Grand Valley State University
The Padnos School of Engineering
Modeling the door closer
EGR 345 Dynamic Systems Modeling and Control
Eric Fleischmann
October 7, 1999
Lab Partners
David Nagy
Becky Engel
Heather Boeve
Fall 1999
Table of Contents *
List of Figures and Tables *
Executive Summary *
1. Introduction *
2. Modeling Motion *
3. Apparatus *
4. Procedure and Results *
4.1 The Spring Constant: *
4.2 Damping Coefficient: *
4.3 Damper and Spring Combination: *
5. Analysis and Interpretation *
6. Conclusions and Recommendations *
Appendix A Mathcad *
Appendix B Working Model *
Table 1 Equipment *
Table 2 Spring Constant Values *
Table 3 Damping Coefficent Values *
Table 4 Spring Deflection Comparison *
Table 5. Damper Velocity Compairison *
Figure 1. The Door Closer *
Figure 2. Door Closer Positions *
Figure 3. Door Closer Position vs. Time *
The objective of this experiment was to obtain a model for the given door closer. Door closers are often used on screen doors to pull the door shut slowly enough to avoid slamming, which could cause damage to the door itself. The door closer was composed of a spring and a damper. In order to model the system the door closer was disassembled and each component was measured independently.
The spring acted in compression so as to push the plunger back to its original position from a deflected position, caused by opening the door. The damper then opposed the motion of the spring, slowing the motion enough to avoid slamming the door. Each component was measured as if the door closer was fully extended. The spring coefficient was found by measuring the displacement caused by a constant force. The damping coefficient was found by measuring the velocity given a constant applied force.
Using the spring coefficient and the damping coefficient a mathematical model of the door closer was found. In order to test this model the displacement given by the model was compared to the actual displacement of the plunger, see figure 3. In addition, Working Model was used to calculate the position, and was then compared to the mathematical model. These two models agreed to within 6.6% of each other. The values for the velocity in the damper agreed with the model in between 1.48 and 10.52%.
In the future, the friction in between the plunger and the inner wall of the closer should be taken into account. More sophisticated equipment, such as proximity sensors, can also be used to find the position and the velocity. These tools can be used to measure the spring and damping coefficients much more accurately.
Door closers are all around us. Just about every entry door in an industrial application contains a door closer of some kind. Door closers are used to gently pull shut doors and avoid slamming. These amazing machines are comprised of a perfect balance of two opposing components. The first of these two components is a spring. The spring supplies the force to pull shut the door. The second component is a damper. This component slows the door so that it will gently glide into a closed position. These two systems are set up in opposition to each other. Just a spring alone would slam the door shut, just as a damper would close the door on its own. It takes a perfect balance of these two components to correctly close a specified door.
The door closer used in this lab is common to most residential area screen doors. It has been chosen because it is so common that most people have seen one in their lifetime. Also it can easily be obtained at a local hardware store for a minimal cost. Since the door closer is readily available we wish to do some backward design so that a mathematical model of the door closer can be obtained. In order to model a door closer three values must be known, the spring constant (K_{s}), damping coefficient (B), and the force (F) that needs to be applied to close the door. In this lab different forces will be applied to the door closer to measure K_{s} and B. The door closer will be disassembled in order to measure each component accurately. Then these values will be plugged into the derived model of motion and compared to the actual motion. Friction will not be taken into account so some differences between the real and calculated values are to be expected.
2. Modeling Motion
The door closer that we will study is a spring and damper system. The spring provides the force to close the door, and the damper will slow the closing down enough to avoid slamming it. In order to model the system a couple of things must be known. The first is the spring constant. We know that the force applied by the spring is related to the spring constant and the displacement.
( 1 ) |
or
( 2 ) |
Where:
F: = force of spring
K: = spring constant
x: = position
The above equation can be used to calculate the spring constant using different forces and their resulting differences. A linear regression worked best for this. Also recall that if a spring is preloaded it will not deflect if to small of a force is applied. Keeping with linear regression, the intercept will be smallest force that will result in a deflection. Forces higher than this must be used to calculate the spring constant.
The second value needed before molding can begin is the dampening coefficient. As you may recall, damping is directly proportional to velocity. By using an applied force and measuring the resulting velocity we can solve for the dampening coefficient (B).
( 3 ) |
Calculating for the damping coefficient is much the same as calculating for the spring constant.
( 4 ) |
Where:
B: = the damping coefficient
v: = the velocity
By taking position and time measurements, velocities can be found again using linear regression to find the dampening constant B. The slope of the line will yield B. Finally, the differential equation can be derived to model the motion of the door closer. First the forces are summed.
( 5 ) |
or
( 6 ) |
By using Mathcad and the Runge Kutta method differentiation can then be performed to find the position vs. time, see figure 3.
3. Apparatus
A limited number of apparatuses were used in performing the above experiment. Most were simple tools, which can be found in any lab or shop.
Item |
Manufacturer |
Model |
Serial Number |
Range |
Resolution |
Ring Stand |
-- |
-- |
-- |
-- |
-- |
Clamp |
-- |
-- |
-- |
-- |
-- |
Mass Set |
Nation Standard |
-- |
-- |
1kg |
.1g |
Meter Stick |
-- |
-- |
-- |
1m |
1cm |
Computer |
National Controls |
XPS PII 266 |
PSE 121 |
-- |
-- |
Stop Watch |
-- |
-- |
-- |
¥ |
.01s |
Below is a drawing of the door closer studied in this lab. The spring and the damper are also identified.
The spring was removed from the system and placed in compression. Three different masses were set on the spring and the deflection was measured. A plot was constructed of these masses and deflections they caused. The slope, or a line of best-fit function, was used to find the spring coefficient, see equation (2). These values are tabled below in table 2.
Table 2 Spring Constant Values
These values were also entered into Mathcad and analyzed using the line of best-fit function. The spring constant values were then averaged and the final spring constant was decided to be 1064 kg/s ± 10%.
The door closer was allowed to operate without the spring in place. Masses were added to the damper and the door closer was fully extended. The mass, acting on the closer in compression, forced the plunger back to its original position. The time for this to happen divided by the distance the plunger traveled gives the velocity. Three different masses were used and the forces along with the velocities and their damping coefficients are tabled below in table three.
Table 3 Damping Coefficient Values
The velocity and the different masses were plotted together. The slope, or line of best-fit function, gives the damping coefficient, see equation 4. The final damping coefficient was an average of the three taken and was 505.139± 10% kg/s
4.3 Damper and Spring Combination:
The door closer was reassembled and set into the clamp. The closer was then fully extended and a mass was hung on the end of it. The mass was released and the spring-damper system returned to its original position, see figure 2 below.
Figure 2. Door Closer Positions
In order to get the most accurate results, the mass of the damper was added into the total mass. Gravity acting on the mass will oppose the force of the spring. The damper will also oppose the motion of the door closer, and bring the system slowly to rest, see figure 2. In order to model this action three marks were made on the door closer. A mass was attached to the door closer and the time was taken at the different positions after the mass was released. The line is the result of solving the differential equation of motion in Mathcad (equation 6) and inserting the values found for the spring constant (K_{s}), the damping coefficient (B), and the total mass applied to the door closer. These results along with the measured values were plotted together to see the congruency. This plot is below in figure 3.
Figure 3. Door Closer Position vs. Time
5. Analysis and Interpretation
The results for the spring mass system showed a discrepancy between Mathcad and Working Model of roughly 6.6%, average. This discrepancy was entirely plausible considering the possible sources of error in the laboratory environment. Sources such as human error and friction, both between the plunger and the wall of the cylinder and between the spring and the wall of the cylinder. The measured deflection and the Working Model values are listed below with their respective error.
Table 4 Spring Deflection Comparison
Measured Deflection (m) |
Working Model Deflection (m) |
% Error |
.023 |
.025 |
8.0% |
.029 |
.028 |
3.8% |
.040 |
.037 |
8.1% |
For the damper mass system there was a wee bit more difference between the Mathcad and the Working Model results than for the spring mass system. These differences ranged from 1.48% to 10.52%. This was most likely due to the different measuring methods that were employed to complete this part of the experiment as opposed to the methods that were used for the spring mass system. The major source of the discrepancy was human error when measuring the time values for the damper mass system. The measured values for velocity and the corresponding values from Working Model are listed below with their respective error.
Table 5. Damper Velocity Comparison
Measured Velocity (m/s) |
Working Model Velocity (m/s) |
%Error |
.009852 |
.010 |
1.48% |
.016 |
.015 |
6.67% |
.017 |
.019 |
10.52% |
As shown above in the Mathcad graph the actual experimental values (dots) for the spring damper system were below the values calculated by Mathcad (line). The Working Model graph showed the same tendency for the position as the calculated Mathcad graph did. The actual values obtained in lab were below both Mathcad and Working Model.
6. Conclusions and Recommendations
This lab showed that studying simple systems such as a spring mass system and damper mass system can give real insight as to how a more complicated system like a spring damper system works. The analyses of the parts of a more complicated system to get a model of that system is often easier that looking at and modeling that system as a whole. After performing the lab the measured values were fairly close to the experimental values that had been calculated but there was a discrepancy between the two. For the spring mass system the difference was 6.6% for the damper mass system the maximum difference was 10.52%. This is very close and still acceptable considering the ± 10% added to the measured coefficients. If more sophisticated measuring techniques had been applied in lab the measured and calculated values would have more closely matched. Also, if the cylinder system that we used in the experiment had contained less friction than it did in reality that would have also dramatically helped close the gap between the experimental and measured values.
Appendix B Working Model