Damped Forced Vibrations Analysis Using CAMP-G® and Simulink® Modeled Solutions to Problem 19.169 (http://gaia.csus.edu/~grandajj/me114/PR19-164.PDF) California State University, Sacramento Mechanical Engineering 114 (Vibrations and Control Systems) Fall Semester 2006 By Matt Rooks

Problem Statement

Engineering Model and System Parameters The motion of an object or system that is experiencing damped forced vibrations can be defined by the following differential equation: Therefore, based on the problem statement, the system was simplified to include one equivalent spring (k), the damper (c), and the combined mass of the motor and unbalanced rotor (me), and the following parameters were determined: System Parameters:

CAMP-G Solution In order to utilize the CAMP-G software, the engineering model had to first be converted to a Bond Graph model (shown below). A brief comparison of the engineering and Bond Graph notations is presented. Engineering Model Notation: me = 400.023 kg Pm = 16.142 N k = 600,000 N/m c = 6,500 N-s/m x = displacement v = velocity Bond Graph Model Notation: I = Inertia Element (me) C = Compliance Element (1/k) R = Resistive Element (c) SE = Source Effort (Pmsin(ωft)) q = displacement p = momentum e = force f = velocity Bond Graph Model: Simplified Engineering Model:

CAMP-G Domain The Bond Graph was constructed in CAMP-G using the available tools on the CAMP-G tool bar shown on the left. First the SE, I, C, and R elements were inserted. Then, the elements were connected using the arrow tool. Note the direction of the arrows.

CAMP-G Domain Using the Interface drop down menu, the CAMP-G Bond Graph model was interfaced with MATLAB.

MATLAB Domain Default output file (campgequ.m) from CAMP-G Modifications made to campgequ.m: Defined driving force (SE1) as Pmsin(ωt) parameters. Unsuppressed TIME, EFFORTS, and FLOWS vectors and defined them according to the Bond Graph. Note that the force on the foundation is the sum of the efforts on the damper and spring (e4+e3). Defined FORCING vector as the effort of the SE1 (e1). Default output file (campgequ.m) from CAMP-G

MATLAB Domain Default output file (campgmod.m) from CAMP-G Note: All “?” needed to be updated accordingly. Modified the following in campgmod.m: Initial conditions System Physical Parameters Simulation Time Control Added FORCING to global variables list (explained later)

MATLAB Domain Default output file (campgmod.m) from CAMP-G Unsuppressed and modified sample figure in campgmod.m to plot: The force on the foundation vs. time Displacement of motor vs. time. Unbalanced rotor force vs. time. Default output file (campgmod.m) from CAMP-G

CAMP-G/MATLAB Modeling Results Results obtained from running campgmod.m

CAMP-G/MATLAB Modeling Results Using the zoom tools, the approximate magnitudes of the results can easily be viewed.

CAMP-G/MATLAB Conclusions Using the magnification factor, the amplitude of the force on the foundation (Fm) was calculated to be 5.75 Newtons and the amplitude of the vertical motion of the motor was calculated to be 7.10x10-6 meters. As shown on the previous slides, the results of the CAMP-G/MATLAB modeling effort compared very well to the calculated values.

Simulink Solution Using the Simulink Library Browser shown on the right, the Simulink Model shown below was constructed.

Simulink Solution The parameter input boxes for each of the Simulink Blocks used in the model shown on the right are shown below. The parameters were updated according to the system parameters stated earlier. Scopes were placed on the model at various locations to obtain graphical results of the: Force on Foundation; Vertical Displacement; and the Forcing Function. Results are shown on the following slides.

Simulink Modeling Results After double-clicking the “Force on Foundation” scope, the Autoscale tool was used to scale the entire plot into view.

Simulink Modeling Results Next the zoom tool was used to zoom in on one of the peaks of the plot. As shown below the results show that the magnitude of the modeled force on the foundation was approximately 5.75 Newtons. The following slides again use the Autoscale and zoom tools to obtain the results of the vertical motion and the force of the unbalanced rotor.

Simulink Modeling Results Modeled vertical motion of the motor.

Simulink Modeling Results As shown below the results show that the magnitude of the modeled vertical motion of the motor was approximately 7.1x10-6 meters.

Simulink Modeling Results Modeled force of the unbalanced rotor.

Simulink Modeling Results As shown below the results show that the magnitude of modeled force of the unbalanced rotor was approximately 16.14 Newtons.

Simulink Modeling Conclusions As mentioned earlier, the magnification factor was used to calculate the amplitude of the force on the foundation (Fm= 5.75 Newtons) and the amplitude of the vertical motion of the motor (xm = 7.10x10-6 meters). As shown on the previous slides, the results of the Simulink modeling effort compared very well to the calculated values.