Use of Symbolic Mathematics in Teaching Vibration Analysis

Javed AlamCivil/Environmental and Chemical Engineering Department

Youngstown State UniversityYoungstown, Ohio, 44555

ABSTRACT

The vibration analysis and structural dynamics are senior and graduate level courses in Civil, Mechanical and Aeronautical Engineering programs. The course topic deals extensively with higher level mathematics involving symbolic calculations. This paper demonstrates the use of an open source freely available symbolic mathematics package MAXIMA to solve the problems related to the vibration analysis.

INTRODUCTION

The vibrations in structures can lead to severe usability problems and catastrophic failures. Engineers learn about this subject matter in a vibration analysis or structural dynamics course [1] offered in their senior year or at the graduate level within Civil, Mechanical and Aeronautical Engineering programs. The subject matter deals with the responses of the structural systems to the transient loading, idealized in the form of systems of both single degree and multi degree freedom systems. The solution to these idealized structural problems involves higher level of mathematics including advanced calculus, matrix operations and second order homogeneous and non-homogeneous differential equations.

In their original form these equations are solved by using symbolic algebra or numerical techniques. The solution in symbolic form provides a more general solution to this class of problems. There are many commercially available software packages such as Mathematica, Maple and Mathcad [2] that can be used to solve the resulting equations. This approach allows solving a wider class of problems as compared to the old approach of manual solution techniques. These packages allows the instructors to tailor their lectures with more computer assisted course material and assign problems that are closer to the real life problems, However, these packages are not widely available at all the Universities. Also, once the students leave the university they may not have access to these packages at their workplace. Therefore, the use of an open source software program MAXIMA[3] that is widely available for Windows and Unix operating systems at zero cost to users by a simple download from the INTERNET is explored in solving problems related to vibration analysis in symbolic algebra mode.

MAXIMA, a Computer Algebra System and its Use in Vibration Analysis

MAXIMA is a software package that allows solution of complex algebra problems symbolically as well as numerically. It originated from the original Macysma system that was developed at MIT in 1960. The software is available free of cost on the INTERNET under GPL license [4]. The software has extensive set of modules that allows the solution of very complex algebraic problems. For vibration analysis the capabilities of symbolic differentiation, indefinite and definite integrals, matrix operations and the solution of second order homogenous and non-homogeneous differential equations are the most relevant. These capabilities will allow solving any vibration problem that has a close form solution to be solved symbolically using this program. In addition to that numerical solution can also be obtained by using the programming capabilities built into the program. This approach provides a good blend of symbolic algebra based calculation with purely numerical techniques where they are needed to obtain the solution to the structural vibration problems.

Four representative problems from vibration analysis were chosen to demonstrate the capabilities and the ease of use for MAXIMA program. The first problem uses the symbolic integration capabilities of the MAXIMA to calculate the response of a single degree of freedom system. The second problem uses the symbolic integration and symbolic matrix operations to find frequencies of a tapered beam using Rayleigh-Ritz method. The third problem uses the symbolic ordinary differential equation solver to find a response to a single degree of freedom system under over-damped, critically damped and under-damped conditions. Finally, the fourth problem uses the symbolic ordinary differential equation solver to find a response of a damped single degree of freedom system to a sinusoidal loading. The last problem demonstrates the capabilities of MAXIMA for dealing with non-homogeneous second order ordinary differential equations with initial values.

The next section provides the details of these problems, the maxima batch file and the solution obtained using the MAXIMA program. A MAXIMA batch file is simply a computer file in plain ASCII text format that contains all the necessary MAXIMA commands needed to obtain the solution to the problem. The MAXIMA can also work in the interactive mode where each command is typed separately. However, the use of batch files allows solving larger problems with easier debugging possibilities. The MAXIMA reference manual [5] provides the details of all the available commands within MAXIMA program.

Forced Vibration Response of an Undamped Single Degree of Freedom(SDOF) System Using Duhamel’s Integral

Figure 1. Idealized undamped SDOF and the applied loading

The governing differential equation of the system is:

Where

Where k represents the stiffness of the spring, m represents the attached mass and represents the natural frequency of the system. The x(t) is the displacement response of the SDOF system.

The Duhamel integral response for zero initial displacement and velocity for the system is shown below:

For t < td

And

For t > td

And

The Maxima batch file that provides the solution is:

assume ( t>0);assume( omega >0);assume (cos(omega*t) >0);assume( td >0);assume( td > t);assume(cos(omega*td-omega*t) >0 );x1:integrate(F0*Tau*sin(omega*(t-Tau)),Tau,0,t)*(1/(m*omega*td));integrate(F0*Tau*sin(omega*(t-Tau)),Tau,0,td)*(1/(m*omega*td))+integrate(F0*sin(omega*(t-Tau)),Tau,td,t)*(1/(m*omega));x2:factor(%);

Table 1. The maxima batch file dumamel.mac

The maxima solution is shown in Figure 2:

Figure 2. Maxima solution for the undamped SDOF system using Duhamel’s Integral

The Natural Frequencies of a Continuous Tapered Beam by Raleigh-Ritz Method

Figure3. Continuous tapered beam

The young’s modulus for the beam is represented by E, the moment of inertia and mass per unit length are given by:

And

Where ρ is the mass density of the material of the beam.

The stiffness and mass matrix for the beam are:

Where

i,j=1,2 and

i,j=1,2

The mode shape functions are given by:

And

If only one mode shape function is used, the natural frequency of the beam is given by:

If both of the mode shape functions are used the following eigen-value problem is formed:

That allows the solution for the first two frequencies of the tapered beam shown in figure 3.

The Maxima batch file that provides solution to the above problem is shown below:

kill(all);phi1:(1-x/L)^2;phi2:(x/L)*(1-x/L)^2;I:(2/3)*(b/L)^3*x^3;mass:rho*2*b*x/L;dphi1:diff(phi1,x,2);k11:integrate((dphi1)^2*E*I,x,0,L);m11:integrate(mass*phi1^2,x,0,L);omega[11]=sqrt(k11/m11);dphi2:diff(phi2,x,2);k12:integrate(E*I*dphi1*dphi2,x,0,L);k22:integrate(E*I*dphi2*dphi2,x,0,L);m12:integrate(mass*phi1*phi2,x,0,L);m22:integrate(mass*phi2^2,x,0,L);K:matrix([k11,k12],[k12,k22]);M:matrix([m11,m12],[m12,m22]);a:K - zeta^2*M;b:determinant(a);float(solve(b=0,zeta));

Table 2. The Maxima batch file frequency.mac

The Maxima solution for the frequencies are shown in figure 4

Figure 4. The Maxima solution for the frequencies of a tapered beam

Free Vibration of a SDOF Damped System

Figure 5. Idealized damped SDOF system

The governing differential equation of motion for the structural system shown in figure 3 is:

Where and

The k represents the stiffness of the spring; m represents the attached mass, represents the natural frequency of the system and ς represents the damping ratio. The x(t) is the displacement response of the SDOF system.

The initial conditions are x0 = 1 and v0 = 0

The Maxima batch file for the problem

assume(omega>0);'diff(x,t,2) + 2*zeta*omega*'diff(x,t) +omega^2*x = 0;ode2(%,x,t);ic2(%,t=0,x=x0,diff(x,t)=v0);y:ev(rhs(%),zeta=2.0,x0=1,v0=0,omega=1);plot2d([y],[t,0,25]);

Table3.The maxima batch file dampedfreevibration.mac

Case I Overdamped system ( ς = 2.0 )

The Maxima solution for non-zero initial displacement and zero initial velocity is shown in figure 6.

Figure 6. The displacement response of an overdamped SDOF system

CaseII Critically damped system ( ς = 1.0 )

The Maxima solution is shown in Figure 7.

Figure 7. The displacement response of a critically damped SDOF system

Case III Underdamped system (ς = 0.1 )

The displacement response for the under damped SDOF system is shown in Figure 8

Figure 8. The displacement response of an under damped SDOF system

Forced Vibration of a Damped SDOF System Subjected to Sinusoidal Excitation

Figure 9. The damped SDOF system under sinusoidal loading

The governing differential equation of motion is:

Where m is the mass, c is the damping coefficient and k is the stiffness of the system shown in Figure 9.

The Maxima batch file is shown below:

assume((4*k*m -c^2) >0);m*'diff(x,t,2) + c*'diff(x,t)+k*x = F0*sin(p*t);ode2(%,x,t);ic2(%,t=0,x=0,diff(x,t)=0);y:ev(rhs(%),k=1,m=1,c=.1,F0=1,p=2);plot2d([y],[t,0,25]);

Table 4. The Maxima batch file dampedforcedvibration.mac

The Maxima solution to the problem is shown below:

Figure 10. The Maxima solution for displacement response for an under damped SDOF system subjected to sinusoidal excitation

Discussion of Results

The cells D8 and D10 in Figure 2 show the closed form displacement response of the SDOF shown in Figure 1. The cells D8 and D18 in Figure 4 show the natural frequencies of a tapered beam in symbolic form. The cell D75 in Figure 6 shows the closed form free vibration response of an over damped SDOF system. The response is evaluated for the damping ratio value of ς=2.0, initial displacement value of x0=1, initial velocity value of

v0=0, and a natural frequency value of ω=1. The resulting displacement response is shown in cell D76 and a plot of this displacement response is shown on the top right corner of the same figure. The cell D107 in Figure 7 shows the closed form free vibration response of a critically damped SDOF system. The response is evaluated for a damping ratio value of ς=1.0, initial displacement value of x0=1, initial velocity value of v0=0, and the natural frequency value ω=1. The resulting displacement response is listed in cell D108 and a plot of the response is shown on the top right corner of the same figure. The cell D121 shown in Figure 8 lists the closed form free vibration response of an under damped SDOF system. The response is evaluated for the damping ratio value of ς=0.1, initial displacement value of x0=1, initial velocity value of v0=0, and the natural frequency value of ω=1. The resulting response is shown in cell D122 and a plot of the response is shown on the top right corner of the same figure. Finally, the cell D47 in Figure 10 shows the closed form forced vibration response of an under damped SDOF system subjected to a sinusoidal loading with zero initial displacement and velocity. The response is evaluated for a stiffness value of k=1.0, a mass value of m=1, the damping coefficient value of c=0.1, a force value of F0=1, and a forcing frequency value of p=2. The resulting response is listed in cell D48 and a plot of the response is shown in the same figure.

Conclusions

The solution of four representative vibration analysis problems shows that MAXIMA program is capable of handling the solution to these types of problems. It can produce results in symbolic algebraic form as well as evaluate them numerically. The procedure discussed in here allows the solution of the problems that involve higher mathematics. They can easily be solved with a few appropriate MAXIMA commands that would require a larger effort if the solutions were done manually. This approach offers the opportunity to carry out a parametric study of the vibration problems providing an in depth knowledge of the subject matter to the student with reduced efforts on their part. The use of MAXIMA as a learning support software also provides more efficient way for the instructors to tailor their lecture material for better presentation of the subject material.

The MAXIMA is a free program and it is distributed on the INTERNET under GPL license. Therefore, it is available to everyone without the licensing fee requirement of the equivalent commercial software. The adoption of MAXIMA software and its cost free nature presents a way for the instructors at the economically challenged institutions to offer close to similar teaching and learning experience to their students that is possible at other institutions of higher learning. The choice of MAXIMA program as an algebraic and numerical compute engine to teach vibration analysis will enable the lecturers, students and the engineers to work in an efficient computing environment that allows a reduction in the cost of computing along with the full fledged capabilities to perform structural dynamics simulation studies.

Reference

1. Paz, M, “Structural Dynamics”. Van Nostrand Rheinhold, New York, 19802. Wester, M. The review of Computer Algebra System

http://www.math.unm.edu/~wester/cas_review.html3. Maxima Website http://maxima.sourceforge.net/4. GNU General Public License http://www.gnu.org/copyleft/gpl.html5. MXIMA reference Manual http://maxima.sourceforge.net/docs.shtml

Biographical Sketch

Alam, Javed

Department of Civil /Environmental and Chemical Engineering, Youngstown State University, Youngstown, OH 44555, USA e-mail: jalam@cc.ysu.edu www: http://www.eng.ysu.edu/~jalam

Javed Alam is a professor of Civil and Environmental Engineering at Youngstown State University. He obtained his B.S. in Civil Engineering from Indian Institute of Technolgy at Kanpur, India and received his M.S. in Structural Engineering from Asian Institute of Technology at Bangkok, Thailand. He pursued further studies at Case Western Reserve University in Cleveland, Ohio to obtain a Ph.D. degree in Civil Engineering. His research interests are in the area of Structural Mechanics, Application of the Artificial Intelligence in solving Engineering problems and Computer Applications in teaching and learning with an emphasis on e-learning.