Damped or Forced Harmonic Motion (Edexcel A Level Further Maths): Revision Note

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

• 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