David Van Eerden
Engineering 345 Analysis Project
December 7,1998
 
 
 
 
 
 
 
 

 

 
 
 
 
 

1: Executive Summary:

The objective of this analysis is to find the critical response of the system using mathematical methods. The critical response is the fastest possible output without for a given input without any oscillation or overshoot. The design variable is the radius of the disk. The results show that the system can not achieve a critical response without changing more than just the radius of the disk. Making the radius large will make the oscillations small, but the oscillations could not be totally removed. A damper must be added to remove the oscillation. The system has only a spring between the two masses. There is nothing in the system to remove the oscillations caused by the spring without a damper between the two masses.

 

2: Introduction and Problem Review

 
Figure 1: Diagram of the System

The objective of this paper is to find a relationship between the input and the output for the system shown in figure 1. The input to the system is the displacement X1 and the output is the rotation q 1. This paper will derive differential equations to predict the output for a given input. The equations will be solved using the Runge-Kutta method. The parameter R1 will be changed to obtain a critical response from the output. All other parameters are to be held constant. The system is in critical response when mass 1 displaces as quickly as possible without oscillation or overshoot.

Table 1: Given System Parameters
 
Parameter
M1
M2
Ks1
Ks2
Kd1
R1
q2
m
Value
10 Kg
5 Kg
100 N/m
100 N/m
100 (N*s)/m
.1 m
40 deg
0.3
 

The parameters for the system are shown in table 1. Mass 2 is displaced 0.1m down the ramp and held there. The spring that attaches the two masses will push mass 1 down the ramp. Since the input to the system is the displacement of mass 2, the spring and damper attached to mass 1, the friction of mass 1, and the mass of block 1 can all be ignored. If the input were a force then the damper, spring, mass, and friction would have an effect. However, since the input is a displacement, the block will be displaced the same regardless if the damper, spring, mass, and friction are there or not.

Figure 2: Free Body Diagram if Mass 1
 

3: Modeling and Analysis 
A free body diagram of mass 1 is shown in figure 2. The forces in the x-direction are summed and result in equation 1.

                                                                               (1)
Summing the moments about the origin gives equation 2.
                                                                                                          (2)
But and , substituting in to equation 2 and solving for F gives equation 3.
 
                                                                                                   (3)
Substituting and equation 3 into equation 1 gives equation 4.

 

                                                                                  (4)

Equation 4 is a relationship between the input, x1, and the output, q 1. Mathcad can now be set up to use the Runge-Kutta method to solve the differential equation. Figure 3 shows how Mathcad is set up to do this.

 

Figure 3: Setting up Mathcad to do the Runge-Kutta method

Now a plot of q 1 vs. time can be produced.

 

 

Figure 4: Plot of q 1 vs. time for R=0.1m

This shows that mass 1 is oscillating. Figure 5 shows R=10.0 to see if a larger radius will remove the oscillation.


 

Figure 5: Plot of q 1 vs. time for R=10.0m

Figure 6: Plot of q 1 vs. time for R=0.0001m

The oscillation has the same frequency as before, but has smaller amplitude. Figure 6 shows R=.0001 to see if a smaller radius will remove the oscillation. The frequency of oscillation remains the same, but has larger amplitude. Without something to remove the kinetic energy, the oscillation can not be stopped.

4: System Design

Once a spring is in oscillation, it will remain oscillating unless something is there to remove the kinetic energy. This system has no damper between the two masses and has no friction in mass 1. The best design for the system, by only changing the given parameter, would be with a large radius. A large radius will produce oscillations with the smallest amplitude. A critical response can not be achieved. But small amplitudes are the most stable.

Figure 8: The system with Kd2 added

The system would have to be altered beyond the design parameters to remove oscillations. If a damper was added between the two masses as shown in figure 8, then the system could reach a critical response.

 

 

 

 

 
 

Summing the forces in the x-direction now gives equation 5.

                                                                                               (5)
Following a similar analysis to what was done in the analysis section yields equation 6.
                                                                    (6)
Figure 9 shows a plot of the q 1 vs. time with a damping coefficient of 75 Nm/s and a radius of 10 cm.
 
Figure 9: Plot of q 1 vs. x1 for R=0.1m and Kd2=75Nm/s

The system is now in critical response. The disk quickly displaces without oscilation or overshoot.

5: Summary

The system that was given can not perform as required. The best that can be done by changing the given design parameter is to mase the oscillatons as small as possible. A radius of 10m will produce a very small socillation. A damper should be added between the masses to obtain a critical response. When a damper is added, a critical response is easily reached.