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The impulsive response of the fractional vibration equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>z</mi><mrow><mo>′</mo><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>b</mi><msubsup><mi>D</mi><mi>t</mi><mi>α</mi></msubsup><mi>z</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mi>z</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>F</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>></mo><mn>0</mn><mo>,</mo><mi>c</mi><mo>></mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>≤</mo><mi>α</mi><mo>≤</mo><mn>2</mn></mrow></semantics></math></inline-formula>, is investigated by using the complex path-integral formula of the inverse Laplace transform. Similar to the integer-order case, the roots of the characteristic equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><msup><mi>s</mi><mi>α</mi></msup><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> must be considered. It is proved that for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>∪</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>, the characteristic equation always has a pair of conjugated simple complex roots with a negative real part on the principal Riemann surface. Particular attention is paid to the problem as to how the couple conjugated complex roots approach the two roots of the integer case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, especially to the two different real roots in the case of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>c</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. On the upper-half complex plane, the root <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>(</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> is investigated as a function of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> and with parameters <i>b</i> and <i>c</i>, and so are the argument <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>(</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula>, modulus <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>(</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula>, real part <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>(</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> and imaginary part <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ω</mi><mo>(</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> of the root <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>(</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula>. For the three cases of the discriminant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>c</mi></mrow></semantics></math></inline-formula>: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>, variations of the argument and modulus of the roots according to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> are clarified, and the trajectories of the roots are simulated. For the case of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>c</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>, the trajectories of the roots are further clarified according to the change rates of the argument, real part and imaginary part of root <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>(</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. The solution components, i.e., the residue contribution and the Hankel integral contribution to the impulsive response, are distinguished for the three cases of the discriminant.