- 1. College of Engineering and Computer Science Department of Mechanical Engineering ME 384 Dr. C. T. Lin Computer Project #2 Rachel Foreman Due: 25 Nov 2015 Submitted: 25 Nov 2015
- 2. 1 Introduction A simple vehicle suspension “quarter car” model can be seen in Figure 1. The car of mass m is initially at rest when it begins moving along a road of sinusoidal shape. Based on a given spring constant, k, damping constant, b, and input velocity, vr(t), one can find the forces acting on the mass as well as its motion as a function of time. The system can be more easily analyzed by creating a block diagram and using MATLAB Simulink to simulate the behavior of the model. Figure 1 Model of quarter car suspension Method Figure 2 shows the free body diagram of the quarter car suspension. In this diagram, the inertial force in acting in the direction of the motion of the mass, while the spring and damper are resisting the motion of the mass. Based on the initial model, xr denotes the motion applied to the system by the sinusoidal road, while xm represents the motion of the mass. Figure 2 Free body diagram of mass, m
- 3. 2 Eqn. 1 Eqn. 2 Eqn. 3 Eqn. 4 Eqn. 5 Eqn. 6 Eqn. 7 Eqn. 8 Eqn. 9 In order to solve for the Eigen values, natural frequency, and damping ratio of the system, the equation of motion must first be determined. This is done by applying Newton’s Law in the direction of the motion (Eqn. 1). ∑ 𝐹 = 𝑚𝑥̈ 𝑚 −𝑘(𝑥 𝑚 − 𝑥 𝑟) − 𝑏(𝑥̇ 𝑚 − 𝑥̇ 𝑟) = 𝑚𝑥̈ 𝑚 𝑚𝑥̈ 𝑚 + 𝑏𝑥̇ 𝑚 + 𝑘𝑥 𝑚 = 𝑏𝑥̇ 𝑟 + 𝑘𝑥 𝑟 Eigen Values, Natural Frequency, and Damping Coefficient Since the system is given to be initially at rest, the initial values of the velocities and displacements of both the road and the mass are equal to zero when time t = 0 [sec]. By taking the Laplace Transform of the equation of motion (Eqn. 3) and normalizing the denominator, the system transfer function in the s domain is determined (Eqn. 4). 𝑋 𝑚(𝑠) 𝑋 𝑟(𝑠) = 𝑏 𝑚 𝑠 + 𝑘 𝑚 𝑠2 + 𝑏 𝑚 𝑠 + 𝑘 𝑚 The denominator of the transfer function in Eqn. 4 can be called the characteristic polynomial of the system. By setting this polynomial equal to zero, it becomes the characteristic equation of the system (Eqn. 5). 𝑠2 + 𝑏 𝑚 𝑠 + 𝑘 𝑚 = 0 Solving the characteristic equation for s leads to the system Eigen values (Eqn. 6). 𝑠1,2 = − 𝑏 2𝑚 ± 𝑗 √4𝑘𝑚 − 𝑏2 2𝑚 [𝑠𝑒𝑐−1 ] Substituting the given mass of 1.5 [kg], spring constant 39.478 [N/m], and damping constant 7.5 [N-s/m] yields the specific Eigen values 𝑠1,2 = −2.5 ± 𝑗4.480 [𝑠𝑒𝑐−1 ]. By comparing the characteristic equation of the physical system (Eqn. 5) to a standard system equation in terms of natural frequency, 𝜔 𝑛, and damping coefficient, 𝜁 (Eqn. 7), the natural frequency of the system can be determined (Eqn. 8). 𝑠2 = 2𝜁𝜔 𝑛 𝑠 + 𝜔 𝑛 2 = 0 𝜔 𝑛 = √ 𝑘 𝑚 [𝑠𝑒𝑐−1 ] Similarly, by comparison, the damping coefficient can be determined (Eqn. 9). 𝜁 = 𝑏 2√𝑘𝑚
- 4. 3 Eqn. 10 For the particular m, k, and b given in this system, 𝜔 𝑛 = 5.13 [𝑠𝑒𝑐−1] and 𝜁 = 0.487. Block Diagram and Simulink To aid with constructing the block diagram of the system, the equation of motion was rearranged to solve for 𝑥̈ 𝑚 (Eqn. 10). 𝑥̈ 𝑚 = 𝑏 𝑚 𝑥̇ 𝑟 − 𝑏 𝑚 𝑥̇ 𝑚 + 𝑘 𝑚 𝑥 𝑟 − 𝑘 𝑚 𝑥 𝑚 From this equation, the block diagram could be established (Fig. 3). Figure 3 Block diagram of quarter car system The block diagram was then used to construct the Simulink model (Fig. 4). Figure 4 Simulink model of quarter car suspension
- 5. 4 Eqn. 11 Eqn. 12 Eqn. 13 Displacement and Velocity of the Mass Specific commands were added to the Simulink model to output desired quantities to the MATLAB workspace. In this case, the displacement and velocity of the mass could be pulled directly from the Simulink diagram without any additional calculations being done. By calling the Simulink model within a MATLAB script, the displacement and velocity of the mass were plotted versus time (Fig. 5). Figure 5 Plot of velocity and displacement of the mass as a function of time Inertial, Damping, and Spring Forces In order to obtain plots of the inertial, damping, and spring forces acting on the mass, additional branches and blocks needed to be added to the Simulink model. Per the free body diagram (Fig. 2), the inertial, damping, and spring forces can be described by equations 11, 12, and 13, respectively. 𝑖𝑛𝑡𝑒𝑟𝑡𝑖𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 = 𝑚𝑥̈ 𝑚 [𝑁] 𝐹𝑑 = 𝑏(𝑥̇ 𝑚 − 𝑥̇ 𝑟) [𝑁] 𝐹𝑠 = 𝑘(𝑥 𝑚 − 𝑥 𝑟) [𝑁] To define the inertial force, a branch outputting 𝑥̈ 𝑚 was multiplied by 𝑚 before being sent to the MATLAB workspace. For the damping force, the velocity input from the road was first subtracted from the velocity of the mass, then a gain of 𝑏 was applied to the quantity. Similarly, the displacement input from the road was subtracted from the displacement of the mass, and the
- 6. 5 Eqn. 14 resulting quantity was multiplied by 𝑘 in order to output the spring force. These outputs were called within the MATLAB script to produce a plot of all three forces versus time (Fig. 6). Figure 6 Plot of inertial, damping, and spring forces as functions of time Relation Between Forces By inspection of the free body diagram (Fig. 2), one can determine that the sum of the inertial, damping, and spring forces should equal zero (Eqn. 14). 𝑚𝑥̈ 𝑚 + 𝐹𝑑 + 𝐹𝑠 = 0 To verify this, the plot was analyzed at a time t = 4 [sec]. At this time value on the graph, the values of inertial, damping, and spring forces are 5.197 [N], 2.818 [N], and -8.015 [N], respectively. When added, these three values equal zero, as predicted in Eqn. 14. By further analysis, one can also determine that Eqn. 14 should hold true for all values of time in the plot.
- 7. 6 Appendix MATLAB Code: Plot Displacement and Velocity %Call Simulink model SimOut = sim('QuarterCarDiagram'); %Plot displacement and velocity plot(t,x_m,'+',t,v_m,'-','linewidth',2) grid title('Displacement & Velocity of the Mass','Fontsize',20') xlabel('t [sec]','Fontsize',14') ylabel('x_m(t) [m], v_m(t) [m/s]','Fontsize',14') legend('x_m(t)','v_m(t)')
- 8. 7 MATLAB Code: Plot Inertial, Damping, and Spring Forces %Call Simulink model SimOut = sim('QuarterCarDiagram'); %Plot inertial, damping, and spring forces plot(t,inertial,'+',t,damper,'-',t,spring,'.','linewidth',2) grid title('Inertial, Damping, and Spring Forces','Fontsize',20') xlabel('t [sec]','Fontsize',14') ylabel('mx`` [N], F_d [N], F_s [N]','Fontsize',14') legend('mx``','F_d','F_s')