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Vibration of a Damped Single Degree of Freedom System
A vibrating single spring mass system has several applications. One of the applications is to determine the Harmonic motion of the vibrating bodies under damp or undamped conditions. These can furthermore have the utility in creating stable constructions. Specifically, in this lab report, three set of experiments are undertaken – study the vibrations of the spring & mass system and damper under undamped , damped and free fall condition. The three experiments’ procedures are explained in this lab report. The expected plots for the first experiment, that is, determining the natural frequency of vibrations of a simple spring mass system are illustrated as well as is determined. Similarly, the expected plots for other 2 experiments – damping coefficient of oil dashpot damper having the spring and during the free fall piston, are also illustrated and evaluated. There are slight deviation observed from the expected plots, based on theoretical evaluation and the evaluated plots through the experiment due to the recoded errors.
When motion is confined to one independent degree-of-freedom, there is a linearized equation that governs the motion. The system’s free motion (f = 0) is analyzed first and then its forced motion. The analysis performed for the free motion is called a transient analysis. The transient analysis is independent of the non-homogeneous term that appears on the right side of the differential equation. The non-homogeneous term is often a force but it can also arise as a result of prescribing the displacement at a point in the system. Next, we consider the harmonic excitation. Systems that contain unbalanced rotating elements, like a washing machine, milling machines, and rotating shafts, are systems that are acted on by harmonic excitations.
For the damped motion, we also have three levels of damping. To distinguish between three levels of damping depending on whether the quantity under the square root is positive, zero, or negative. (Damped Harmonic Oscillators, nd). The roots are :
We will study in more details, the nuances between the damped, underdamped, critically damped and undamped vibrations in mode details in theory section. (Kelly, 2011)
In this experiment, I will demonstrate 1-degree-of-freedom systems to a step (square wave) pulse applied force. Three systems will be studied: 1) under-damped (=0.1), critically damped ( =1.0), and over-damped ( =2.5). From the response of the under-damped system, I will determine the natural frequency and damping ratio and compare them to theoretical values.
a. Calculate the theoretical natural frequency of the undamped vibration
b. Solve the differential equations
Verify that the system has weights on the first carriage and that the medium spring is attached from the support structure to the first carriage.
Set the damping constant c to the value calculated in your prelab for the underdamped system using Force+Spring+Damper in the Setup Driving Function dialog box. Set k=0 within the driving function. Since you have a real spring attached to the system, you do not need the system to simulate a spring.
Set up a unidirectional Step input shape of 9 N and 2?? second duration. Set the data acquisition function to collect Encoder 1 and Drive Input data every 4 servo cycles. Note that a servo period Ts=0.0042 s.
Execute the step input and plot and save your Encoder 1 Position data. Be sure to choose cm as the output units for the displacement.
Check that the results are in close agreement with your expectations and prelab plots. If they are not, verify that all settings are correct.
For the underdamped system only, change the data acquisition function to collect data every 40?? servo cycles. Execute the step input and plot your Encoder 1 Position data.
Repeat for the damping coefficients corresponding to the critically damped and overdamped systems. (Adhikari, 2000)
The harmonic excitation causes a system to undergo harmonic motion. Free Un-damped Motion can be represented in the below figure:
The mass-spring-damper system
The linearized equation that governs the motion is
First consider the free undamped system (See 1). Letting f = 0 and c = 0 in Eq. (1) yields
Equation (2) is a homogeneous constant-coefficient linear differential equation. As with any constant-coefficient linear differential equation, the solution is a combination of complex exponential functions. We start by looking at the single complex exponential function
where s is a complex number that needs to be determined. Substitute Eq. (3) and its second time derivative into Eq. (4 – 2) to get
The values of s for which x = satisfies the differential equation are
where The two solutions are
These two solutions may seem a bit odd; after all they’re complex. The complex solutions are actually just building blocks from which the real solution is constructed. Recall that and in Eq. (5) are complex harmonic functions of the form
Now let us consider the free damped system. Letting f = 0 in Eq. ( 1) yields
The procedure for solving Eq. (12) is the same as the procedure followed for solving Eq. ( 2). Start by assuming a solution in the form of a complex exponential function. Substitute Eq. (3) into Eq. (12) to get
Dividing by yields the quadratic equation
Its roots are
Let’s now distinguish between three levels of damping depending on whether the quantity under the square root is positive, zero, or negative. The roots are
(4 – 15)
We see in Eq. (17) that the transient solution of a free under-damped system is a damped harmonic function. Its natural damping rate is and its damped natural frequency is d (See Fig. 4 – 5). The damped natural period is
wn = 2πnsl equation 1
Where wn is the natural frequency of the system, n is the number of the cycle and s is the speed of the paper and l is the length over the number of the cycles. The natural frequency is found from Eq.1 aboive. Its standard units are rad/s. The natural frequency is also sometimes expressed in terms of cycles per seconds which is the same as a Hertz, abbreviated Hz. Since 1 cycle is 2 radians it follows that and so 1 Hz is about six times larger than 1 rad/s.
δ =ln x1x2 equation 2
Where δ the logothermical decrement, x1 is the first peak and x2 is the second peak.
ζ = 14π2δ2+1 equation 3
Where ζ is the damping ratio. The damping ratio for
critically damped ( =1.0), and
over-damped ( =2.5).
cc = 2mwn equation 4
m is the mass which is 3.7 kg and cc is the critical damping coefficient.
c = cc ζ equation 5
c is the damping coefficient
v = h*sd equation 6
Where v is the velocity of the piston in the oil, and h and d are given in figure below:
c = mgv equation 7
Where g is the gravity, m is the mass of the total mass of the system and c is the damping coefficient in the free fall. The laminar flow of the oil through the perforations as the piston moves
causes a damping force on the piston. The force is proportional to the velocity of the piston in a direction opposite that of the piston motion. This damping force has the form:
where c is a constant of proportionality related to the oil viscosity. The constant c, called the damping coefficient, has units of Ns/m, or kg/s.
2 slotted discs (total mass 2 kg) attached to the spring machine
Frame of mass 1.7 kg to move vertically in roller guides with spring stiffness k = 3.3 kN/m
Pen attached to the vibrating frame
A paper strip driven by synchronous motor
Dampening provided with piston moving in a cylinder filled with oil
Figure 1 the apparatus used in this experiment
The following procedures are taken from the given note and the references related to the lab report. and the rectilinear system manual.
Spring stiffness k= 3.3kn/m
spring constant (soft spring) = 175 N/m
spring constant (medium spring) = 400 N/m
spring constant (hard spring) = 800 N/m
mass (brass weights) = 500 g each
carriage mass = 500 g
The equipment used has a frame of mass of 1.7 Kg. Slotted mass are of each 1 kg. Dampening is provided by the piston moving in the cylinder filled with the oil. Pen is attached to the vibrating frame with a paper strip driven my synchronous motor.
The upper string is locked to the top plate. Clamp the dashpot at a certain distance h along the beam and pull down on the free end of the beam and release it. Draw the decaying curve on the recording drum, while it is rotating. Vary the damping characteristics of the system by moving the dashpot to a new position and obtain a new record of the decaying curve. Select a spring and attach to the upper end of the locked support. Vary the damping characteristics of the system by moving the dashpot to a new position and obtain a new record of the decaying curve. The size of thumb nuts can be adjusted at the upper end of the spring. Repeat steps to get the records for 8 different positions of the dashpot. From the resulting plot, obtain the undamped natural time period n and based on that calculate the natural (undamped) frequency of the system ωn=2π/τn. A plot similar to what is shown below should be produced:
Dashpot is connected to the mass carriage by thumb screw. Dashpot piston is checked. Damper is closed Carriage is pulled down and the recorder is witched on and carriage is released. The damper is opened a quarter turn and the steps are repeated. The plots are taken in the same manner and the plots should represent as follows:
Spring is removed to enable free falling leaving the dashpot just connected. One disc is placed and the carriage is closed. Carriage is lifted to max limits and the pen recorder is switched on. Then it is switched off when carriage comes to lowest limits. The experiment is repeated and we should get the following plots:
A total of six test was conducted to find the average value of the natural frequency in the un-damped system using equation 1. The recordings are tabulated as given below:
The plot is as shown below and it represents the figure which was earleir mentioned in the procedure.
Figure 2 results from one of the test conducted.
Sample Standard Deviation of (wn) s = i=1N(xi-30.64)6-1 = 0.9524
Using equation 2, 3, 4, 5 the chart is created. The detailed calculations are given in the appendix. Using equation 5 the critical damping coefficient was 226.74 Ns/m.
Using equation 6 and 7, the following chart is created. The details of the calculations are given in the later sections.
The propagation error
The three set of results were noted – one with damped system, other with undamped and the third set was with free falling system. We started the experiment with the expected results. The expected plots were also illustrated in the procedures. The deviation was observed in the plots. For instance for the undamped system, it was expected that the trace will result in to the following plot:
However, the actual plot which was observed was :
Similarly, for the damped systems, the set of points can be evaluated theoretically based on the given formula. The plots are taken and are represented as follows:
However, from the experiment, where the Dashpot piston is used, the Damper is closed as per the process. The Carriage is pulled down and the recorder is switched on and carriage is released. The damper is opened a quarter turn and the steps are repeated. The plots are taken in the same manner and the plots come out as follows:
As observed, there is only slight deviation from the original, as more or less these are representing the same. The errors can be due to the timings in pulling down the Carriage and the recordings are the time of pulling down. If there is time lag, there would be more deviations noticed in the recorded plots.
For the mass which is 3.7 kg and cc is the critical damping coefficient is given by
c = cc ζ
Using equation 5 the critical damping coefficient was 226.74 Ns/m. However, from the formulae the damping coefficient comes as 254.5. There is a slight deviation here
In the same manner, the free flow plots theoretical and formula based plots are represented as follows:
In the actual recording of the data, the plot is as shown below:
There is remarkable deviation during the taking of free fall observations. Obviously the lines are not straight as those should be in free fall. This could be due to the fact that there is some amount of friction which plays into the free fall process for the piston. This friction can be the reason that there is remarkable deviation from the expected plots. Using low viscosity oil which offers lower friction is one solution to get the expected plots.
The three experiments’ procedures are explained in this lab report. The expected plots for the first experiment, that is, determining the natural frequency of vibrations of a simple spring mass system are illustrated as well as is determined. Similarly, the expected plots for other 2 experiments – damping coefficient of oil dashpot damper having the spring and during the free fall piston, are also illustrated and evaluated. There are slight deviation observed from the expected plots, based on theoretical evaluation and the evaluated plots through the experiment due to the recoded errors
Adhikari S. (2000). Damping Models for Structural Vibration. Cambridge University. Available from http://www-g.eng.cam.ac.uk/dv_library/Theses/sondiponthesis.pdf [Accessed March 05. 2015]
Damped Harmonic Oscillators. (nd). Under, Over and Critical Damping. MIT Open Courseware. Massachusetts Institute of Technology. Available from http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-ii-second-order-constant-coefficient-linear-equations/damped-harmonic-oscillators/MIT18_03SCF11_s13_2text.pdf [Accessed March 05. 2015]
Kelly G. S. (2011). Mechanical Vibrations: Theory and Applications, Cengage Learning. Available from https://books.google.co.in/books?id=5ToJAAAAQBAJ&pg=PA384&lpg=PA384&dq=underdamped,+critically+damped+and+undamped+vibrations+in+mode+details+in+theory+section.&source=bl&ots=YmMLbUH1Ni&sig=DaXdEK5HOSYsdwH6MUyBT9R9WI0&hl=en&sa=X&ei=vKX6VOmvJMm-uATepoCIBQ&ved=0CEQQ6AEwBg#v=onepage&q=underdamped%2C%20critically%20damped%20and%20undamped%20vibrations%20in%20mode%20details%20in%20theory%20section.&f=false [Accessed March 05. 2015]
Thomson W.T. (1988). Theory of Vibration with Applications. Allen Unwin Ltd.
Church A.H. (1973). Mechanical Vibrations. 2nd Ed. John Willey & Sons.
Roa S.S. (1986). Mechanical Vibrations. Addison Wesley Publishing co.
Seto W.W. (1964 ). Mechanical Vibrations. Schaum Outline Series, McGraw-HilI, USA
Theoretical value of the natural frequency wn = km = 3.3×1033.7 = 29.9 rad/s
Theoretical value of the critical damping coefficient cc = 2mwn = 2*29.9*3.7 = 221.26 rad/s
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