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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

       (1.111)

      Using this in Equation (1.109) gives also

       (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

       (1.113)

      and get

(1.114)

(1.115)

      for the modal mass m_n and stiffness k_n with the following relation to the modal frequency

(1.116)

      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:

       (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

       (1.118)

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

       (1.119)

       (1.120)

       (1.121)

The prime (′) denotes modal coordinates or matrices and [I] is the unit matrix. Due to the orthogonality of the modes every solution q can be expressed as

(1.122)

      and

       (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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