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A precision instrument package (m = 1 kg) needs to be mounted on a surface vibrating at 60 Hz. It is desired that only 5% of the base surface vibration amplitude be transmitted to the instrument. Assuming that the isolation is designed with its natural frequency significantly lesser than 60 Hz, so that the effect of damping may be ignored. The stiffness (in N/m) of the required mounting pad is ________.
(Important - Enter only the numerical value in the answer)
Correct answer is '6767.6'. Can you explain this answer?
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Given information:
- Mass of the instrument, m = 1 kg
- Base surface vibration frequency, f = 60 Hz
- Desired transmission ratio, T = 5%
- Assumption: Natural frequency of isolation system is significantly lower than 60 Hz, so damping can be ignored.
To calculate:
Stiffness of the required mounting pad.
Solution:
The transmission ratio, T, is given by the equation:
T = (2πf)^2 / ωn^2
where ωn is the natural frequency of the isolation system.
Since the natural frequency of the isolation system is significantly lower than the base surface vibration frequency, we can assume that the natural frequency is close to zero. Therefore, we can rewrite the equation as:
T = (2πf)^2 / (0)^2
Simplifying further, we get:
T = ∞
This means that the transmission ratio is infinite, which implies that no vibration will be transmitted to the instrument. This is not possible in reality, so there must be some mistake in the equation or assumptions made.
Let's try a different approach to solve the problem.
Alternate approach:
The transmission ratio can also be calculated using the equation:
T = 1 / (1 + (ω / ωn)^2)
where ω is the angular frequency of the base surface vibration, given by ω = 2πf.
Substituting the values in the equation, we get:
T = 1 / (1 + ((2πf) / ωn)^2)
Since we are given that T = 5%, we can rewrite the equation as:
0.05 = 1 / (1 + ((2πf) / ωn)^2)
Simplifying further, we get:
(2πf / ωn)^2 = 1 / 0.05 - 1
(2πf / ωn)^2 = 1 / 0.95
Taking the square root of both sides, we get:
2πf / ωn = √(1 / 0.95)
ωn = (2πf) / √(1 / 0.95)
Substituting the values, we get:
ωn = (2π * 60) / √(1 / 0.95)
ωn = 376.991
The stiffness of the mounting pad, k, is given by the equation:
k = m * ωn^2
Substituting the values, we get:
k = 1 * (376.991)^2
k ≈ 14183.2 N/m
Therefore, the stiffness of the required mounting pad is approximately 14183.2 N/m.
The correct answer is '6767.6'.
- Mass of the instrument, m = 1 kg
- Base surface vibration frequency, f = 60 Hz
- Desired transmission ratio, T = 5%
- Assumption: Natural frequency of isolation system is significantly lower than 60 Hz, so damping can be ignored.
To calculate:
Stiffness of the required mounting pad.
Solution:
The transmission ratio, T, is given by the equation:
T = (2πf)^2 / ωn^2
where ωn is the natural frequency of the isolation system.
Since the natural frequency of the isolation system is significantly lower than the base surface vibration frequency, we can assume that the natural frequency is close to zero. Therefore, we can rewrite the equation as:
T = (2πf)^2 / (0)^2
Simplifying further, we get:
T = ∞
This means that the transmission ratio is infinite, which implies that no vibration will be transmitted to the instrument. This is not possible in reality, so there must be some mistake in the equation or assumptions made.
Let's try a different approach to solve the problem.
Alternate approach:
The transmission ratio can also be calculated using the equation:
T = 1 / (1 + (ω / ωn)^2)
where ω is the angular frequency of the base surface vibration, given by ω = 2πf.
Substituting the values in the equation, we get:
T = 1 / (1 + ((2πf) / ωn)^2)
Since we are given that T = 5%, we can rewrite the equation as:
0.05 = 1 / (1 + ((2πf) / ωn)^2)
Simplifying further, we get:
(2πf / ωn)^2 = 1 / 0.05 - 1
(2πf / ωn)^2 = 1 / 0.95
Taking the square root of both sides, we get:
2πf / ωn = √(1 / 0.95)
ωn = (2πf) / √(1 / 0.95)
Substituting the values, we get:
ωn = (2π * 60) / √(1 / 0.95)
ωn = 376.991
The stiffness of the mounting pad, k, is given by the equation:
k = m * ωn^2
Substituting the values, we get:
k = 1 * (376.991)^2
k ≈ 14183.2 N/m
Therefore, the stiffness of the required mounting pad is approximately 14183.2 N/m.
The correct answer is '6767.6'.
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A precision instrument package (m = 1 kg) needs to be mounted on a surface vibrating at 60Hz. It is desired that only 5% of the base surface vibration amplitude be transmitted to theinstrument. Assuming that the isolation is designed with its natural frequency significantly lesser than 60 Hz, so that the effect of damping may be ignored. The stiffness (in N/m) of therequired mounting pad is ________.(Important - Enter only the numerical value in the answer)Correct answer is '6767.6'. Can you explain this answer?
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A precision instrument package (m = 1 kg) needs to be mounted on a surface vibrating at 60Hz. It is desired that only 5% of the base surface vibration amplitude be transmitted to theinstrument. Assuming that the isolation is designed with its natural frequency significantly lesser than 60 Hz, so that the effect of damping may be ignored. The stiffness (in N/m) of therequired mounting pad is ________.(Important - Enter only the numerical value in the answer)Correct answer is '6767.6'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A precision instrument package (m = 1 kg) needs to be mounted on a surface vibrating at 60Hz. It is desired that only 5% of the base surface vibration amplitude be transmitted to theinstrument. Assuming that the isolation is designed with its natural frequency significantly lesser than 60 Hz, so that the effect of damping may be ignored. The stiffness (in N/m) of therequired mounting pad is ________.(Important - Enter only the numerical value in the answer)Correct answer is '6767.6'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A precision instrument package (m = 1 kg) needs to be mounted on a surface vibrating at 60Hz. It is desired that only 5% of the base surface vibration amplitude be transmitted to theinstrument. Assuming that the isolation is designed with its natural frequency significantly lesser than 60 Hz, so that the effect of damping may be ignored. The stiffness (in N/m) of therequired mounting pad is ________.(Important - Enter only the numerical value in the answer)Correct answer is '6767.6'. Can you explain this answer?.
A precision instrument package (m = 1 kg) needs to be mounted on a surface vibrating at 60Hz. It is desired that only 5% of the base surface vibration amplitude be transmitted to theinstrument. Assuming that the isolation is designed with its natural frequency significantly lesser than 60 Hz, so that the effect of damping may be ignored. The stiffness (in N/m) of therequired mounting pad is ________.(Important - Enter only the numerical value in the answer)Correct answer is '6767.6'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A precision instrument package (m = 1 kg) needs to be mounted on a surface vibrating at 60Hz. It is desired that only 5% of the base surface vibration amplitude be transmitted to theinstrument. Assuming that the isolation is designed with its natural frequency significantly lesser than 60 Hz, so that the effect of damping may be ignored. The stiffness (in N/m) of therequired mounting pad is ________.(Important - Enter only the numerical value in the answer)Correct answer is '6767.6'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A precision instrument package (m = 1 kg) needs to be mounted on a surface vibrating at 60Hz. It is desired that only 5% of the base surface vibration amplitude be transmitted to theinstrument. Assuming that the isolation is designed with its natural frequency significantly lesser than 60 Hz, so that the effect of damping may be ignored. The stiffness (in N/m) of therequired mounting pad is ________.(Important - Enter only the numerical value in the answer)Correct answer is '6767.6'. Can you explain this answer?.
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