State Youngstown University College of Engineering & Technology
Civil & Environmental/Chemical Engineering Program ENGR6924: Computer Based Tools for Engineers Problem no. 1 Forced Vibration Response of an Undamped Single Degree of Freedom(SDOF) System Using Duhamels Integral
Figure 1. Idealized undamped SDOF The governing differential equation of the system is: d 2 x (t ) f (t ) + 2x = 2 dt m Where = k m
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 >0 f (t ) = F0 (t ) And x(t ) =
t
1 f ( )sin (t )d m 0
Use the following forcing functions to determine the response of the SDOF of system shown in the figure above:
1. f (t ) = F0 sin( pt ) 2. f (t ) = F0 cos( pt ) 3. f (t ) = F0 Evaluate these displacement responses at m = 1, = 1, F0 = 1.0 . Use three different values of p given as follows p=2, p=0.9 and p=0.9999 Also use print and plot2d to commands print and plot the responses. Hint: Develop a sdofresponse function that reads the forcing function as input by using the read command as a console input. Set up the integration for the response and use ev command three times for three different p values to evaluate the response for the given parameters. Also use the print and plot2d commands with each ev command to get the desired output printed as well as plotted. The integration uses f ( ) instead of f (t ) therefore the forcing functions need to be changed as a function of . For example f (t ) = F0 sin( pt ) will transform to f ( ) = F0 sin( p ) . Use the range t = 0, 100 for p=2 and 0.9999 a...