代写ENG3015、R编程设计代做

日期：2024-04-05 06:03

University of Exeter

ENG3015 Structural Dynamics

Year: 2023/24 and onwards

Coursework 1

Q1: Figure Q1 shows a structural frame with rigid cap beam AB vibrating horizontally. The

beam is supported vertically by three columns (AE, GF and BH) and restrained horizontally

by a system of horizontal steel tubes (BC and CD) supported by another vertical column (CI).

Apart from the steel tubes, all other structural members may be assumed not to deform

axially.

Figure Q1

a) Sketch the development of the equivalent SDOF system for horizontal vibration of

beam AB

b) Using the notation provided, derive expression for the undamped natural frequency

corresponding to the horizontal vibration of the cap beam.

c) Using the developed expression and the following design data, calculate the natural

frequency of this SDOF system in the horizontal direction

m’=12,000 kg/m

E=210 GPa

I=104170 cm4

D=0.5 m; d=0.4 m

h1=4 m; h2=6 m

l1=2 m; l2=4 m

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Q2: The system shown in Figure Q2 may be assumed to have rigid levers, no mass and

frictionless pivot. For small vertical vibrations of the point mass:

Figure Q2

a) Derive expression for the damping ratio.

b) Derive expression for damped natural frequency.

c) Derive expression for the critical damping coefficient.

Note: all expressions should be given in terms of variables m, c and k, describing the physical

properties of the system, and p, q and r relating to the geometry of the system. There are no

specific numerical values associated with these six variables.

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Q3: In a type approval test, a helicopter (Figure Q3) is dropped under gravity from a height,

?, with its rotors stationary. The pilot's seat is mounted on springs of combined stiffness ??,

and of dampers of combined rate ??. The combined mass of the (dummy) pilot and seat is ??.

Figure Q3

a) Using clear sketches and maximum 100 words, show that the equation of motion of the

pilot and seat after impact with the ground, which occurs at t=0 s, is:

?????(??) + ?????(??) + ????(??) = 0

with the following initial conditions: ??(0) = ?

????

??

and ???(0) = √2???, the positive

coordinate direction being as shown in Figure Q3. In the given equation of motion ??(??) is

the dynamic displacement from the static equilibrium position of mass ?? after the

helicopter strikes the ground.

b) Consider the case where m=100 kg, k=2.5x105 N/m, c=3000 Ns/m and h=5 m. Using NDOF

software, calculate the maximum displacement, velocity and acceleration experienced by

the pilot after the helicopter strikes the ground. Present relevant annotated NDOF plots

indicating maximum values calculated during the first 1 s after the helicopter strikes the

ground.

c) Compare the acceleration calculated under b) with that which would be obtained without

dampers. Use a pair annotated plots NDOF for this comparison. Explain in no more than 50

words the effect of damping and lack of it for this particular structural dynamics design

problem.

d) What would be the maximum acceleration of the pilot if the spring stiffness were reduced

to 6x104 N/m with no dampers fitted? Use NDOF plots to compare the undamped responses

under c) and d). In no more than 50 words describe advantages and disadvantages of using a

softer spring?

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University of Exeter

ENG3015 Structural Dynamics

Year: 2022/23 and onwards

Coursework 2

Q1: A two-storey frame is shown in Figure Q1. The horizontal beams are infinitely sMff

relaMve to the verMcal columns.

Figure Q1

Design data:

? Bending sMffness of columns: EI=81.35 MNm2

.

? Mass per unit length of the horizontal beams: m’=10,000 kg/m.

? SMffness of the horizontal spring restraining the top floor: k=50 MN/m.

? It can be assumed that damping raMo is 2% for all relevant modes of horizontal

vibraMon.

For the structural frame shown in Figure Q1:

a) Calculate the mass and sMffness matrices of this system.

b) Calculate the natural frequencies of the system.

c) Determine and sketch the unity-scaled 2nd mode shape of this system.

d) Calculate the modal mass corresponding to the unity-scaled 2nd mode shape calculated

in c) above.

e) Which modelling parameter needs changing and how to enable modelling of this system

in the NDOF so_ware, considering the limitaMons of that so_ware?

f) Introduce the assumpMon made in e), sketch the new system and using the same

degrees of freedom as shown Figure Q1 re-calculate:

i) mass and sMffness matrices of the system; check the calculaMons using NDOF so_ware.

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ii) natural frequencies of the system; check the calculaMons using NDOF so_ware.

iii) unity scaled 2nd mode shape of this system; check the calculaMons using NDOF

so_ware.

iv) modal mass corresponding to the unity-scaled 2nd mode shape calculated; check the

calculaMons using NDOF so_ware.

g) Compare modal masses calculated under d) and under f-iv) and explain the difference, if

any. Which physical feature of the structure is causing the difference and why, if any?

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Q2: A three-storey frame shown in Figure Q2 is subjected to the harmonic force . The

horizontal beams are infinitely sMff relaMve to the verMcal columns.

Figure Q2

.

Modal damping raMos for the system are: .

Using NDOF and EI=4.5x106 Nm2 answer following quesMons.

a) Calculate natural frequencies and unit normalised mode shapes.

b) Determine modal masses. Check the obtained modal mass for the 3rd mode by hand.

c) Plot the forced displacement response of the system in the first 15 s.

d) The forced displacement response plot produced under c) has some specific clearly

observable features which determine the type of the response. How is response

plofed called?

a. Transient response,

b. BeaMng response,

c. Resonant response?

Select one answer. Explain your answer referring to the relaMonship between the excitaMon

frequency and natural frequencies of the system.

University of Exeter

ENG3015 Structural Dynamics

Year: 2022/23 and onwards

Coursework 3

Figure Q3(a) shows a finite element model of a floor structure featuring six floor panels,

each panel being 8 m long and 4.5 m wide. Therefore, the total length of the floor plate is

27 m (6x4.5 m).

Figure Q3(a).

Figure Q3(b) shows modal properSes for the first four modes of vibraSon.

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Figure Q3(b)

Using Appendix G - Concrete Society Technical Report 43 (2nd EdiSon) answer the following

quesSons related to the vibraSon serviceability of this floor structure.

a) Is this a low- or a high-frequency floor? Explain the reasoning behind your answer and

what kind of vibraSon response does this floor need to be design for under human

walking: transient, random, free or resonant vibraSon?

b) Calculate maximum response factor for verScal vibraSon at point 2 due to single person

walking at point 1 of this floor. Show all calculaSons and clearly state the assumpSons

made. Data needed for calculaSons:

? Pedestrian weight is 700 N.

? The minimum pacing rate is 1.4 Hz.

? The maximum pacing rate is 2.0 Hz.

? Modal damping raSo is 1.5% in all relevant modes of vibraSon.

? Assume that the walking path is 8m long across the floor through point 1.

Mode 1: f1=3.8Hz, m1=37,000kg Mode 2: f2=5.1Hz, m2=25,000kg

Mode 3: f3=7.1Hz, m3=20,000kg Mode 4: f4=9.0Hz, m4=18,000kg

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? Mode shape amplitudes at point 1 are:

o For mode 1: 0.7

o For mode 2: -0.8

o For mode 3: 0.3

o For mode 4: -0.6

? Mode shape amplitudes at point 2 are:

o For mode 1: 1.0

o For mode 2: 0

o For mode 3: 0

o For mode 4: -0.5

c) Check your floor vibraSon response calculaSons using independently the ‘App G’ opSon

in the NDOF sodware. Submit screenshot of the acceleraSon response envelope as a

funcSon of the pacing frequency with annotated interpretaSon of the relevant maximum

response in terms of R-factor and compare with the calculaSons under b) above.

d) Can this floor be used as a workshop?

Note:

Some design data provided my not be needed in calculaSons. If any design data is missing,

make reasonable assumpSons and clearly state such assumpSons.