Calculate the natural frequencies and the mode shapes

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Reference no: EM133536620

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Question 1

An unbalanced engine mounted on an elastic support imposes a periodic vertical force on the support at the rotational frequency of the engine. In order to determine the relationship between the engine rotational frequency and the resonance frequency of the elastically supported engine, the engine run speed was slowly increased from its normal value of 600 rpm to 840 rpm, which resulted in a trebling of the amplitude of the measured displacement. No resonance was observed in the displacement response in the range of speeds investigated.
Assuming that the damping present in the existing elastic support is negligible, determine:

The natural frequency of the elastically supported engine _______ rad/s. Your answer should be acurate to +/- 0.1.

It is proposed to reduce the vibration displacement amplitude at the normal run speed of 600 rpm to 80 % of its current value by inserting a damping device.

The required damping ratio of the device is
Your answer should be acurate to +/- 0.01.

If the mass of the machine is 600 kg, the damping coefficient of the damping device is ______ kNs/m
Your answer should be acurate to +/- 1.

Question 2:

A beam of length 0.5 m, with circular cross-section of uniform radius 30 mm is made of an alloy material with material properties:

density = 3,000 kg/m^3 Young's modulus = 70 GPa Poisson's ratio = 0.23

The natural frequency of vibration for clamped-pinned boundary conditions for transverse mode 4 is:

(a) 51667.5 rad/s

(b) 3869.7 rad/s

(d) 81159.1 rad/s

(e) 227.3 rad/s

The natural frequency of vibration for fixed-fixed boundary conditions for longitudinal mode 5 is:

(a) 303506.7 rad/s

(b) 113815.0 rad/s

(d) 455260.0 rad/s

(e) 227630.0 rad/s

The natural frequency of vibration for fixed-free boundary conditions for torsional mode 6 is:

(a) 106429.8 rad/s

(b) 212859.7 rad/s

(d) 319289.5 rad/s

(e) 79822.4 rad/s

Question 3:

2045_wing vibration.jpg

The vibration in the vertical direction of an airplane and its wings can be modelled as a three-degree-of-freedom system with one lumped mass corresponding to the right wing and engine, one lumped mass for the left wing and engine, and one mass for the fuselage. The stiffness connecting the three masses corresponds to that of the wing and is a function of the modulus, E, the second moment of area,
Model-Z aircraft fuselage is Z m = 4.5 m, where m is the combined mass of each wing and engine. I, and the length, l, of the wing. A model for the airplane assuming the wings are cantilevers is shown in Figure 1. The mass of the Derive the equations of motion of the airplane in matrix form for free vibration, for unknown values of E, I, l and m which are to remain variables and be represented as symbols.

Solve the symmetric eigenvalue problem, and use Matlab to calculate the natural frequencies and the mode shapes as a function of of E, I, l and m without using Matlab's symbolic calculation features. [Hint: this is possible by rearranging or normalising the characteristic equation so that the transformed stiffness matrix is only comprised of numbers.]
The three natural frequencies of the airplane model (in order from lowest to highest) are:

----
Mode 1: _____ x √EI/ml³ rad/s

Mode 2: _____ x √EI/ml³rad/s

Mode 3: _____ x √EI/ml³rad/s

The three vibration mode shapes corresponding to the three natural frequencies of the airplane, normalised by setting the first element to
unity, i.e. xi₁ = 1.

For Mode 1:

x₁₁ =

x₁₂ =

x₁₃ =

For Mode 2:
x₂₁
x₂₂
x₂₃

For Mode 3:
x₃₁ =

x₃₂ =

x₃₃ =

Reference no: EM133536620

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