6ẍ + 9ẋ + 27x = 0 is the equation of motion for a damped vibration. The damping factor shall be:
0.35
The given equation of motion for a damped vibration is:
\[ 6\ddot{x} + 9\dot{x} + 27x = 0 \]This equation represents a second-order linear homogeneous differential equation, which is commonly used to model damped vibration systems. The standard form of such an equation is given by:
\[ m\ddot{x} + c\dot{x} + kx = 0 \]where:
- \( m \) is the mass of the system.
- \( c \) is the damping coefficient.
- \( k \) is the stiffness of the system.
By comparing the given equation \( 6\ddot{x} + 9\dot{x} + 27x = 0 \) with the standard form \( m\ddot{x} + c\dot{x} + kx = 0 \), we can identify the coefficients:
- Mass, \( m = 6 \)
- Damping coefficient, \( c = 9 \)
- Stiffness, \( k = 27 \)
The question asks for the damping factor. The damping factor, also known as the damping ratio, is denoted by \( \zeta \) (zeta). It is defined as the ratio of the actual damping coefficient \( c \) to the critical damping coefficient \( c_c \).
\[ \zeta = \frac{c}{c_c} \]The critical damping coefficient \( c_c \) is the minimum damping required to prevent oscillations. It is given by the formula:
\[ c_c = 2\sqrt{mk} \]Now, we can calculate the critical damping coefficient using the values of \( m \) and \( k \) identified from the given equation:
\[ c_c = 2\sqrt{(6)(27)} \] \[ c_c = 2\sqrt{162} \]We can simplify the square root term \( \sqrt{162} \):
\[ \sqrt{162} = \sqrt{81 \times 2} = \sqrt{81} \times \sqrt{2} = 9\sqrt{2} \]So, the critical damping coefficient is:
\[ c_c = 2 \times 9\sqrt{2} = 18\sqrt{2} \]Now we can calculate the damping factor \( \zeta \) using the calculated value of \( c_c \) and the identified value of \( c \):
\[ \zeta = \frac{c}{c_c} = \frac{9}{18\sqrt{2}} \] \[ \zeta = \frac{1}{2\sqrt{2}} \]To get a numerical value, we can approximate \( \sqrt{2} \approx 1.414 \):
\[ \zeta = \frac{1}{2 \times 1.414} = \frac{1}{2.828} \]Calculating the value:
\[ \zeta \approx 0.3535 \]Let's compare this calculated damping factor with the given options:
- Option 1: 0.35
- Option 2: 0.65
- Option 3: 0.5
- Option 4: 0.25
The calculated value \( 0.3535 \) is closest to \( 0.35 \).
Therefore, the damping factor for the given damped vibration equation is approximately 0.35.
Important Questions from Damping Coefficient and Damping Ratio
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______ is defined as the ratio of the actual damping coefficient to a critical damping coefficient.
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