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1.0 Introduction
The stalk of a grain plant, which is blown and turned away by the wind, does not bend over because it retreats elastically and, when the wind subsides, it causes oscillations. Did nature accidentally design this way or is there a principle behind it that is not recognizable at first glance?
We encounter vibrations at many places in daily life. Most people know from the fair the little devils that swing up and down on spiral springs. If you observe closely, you will see that the water surface, which was crushed by a stone's throw, shoots above its normal height. On all musical instruments, barely visible components vibrate, which stimulate the air to sound waves and create the auditory impression in our ears. High-rise buildings are equipped with an elastic steel skeleton so that they can give way like the grain plant during storms or earthquakes. In the watches, balance, pendulum or an oscillating quartz work for an exact time, the driving comfort in the vehicle is increased by oscillating damped suspension. Even molecules, the atoms in a crystal lattice and even the atomic nuclei and their elementary particles carry out vibrations and thus store energy.
The principle of oscillation is always the same: elastic objects that are brought out of equilibrium by an external force vibrate when they react to the external excitation with a backward, location-dependent force. Backward means that the force exerted by the object acts opposite to the direction of the stimulating force. Location-dependent means that as further the object is deflected from its equilibrium, as greater the reaction force is.

With the Parabolic-dish scooter you can experimentally measure different physical quantities that determine a vibration. In this experimental setup you will learn a lot about the concept of vibration. And once you understand the general principle, you can apply it to many different areas of physics.
You will learn important things about measurement technology while experimenting. The sensible execution of the experiments is a prerequisite for good results. With every measurement, measurement inaccuracies occur that you cannot avoid. For example, if you let several people measure the height of a tabletop, you get different results (77.3 cm, 77.2 cm, 77.7 cm, 77.55 cm, 76.9 cm...). The measured values scatter around a certain value, the average value, which here is 77.33 cm. The range in which the measurement results occure, here ± 0.4 cm, is called measurement uncertainty. If you can specify the range of measurement uncertainty, you are able to make a statement about the accuracy of your measurement. certainly, one goal is to make this value as small as possible.

2.0 Experimental setup
The experimental setup consists of a parabolic-antenna dish with axis lengths of 72 cm and 83 cm. On a board, the bowl is stored on four height-adjustable tips. In the parabolic dish you can run different balls and cylinders back and forth. If you start the "scooters" at the edge, they perform vibrations and remain after some time in the lowest point of the bowl. For the experiments you need different objects from the following table. Please check at the beginning of your experiments if everything is there.
  • 2.1 4 adjusting screws with fixed knurled nuts
  • 2.2 1 tennis ball (diameter 60 mm)
  • 2.3 1 ball of a computer mouse (diameter 22 mm)
  • 2.4 3 marbles (diameter 16.9; 16.1 and 11 mm)
  • 2.5 4 cylinders (diameter 50 mm, 2x 35 mm, 14 mm)
  • 2.6 3 hollow cylinders (diameter 2x 50 mm and 35 mm)
  • 2.7 several length measuring devices (caliper, scale...)
  • 2.8 Spirit level (length 1 m, 2 dragonflies)
  • 2.9 Stopwatch (accuracy 0.1 s)
  • 2.10 Balance 0 - 500 g (accuracy 0.1 g)
  • 2.11 Motor with 2 eccentric discs
  • 2.12 2 springs (attached to the bowl for counterforce)
  • 2.13 1 tensioning screw with hole
  • 2.14 adjustable DC voltage source (0 - 20 V=, 1 A)
  • 2.15 Multimeter (0 - 20 V) with a 3-digit display
  • 2.16 Multi-purpose grease
3.0 Definitions and terms
A vibration is a periodic movement. In this example, a rolling object performs a vibration in the parabolic dish. There is a fixed physical quantity for this movement: The oscillation duration, which describes the movement and depends on the structure of the bowl and the properties of the materials used. The oscillation duration is the time it takes for the deflected scooter to return to the same state. These can be any places i, but also the maximum end positions in the bowl. The number of oscillations that the scooter performs in one second is called frequency and is measured in units [1/s] or Hertz [Hz]. It can be calculated as a reciprocal from the oscillation duration. The frequency of the satellite roller can also be less than 1 Hz. If the system is left to itself, it usually oscillates at a very characteristic frequency, such as a pendulum. This frequency is called natural frequency. The largest deflection of an oscillation is called amplitude.
The rolling speed is not constant in vibration and is affected by friction between the roller and the bowl surface. The friction causes the amplitude to decrease over time. After some time, the state of equilibrium, namely the resting position, usually at the lowest point of the bowl, occurs.

4.0 First observations PIC2
It is best to do a trial phase first. Bring the bowl with the adjusting screws and the spirit level into a horizontal position. You have to make sure that the screws do not protrude from the board below. The maximum adjustment travel is determined by the board thickness and the height of the mounting blocks. Now let a scooter of each type roll from certain heights (deflection) or from the edge of the bowl into the rest position. Limit yourself to the 2 axes. How many vibrations can you observe on which scooter? How long does a vibration last? Where do the scooters stop? Is the magnitude of the deflection related to the oscillation duration or the number of oscillations?

You now have an idea of what happens to the scooters. But now your experiment should be more precise. Good scientists first describe their experimental setup and then what and how they have measured. An important scientific tenet is the verifiability of your results by others. So you should now be able to generate real measured values and also say how good they are.

5.0 General information
Quantitative measurements make experiments comparable. This also makes it possible to check predictions from theory. With every measurement, however, measurement inaccuracies arise, which of course have nothing to do with incorrect work. Only by specifying the size of these inaccuracies the quality of the measured values can be assessed.

5.1 The following measurement inaccuracies may occur:
  • 5.1.1 Systematic measurement errors: errors resulting from the experimental set-up that the measuring instrument brings with it as well as systematic reading errors. All these errors can be reduced.
  • 5.1.2 Random errors: errors due to external disturbances, random reading errors. These errors are difficult to reduce.
5.2 Possibilities for error limitation and error estimation:
  • 5.2.1 Carry out as many measurements as possible, which are allowed by time planning and the experimental setup.
  • 5.2.2 Create a list of possible disturbances.
  • 5.2.3 Organization of the measurement: Create measurement tables, select task-adapted measurement methods, determine the distribution of tasks in the group before the measurement.
  • 5.2.4 Mathematical error calculation, averaging, double calculation, etc.
5.3 General tasks PIC3
  • 5.3.1 Measure your rolling bodies: diameter (outside and inside of the hollow cylinders), length and mass. Determine their material, their density and make a statement about their surface. Summarize your results in the form of a table.
  • 5.3.2 Measure the geometry of the parabolic dish. To do this, place the spirit level on the edge of the bowl and measure the depth of the bowl in several places with the caliper on both axes.
  • 5.3.3 At what intervals should you measure the depth?
  • 5.3.4 At what point do you start measuring?
  • 5.3.5 Make a statement about the accuracy of this measurement.
  • 5.3.6 As a result, you get the so-called potential function on the axes of the bowl (its meaning is discussed in more detail in chapter 8, energy considerations).
  • 5.3.7 Create the graph of this function
  • 5.3.8 Which calculation do you have to perform so that you can determine a function from your measurement that describes the parabolic dish in a meaningful way (lowest point = zero point = intersection of the coordinate axes)?
6.0 Information on vibration duration>
It was already mentioned in the introduction: The oscillation duration is the time that the deflected scooter needs to meet the same place again. Depending on the starting location, these can be any places in the bowl. The axles are of course predestined.

6.1 Vibration duration tasks PIC4
Since these measurements are time-critical, it makes sense that you think about the process before the measurement. Which physical quantities are measured? What range of values can they take? What preparations are important before the measurement? How are the tasks distributed? How should the measured value table be created?
  • 6.2.1 First start with the sphere from the computer mouse, start it at a fixed point on an axis (e.g. the shorter transverse axis) and describe the observation qualitatively. Use the terms speed, oscillation duration, amplitude and zero crossing or rest position
  • 6.2.2 Start the ball at the highest point of the transverse axis and measure the oscillation duration with the stopwatch. Note: Most mobile phones have a fairly accurate stopwatch.
  • 6.2.3 At which deflections can the oscillation duration be measured best?
  • 6.2.4 Can you make a statement about the error of your measurements?
  • 6.2.5 How can you get more accurate measurements?
  • 6.2.6 What assumption must be fulfilled for this?
  • 6.2.7 On which variables could the oscillation duration depend?
  • 6.2.8 So now carry out measurements with different scooters on both the longitudinal and transverse axes. Create a design beforehand, what will be measured how and how exactly, in order to estimate the time required. Since you gain new information from your measurements, it may make sense to adjust the experimental design accordingly.
  • 6.2.9 Develop a theory that allows you to calculate the duration of oscillation. From the balance of forces of the downhill force and the inertial force, you can establish the distance-time law of oscillation. Use your potential function and the adjacent drawing.
7.0 Information on vibration amplitude PIC5
During the first measurements on the parabolic dish, you noticed that the amplitude of the oscillation decreases over time. This oscillation is also called damped oscillation. In the adjacent figure (from literature) you can see an example of how the amplitude of a scooter vibration decreases with time, which was initially deflected by the value x. The reason for the amplitude decrease of the oscillation is the friction of the roller on the bowl surface. Due to friction, vibrational energy is lost, the amplitude of the oscillation is dampened. Depending on the degree of attenuation, three cases can be distinguished:
  1. 7.0.1 Weak damping (oscillation fall)
    The movement of the scooter contains enough energy to perform one oscillating movement, i.e. several zero crossings, or to take amplitudes. The amplitudes of the oscillation decrease until the resting position is finally taken.
  2. 7.0.2 Mean attenuation (aperiodic limit case)
    The friction is just enough that the scooter returns to the resting position after the initial deflection within the first oscillation. The final value is reached the fastest of all three cases.
  3. 7.0.3 Strong damping (creep)
    The scooter is braked by the friction so much that no vibration occurs. It creeps slowly from the first deflection into the resting position.
7.1. Vibration amplitude tasks
  • 7.1.1 Measure the amplitudes of the mouse ball for multiple oscillation passes
  • 7.1.2 Graphically represent the measured amplitudes as a function of time and describe the result qualitatively. Note: Use a spreadsheet program, such as Microsoft Excel with the chart feature. Please keep in mind that MS - Excel can only map whole values on the ordinate. How can you get around the problem?
  • 7.1.3 Find out about the different types of friction: static friction, sliding friction and rolling friction.
  • 7.1.4 Perform amplitude measurements with various scooter devices. To do this, create a test plan beforehand to estimate the time required.
  • 7.1.5 Determine the dependence of the amplitude on the geometry of the rolling equipment
  • 7.1.6 Make a statement about the accuracy of your measurements
  • 7.1.7 Is the aperiodic limit case feasible with the rolling bodies?
8.0 Energy considerations at the vibration of the Parabolic-dish scooter PIC6
8.0.1 A harmonic oscillation occurs when - as already explained in the introduction - location-dependent, driving forces are present in a system. In the satellite scooter, the downhill force, a component of gravity, is involved as a driving force. After starting the scooter at the edge of the parabolic-antenna dish (excitation of the oscillation), the positional energy (potential energy) of the scooter is converted several times into kinetic energy.
The potential energy follows from the potential function, which you have already determined except for a constant factor (weight force of the rolling equipment) when measuring the bowl surface. From a physical point of view, a potential is a rule from which energy values can be calculated. If you move an object on the key surface between two points of different height (different potential), you must either move it up and do work, or move it down, freeing up work. This work is equal to the potential energy that can also set the scooter in motion if you let it roll freely. The downhill force generates a rotational movement via the lever arm center of gravity - contact point (radius R) a rotary movement. The scooter rotates around its center of gravity, which moves at a distance R above the bowl surface to its lowest point. The rotation can be understood as kinetic energy of the scooter. Analogous to the inertia of a linear motion, the moment of inertia acts in the rotational motion of a body Θ.
If you own a swivel chair, you can perform a simple experiment to experience the effect of the moment of inertia. Get on a swivel chair, stretch out your arms and legs to get some momentum and rotate yourself (right picture). Then you pull your arms and legs close to your body (left picture). What do you observe? To repeat, you can stretch your arms and legs again.
The moment of inertia is a quantity that takes into account the distribution of mass with respect to its axis of rotation (outstretched, applied arms or legs). To determine the moment of inertia, each mass particle is weighted with the square of the distance from its axis of rotation. If one piece of mass is considered at a distance R1 and another piece of the same size at a distance of 2 * R1 from its axis of rotation, the latter mass has 4 times the moment of inertia. The greater the moment of inertia of a body, i.e. as further away its mass is from its axis of rotation as harder it is to set a body into rotation. If two bodies of the same mass but with different moment of inertia have the same rotational energy, the body rotates more slowly around its axis of rotation with a greater moment of inertia.
The third energy component is the loss energy. If the ball or cylinder rolls over the bowl surface, part of its energy is converted into thermal energy at each oscillation at the support point with each oscillation. This is released into the environment and is lost to the system. The loss is reflected in the amplitude decrease that you have already observed. The ratio of stored energy to loss energy is called quality, Q of the vibrating system.

8.1.0 Tasks relating to the different forms of energy of scooter vibration
  • 8.1.1 Describe the driving and inhibiting forces of scooter vibration.
  • 8.1.2 What causes the amplitude decrease in damped oscillation?
  • 8.1.3 Determine the damping constant γ from the amplitude-time diagram. To do this, you first form the natural logarithm from the amplitude ratios
    An : A1 (An = measured amplitude, A1 = initial amplitude). Then you graphically represent these values as a function of time and determine their slope. You can save logarithmizing if you use special drawing paper in which the y-axis is logarithmically divided.
  • 8.1.4 Qualitatively describe the different energy forms of scooter vibration.
  • 8.1.5 Find out in the literature or on the Internet how to calculate the moment of inertia of the various rolling bodies (sphere, cylinder, hollow cylinder).
  • 8.1.6 What is the temporal relationship between potential energy, forces and speeds
  • 8.1.7 Can you calculate the energy loss resulting from the attenuation for two consecutive amplitudes?
  • 8.1.8 The ratio of the energy stored in the oscillation to the loss energy is called quality Q of the vibrating system. Determine the quality of the system for the different scooter types using the equation Q = ω o / γ (with γ = damping value and ω o is the natural frequency of the oscillation).
9.0 Information on excited vibration PIC8
9.0.1 The parabolic dish is mounted on two tips on its longitudinal axis and is driven on the transverse axis on one side by the motor and the eccentric wheel, which converts a rotary movement into a linear movement. A preloaded spring on the opposite side ensures that the bowl always rests against the eccentric wheel. The number of revolutions of the motor can be varied by changing the motor voltage within a sufficient range.
With this arrangement, the parabolic dish can be periodically moved up and down around an axis and the vibration of the rolling equipment can be maintained over a longer period of time. The parabolic dish, which can be tilted in one direction, and the cylinders or spheres rolling in it in the same direction form two oscillating systems, each of which can move with different frequencies or oscillation durations.
According to the superposition principle, these two oscillations overlap, the deflections add up at any time. If you start the scooter at any point of the bowl, it oscillates in its natural frequency, which experiences an amplitude decrease due to the damping. When all the vibration energy is used, the rolling device remains at the lowest point of the parabolic dish (decay process: Fig. a).
9.0.2 If the drive is switched on in the meantime, the natural oscillation of the scooter overlaps with the excitation by the motor and the scooter performs complicated forms of movement, which are also referred to as the transient process of a forced oscillation (see Fig. b).
9.0.3 If the initial energy of the scooter is dampened away and the periodic drive continues to work uniformly, a stationary behavior occurs (right section in Fig. b).
9.0.4 Whenever the amplitude, phase or frequency of one of the oscillating systems changes, these transient or decay processes occur; if you wait a while afterwards, you have a stationary behavior again.
9.0.5 If the speed of the motor is increased from low to higher values, the scooter oscillates stationary after the corresponding waiting time with an amplitude dependent on the excitation. If the natural frequency of the bowl-roller system is reached, the amplitude assumes a maximum.
9.0.6 This is referred to as resonance.
The lower the attenuation or the greater the quality Q of the system, the higher and the narrower the amplitude curve as a function of the excitation frequency (see Fig. c; resonance curve).
9.0.7 If the excitation frequency is significantly higher than the natural frequency of the scooter, then the amplitude drops again.
9.0.8 In addition to resonance, there is another special stationary state: beating. Beating occurs in the event that the frequency difference between the excitation and the rolling movement is small and the quality Q of the system is particularly high.
9.0.9 In the adjacent figure, two oscillations are superimposed with the ratio T1/T2 = 7/6. The result is an oscillation whose amplitude is modulated with the oscillation duration. As smaller the distance between excitation frequency and natural frequency, as greater the oscillation duration TS of the beat is. At the same time, the enveloped oscillation duration TR of the scooter vibration changes proportionally to the reciprocal of the sum of excitation and natural frequency.

10.1 Tasks for excited scooter vibration
  • 10.1.0 Measure the number of revolutions (per minute) of the motor as a function of its operating voltage and create a graph showing the motor frequency as a function of the motor voltage.
  • 10.1.1 Periodically move the Parabolic dish at different engine speeds. Create the amplitude-frequency diagrams for the different spheres. At what frequency does resonance occur?
  • 10.1.3 Investigate excitation in more detail. Which mechanism transfers the excitation energy to the scooters?
  • 10.1.4 Measure the amplitude of a cylinder vibration over a longer period of time. To do this, use a frequency that is close to the natural vibration of the scooter. Start the scooter at a medium height of the satellite dish. It is important that you determine both: The frequency of the excitation and the frequency of the scooter.
  • 10.1.5 Qualitatively describe the time course (phase relationship) between excitation and cylinder movement in the steady state. To do this, note the position of the eccentric wheel in the reversal points of the oscillation and when the scooter is in zero crossing.
10.0. Literature

  1. 10.1 Bergmann; Schaefer: Lehrbuch der Experimentalphysik Band 1: Mechanik, Relativität, Wärme, de Gruyter Berlin; New York 1998, ISBN: 3110128705
  2. 10.2 Tobias Schmidt, Dirk Schlender: Untersuchung zum saisonalen Reifenwechsel unter Berücksichtigung technischer und klimatischer Aspekte; Projektbericht, Bergische Universität Wuppertal Sicherheitstechnik Fachgebiet Verkehrssicherheitstechnik, Wuppertal 2003
  3. 10.3 Niemann, Gustav, Höhn, Bernd-Robert, Winter, Hans: Maschinenelemente, Konstruktion und Berechnung von Verbindungen, Lagern, Wellen; Springer-Verlag: Berlin, Heidelberg 2005, Online-Ausgabe, ISBN: 9783540264248
  4. 10.4 Vöth, Stefan: Dynamik schwingfähiger Systeme, Von der Modellbildung bis zur Betriebsfestigkeitsrechnung mit Matlab/Simulink, Vieweg Verlag 2006, ISBN: 3834801119
  5. 10.5 Wikipedia, die freie Enzyklopädie:
  6. 10.6 Tross, Arnold: Über den Mechanismus der Rollreibung und ihres Kraftschlusses, aus: Beiträge zur Klärung der Mechanismen von Festigkeit, Reibung und Verschleiß, 2. Serie, 8. Bericht, München 1966
  7. 10.7 Föppl, Ludwig: Die strenge Lösung für die rollende Reibung, Leibnitz-Verlag, München 1947
  8. 10.8 Hakenjos, Volker: Untersuchung über die Rollreibung bei Stahl im elastisch – plastischen Zustand, Dissertation an der Technischen Hochschule Stuttgart, 1967
  9. 10.9 Czichos, H.: Die Rollreibung von Metallen bei kleinen Belastungen, Abschlußbericht zur Forschungsaufgabe DFG Ki 66/11, Bundesanstalt für Materialprüfung (BAM), Fachgruppe 5.2 (Rheologie & Tribologie), Berlin-Dahlem, 1971
  10. 10.10 Czichos, H.: Die Energieverlustmechanismen der Rollreibung, Schmiertechnik und Tribologie 16, Berlin-Dahlem 1969, Heft 2, S 62 – 68
  11. 10.11 Czichos, H.: Ein photoelektronsiches Torsionsdynamometer zur Messung kleiner Drehmomente, Carl Hanser Verlag München, 1969, Sonderdruck aus der Zeitschrift Feinwerktechnik, 73. Jahrgang 1969, Heft 9
  12. 10.12 Böge, Alfred: Technische Mechanik; Statik - Dynamik – Fluidmechnaik – Festigkeitslehre, Vieweg Verlag Braunschweig, 2001, ISBN: 3-528-05010-1, S. 134 ff