damped-or-forced-harmonic-motion

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 <img alt=”fraction numerator negative k plus-or-minus square root of k squared minus 4 omega squared end root over denominator 2 end fraction” data-mathml='<math ><semantics><mfrac><mrow><mo>-</mo><mi>k</mi><mo>±</mo><msqrt><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><msup><mi>ω</mi><mn>2</mn></msup></msqrt></mrow><mn>2</mn></mfrac><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“language”:”en”,”fontFamily”:”Times New Roman”,”fontSize”:”18″}</annotation></semantics></math>’ height=”52″ role=”math” src=”data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2252%22%20width%3D%22129%22%20wrs%3Abaseline%3D%2235%22%3E%3C!–MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmfrac%3E%3Cmrow%3E%3Cmo%3E-%3C%2Fmo%3E%3Cmi%3Ek%3C%2Fmi%3E%3Cmo%3E%26%23xB1%3B%3C%2Fmo%3E%3Cmsqrt%3E%3Cmsup%3E%3Cmi%3Ek%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsup%3E%3Cmo%3E-%3C%2Fmo%3E%3Cmn%3E4%3C%2Fmn%3E%3Cmsup%3E%3Cmi%3E%26%23x3C9%3B%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsup%3E%3C%2Fmsqrt%3E%3C%2Fmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmfrac%3E%3C%2Fmath%3E–%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A’math180d3ca3b3827ec71585593b866’%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAADxjdnQgDVUNBwAAAVgAAAA6Z2x5ZoPi2VsAAAGUAAABDmhlYWQQC2qxAAACpAAAADZoaGVhCGsXSAAAAtwAAAAkaG10eE2rRkcAAAMAAAAADGxvY2EAHTwYAAADDAAAABBtYXhwBT0FPgAAAxwAAAAgbmFtZaBxlY4AAAM8AAABn3Bvc3QB9wD6AAAE3AAAACBwcmVwa1uragAABPwAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACgAAAAGAAQAAQACALEiEv%2F%2FAAAAsSIS%2F%2F%2F%2FUN3wAAEAAAAAAAAAAAFUAywAgAEAAFYAKgJYAh4BDgEsAiwAWgGAAoAAoADUAIAAAAAAAAAAKwBVAIAAqwDVAQABKwAHAAAAAgBVAAADAAOrAAMABwAAMxEhESUhESFVAqv9qwIA%2FgADq%2FxVVQMAAAIAgP%2F%2FAoACqwALAA8AZRgBsBAQsA%2FUsA8QsAA8sAAQsAHUsAEQsATUsAQQsAXUsAEQsAo8sAQQsAc8sAUQsA48ALAQELAP1LAPELAM1LAMELAJ1LAJELAK1LAKELAB1LABELAC1LABELAEPLAKELAHPDAxEzM1MxUzFSMVIycHESEVIYDWVdXVVQHVAgD%2BAAHV1tZW1NUB%2FtVVAAEAgAFVAtUBqwADADAYAbAEELEAA%2FawAzyxAgf1sAE8sQUD5gCxAAATELEABuWxAAETELABPLEDBfWwAjwTIRUhgAJV%2FasBq1YAAAABAAAAAQAA1XjOQV8PPPUAAwQA%2F%2F%2F%2F%2F9Y6E3P%2F%2F%2F%2F%2F1joTcwAA%2FyAEgAOrAAAACgACAAEAAAAAAAEAAAPo%2F2oAABdwAAD%2FtgSAAAEAAAAAAAAAAAAAAAAAAAADA1IAVQMAAIADVgCAAAAAAAAAACgAAADEAAABDgABAAAAAwBeAAUAAAAAAAIAgAQAAAAAAAQAAN4AAAAAAAAAFQECAAAAAAAAAAEAEgAAAAAAAAAAAAIADgASAAAAAAAAAAMAMAAgAAA