Engineering Mathematics and Computation: Lab #2 Solved

One of the main issues in the design of structures for dynamic loads, such as earthquakes or wind forces, is to establish a good estimate of the natural period or periods of vibration. In earthquake engineering, the first fundamental mode of vibration is used to estimate the acceleration demand, and in turn, the force demand on a structure. The structure acts like a filter and filters the ground motion signal until it reaches the masses. The dynamic response of the system depends on its physical characteristics, e.g. its mass and stiffness. Elongating the period (reducing the natural period of the structure) is a good way to avoid damage from earthquakes, since the frequency content of earthquake is quite high, and this is accomplished by modifying the design of the structure.

Typically the number of storeys a building has will dictate an approximate number of modes of vibration the structure will be excited in. The higher modes require far too much energy to excite them and usually only the first 2 to 5 modes (or perhaps 10 if the structure is very tall) are important.

For a two-storey structure the problem condenses to two degrees of freedom described by a set of coupled, second order differential equations. The derivation of this model comes directly from free body diagrams of the masses. A two-storey building can be modelled as seen in the figure on the next page, with the mass concentrated on each floor, and the stiffness concentrated on end columns.


M2





k2/2



k2/2






M1










k1/2

k1/2








Figure 1 - Model of a two-storey building

By lumping the stiffness on each floor, the model can be further simplified into a system of two masses coupled by springs, or what is commonly referred to as the “Lollipop Model”.
M2    x2




k2

M1



k1





x1










Figure 2 – Lollipop model of a two-storey building


Four forces are considered when taking a free body diagram of each mass:

    • An inertial force from the acceleration of the mass

    • Damping forces from the viscous damping associated with velocity

    • Spring forces as described by Hooke’s law

    • A time-varying externally applied force

In this lab, you are asked to simulate the building’s behaviour when an externally applied force is assumed not to be present, i.e. under what is called free vibration. We will also choose to ignore the viscous damping forces. Under these assumptions, the mathematical model of the building consists of the following two, coupled, second order differential equations:
  ̈= −      −    (   −    )
11    11    2    1    2

  ̈ = −   (   −    )
22    2    2    1
1 =  2 = 4.66 kN/mm
1 = 0.0917 kNsec2/mm
2 = 0.0765 kNsec2/mm

Formulate the state matrix    for the mathematical model of the two-storey building (hint:
you should end up with a 4x4 matrix). Using initial velocities of zero and    1(0) =

100 mm and 2(0) = 50 mm, numerically solve this IVP using the Euler method for the horizontal displacements of the first and second storeys, i.e. 1(  ) and 2(  ).

Develop a figure showing the resulting numerical solutions for ∈ [0,10       ], using different line types for different time steps N, where = 10/Δ  . Use the subplot command to plot 1(  ) versus t on the top plot and 2(  ) versus t on the bottom plot. Label the axes and add a legend showing the time step N associated with each line type. By studying the approximate solutions using different numbers of time steps, determine how many time steps are needed in order to obtain a reasonably accurate solution.

The final part of the lab is to perform some analysis on the simulation results and relate these results to our earlier work on matrices and their eigenvalues.

By examining your simulation results, observe the resonant frequencies. Try to estimate these resonant frequencies directly from the data by estimating their periods and converting these periods to angular frequencies (radians/sec).

Next, open the System Identification App in MATLAB. Select the drop down menu for importing data and select Time Domain Data. Under the Workspace Variables, enter the name of the vector containing 1(  ) as the Output, set the Starting Time to zero, and set the Sample Time to the Δ value that goes with 1(  ). Click “Yes” when the next window appears asking if a time series should be created from the output variable. When you do this, the data will appear in the Data Views. If you check the Time plot, you will see a time domain plot of the data. When you check the Data spectra, you will see a periodogram of the data which is the absolute square of the Fourier transform of the data. This is a frequency domain plot that clearly shows the resonant frequencies contained in 1(  ). Determine the frequencies at which the system is resonating. It will help if you add a grid to the plot from the Style menu and recognize that the periodogram is a log-log plot. Repeat by examining the resonant frequencies associated with 2(  ).

Now for the pièce de résistance (French for the most important or remarkable feature). Determine the eigenvalues of the state matrix using the built-in function in MATLAB called ‘eig’. You will see that the eigenvalues come in complex conjugate pairs and correspond to the resonant frequencies or natural frequencies of the system. Voilà! Lots more to come when you take MAT292F Ordinary Differential Equations in second year.

You may also enjoy watching the following YouTube video:

https://www.youtube.com/watch?v=OaXSmPgl1os

This video was produced by Professor Kwon of our Civil Engineering Department when he was at Missouri University of Science and Technology. It shows a forced vibration test for a physical model of a three-storey building with an external force applied at the base. Wait and watch for the first two resonant frequencies (the system would have a third resonant frequency being a three-storey structure but it is probably higher than the machine can generate or is too small to observe.


    i Special thanks to Giorgio Proestos and Allan Kuan of CIV102F for helping put this exercise together for ESC103F. Both Giorgio and Allan are EngSci alumni. Giorgio was the head TA for CIV102F before Allan and is currently an Assistant Professor at NC State University in Raleigh, North Carolina.