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Examination Department of Mechanical Engineering ___________________________________________________________________________

Course: MT2562 (MT2529), Structural Analysis

Date, time: 2019-03-23, 09:00 – 14:00

Examiner: Ansel Berghuvud

Means: Writing materials, pocket calculator

Grades: 0 – 9 = F, 10 - 12 = E, 13 -15 = D, 16 - 18 = C, 19 -21 = B, 22 - 24 = A

Complete solutions in English must be submitted ___________________________________________________________________________ 1. A car deck on a bridge is fastened by a hinge in one end and is resting on rollers in the other end. The deflection of the bridge when subject to the static weight of a car is to be estimated. A scaled physical model of the system is setup in a laboratory. The system is also modelled as a simply supported Euler-Bernoulli beam. Measurements on the physical model are carried out and the behaviour of the analytical model is computed by a numerical method. Exemplify, explain, and motivate at least six types of errors that can be expected to get involved in these estimates of the bridge deflection. (3 p)

(Answer, purchase paper to get full solution)

1. 1. The boundary conditions may not be fulfilled. Meaning that, there might be deflections at the ends of the beam
   2. Systematic errors – Analog signals will provide parallax errors when measuring the data from the scales.
   3. Random human errors while measuring. This can be avoided by taking average of a repeated measurements.
   4. Mass of the beam is neglected in Euler-Bernoulli beam theory, but in reality, it will cause small deflections along the beam.
   5. Non-Uniformity in the beam. This can be avoided by taking an average along the cross sections of the beams at various position along the length of the beam.
   6. Errors in Numerical model analysis
   7. The beam has the tendency to move in sideways along the boundary conditions.
   8. The force used in the experimental model cannot be exactly represented in the numerical model.

2. In the derivation of Euler-Bernoulli beam theory applied on homogeneous linear elastic isotropic material, the constitutive relation produces a contradiction.

a) What is this contradiction, and how is it handled in Euler-Bernoulli beam theory? (2 p)

b) Give a plausible explanation to why the above mentioned material properties are popular in engineering analyses. (1 p)

3. Consider an initial value problem governed by the equation f(x,y) given in equation 3.1.

 a) Determine the function value ????(0.05) for the equation with an error of O(h3 ) using the Euler method combined with Richardson extrapolation. (2 p)

b) Specify the total number of f-calculations needed, and produce an estimate of how many steps that would be needed for similar accuracy if using the Euler method only. (1 p)

4. A supported beam shown in Figure 4.1 has a Young’s modulus E = 2.1×10¹¹ N/m² , a length L = 4.5 m, and a moment of inertia I = bh³ /12 m⁴ where it ?s rectangular cross section width is b = 0.15 m and the cross section height is h =  0.13 m. The beam carries the total weight Q =15×104 N which can be considered as a constant distributed load q(x)= Q L.

According to the Euler-Bernoulli beam theory the problem is governed by equation 4.1.

a) Sketch the expected deflection shape of the beam. (0.5 p)

b) Specify and motivate which of the given data that gives the largest uncertainty in the result. (0.5 p)

c) Determine the static deflection of the beam at x = L/3. (2 p)

5. A signal h(n) recorded during one second and is shown in Figure 5.1. Estimate the following parameters of h(n):

a) What is the RMS amplitude of the AC component of the signal? (1 p)

b) What is the sampling frequency the signal was sampled with? (1 p)

c) What is the phase in radians of the signal? (1 p)

Detailed description of the estimation process is required. Figure 5.1

6. (3 p) The one sided RMS spectrum of the signal y(n) is shown in the Figure 6.1.

Number of samples N = 512 with Fs = 256 Hz were acquired. The Matlab command fft() produced the spectrum Y(k), where k = 0,…,511.

Estimate the magnitudes and the frequencies (in Hz) of the signal and write the mathematical expression for the time signal y(t). Detailed description of the estimation process is required.

7. You have studied the Matlab function freqz, that can returns the complex frequency response H(jω) (Laplace transform) of a system h(n). In Figure 7.1 the frequency response H(jω) of the unknown system h(n) is shown.

In what frequency range(s) the system h(n):

a) Attenuates the input signal (1 p)

b) Amplifies the input signal (1 p)

c) Does not modify the input signal (1 p)

Detailed motivation of answer is required.

8. Assume that the input-output relationship can be modelled as a single-degree-of-freedom system with M = 5 [kg], as shown in Figure 8.1. In this model, f(t) is the applied force at the free end of the cantilever beam and x(t) is the displacement response at the free end.

Use the experimental results shown in Figure 8.2 to determine appropriate value on K [unit ?]. (1.5 p)

The resonance frequency of the system in the Figure 8.1 is to be increased to a new value, which is two times greater than the current resonance frequency. What can be modified and how much? Give two options. For each option state the numerical value and correct unit. (1.5 p)

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Last updated: Jun 08, 2020 06:59 AM