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UNIVERSITY OF GLASGOW

Degrees of MEng, BEng, MSc and BSc in Engineering

DYNAMICS 5 (ENG5299)

2017
90 minutes


Attempt any three questions

The numbers in square brackets in the right-hand margin indicate the marks allotted to the
part of the question against which the mark is shown. These marks are for guidance only.




An electronic calculator may be used provided that it does not have a facility for either
textual storage or display, or for graphical display.






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Q1 The wings of an aircraft are slender light structures designed to safely sustain the weight
of the aircraft. During operation, the aerodynamic load and the weight of the fuselage
induce continuous vibrations in the wings. A wing of an aircraft can be modelled in a
very rough approximation as a cantilever beam with a mass at the tip (the engine).
(a) Using Lagrange equation, derive the equation of motion for a system featuring
a mass at end of a cantilever beam. The mass is loaded with a sinusoidal force.
Assuming large deformation, the vertical displacement at the end of the beam
can be related to the axial contraction at the same point through the formula ! = "#$#(, ) ) %&!*, where l is the length of the beam. The potential energy
can be written as = )'()!#!" * *#(, ) where E is the Young modulus of the
material and Iz the second moment of inertia with respect to axis of the beam.
Consider that the wing is massless and that all the mass is concentrated at the
tip of the wing. [6]
(1) In addition to the potential energy, to apply Lagrange formula we need the kinetic
energy of the mass. This can be written as
= 12(̇!# + ̇!#) = 123̇# + 36#̇#25# 7
Applying the Lagrange equation (1 point for the equation) 9̇!; − ! + ! = +#
After substituting the expressions for the kinetic energy and the potential energy (4 =
1 point for each correctly derived term in the Lagrange equation), we obtain
>̈! + 9 3625#; ̈!!# + 9 3625#;!̇!# + 93,' ; !B = ()
(b) Discuss the order of magnitude of the terms appearing in the equation and how
you would order the terms to apply the method of multiple scales. (Assume a
time scale = Ω, and write ! = and a small force () = () [4]
Write the derivative in respect with ∎ = Ω ∎ = Ω∎′ ##∎ = Ω# ##∎ = Ω#∎′′

Substituting we have

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>Ω#!′′ + 9 3625#;Ω#!′′!# + 9 3625#; !(Ω!-)# + 92,' ; !B = ()
(2 points)
Assuming ! = and () = () we have >Ω#′′ + 9 3625#;Ω#'--# + 9 3625#; '-# + 92,' ; B = ()
By using the result derived, the equation of motion can be written as
-- + 9 3625#; [#--# + #-#] + .#Ω# = cos ()Ω#
(2 points)

Q2 The equation of motion of a structure usually come in the form ̈ + ̇ + + Y, ̇, ̈Z = () (2.1)
Where M,C,K are the mass, damping and stiffness matrix, X is the vector of the
deformations, N is a vector of nonlinear terms and F the external force.
(a) Assuming that the system is linear (N=0) and has a single degree of freedom,
how can we write equation 2.1 in state space? [1]

In the case of single degree of freedom the state vector is made of one single variable
and the equation of motion becomes ̈ + ̇ + = ().
>̇"̇#B = ^− − 1 0 ` a"#b + a()0 b ℎ ḟ = " = #
(b) How does the representation changes if the system has multiple degrees of
freedom? Can we write the state matrix in terms of the matrices M,C, and K?
[2]
If the system MDOF the representation is very similar. The state vector now is made
of the displacement vector in the lower part and in the velocity vector in the upper
part. By using this change of variable, it is possible to write the state matrix in terms
of the Mass, stiffness and damping matrix as follow

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ġ"̇#i = >−/ −/ B >"#B + a() b ℎ ḟ = " = #
Any state space representation (1 point)
The state space representation in terms of the M, C and K matrix (1 point)
(c) Write the equation of motion in state space for the two degree of freedom
system in figure 2.1. The notation (k1, 1) means that the elastic force of the
spring is the sum of a linear component k1 Δ and a nonlinear component Δ'. The system is damped with three linear dashpots and excited with forces
F1 and F2. [3]

Figure 2.1 Two DOF system.
The state space representation for this system is as follow
q̇"̇#'1r = ⎣⎢⎢
⎢⎢⎡− " + # ## −" + #
−" + # ## −" + #1 00 1 0 00 0 ⎦⎥⎥
⎥⎥⎤ y"#'1z
+ q−"'' − #(' − 1)'−"1' − #(1 − ')'00 r + q
"()#()00 r ℎ {
̇" = "̇# = #" = '# = 1
Any correct state space representation (3 points)

(d) Write the mode shapes of the underlying linear system in vector form. Note that
the system is symmetric. [2]
The modes can be derived from the eigenvalue problem

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("# −/") = |g2# 00 2#i − q
" + # −#−# " + # r} a"#b = 0
From this we obtain
~2# − " + # ## 2# − " + # ~ = 0
Which leads to "# + 2"# + (−2" − 2#)2# +#21 = 0
From which the solutions are
"# = " ,"# = " + 2#
Substituting in the eigenvalue problem we obtain for "#
q−# ## −#r a"#b = 0 ⇒" = # ⇒" = [1 1]
Substituting in the eigenvalue problem we obtain for ##
q−# ## −#r a"#b = 0 ⇒" = −# ⇒# = [1 − 1]
By noting that the system is 2DOF and symmetric, the eigenvectors can alternatively
be written by considering that in this case the masses will move of the same quantity
either in phase (1st mode) or in anti-phase (2nd mode).
1 point for each mode
(e) Write the nonlinear elastic forces in terms of the modal coordinates. [2]
The underlying linear system has two eigenvector: "and #. Remembering the
definition of mode

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4 = ‚242.2
We can write " = " + # # = " − #
The nonlinear elastic forces on m1 and m2 are respectively 1: ""' + #(" − #)'2: "#' + #(# − ")'
Substituting the modal coordinates, we obtain (q" + q#)'" + 8q#'# (q" − q#)'" − 8q#'#
Q3 Everything in nature is ultimately nonlinear, nonetheless in engineering we often
assume linear behaviour.
(a) Describe the rationale behind linear approximation in structural dynamics [1]
Although all structures are inherently nonlinear, the assumption of linear behaviour
is valid for a vast range of applications. This depends on the fact that many materials
used in engineering exhibit a range in which their constitutive laws are linear and
also in many applications it is reasonable to assume that the deformations
experienced by the structure are small.
(b) Name two cases in which the linear approximation cannot capture the dynamics
of the system. [2]
Any two examples are fine: friction, impacts (discontinuous forces), imperfect
constraints (nonlinear boundary conditions), geometric nonlinearities …
(c) How can one ascertain if one structure is behaving in a linear manner? [2]
Any suitable answer will be considered valid: checking if doubling the force
amplitude, the response doubles, looking at the correlation between input and output,
checking that the resonance frequency do not depend on the amplitude of excitation
…

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(d) What is the superposition principle and how is it related to modal analysis? Why
is it not valid for nonlinear structures? [2]
The superposition principle is a principle valid for linear systems that states that the
response of a system to an external input, which is the sum of several inputs, is the
sum of the separate responses to each input. In modal analysis this principle is used
to write the response of a system as the sum of the response of multiple SDOF
systems. In nonlinear structures, this principle cannot be used because, by definition,
this structures are governed by nonlinear equations (i.e. equations for which the
either/both the additivity or/and homogeneity properties are not valid)

(e) What is a bifurcation? [2]
A bifurcation is a topological change of the phase portrait of a dynamical system
under the change of a parameter.
(f) Name and describe two types of bifurcations that can be observed in nonlinear
systems [1]
Hopf-bifurcation: two eigenvalues cross the imaginary axis parallel to the real axis.
Fold bifurcation: two fixed point, one stable and one unstable, collide and annihilate
each other. This is also called a saddle node bifurcation.
Pitchfork bifurcation: one fixed point split in three fixed point two of which maintain
the same stability, the third one has opposite stability.
…


Q4 Multiple Scales (MS): this is one of the methods that can be used to find an approximate
solution to nonlinear ordinary differential equations.
(a) What class of methods do the Multiple Scales belong to? [1]
The Multiple Scales belong to a class of methods called the perturbation methods.
(b) How do you expand the state variables in the MS? Please discuss the order of
magnitude of the terms in the expansion. [1]
In the MS, one of the fundamental assumption is that the state variable can be written
as the sum of a series of terms. Each term in the series is one order of magnitude
smaller that the preceding term.

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= ‚22526*

(c) What hypothesis do you make on the time? How does this assumption affect
the first and second derivative? (Write a suitable time expansion and
accordingly, the first derivative and second derivative up to the first order.) [2]
Multiple time scales are used in this method (from which the name). We assume
that each time scale is one order of magnitude slower than the preceding = (*, ", … , .) ℎ 27" = 2
The first derivative is written as = * + "
And the second ## = #*# + 2 * "



(d) The equation of a pendulum suspended from a horizontally moving point can
be written as:
̈ + 2.̇ + .# sin() = $Ω# cos(Ω) cos()
i) Write the ordered form for this equation (use Taylor expansion up to the
third order). [2]
After expanding the functions depending on state variables up to the third order, the
ordered form for the equation can be written as
̈ + 2 Š.̇ + .# − ℎ‹' = Š$Ω# cos(Ω) − Š$Ω#2 #cos (Ω)
Assuming

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ℎ‹ = ℎ = .#6 Š$ = $
ii) Write the zeroth and the first perturbation equations [4]

To write the zeroth and first perturbation equation:
1- Substitute = ! + "in the ordered equation
2- Apply the derivative in respect to T0 and T1 to the ordered equation
3- Collect the terms with order ! and "
!: !#! +$# ! = 0 ": *#" + .# " = −2*"* − 2.Š ** + ℎ‹*' + Š$Ω# cos(Ω)
(2 points for the procedure, 1 point each for the correct equations)









































































































































































































































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