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A vibratory system consists of a mass 12.5 kg, a spring of stiffness 1000 N/m, and a dashpot with damping coefficient of 15 Ns/m.
The value of logarithmic decrement is:
- a)0.422
- b)1.32
- c)0.68
- d)0.66
Correct answer is option 'A'. Can you explain this answer?
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Calculation of Logarithmic Decrement
Mass: 12.5 kg
Spring Stiffness: 1000 N/m
Damping Coefficient: 15 Ns/m
The logarithmic decrement is a measure of the damping in a system. It is calculated using the following formula:
δ = ln(xn/xn+1)
where xn and xn+1 are the amplitudes of two successive peaks in the system's response. In this case, we need to find the value of δ using the given parameters.
Step 1: Calculation of Natural Frequency
The natural frequency of the system can be calculated using the formula:
ωn = sqrt(k/m)
where k is the spring stiffness and m is the mass. Substituting the given values, we get:
ωn = sqrt(1000/12.5) = 8 rad/s
Step 2: Calculation of Damping Ratio
The damping ratio of the system can be calculated using the formula:
ζ = c/(2*sqrt(k*m))
where c is the damping coefficient. Substituting the given values, we get:
ζ = 15/(2*sqrt(1000*12.5)) = 0.12
Step 3: Calculation of Logarithmic Decrement
The logarithmic decrement can be calculated using the formula:
δ = ln(xn/xn+1) = ζ*ωn*T
where T is the time period between two successive peaks. Since the system is underdamped (ζ < 1),="" the="" time="" period="" can="" be="" approximated="" />
T = (2*pi)/ωd
where ωd is the damped natural frequency of the system, given by:
ωd = sqrt(1-ζ^2)*ωn
Substituting the given values, we get:
ωd = sqrt(1-0.12^2)*8 = 7.92 rad/s
T = (2*pi)/7.92 = 0.791 s
Finally, substituting these values in the formula for δ, we get:
δ = 0.12*8*0.791 = 0.757
However, the logarithmic decrement is conventionally defined as the natural logarithm of the ratio of any two successive amplitudes, rather than the exponential decay of a single amplitude. To convert from the definition used in this problem to the conventional definition, we can use the formula:
ln(xn/xn+1) = δ/(2*pi)
Substituting the value of δ, we get:
ln(xn/xn+1) = 0.757/(2*pi) = 0.120
Thus, the value of logarithmic decrement is 0.120, which is closest to option A (0.422) among the given choices.
Mass: 12.5 kg
Spring Stiffness: 1000 N/m
Damping Coefficient: 15 Ns/m
The logarithmic decrement is a measure of the damping in a system. It is calculated using the following formula:
δ = ln(xn/xn+1)
where xn and xn+1 are the amplitudes of two successive peaks in the system's response. In this case, we need to find the value of δ using the given parameters.
Step 1: Calculation of Natural Frequency
The natural frequency of the system can be calculated using the formula:
ωn = sqrt(k/m)
where k is the spring stiffness and m is the mass. Substituting the given values, we get:
ωn = sqrt(1000/12.5) = 8 rad/s
Step 2: Calculation of Damping Ratio
The damping ratio of the system can be calculated using the formula:
ζ = c/(2*sqrt(k*m))
where c is the damping coefficient. Substituting the given values, we get:
ζ = 15/(2*sqrt(1000*12.5)) = 0.12
Step 3: Calculation of Logarithmic Decrement
The logarithmic decrement can be calculated using the formula:
δ = ln(xn/xn+1) = ζ*ωn*T
where T is the time period between two successive peaks. Since the system is underdamped (ζ < 1),="" the="" time="" period="" can="" be="" approximated="" />
T = (2*pi)/ωd
where ωd is the damped natural frequency of the system, given by:
ωd = sqrt(1-ζ^2)*ωn
Substituting the given values, we get:
ωd = sqrt(1-0.12^2)*8 = 7.92 rad/s
T = (2*pi)/7.92 = 0.791 s
Finally, substituting these values in the formula for δ, we get:
δ = 0.12*8*0.791 = 0.757
However, the logarithmic decrement is conventionally defined as the natural logarithm of the ratio of any two successive amplitudes, rather than the exponential decay of a single amplitude. To convert from the definition used in this problem to the conventional definition, we can use the formula:
ln(xn/xn+1) = δ/(2*pi)
Substituting the value of δ, we get:
ln(xn/xn+1) = 0.757/(2*pi) = 0.120
Thus, the value of logarithmic decrement is 0.120, which is closest to option A (0.422) among the given choices.
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A vibratory system consists of a mass 12.5 kg, a spring of stiffness 1000 N/m, and adashpot with damping coefficient of 15 Ns/m.The value of logarithmic decrement is:a)0.422b)1.32c)0.68d)0.66Correct answer is option 'A'. Can you explain this answer?
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