A machine, with an unbalanced rotating shaft, is mounted onto a fixture supported on isolators. The effective mass of the motor and fixture is . In an experiment on the machine, a maximum amplitude of was recorded at frequency of . It was also found that the system has a damping constant of . Determine the: (a) natural frequency in , of the vibrating system, [125.664 ] (3 marks) (b) combined stiffness of the isolators, [1105.4 kN/m] (3 marks) (c) damping ratio of the isolators, [0.0327] (3 marks) (d) amplitude of vibration of the machine at a frequency of , (7 marks) (e) amplitude of vibration of the machine when the system eventually approaches a very high speed, and (4 marks) (f) the maximum dynamic force transmitted to the floor. [15.509 kN] (5 mark
The Correct Answer and Explanation is:
To solve this problem, we need to break it down into the individual parts based on the information provided.
Given Data:
- Effective mass of the motor and fixture = m
- Maximum amplitude recorded at frequency = AmaxA_{max}Amax
- Frequency of maximum amplitude = fff
- Damping constant = ccc
(a) Natural Frequency ωn\omega_nωn (in rad/s):
The natural frequency of a vibrating system depends on the mass (m) and the stiffness (k) of the system, and it can be calculated using the formula:ωn=km\omega_n = \sqrt{\frac{k}{m}}ωn=mk
Where:
- kkk = stiffness (which we need to find)
- mmm = mass (which is also not explicitly given but assumed to be part of the system information)
Given that ωn=125.664 rad/s\omega_n = 125.664 \, \text{rad/s}ωn=125.664rad/s is provided, we can use this as the natural frequency.
(b) Combined Stiffness of the Isolators kkk (in N/m):
The stiffness of the isolators can be calculated from the relationship between the natural frequency and mass:k=m⋅ωn2k = m \cdot \omega_n^2k=m⋅ωn2
(c) Damping Ratio ζ\zetaζ:
The damping ratio is given by the formula:ζ=c2km\zeta = \frac{c}{2\sqrt{km}}ζ=2kmc
Where:
- ccc is the damping constant (which is given)
- kkk is the stiffness (which we can calculate in part b)
- mmm is the mass (which is assumed to be given or can be inferred)
(d) Amplitude of Vibration at Frequency fff:
The amplitude of vibration at a given frequency is determined by the resonance effect, where the amplitude is maximum at the natural frequency. The formula for the amplitude AAA at a frequency fff is given by:A=Amax(1−(ffn)2)2+(2ζ⋅ffn)2A = \frac{A_{max}}{\sqrt{(1 – (\frac{f}{f_n})^2)^2 + (2\zeta \cdot \frac{f}{f_n})^2}}A=(1−(fnf)2)2+(2ζ⋅fnf)2Amax
Where:
- AmaxA_{max}Amax is the maximum amplitude
- fff is the frequency at which the amplitude is being calculated
- fnf_nfn is the natural frequency in Hz (converted from rad/s by dividing by 2π2\pi2π)
- ζ\zetaζ is the damping ratio
(e) Amplitude at Very High Speed:
At very high speeds, the system will tend to a steady state where the amplitude of vibration stabilizes. As the frequency increases, the system moves away from resonance and the amplitude approaches a minimum. This can be calculated by considering the response of the system at a high frequency, where the denominator of the amplitude formula becomes large, effectively reducing the amplitude.
(f) Maximum Dynamic Force Transmitted to the Floor:
The maximum dynamic force transmitted to the floor is given by:Fmax=m⋅Amax⋅ωn2F_{max} = m \cdot A_{max} \cdot \omega_n^2Fmax=m⋅Amax⋅ωn2
Where:
- AmaxA_{max}Amax is the maximum amplitude
- ωn\omega_nωn is the natural frequency (in rad/s)
- mmm is the effective mass of the system
Using the values provided, we can directly plug in the known values into these equations for each of the parts. If you can provide more details such as the damping constant, effective mass, and the actual values of the frequencies involved, I can help calculate the precise answers for each part of the problem.
