The response \(x(t)\) of a freely vibrating single degree of freedom underdamped system is given below. In the equation, \(A\) and \(\phi\) are constants. The damping ratio of the system is _______ (rounded off to 3 decimal places).
\[ x(t) = A e^{-5t}\sin(10t+\phi) \]
Show Hint
Compare \(x(t)=Ae^{-\zeta\omega_n t}\sin(\omega_d t+\phi)\) with the given response: \(\zeta\omega_n=5\), \(\omega_n\sqrt{1-\zeta^2}=10\), then solve the two equations together.
Step 1: Match the response with the standard underdamped form.
The free response of an underdamped single degree of freedom system is written in the standard form
\[ x(t) = A e^{-\zeta \omega_n t}\sin(\omega_d t + \phi) \]
where \(\zeta\) is the damping ratio, \(\omega_n\) is the undamped natural frequency, and \(\omega_d=\omega_n\sqrt{1-\zeta^2}\) is the damped natural frequency. Comparing this with the given response
\[ x(t) = A e^{-5t}\sin(10t+\phi) \]
term by term gives two equations: the exponential decay rate \(\zeta\omega_n = 5\), and the oscillation frequency \(\omega_d = \omega_n\sqrt{1-\zeta^2} = 10\).
Step 2: Eliminate \(\omega_n\).
From the first equation, \(\omega_n = 5/\zeta\). Substitute this into the second equation:
\[ \frac{5}{\zeta}\sqrt{1-\zeta^2} = 10 \]
\[ \sqrt{1-\zeta^2} = 2\zeta \]
Step 3: Solve for the damping ratio.
Square both sides (valid since \(0<\zeta<1\) for an underdamped system, so both sides are positive):
\[ 1-\zeta^2 = 4\zeta^2 \]
\[ 1 = 5\zeta^2 \]
\[ \zeta^2 = \frac{1}{5} = 0.2 \]
\[ \zeta = \sqrt{0.2} = 0.4472 \]
Step 4: Round to the required precision.
Rounded off to 3 decimal places, \(\zeta = 0.447\). This is comfortably inside \((0,1)\), confirming the system is indeed underdamped, consistent with the oscillatory \(\sin(\cdot)\) form given.
Final Answer:
The damping ratio of the system is 0.447.
\[ \boxed{\zeta \approx 0.447} \]
