This assignment is to carry out a structural vibration analysis of an airplane wing model and
investigate structural integrity assessment through vibration tests.
Why are we asking you to do this?
Structural vibration is the fundamental cause of many structural failures. This can be due to the high
level of stress/strain in the material when a structure is subject to extreme loading conditions or
experiences a resonant condition. More commonly a structure fails unexpectedly due to fatigue after
the material undergoes cyclic stress/strain over a long period of time when the system vibrates within
its normal operating environment. The apparent ‘sudden fracture’ is the result of fatigue damage
being accumulated to a critical level and hence by logic monitoring the fatigue damage accumulation
will allow predictive maintenance of the structure and avoid the eventual fatigue failure of the
structure. In industry, various methods of predictive maintenance have been researched and
developed continuously and in the era of Industry 4.0, this is becoming an integral part of PLM
(product lifecycle management). One important group of techniques uses vibration and this is mainly
due to that vibration characteristics relate directly to the structural integrity status, and also because
vibration responses can be measured easily and reliably in general.
To develop the analytics to establish the relationship between the vibration responses and the
structural integrity condition, an understanding of the input/output relationship of the vibrating
structure through vibration analysis is important. For simple systems, theoretical analysis can be
utilised to solve the governing equation of motion. For real-world engineering structures, vibration
analysis is carried out typically by numerical simulations. Furthermore, the knowledge and skills of
digital signal processing are of great importance for developing methods that can be used together
with other PLM tools in the digital era.
An airplane wing is a representative example of structures that works in a dynamic loading
environment over a long period of time and therefore its structural integrity status needs to be
assessed periodically, especially due to its safety critical nature. In this assignment, an airplane wing
will be modelled as a multi-DOF system. Through doing this assignment, you gain the experience of
carrying out a representative structural vibration analysis of a multi-DOF system involving theoretical
analysis, numerical simulation and digital signal processing. An emphasis has been placed on the
verification of the results obtained by different solution methods. This is a good practice for increasing
your confidence in your results as well as developing the critical thinking ability and professional
attitude that will greatly benefit your future careers whatever they will be.
Task description
A twin-engined airplane is shown in Figure 1(a). The wings have a cantilevered structural configuration
and experience dynamic stress/strain due to dynamic loading during flight. In order to develop
analytics for assessing the structural integrity of the wing, the relationship between the equivalent
stiffness and the vibratory behaviour of the wing needs to be established. In this coursework, a
mathematical model of a two-DOF mass-spring-damper system is considered that will allow
investigations to be carried out in the low frequency range covering the the first two resonant
frequencies. A corresponding computer model using Matlab will be constructed to simulate a
vibration test scenario as shown in Figure 1(b). The Matlab program will first be verified by theoretical
solutions. Then the Matlab program will be used to simulate vibration responses to more realistic and
complex types of excitation. The simulated input (excitation) and output (response) will be used to
represent the measured signals of the vibration tests and British Standards regarding digital signal
processing of vibration measurements will be used.
Figure 1 (a) A twin-engine airplane DA42-VI
Figure 1 (b) Idealised 2-DOF models of one wing in the vertical plane (without
damping) and a vibration testing scenario
Table 2 Task descriptions with details
1 Setting up equations of motion
1.1 State the assumptions that are required to idealise the system in Figure 1(a) to
obtain the 2-DOF model in Figure 1(b).
1.2 Apply the following methods to set up equations of motion:
(a) Newton’s 2nd law method
(b) Lagrange’sequations
2 Carrying out modal analysis
2.1 Determine the natural frequencies and the normal modes using the following
• Manual solution by the matrix iteration method
3 Calculating the vibration response under sinusoidal excitation by
the modal superposition method
3.1 Obtain the time histories of vibration responses by
(a) Manual solution by modal superposition method
4 Investigating the sensitivities of the resonant frequencies to the change of the
effective stiffness
4.1 Determine and record the changes in the two resonant frequencies corresponding
to the changes of k1 and k2
Table 1: Tasks
Initial variables:
m1 (kg) m2 (kg) k1 (N/m) k2 (N/m) F0 (N) 𝜁” 𝜁#
280.2618 70.42342 1.07E+08 3.23E+06 5.28E+03 0.05 0.03
Table 2: Initial variables
Default duration of simulation: T = 40 s (This can be changed with justification)
Range of values of k1 and k2 for sensitivity study:
from 0.5 to 1.5 (i.e., from 50% to 120%) times the initial value
Further notes:
1. w1 and w2 are the two natural frequencies of the 2-DOF model.
2. For Task 3.1, the excitation force f(t) is a sinusoidal signal.
𝑓(𝑡) = 𝐹*sin (𝜔𝑡)