Samlade Flashcards
(31 cards)
What does the following two Matlab functions do (1) eig(a,b) and (2) fft(x)?
Eig(a,b) solves the eigenvalue problem.
fft(x) is the fast fourier transform → for a given signal in time-domain it transforms it to the frequency domain.
What numerical methods can be used to solve the equation of motion?
Newmark method (numerical solution), central difference method (numerical solution) or duhamel’s integral (analytical solution).
(Newmark is an implicit method and CDM is an explicit method)
Give 3 solutions to build earthquake-resistant structures.
Reinforce the building structure with: Shear walls and cross braces and horizontal frames (diaphragms)
Use of earthquake resistant materials: Lightweight materials (as wool), structural steel (various shapes allow material to bend without breaking)
What is the dynamic magnification factor (or dynamic response factor)? Make a typical
plot to show how it looks for a single-degree-of-freedom system exposed to a harmonic
load. Show schematically how this curve changes when damping is increased (1,5 point)
The dynamic response factor = The dynamic magnification factor quantifies the ratio of the system’s maximum steady-state response to the static displacement caused by the same load.
What are the advantages/disadvantages of using a lumped mass matrix?
The mass of a structure or system is concentrated (lumped) at the discrete points (usually nodes) of the system, rather than being distributed continuously across the elements.
Advantages - Reduces complexity and simplifies matrix operations. For time integration schemes like explicit methods (e.g., in dynamic analysis or crash simulations), the lumped mass matrix leads to better numerical stability.
Disadvantages - The lumped mass matrix is a simplified approximation and may not accurately represent the actual mass distribution of the system, this can lead to less accurate results. Not suitable for systems with complex geometry or mass distribution, the lumped mass approach may be insufficient.
What are Transient Vibrations and Steady-State Vibrations? Show with a simple graph.
(1 point)
Two types of forced vibration:
- Transient vibrations are temporary vibrations that occur in a system due to a sudden disturbance or force, and they gradually decrease in amplitude over time. The system moves toward an equilibrium or steady state as energy is lost through damping (friction, air resistance, etc.)
- Steady-state vibrations occur when a system is subjected to a continuous, periodic force and the system vibrates at a constant amplitude and frequency.
What does the following Matlab functions do (1) eig(a,b), (2) fft(x), and (3) hann(x)?
Eig(a,b) - computes the generalized eigenvalues, describing the systems natural frequencies.
fft(x) - fast fourier transform, the function returns in fourier coefficients, which are complex numbers representing the amplitude and phase of different frequency components of the signal.
hann (x) - generates a hann window, that helps minimize discounties at the edges of a time domain signal before performing frequency-domain analysis.
How large is roughly the damping ratio in (1) a welded steel structure, (2) a riveted steel
structure, (3) a reinforced concrete structure, and (4) a pre-stressed concrete structure?
Give approx. values in %. (1 point)
1.≈ 0.5%
2.> 1 %
3.≈ 2 %
4.< 3 %
Describe very briefly the concept of mode-superposition, advantages & disadvantages and its limitation (1 point)
Mode-superposition is used in structural dynamics to analyze the response of a system under dynamic loading. It is a method used to determine the response of a multi degree of freedom system against an applied load. The idea is to transform the MDOF into a series of SDOF systems.
One advantage is that the superposition requires less CPU (computational effort) usage because it only focuses on solving for the most significant modes of the structure.
Disadvantage is that you only can use it for systems with elastic behavior and another is that for bigger problems the computational matrix becomes big and reduces the CPU effectiveness.
Limitations:
1. Superposition considers linearity of the system (linear assumption)
2. Limited accuracy for higher modes
What is aliasing (or folding) in the concept of signal analysis? Give an example (1 point)
Aliasing is under sampling, meaning that the max-frequencies in the signals are higher than half of the Nyquist frequency (the sampling frequency).
Example: imagine an audio signal with frequencies up to fmax = 15 Hz and if we use Nyquist sampling frequency as fs=20Hz, all the frequencies above 10 Hz will incorrectly appear as lower frequencies, resulting in a distorted and unclear audio signal.
Give 3 factors which can influence the level of damping in a railway bridge (1 point)
- Ballasted/non ballasted (higher damping if ballasted due to friction)
- Temperature (stiffness can increase → lower damping)
- Friction at e.g. supports
When the damping is assumed to be proportional to the speed of vibration, what is such damping usually called in other words? (0.5 point)
Viscous damping is damping that is proportional to the velocity of the system.
The train operator SJ is planning to procure new high-speed trains that can travel at 300 km/h. The train coaches have axel distance of 20 m. Bridges with what frequency will be experiencing resonant vibrations from this train? (1 point)
f=4.167 Hz
What simple methods can be used to estimate damping from a measured signal?
We have two methods, logarithmic decrement method and half-power bandwidth method that can be used to estimate damping ratio of a signal. Logarithmic decrement is used in the time domain and half power bandwidth method is used in the frequency domain.
A railway bridge with a fundamental frequency of 3.5 Hz carries a high-speed train having an axel distance of 20 m. What is the expected critical speed at which resonance will occur?
vcr = 252 km/h
Give expressions for the Modal mass and Modal stiffness.
Modal mass, Mn = ΦnT * M * Φn
Modal stiffenss, Kn = ΦnT * K * Φn
Describe the difference between explicit and implicit time stepping methods.
In explicit time stepping methods the equations of motion are evaluated at the current time step t=ti and assumptions are used to determine the displacement ui+1.
While in the implicit methods the equations of motion are evaluated at the current time step and the next (future) (t=ti+1) and assumptions used to determine the displacement ui+1.
Describe one common method used for constructing the damping matrix C in FEM
You can use the Rayleigh (or proportional damping). The damping matrix C can be expressed as a linear combination of the mass M and the stiffness K.
It formulates damping proportional to M and K as C = a0M+a1K
What does the Nyquist sampling theorem state (answer very briefly)?
Describes how you can correctly reconstruct a signal if the sampling frequency is at least twice as high as the highest frequency in the signal. It states that to avoid Aliasing when sampling you should sample with a frequency at least 2 times higher than the maximum frequency.
Expressed as: fs>2*fmax
What are the symbols (ω, ωn, φ, ξ, p0/k) in the displacement formula for the SDOF-system representing?
ω - omega - circular frequency of the load
ωn - the natural frequency of the system
φ - phi - the phase angle, representing the phase difference between the applied force and systems respons
ξ - xi - damping ratio, measures the daming in the system (represents the relationship between actual damping and critical damping)
p0/k - the static deformation (p0 = amplitude of the force and k=stiffness of the system)
Explain when the phenomenon of resonance occur
Resonance occurs when: ω=ωn
Resonance occurs when a system is subjected to a periodic force whose frequency is equal or very close to the system’s natural frequency. At resonance, the system can experience large amplitude oscillations.
The axial stiffness of a pile can be represented with a spring. Derive an expression for the spring stiffness k
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Explain with simple graphs what frequencies are removed when applying a) low-pass filter, b) high-pass filter, c) band-pass filter
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Low-pass filter: Allows low frequencies and removes high frequencies.
High-pass filter: Allows high frequencies and removes lower frequencies.
Band-pass filter: Allows frequencies within a certain range, removes both very low and very high frequencies.
(Dessa filter används för att manipulera signaler genom att selektivt blockera eller tillåta vissa frekvenskomponenter. Exempelvis används high-pass filter i mobiltelefoner för att filtrera bort lågfrekventa störningar som vindljud eller buller från vibrationer. Eller band-passfilter används i radioapparater för att lyssna på en specifik radiostation och blockera andra frekvenser.)
When conducting a dynamic analysis of a civil structure, why do we typically just assume a value of damping in the modes?
We usually assume a value for damping in dynamic analysis because precise modeling of damping is complex, while the assumed values generally provide a reasonable approximation, to ensure safety and accuracy in the structural performance.
