in a question may allow you to work out the precise values of any arbitrary constants in your general solution
• If k ≠ 0 then the long-term behaviour of the system will be predominantly determined by the driving force f(t)
  • If k = 0, then the long-term behaviour will be a combination of the effects of the driving force and of the system’s natural oscillation