October 7, 1999

Becky Engel

Table of Contents *

List of Figures and Tables *

Executive Summary *

1. Introduction *

2. Simple Translational Systems *

3. Apparatus *

4. Procedure *

5. Results *

6. Analysis and Interpretation *

7. Conclusions and Recommendations *

The objective of this experiment was to analyze simple translational systems consisting of mechanical components such as masses, springs and dampers. The procedure consisted of three separate systems. In each of these systems, experimental values for displacement versus force were collected. The weight of the mass was used as the force in the systems and the resulting displacement of the mass was measured as a function of this force. The first system analyzed was a spring and mass only. The second consisted of a damper and masses. The final system combined the spring and damper with the masses. Using the experimental values that were obtained, the spring constant and damping coefficient for the system components were calculated. When compared with theoretical values for displacement versus force in the systems, all of the error percentages were below 14%. Though there was some discrepancy between the theoretical and experimental values obtained, this experiment supported the theory behind analyzing these types of simple translational systems. The values for displacement in this experiment were taken with linear measuring devices by hand. More accurate results could be obtained by using more precise and accurate measuring devices such as computer-controlled sensors and timers. Also, collecting more data points may have increased the accuracy of this experiment.

This experiment was conducted in order to analyze simple translational systems consisting of masses, springs and dampers by comparing experimental results to values obtained theoretically, thus supporting the theory behind these systems. Systems of springs, dampers, and the combination thereof are present in many of the everyday objects and mechanisms that people use. They provide the useful purpose of opposing or retarding motion in things like automobile shock systems or door closers. These systems can be analyzed quite simply with the use of equations such as d’Alembert’s Law, Hooke’s Law, the definition of the damping coefficient, and common calculus concepts such as the first and second order derivative.

The theory section will explain how all of these equations and concepts fit together to theoretically analyze systems of springs and dampers. In the procedure, the experiment is conducted on the translational systems and data is collected to be used in these equations. The experimental results are compared to the theoretical results in order to show that the translational systems in the real world behave according to the theoretical descriptions of motion explained in the theory section.

Many mechanical systems can be analyzed by using the basic concept of d’Alembert’s Law, as shown below in equation (1). Equation (1) states that the sum of the forces acting on a body is equal to the mass of the body multiplied by its acceleration.

where,

F: Force (N)

m: Mass (kg)

a: Acceleration (m/s²)

All three of the translational systems analyzed in this experiment use this basic concept.

A system consisting of a mass, M, and a spring with spring constant K_s will oscillate when a force is applied to it. The force exerted by the spring on the mass can be calculated by using Hooke’s law, which is stated in equation (2).

where,

F_s: Force exerted by spring (N)

K_s: Spring constant (N/m)

y: Displacement of spring (m)

Figure 1 below illustrates the translational system of a mass and a spring. Using d’Alembert’s Law and equation (2), equation (3) can be formulated which describes the motion of the mass in this system.

where,

F_s: Force exerted by spring (N)

m: Mass (kg)

g: Acceleration due to gravity (9.8 m/s²)

y: Displacement of the mass (m)

This experiment will involve calculating a value for the spring constant based on measurements in the lab. If two different known forces are applied to the spring, two different displacements will result. By applying the differences in these force values, a value for the spring constant can be calculated. Equation (2) has been modified below in equations (4) through (6) in order to get a useful formula for calculating an experimental value for the spring constant.

where,

F₁: First force applied to the spring (N)

F₂: Second force applied to the spring (N)

y₁: Displacement of spring caused by F₁ (m)

y₂: Displacement of spring caused by F₂ (m)

A system consisting of a mass, M, and a damper with damping coefficient K_d will tend to return to rest after the initial force supplied by the weight of the mass is applied because the damper dissipates energy. The force exerted by the damper on the mass can be calculated by using equation (7).

where,

F_d: Force exerted by damper (N)

K_s: Damping coefficient (Ns/m)

y: Displacement of spring (m)

Figure 2 below illustrates the translational system of a mass and a damper. Using d’Alembert’s Law and equation (7), equation (8) can be formulated which describes the motion of the mass in this system.

where,

F_d: Force exerted by damper (N)

m: Mass (kg)

g: Acceleration due to gravity (9.8 m/s²)

y: Displacement of the mass (m)

This experiment will involve calculating a value for the damping coefficient based on measurements in the lab. The definition of the first derivative can be used to estimate the value for the first derivative required in order to solve for the damping coefficient as shown in equation (7). The weight of the mass will act as the known damping force, F_d, and the displacement of the mass on the damper will be measured initially and after a known amount of time has passed. This formula for calculating the damping coefficient is derived in equations (9) and (10).

where,

F_d: Force exerted by damper (N), in this case it is the weight of the mass (N)

y(t): Initial displacement of the mass (m)

y(t + D T): Displacement of the mass after D T seconds (m)

D T: Time between displacement values (s)

A spring and damper are often used in combination in a system, usually by placing the spring inside of the cylinder of the damper. This type of system can be analyzed as if the spring and damper mechanisms were assembled in parallel, as shown in Figure 3. Using d’Alembert’s Law and equations (3) and (7), an equation that describes the motion of the mass in the system of Figure 3 can be formed as shown in equation (11).

where,

F_d: Force exerted by damper (N)

F_s: Force exerted by spring (N)

m: Mass (kg)

g: Acceleration due to gravity (9.8 m/s²)

y: Displacement of the mass (m)

Most of the equipment used in this experiment are common elements used in everyday life. Table 1 below lists these items and descriptions individually.

The surface clamp and metal rod were used to secure the mass-spring system

as shown in Figure 1. Three different masses were place on the top of the spring and the displacement was measured for each mass.

The spring was then replaced with the damper and three different masses were placed on top of the damper as shown in Figure 2. The velocity was calculated as a function of time by measuring the time required to displace the damper a fixed distance.

The spring was then placed inside the damper and secured onto the surface

clamp as shown in Figure 3. The amount of spring compression as a result of adjusting the damper to its neutral position was measured. The velocity was calculated as a function of time by measuring the time required to displace the damper for three equal, consecutive fixed distances with a single mass. The response of the system was calculated using Mathcad.

Throughout the experiment, each lab partner verified the displacement values in order to reduce the risk of human error. Successively larger forces should have produced larger distance displacement/smaller time differences and so this property of the systems was taken into consideration when evaluating whether or not the collected data seemed reasonable.

The force and displacement data for the mass-spring system are tabulated below in Table 2. These values were used in equation (6) in order to evaluate the experimental value of the spring constant. The individual spring constants were averaged and the spring constant was found to be 1064 kg/s^{2 ±} 10%.

For the mass-damper system, the first derivative of the displacement, or velocity, had to be measured. Since the distance of displacement was fixed, the time for the system to move through this displacement was the only experimental data collected. The force and time data for this system are tabulated below in Table 3. These values were used in equation (10) in order to evaluate the experimental value of the damping coefficient. The individual damping coefficients were averaged and the damping coefficient was found to be 505.139 kg/s ± 10%. 

For the mass-spring-damper system, experimental values of position in the y-direction (m) versus time (s) were collected. These values are plotted below in the graph of Figure 4. Note that the system was released at a position of –0.06 m.

In order to provide for some means of comparison between theoretical and experimental values, the Working Model 2D program was used. The first two systems in the procedure system were created in the program and the experimental spring/damping constants and known masses were applied as inputs into the systems. Figures 5 and 6 below show the mass-spring and mass-damper systems, respectively, which were created in Working Model.

The outputs of displacement for the mass-spring system, and time for the mass-damper system from Working Model were compared to the experimental values obtained. Tables 4 and 5 below list these theoretical and experimental values and the percentage errors between them.

For the mass-spring-damper system, a second order differential equation was obtained by rearranging equation (11). The solution to this equation is the theoretical motion of the mass in the combined system. Mathcad was used to solve and plot the solution as a function of time. The data points collected for this part of the experiment were superimposed on this graph to provide a means of comparing experimental and theoretical values for the system. This graph is shown below in Figure 7.

The results of the mass-spring system produced errors ranging from 4 – 8% between the Mathcad calculations and the Working Model predictions. Possible sources of error in this part of the experiment include human error when measuring the exact displacement of the mass, and friction from contact between 1) the walls of the cylinder, which the spring was encased in and 2) the plunger which rested on top of the spring. We had limited control over these types of error sources, especially human, and so the discrepancies between experimental and theoretical values are reasonable.

The results of the mass-damper system produced errors of 1.5%, 3.5%, and 13.7% between the Mathcad calculations and the Working Model predictions. Human error when measuring the time values is the most likely cause for these errors. A quarter of a second hesitation, the typical response time of human data collection, would produce large errors.

Section 2.1 of the theory explained the system model for a mass-spring system, and section 2.3 explained the system model for a mass-damper system. Without conducting these parts of the experiment, damping and spring coefficients could not have been determined for later use in analyzing the theoretical response of the mass-spring-damper system. Likewise, any errors incurred in these two parts of experiment would effect the theoretical response of the mass-spring-damper system. The graph of the response of the mass-spring-damper system was created in Mathcad. This is indicated by the smooth line in Figure 7 that approaches zero from the original displacement and levels off near zero after about 3 seconds. The dots on this graph are the actual experimental values obtained for this part of the experiment. Though the dots proceed in time with the same tendency towards zero, they are all situated well below the graph of the smooth line generated by the numerical integration rkadapt function. The largest source of error in this part of the experiment is again most likely due to human error in the time measurements.

Three types of simple translational systems, mass-spring, mass-damper, and mass-spring-damper, were evaluated in this experiment. The experimental results obtained for these systems support the theories that were explored. Though there was some discrepancy between theoretical and experimental values of displacement and time, the calculated values for spring and damper coefficients seemed reasonable based on this experience in the real world with such systems. It is concluded that the analytical models agree with the observed behavior of the system within 15% with a confidence level of 90% for the conditions of this experiment. More accurate results for all parts of this experiment could be obtained by utilizing more precise measuring devices such as computer-controlled sensors and timers. Taking additional experimental data points may also have led to more conclusive results.