| Summary: | We present two novel results for small damped oscillations described by the vector differential equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mo>+</mo><mi>C</mi><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mo>+</mo><mi>K</mi><mi>x</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, where the mass matrix <i>M</i> can be singular, but standard deflation techniques cannot be applied. The first result is a novel formula for the solution <i>X</i> of the Lyapunov equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>A</mi></mrow><mi>T</mi></msup><mi>X</mi><mo>+</mo><mi>X</mi><mi>A</mi><mo>=</mo><mo>−</mo><mi>I</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>=</mo><mi>A</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> is obtained from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>,</mo><mi>C</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></mrow></semantics></math></inline-formula>, which are the so-called mass, damping, and stiffness matrices, respectively, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>rank</mi><mo>(</mo><mi>M</mi><mo>)</mo><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Here, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> is positive semidefinite with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>rank</mi><mo>(</mo><mi>C</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Using the obtained formula, we propose a very efficient way to compute the optimal damping matrix. The second result was obtained for a different structure, where we assume that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">dim</mo><mo>(</mo><mi mathvariant="script">N</mi><mo>(</mo><mi>M</mi><mo>)</mo><mo>)</mo><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> and internal damping exists (usually a small percentage of the critical damping). For this structure, we introduce a novel linearization, i.e., a novel construction of the matrix <i>A</i> in the Lyapunov equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>A</mi><mi>T</mi></msup><mi>X</mi><mo>+</mo><mi>X</mi><mi>A</mi><mo>=</mo><mo>−</mo><mi>I</mi></mrow></semantics></math></inline-formula>, and a novel optimization process. The proposed optimization process computes the optimal damping <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> that minimizes a function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>↦</mo><mi>trace</mi><mo>(</mo><mi>Z</mi><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula> (where <i>Z</i> is a chosen symmetric positive semidefinite matrix) using the approximation function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>c</mi><mi>v</mi></msub><mo>+</mo><mfrac><mi>a</mi><mi>v</mi></mfrac><mo>+</mo><mi>b</mi><mi>v</mi></mrow></semantics></math></inline-formula>, for the trace function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>≐</mo><mi>trace</mi><mo>(</mo><mi>Z</mi><mi>X</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula>. Both results are illustrated with several corresponding numerical examples.
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