This report describes the analysis and design of a lever arm system, the object of which is to achieve the fastest output response while avoiding oscillation and overshoot. The report also includes the structures, parameters, and components necessary for this analysis and design.
Comparing the three cases from design section, we see that case 1 gives the fastest response and produces the smallest oscillation. The graph below shows the output response of the force F. In this graph, the force F indicates the magnitude only.
The average output of the force F is 12 N. The time of the response is 5 seconds. This graph indicates that the system has a fast response and no oscillation. However, we can modify the lever arm system by simply adjusting the parameters to reflect a real application.
In order to achieve a lever arm system with the fastest output response without overshoot and oscillation, it is important to understand the mechanical system and the design objectives. For this project, the moment inertia J is the variable. The displacement x1 is the input and the force F is the output response. To develop the equations we use analysis techniques. For example, the free body diagram is one of the most common techniques for developing the equations. Also, selecting the state variable equations will help to formulate the system model and the Runga-Kutta method will be used for solving state variable equations. Finally, it is important to select the value for the moment inertia J of the bar.
2. Problem Review and Design Objectives
In this system, (see Figure 1) the lever arm has a spring Ks2 on the left side and a spring-damper combination Ks1-Kd with a suspended mass M on the right side. The pivot is displaced R1 meter from the spring force Ks2x2. As the lever arm on the right side move downward, the left side of the lever arm tends to move up. This movement causes displacements: x1, x2, x3, and the displacement angle q. Notice that if we chose x1 and x2 downward as the positive direction, the x3 is in the negative direction. This movement also changes the values of the spring forces Ks1x1, Ks2x3, and the damper force Kdv1. In order to make the lever arm system more useful, the objective for this design is to find how much the design variable moment inertia J affect the lever arm system. As we change the moment inertia J, the lever arm system will behave differently. The objective is to find a good result so that the system gives the fastest response without overshoot and avoids as much oscillation as possible.
3. Modeling and Analysis
In Figure 1, let x1, x2, and x3 denote displacement measured with respect to reference positions that correspond to a single equilibrium condition of the system. The respective elongations of Ks1, and Ks2 are x1-x2, and x3, which are related to the potential energy, and velocity v2 of the lever arm, which is kinetic energy. Therefore, respective forces of the springs and damper are Ks1 (x1-x2), Ks2x3, and Kdv2. By analyzing the lever arm system, we can develop two equations, one from the lever arm and one from the mass M. Therefore, only two unknowns (x1 and x2) are allowed to appear in the equations. Since x1 is the input. The derivative of x1 is zero. In this case, the equation for the mass M is negligible. Since we have only one equation. The subtitude method is used for reduce the variables. The relationship between x2 and x3 can be found in Appendix B.
The distance d is calculated as follows.
The lever arm (Mbar) is calculated from the moment inertia J. Form the parallel axis theorem, the moment inertia of the lever arm should be determined about its centriodal axis, which is parallel to the reference axis FN.
From the free body diagram in Figure 2, the output of the force F can be determined by the equation F= Ks2 (-x3). The displacement –x3 is found in Equation 1(see Appendix B). At the pivot, FN is the normal force that keeps the lever arm from falling. In the Equation 4, the weight Mbar (g) also pulls the lever arm downward. On the other side of the lever arm, Kdv2 and Ks (x1-x2) are the vertical component forces of the spring and damper. Furthermore, in Figure 1 the mass M and the lever arm are connected by the spring Ks1. Therefore, the magnitude of the forces on the lever arm and the mass M must have equal magnitude and opposite direction. By summing the moment on the free body diagram of Figure 2 clockwise direction, we obtain,
From the Equation 4, we can see the relationship between the mass Mbar and the moment inertia J. As the moment inertia J increases, the mass Mbar also increases proportional to the values that we put in for the moment inertia J.
In order to solve the lever arm system above, we need to select a set of state variable equations to describe the effect of the history of the system on its future behavior. The approach for this case is to determine which once are the inputs, outputs, and state variables. Each equation must express the derivative of one of the state variables as an algebraic function of the state variables and inputs. Also, we choose as the state variables the compression x3 of the spring Ks2. This is related to the potential energy of the springs. We choose the derivative of the displacement q in Figure 2 as a state variable for the moment inertia J. The derivative of the displacement x2 is related to the displacement q in the Equation 2 (see Appendix B). As the final variable we choose the velocity v2 of the lever arm for the friction coefficient of the damper. Having these choices, we obtain the state variable equations
The solutions of this system can be found by simply setting the state variable, Equation 7 and Equation 8 into Mathcad. The result yields to the equation 9.
In order to achieve the fastest output response while avoiding oscillation, and overshoot, we needed to test the equations by changing the design variable moment inertia J.
a) Case 1: For the moment inertia J equal 1 kgm^2, the average displacement of x2 is 0.5m. The system did not oscillate. The average output force F was 12N. The time of the output force F response was 5 seconds.
b) Case 2: For the moment inertia J equal 100 kgm^2 the average displacement of x2 is 2.5m. The displacement, velocity, and the output force F were not oscillating much. The average output force F was 100N and the time of the output force F response was 20 seconds.
c) Case 3: For the moment inertia J equal 150 Kgm^2, the average displacement of x2 was 3m. The displacement, velocity, and the output force F were not oscillating much. The average output force F was 120N. The time of the output response was 30 seconds.
Comparison of the three cases shows that, the first case gave us a fastest output response and produced the smallest oscillation for the output force F. But the output force F was small compare to the second case and third case. Inspection of the output force F response, shows that as the moment inertia J increases, the output force F was also increased. Again, both the fastest response output and the smallest oscillation happened when the value for moment inertia J was small.
This report focuses mainly focused on the analysis and design for the lever arm system. The design variable is the moment inertia J. The output force F depends on the moment inertia J. Therefore, we can adjust the variable force F for the results that we expect. Notice, in order to avoid system failure we have to change the springs and damper coefficient when we input a large moment inertia J. Also, the strain of the lever arm must be considered.