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alt="left-parenthesis omega Subscript n Superscript 2 Baseline minus omega Subscript m Superscript 2 Baseline right-parenthesis Start 1 By 1 Matrix 1st Row normal upper Psi Subscript m EndMatrix Superscript upper T Baseline Start 1 By 1 Matrix 1st Row upper M EndMatrix Start 1 By 1 Matrix 1st Row normal upper Psi Subscript n Baseline EndMatrix equals 0"/> (1.110)

      Since ωn2≠ωm2 this requires

       Start 1 By 1 Matrix 1st Row normal upper Psi Subscript m Baseline EndMatrix Superscript upper T Baseline Start 1 By 1 Matrix 1st Row upper M EndMatrix Start 1 By 1 Matrix 1st Row normal upper Psi Subscript n Baseline EndMatrix equals 0 for m not-equals n (1.111)

      Using this in Equation (1.109) gives also

       Start 1 By 1 Matrix 1st Row normal upper Psi Subscript m Baseline EndMatrix Superscript upper T Baseline Start 1 By 1 Matrix 1st Row upper K EndMatrix Start 1 By 1 Matrix 1st Row normal upper Psi Subscript n Baseline EndMatrix equals 0 for m not-equals n (1.112)

      Thus, the mode shapes are orthogonal to each other with respect to [K] and [M]. For normalisation we multiply (1.106) from the left with {Ψ}m

       Start 1 By 1 Matrix 1st Row normal upper Psi EndMatrix Subscript m Superscript upper T Baseline Start 1 By 1 Matrix 1st Row upper K EndMatrix Start 1 By 1 Matrix 1st Row normal upper Psi EndMatrix Subscript m Baseline equals minus omega Subscript m Superscript 2 Baseline Start 1 By 1 Matrix 1st Row normal upper Psi EndMatrix Subscript m Superscript upper T Baseline Start 1 By 1 Matrix 1st Row upper M EndMatrix Start 1 By 1 Matrix 1st Row normal upper Psi EndMatrix Subscript m Baseline left-parenthesis m equals 1 comma 2 comma ellipsis upper N right-parenthesis (1.113)

      and get

      for the modal mass mn and stiffness kn with the following relation to the modal frequency

      1.4.4.1 Equation of Motion in Modal Coordinates

      The orthogonality of the mode shapes allows using them as a base for new coordinates that will simplify or condense the equation of motion. It is convenient to chose a normalisation with modal mass unity, thus mn=1, hence:

       Start 1 By 1 Matrix 1st Row normal upper Phi EndMatrix Subscript n Baseline equals StartFraction 1 Over StartRoot m Subscript n Baseline EndRoot EndFraction Start 1 By 1 Matrix 1st Row normal upper Psi EndMatrix Subscript n (1.117)

      {Φ}n is called the mass-normalized mode shape of the system. With Equation (1.116) and writing the mass normalized modes in matrix form

       Start 1 By 1 Matrix 1st Row normal upper Phi EndMatrix equals Start 1 By 4 Matrix 1st Row 1st Column Start 1 By 1 Matrix 1st Row normal upper Phi EndMatrix Subscript 1 Baseline 2nd Column Start 1 By 1 Matrix 1st Row normal upper Phi EndMatrix Subscript 1 Baseline 3rd Column midline-horizontal-ellipsis 4th Column Start 1 By 1 Matrix 1st Row normal upper Phi EndMatrix Subscript upper N EndMatrix (1.118)

      it becomes clear that the normalisation from (1.114) and (1.115) now reads as follows:

       StartLayout 1st Row 1st Column Start 1 By 1 Matrix 1st Row normal upper Phi EndMatrix Superscript upper T Baseline Start 1 By 1 Matrix 1st Row upper M EndMatrix Start 1 By 1 Matrix 1st Row normal upper Phi EndMatrix 2nd Column equals 3rd Column Start 1 By 1 Matrix 1st Row upper I EndMatrix equals Start 1 By 1 Matrix 1st Row upper M EndMatrix prime EndLayout (1.119)

       StartLayout 1st Row 1st Column Start 1 By 1 Matrix 1st Row normal upper Phi EndMatrix Superscript upper T Baseline Start 1 By 1 Matrix 1st Row upper K EndMatrix Start 1 By 1 Matrix 1st Row normal upper Phi EndMatrix 2nd Column equals 3rd Column Start 1 By 1 Matrix 1st Row upper K EndMatrix prime equals diag left-parenthesis omega Subscript n Superscript 2 Baseline right-parenthesis EndLayout (1.120)

       StartLayout 1st Row 1st Column with diag left-parenthesis omega Subscript n Baseline right-parenthesis 2nd Column equals 3rd Column Start 4 By 4 Matrix 1st Row 1st Column omega 1 squared 2nd Column 0 3rd Column midline-horizontal-ellipsis 4th Column 0 2nd Row 1st Column 0 2nd Column omega 2 squared 3rd Column midline-horizontal-ellipsis 4th Column 0 3rd Row 1st Column 0 2nd Column midline-horizontal-ellipsis 3rd Column down-right-diagonal-ellipsis 4th Column 0 4th Row 1st Column 0 2nd Column midline-horizontal-ellipsis 3rd Column 0 4th Column omega Subscript upper N Superscript 2 EndMatrix EndLayout (1.121)

      and

       StartLayout 1st Row 1st Column Blank 2nd Column Blank 3rd Column Start 1 By 1 Matrix 1st Row normal upper Phi EndMatrix 4th Column equals Start 1 By 4 Matrix 1st Row 1st Column normal upper Phi 1 2nd Column normal upper Phi 2 3rd Column midline-horizontal-ellipsis 4th Column normal upper Phi Subscript upper N EndMatrix EndLayout (1.123)

      The vector {q′} with components qn′ is the displacement in modal coordinates. coordinates ! modal Entering this into (1.103) and multiplication from the left with [Φ]T provides

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