Structural Dynamics Education Module
The purpose of this education module is to provide a medium for the integration of classical dynamics, computer modeling, and experimental testing. This goal is achieved by the analysis, modeling, and experimentally testing of a 4story scale structure that was designed, fabricated, and experimentally tested using UCLA NEES equipment. The geometric details of the scale structure are displayed in the first section of the module. In the second section, an analytical model of the scale structure in the form of an idealized shear building will be constructed and analyzed to obtain modal properties and response quantities. In the third section, a computer model will be created using SAP2000 software in which modal properties and response quantities can readily be obtained and compared to results in the second section. In the third section, data from an actual experiment where the scale structure was attached to the mass of a linear shaker will be used to perform system identification, and again, results will be compared to previous sections.
2. Classical Structural Dynamics
2.1. Newton’s Equation of Motion.
2.2. Modal Analysis
2.3. Newmark’s Method of Numerical Integration
2.4. Quantities of Interest
2.5. Relevant Material
2.6. Assignment 1
3.1. SAP2000 Free Download (Educational Version)
3.2. Issues in Modeling
3.3. Suggested Procedure
3.4. Assignment 2
4.1. Introduction to System Identification
4.2. Fourier Transforms
4.3. Transfer Functions
4.4. The ARX Model
4.5. Experimental Testing
4.6. Data Processing
4.7. Relevant Material
4.8. Assignment 3
5. References
The scale structure is shown above attached to the mass of a linear shaker. Although this image includes bracing in the direction of shaking, these braces were removed during testing and should therefore be ignored.
All members (beams and columns) are 5/8 in square aluminum (AL6061) bars with a stiffness of E = 10,000 ksi. The geometric dimensions are displayed in the following figure. In addition, a typical interior joint is displayed for the purposes of joint rigidity considerations (section 2).
Weight Estimate per Floor 








Number per Floor 
Weight (lbs) 
Weight (lbs per Floor) 
Frame 



Columns 
2.25 
0.80 
1.81 
Crossbars 
2 
1.24 
2.48 
Beams 
6 
0.89 
5.33 






Frame Total 
9.62 
Floors 



Floors 
2 
7.13 
14.26 
Masses 
8 
4.66 
37.28 
Sensors 
2 
2.33 
4.66 
Connection Hardware 
0.25 
12.76 
3.19 






Floor Total 
59.38 
2. Classical Structural Dynamics
Structural dynamics is the study of the behavior and response of structural systems to a set of dynamic conditions. A brief review of classical dynamics is the subject of this first section.
2.1 Newton’s Equation of Motion
A detailed introductory look at formulation and solution of the governing Newton’s equation of motion is located here.
2.2 Modal Analysis
A multi degree of freedom structure, such as our scale structure can be idealized as a shear building, which is an analytical representation of a frame in which the beams are considered rigid and therefore, the number of floors equals the number of degrees of freedom (NDOF).
The
system of equations can be written in matrix form:
Instead of solving this intricate system of differential equations, the equations can be uncoupled and solved individually. This is known as modal analysis. At the core of this analysis, is the eigenvalue problem, whose solution yields the modal frequencies and mode shapes of the system.
Once the modal properties are known, the response of a system can then be expressed modal coordinates and the resulting SDOF equations, q(t), can be solved individually.
2.3 Newmark’s Method of Numerical Integration
When dealing with structural responses to ground motion, numerical integration techniques are necessary to solve the governing equations of motion in discrete form. Some of the more commonly used methods include; excitation interpolation methods, central difference method, and Newmark’s method. The method described and used in this module will be Newmark’s Method which is actually a family of timestepping methods based on the following two discretized equations:
The parameters b and g are known as Newmark parameters and are chosen by the user depending on which relationship for acceleration between time steps is assumed. For example;
is chosen for an average acceleration relationship
is chosen for a linear variation of acceleration
The two equations above and the equation of motion at the given time step are simultaneously solve to yield the next state (i+1) from the current state (i).
Newmark’s method has the following stability condition;
Where T_{n} is the natural period of the system.
For the two cases above, the respective stability conditions exist.
&
Implementation of this iterative method can be expressed in a four step algorithm.




2.4 Quantities of Interest
In engineering analysis and design, several response quantities are of special interest. Usually, theses response quantities give insight into the largest forces and deformations that the system will experience in a given event. Four such quantities are base shear, overturning moment, interstory drift, and roof displacement. In most cases, the history of such quantities is not of importance, it is only the peak responses that engineers are interested in.
The peak roof displacement and interstory drift are self explanatory and can be easily computed from the systems displacement history. Base shear can be calculated by multiplying the first story stiffness by the first story drift and the overturning moment is the sum of story shears multiplied by the story heights
2.5 Relevant Material
Newmark Integrator Matlab function NewmarkIntegrator.m
Loma Prieta Ground Acceleration History lphist.txt
SAP2000 is a 3D stateofart finite element modeling and analysis program that is widely used in both academia and industry. An accurate model of the scale structure will provide a basis to which we can compare analytical and experimental results to.
3.1 SAP2000 Free Download (Education Version)
A trial version of SAP2000 can be downloaded at http://www.csiberkeley.com/. This will be a full functioning version with limited capacity of only 100 nodes. Since our 4story scale structure has nine columns, a minimum of 36 nodes is required.
3.2 Modeling Issues
When modeling a real structure with a FEM model such as in SAP2000, certain assumptions can be made to simplify the process. For example, rigid diaphragms will be assigned at each floor level to assume shear building behavior. It is also assumed that our scale structure will remain in the elastic range throughout testing.
From the scale structure details section, a close up of a typical joint is included which gives some insight into how the end offsets should be assigned. Because the angles connecting the beams to the columns are made of steel as opposed to aluminum, joint rigidity needs to be addressed. Initially, it was assumed that the joints were 100% rigid, but this yielded a much stiffer model than actually exists. It makes sense that the joints are not completely rigid but to what degree is an issue that is of current research interest and is not yet completely understood. If the joints are assumed 50% rigid, better results are obtained and therefore, 50% rigidity will be used in this section.
3.3 Suggested Procedure
A detailed step by step procedure is not given, but a suggested guide line is and with the SAP2000 user manuals, also available at the website above, modeling of the scale structure should be relatively easy.
Create Frame. Modeling in SAP2000 is made easy by the use of templates. Most of the frame will be drawn for you from inputting only the bay dimensions. The crossbars in the direction perpendicular to direction of loading need to be manually drawn.
Assign Base Joint Restraints. Assume fixed base condition exists.
Define Materials. Use E = 10000 ksi for AL6061. Be careful of units.
Define & Assign Frame Section. All members are 5/8 in square bars.
Assign End Offsets. Assign rigid joints (50%) due to joint steel connections.
Assign Rigid Diaphragms. Assign rigid diaphragms to all floors to constrain the behavior of the structure to that of a shear building
Assign Mass. Since we are only interest in uniaxial behavior and not torsional, the distribution of mass in not important. Simply assign all the floor mass of each floor to any node on that floor. Be careful of units.
Define Time History Function. Add Loma Prieta acceleration text file, and insert proper information, i.e. time step = 0.005 s.
Define Time History Case. Add acceleration history in direction of interest (long direction). This is where you assign modal damping. Don’t forget to check the “Envelopes” box; you will need that to see response histories.
Set Dynamic Analysis Options. Choose a reasonable amount of modes and check generate output option.
Run Dynamic Analysis.
Analyze Results. You can get modal frequencies and mode shapes from the generated output. For the response histories, you will need to display “Time History Traces”, define appropriate functions (Disp, Accel, ...), and “Print Tables to File.” This way you can plot the time histories and determine response quantities in the same way as in assignment 1.
4.1 Introduction to System Identification
It is usually the goal of classical dynamic analysis to describe a known system and determine that systems’ response to a given excitation. The goal of system identification is to determine the properties of a system from the known response of that system to a given excitation. We have achieved the preceding goal in two different ways in the previous two sections. In this section we will perform the latter procedure.
There are many different methods of system identification. One method that is familiar in the mechanical and control engineering fields and is relatively simple is the Auto Regressive with exogenous input or ARX approach. This method will be introduced in section 4.4 after some preliminary subject matter is addressed. The assignment at the end of this section involves using the builtin ARX function within Matlab’s system identification toolbox. If that toolbox is not available to the students, then this assignment can not be completed.
4.2 Fourier Transforms
The first step in system identification is to take an in depth look at the data one is given and obtain an idea of what to expect. The best way to do this is to take the Fourier transform of the data to express it in frequency domain. This will give you an initial idea of the modal frequencies of the system.
The direct Fourier transform of a given function f(t) is given as,
The inverse Fourier transform of the complexvalued function F(iw) is then given as,
This set of two equations is known as the continuous Fourier transform pair. For discrete functions, such as our earthquake ground motion, a discrete Fourier transform (DFT) is employed.
&
Where, N = number
of data points
4.3 Transfer Function
When dealing with structural systems, it is convenient to describe the system by its transfer function which is the relationship between a source of excitation and the dynamic response of the system, and is derived here. Recall the general equation of motion
Expressed in a it’s modally decomposed form for the nth mode;
Where,
&
Where, N is the number of degrees of freedom. We can express this equation in the frequency domain by taking the Fourier transform of each term.
Note that for a given function, f(t);
Summing over all the modes will yield the total response;
Or
is known as the transfer function and is in the continuous frequency domain.
4.4 ARX Model
The ARX model is a linear discrete time domain approximation of the transfer function derived above and it takes on the following form:
Where is a set of constants to be determined, 2N is the minimum model order, and d represents the time delay of the response, u(t) from load, p(t). In other words, what ARXmodeling assumes is that there exists a set of coefficients to a polynomial of order of at least twice the degrees of freedom that relates the dynamic response of a structure to a given excitation.
If the following definitions are made
The ARX model can be expressed in the following compact form:
Since g(t) is known, the goal of identification is to obtain the unknownq vector. This is done by minimizing the following error function for a given q vector.
The total error, E(T, q) where T is the total number of data points in the recording, can be estimated by taking the sum of squares of errors of all the data points.
The reason why it has to start at t=2N+d+1 is because of the time delay and model order, the method simply requires a history that far back. To minimize the total error, the derivative of E(T, q) is set to zero;
Solving the above differential equation yields the final form of q.
4.5 Experimental Testing
Experimental testing was performed at UCLA using NEES equipment. As seen in the picture in the first section, the scale structure is attached to the mass of a linear mass shaker. In the direction of loading, uniaxial accelerometers were attached at each level (two per floor for redundancy) including the base and DCDT’s were mounted at each floor to record interstory drift (At this time only accelerometer data is available). The sensor layout is displayed below.
A movie of the experiment can be seen here.
4.6 Data Acquisition and Processing
The sensors were connected to Quanterra Q330’s, where data is converted (A/D), GPS time stamped, logged and then transferred to the control laptop running Antelope software. Processing data involves converting to desired units, detrending, and applying a high band pass filter. Both unprocessed and processed data are available for this module but the tools used for processing are not. The processed data comes in a single file with six columns; 1^{st} column is time (s), 2^{nd} column is base acceleration (g), 3^{rd} to 6^{th} columns are story accelerations (g) starting from the 1^{st} story to the 4^{th}. The unprocessed data includes a file for each channel with two columns; 1^{st} column is time (epoch), 2^{nd} column is acceleration (nm/s^{2}).
4.7 Relevant Material
Processed Accelerometer Data data.txt
Unprocessed Base Accelerometer Data base.txt
Unprocessed 1^{st} Floor Accelerometer Data first.txt
Unprocessed 2^{nd} Floor Accelerometer Data second.txt
Unprocessed 3^{rd} Floor Accelerometer Data third.txt
Unprocessed 4^{th} Floor Accelerometer Data fourth.txt
ARX Matlab Function sarx.m
Fourier Transform Matlab Function fftplot.m
Chopra, Anil, K. Dynamics of Structures Theory and Applications to Earthquake Engineering, 2^{nd} Edition. Prentice Hall, New Jersey. 2001.
Safak, Erdal. Identification of Linear Structures Using DiscreteTime Filters. Jrnl of Strct Eng. Vol. 117, No. 10, Oct, 1991. Pg 30643068.
Acknowledgements  This structural dynamics educational module was developed by Derek Skolnik under the supervision of Professor John W. Wallace. This work was supported primarily by the George E. Brown, Jr. Network for Earthquake Engineering Simulation (NEES) Program of the National Science Foundation under Award Number CMS0086596