one degree of freedom 2014-2015

8/10/2019 One Degree of Freedom 2014-2015

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Sheikh Shahir One Degree OF Freedom KEM120702

NAME : Sheikh Shahir Muhaamad Afiful IslamMATRIC NO : KEM120702

TITLE : ONE DEGREE OF FREEDOM

1.0 OBJECTIVE

The experiment consisted of three different objectives:

1. Using the information provided on the stiffness and mass, we are required to find out the undampednatural frequency of the single degree of freedom system.

2. From the transient response of the system, we are required to found the natural frequency of thedamped system, the undamped natural frequency and also the damping ratio.

3. To finish it off, comparison will be made from the results obtained from the above objectives.

2.0 ABSTRACT

The following experiment was carried out to determine the undamped natural frequency, o , of a onedegree of freedom system by making use of the values of stiffness and mass. Having done that, we also

went to determine the damping ratio, which is and the damped natural frequency, d and undampednatural frequency, o, from the transient response of the system. The instruments used in this experimentare a cantilever beam. A cantilever beam is a prime example of a single degree of freedom system. Along

with that we also have a Linear Variable Differential Transformer (LVDT) tool conducted to our laptop.Our experiment consists of two parts. In the first part of our experiment we apply a force to the cantilever

beam and record the displacement of the beam. This is done slowly in steps with the application of a load.By plotting a graph of the load applied versus the change in displacement, we determine the springconstant or stiffness constant, k, of the cantilever beam. In the second part of the experiment we ensure

that a steady load of a constant value is applied to beam. As the beam is displaced, we are able to acquirethe transient response. After that, its possible to plot a graph of ln Y versus time using the data that we

obtained.

3.0 INTRODUCTION AND THEORETICAL BACKGROUND

3.1 INTRODUCTION

First we look into the definition of vibration.

Vibration (also known as Oscillation) is said to be any kind of motion that repeats after a certain intervalof time. Examples of Vibration include the swinging of a Pendulum or the motion of a plucked string.The theory of vibration deals with the study of oscillatory motions of bodies and the forces associatedwith them.

Vibration can be catergorized into the following types types:


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Free vibration- A system is said to be vibrating freely when an initial force or disturbance is applied to

the system and then its left to vibrate freely on its own. An example would be to flick a fixed spring

sideways and watching it oscillate a few times to the left and right before coming to a stop. In this case,

the spring, which is the mechanical system in question, vibrates at one or more of its natural frequency

before its damped enough to come to a stop.

Forced vibration

Forced vibration is extremely different from free vibration. In forced vibration, the mechanical system is

subjected a changing force or motion. An example of forced vibration would be a toy that requires an

objected supported by elastic band. One such toy could be a paddle ball that is suspended from a finger.

When a child moves the paddle by making use of his her finger, he/she is driving the paddle ball at a

certain frequency. The child is applying an external force to vibrate the ball up and downIn a vibratory

system, there is a means for storing potential energy, a means for storing kinetic energy and a means by

which energy is gradually lost. In vibrational motion, there is said to be conservation of energy. For

instance, when we are extending a spring by a certain length, the spring is said to have stored some

potential energy. When we let go off the spring, the spring returns to its initial state. When the spring has

reached its un-stretched state all of its potential energy is transformed into kinetic energy. When the

spring oscillates, energy is transferred to and fro the two different formskinetic and potential energy.

A mass is supposed to oscillate for an infinite number of time in such a system, however in reality the

existence of damping will damping will result in the loss of energy and the system will eventually come

to rest.

Degree of Freedom:

The degree of freedom of a system can be defined as the minimum number of coordinates required to

completely determine the positions of all parts of a system at any instant of time .The spring mass systemshown below is a prime example of a single degree of freedom system. The positions of all parts of the

system at any given time can be determined by a single co-ordinate, which is x. In case of the pendulum,

the motion of the simple pendulum can be stated in terms of the angle, theta, or in terms of the Cartesian

co-ordinates x and y. If the co-ordinates x and y are used to describe the motion, it must be recognized

that these co-ordinates are not independent. They are related to each other through the relation X 2 + Y2=

L2. A system can be supposedly have from one to infinite degrees of freedom.


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Figure 1- Single Degree-of-Freedom (SDOF) systems

Linear Variable Differential Transformer:

This is a tool used for the sake of measuring linear displacement. It consists primarily of an electrical

transformer. The linear variable differential transformer(LVDT) (also called just a differential

transformer) is a type of electrical transformer used for measuring linear displacement (position).

According to Wikipedia, linear variable differential transformer (LVDT in short) serves as a means to

measure continuous displacement. The LVDT consists of three solenoidal coils that are placed around a

tube in an end-to-end pattern. It has a centre coil which is the primary coil and there are two external coils

at the top and bottom which are considered to be secondary. The object whose position is to be measured

is attached to a cylindrical ferromagnetic core which slides along the axis of the tube. The system relieson alternating current to derive the primary coil resulting in the induction of a voltage in the secondary

coils. The frequency of typical LVDTs range from 1 to 10KHz.

Figure 2 :Simple form of linear variable differential transformer (LVDT)


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Figure 3: Various types of linear variable differential transformer (LVDT)

LVDTs have a variety of applications and are commonly used for position feedback in servomechanism,

and for automated measurement in machine tools and many other industrial and scientific applications.

3.2 THEORETICAL BACKGROUND

For an unforced damped SDOF system, thegeneral equation of motionbecomes,

with the initial conditions,

This equation of motion is asecond order,homogeneous,ordinary differential equation (ODE). If all

parameters (mass, spring stiffness, and viscous damping) are constants, the ODE becomes alinear ODEwith constant coefficients and can be solved by the Characteristic Equation method. The characteristicequation for this problem is,

Using our knowledge of differentials, it can be seen that the equation of motion in that we have in our

hand is a homogenous, second order and ordinary differential equation. If we have constants for all thedifferent parameters such as stiffness, viscous damping and mass, this equation becomes a linear ODE

with constant coefficients. Henceforth we can use solve this using the Characteristic Equation.

Which determines the 2 independent roots for the damped vibration problem. The roots to the

characteristic equation fall into one of the following 3 cases:

1.If < 0, the system can be said to be underdamped. The roots are found to be

complex conjugates which corresponds to oscillatory motionwith an exponential
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decayin amplitude.

2.If = 0, it is said the system is critically damped., corresponding tosimple

decaying motionwith at most one overshootof the system's resting position.

3.If > 0, the system is termedoverdamped.Solving the characteristic question

gives repeated roots and the roots of the characteristic equation are purely real and

distinct, corresponding to simpleexponentially decayingmotion.

To simplify the solutions coming up, we define the critical dampingcc, the damping ratio, and the

damped vibration frequencydas,

Where n which is the natural frequecnyis given by,

Note thatdwill equalnwhen the damping of the system is zero (undamped system).

Below are further explanation of underdamped system which is one of the most common type of system

among the three types of system.

Underdamped Systems

When < 0 (equivalent to < 1 or < ), the characteristic equation has a pair of

complex conjugate roots. The displacement solution for this kind of system is,

An alternate but equivalent solution is given by,
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The displacement plot of an underdamped system would appear as,

Figure 4: The displacement plot of an underdamped system

Note that the displacement amplitude decays exponentially (i.e. the natural logarithm of the amplitude

ratio for any two displacements separated in time by a constant ratio is a constant; long-winded!),

where is the period of the damped vibration.

From theory it is known that the most basic vibration system is a one degree of freedom vibration system.

A simple example of this system would be a system consisting of a rigid mass body connected to a spring.


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The mass can only move along the length of the spring therefore its motion is restricted to one directiononly thereby giving it a single degree of freedom.

Figure 5: The equivalent model of the cantilever system used in the experiment

Making use of Newtons second law of motion which states that force is equal to mass times accelerationwe can visualize vibration as according to the following diagram:

Figure 6: Free body diagram of SDOF

Treating the cantilever beam according to above diagram, we can simulate an equation for its motion:

In our experiment no external forces act on the cantilever beam. Therefore f(t) is zero and we substitutethat the differentials for acceleration and velocity in the equation to get:

2

2 0

d x dxm c kx

dt dt

We assume that theres no damping, and therefore the value of c is zero. The figue below shows an

undamped system.

Figure 7: Undamped system


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Natural frequency is the frequency or frequencies at which an object tends to vibrate with amplitudes ofthe vibrations when hit, struck, plucked, strummed or somehow disturbed. Each types degree of freedomhas their distinct natural frequency expressed by o. It also can be undamped or damped and this depends

on whether the system has significant damping. The equation of undamped natural frequency is as below:

o

k

m

For damped natural frequency, d, in mathematical equation, it is equal to product of undamped naturalfrequency with square root of one minus the square of damping ratio.

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d o

Damping ratio, , shows how the oscillations in a system decay after a disturbance. It is a dimensionlessmeasure. This can be expressed by mathematical formula where the level of damping in a system relative

to critical damping.

cr

c

c

Damping can be classified into underdamped, overdamped and critical damping. In this experiment, we

only consider the critical damping. It only occurs when the damping ration is one (=1) where the systemwill fail to overshoot and will not make a complete single oscillation. In critical damping system, it doesnot oscillate and returns to its equilibrium position without any oscillating. It can be expressed as:

2cr

c km

is the decay rate (rad/s). It is the measure of how fast the amplitude of oscillation decay to zero with

the change of time. The equations of decay rate are expressed as:

o

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1d


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The time taken for one complete vibration with returns to its original position and velocity for theoscillator is the period of vibration, T. For this experiment, T Dis the time taken to complete the

oscillation.

2

D

d

T

Figure 8-Graphical representation of displacement against time

The amplitudes x are defined as:

x = Ae-t

logex = logeA + logee-t

logex = logeAt

Figure 9: Graph of logex versus t


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We can determine from the gradient of the graph, hence can be calculated.

4.0 RESULT

4.1 Part I: Stiffness and Mass

Table 1: Load and Displacement

Force (N) Displacement, x (mm)

(i)

Displacement, x

(mm)

(ii)

Displacement, x

(mm)

(iii)

Mean

(mm)

0 0.01 0.01 0.01 0.01

2 0.15 0.16 0.16 0.156

4 0.30 0.29 0.29 0.293

6 0.46 0.46 0.45 0.456

8 0.61 0.61 0.61 0.61

10 0.77 0.77 0.76 0.76


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Graph 1: Force, F (N) against Displacement, x (mm)

We obtain, stiffness, k, from the graphs equation. However we make sure to calculate the effective mass

and undamped natural frequency, o

Based on the graph 1, the equation of the graph is:

13.263x - 0.0511

Stiffness, k = gradient of the Graph of Force versus Displacement

= 13.263 N/mm

= 13263 N/m

Effective Mass, m = 0.4077 kg

Undamped Natural Frequency, 0,1

y = -0.7812x + 0.0223

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Force(N

)

Distance (mm)


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= 180.36 rad/s

= 28.705 Hz

4.2 Part II: Transient Response

Table 2: Transient Response Data

Displacement, Y (mm) ln Y TD(s) Time (s)

0.9434 -0.05826 0.000 0.000

0.7878 -0.2385 0.065 0.5580

0.6986 --0.3586 0.065 0.6230

0.6113 -0.4922 0.065 0.6880

0.5355 -0.6245 0.065 0.7550

0.4563 -0.7846 0.065 0.8200

0.3859 -0.9521 0.065 0.8830

0.3759 -0.978 0.065 0.9480

0.3126 -1.163 0.065 1.0130

0.3044 -1.1894 0.065 1.5900


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Graph 2: lnY against Time

For the calculation part, it is based on the graph 2 above

1. Decay Rate, From the graph of ln Y versus time (s), the equation of the graph is:

Y = -1.233x - 0.2218Therefore, the gradient of the graph is -0.8253 s

-1.

Decay Rate,

= - (gradient of the Graph of ln Y versus Time)

= 1.233 s-1

2. Damped Natural Frequency, d

Period, TD

= 0.065 s

y = -1.2333x + 0.2218

-2

-1.5

-1

-0.5

0

0.5

0 0.5 1 1.5 2

Ln

Y

Time (s)

Y-Values

Linear (Y-Values)


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= 96.66 rad/s

3. Damping Ratio,

= 4. Undamped Natural Frequency, o

0,2 = 96.67

= 15.38Hz

5. Damping Coefficient, c

Critical damping, ccr =

= = 147.06 Ns/mc = ccr

= (x 10-3)(147.06 Ns/m)= 1.0892 Ns/m

5.0 DISCUSSION

From the calculations done based on the data obtained from part I and part II, we found that the

undammed natural frequency in part I is 180.36 rad/s; and for part II, it is 96.67rad/s. By using the

equation shown below, it was found that percentage of error which is 46.396%. The percentage difference

is of a considerable amount therefore there are two different possibilities. Either the experiment was done

incorrectly or the data obtained was incorrect resulting in the great deviations.

Percentage of Difference =100

1.

2.1.

o

oo

= 10036.180

68.9636.180

= 46.396 %


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Also the graphs obtained are skewed in many different ways even though we are supposed to obtain astraight lines. In the first graph we can see a fluctuation pattern that almost represents an exponential

graph. and almost similar to an exponential graph in the 2ndpart of the experiment. There can be some

possible errors in the experiment and we can address them below.

To start off, we completely ignored the mass of the cantilever beam in the equations.. But themass of the cantilever beam cannot be fully neglected as it has substantial mass compared to themass of the hangar. We can solve this issue by including the mass of the cantilever beam in the

equation. However, if we want to make use of the current formula we have to use a much largerbeam mass so that we can completely ignore the mass of the cantilever beam.

We might have some issues with our LVDT probe. In case it wasnt pressed completely, the

displacement readings would not be up to the mark. We must ensure that the LVDT undergoes thehighest possible compression so that its set to zero and gives significant displacement whenweights are placed on the hanging mass. But practically its impossible to give an LVDT

maximum compression as our sensitivity is quite low compared to that of the probe. Thereforewhen we operate the probe and release it after compression it might deviate slightly.

We assume the damping to be constant at all times however in reality, it varies all the time. Its not

practically possible to halt a moving system completely just by having damping.

. In this experiment without having external damping device, the damping of cantilever beam isprobably because of dissipation of energy due to the strain applied to it via the uploading of thehangar. Other external force such as wind and accidental touches from the observers might have

indirectlycause some small changes in damping constant of the system.

The first part of our experiment was done in a static system whereas the second part of our

experiment can be considered to be dynamic. So we can safely assume that the second part of ourexperiment is more accurate as in reality all systems are dynamic. The atoms or molecules of a

substance will be vibrating all the time as it subjected to unseen external forces such as air fromsurrounding, thus it is hard to imagine that a system can be set at static unless it is in vacuum.Dueto this factor, there will be percentage of difference between data from the two parts of the

experiment.

We can assume the weight and the hanger to be completely rigid as rigid bodies are absolutelyunmovable. But as soon as we put the hangar on the beam we ad an external force to the entire

system. Also the beam can be said to be undergoing internal vibration which we cannot see withthe naked eyes. We also have to take into consideration the oscillation and vibrations to other parts

of equipment and this will dissipate energy to surroundings which caused energy loss and causingthe damping to occur faster due to the energy loss.


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6.0 CONCLUSION

In conclusion for the vibrating cantilever system, the undamped natural frequency is 180.36 rad/s asobtained from mass and stiffness information in the first part of the experiment whereas for the 2

ndpart

the undamped natural frequency is 96.67 rad/s as obtained from transient response analysis. They havepercentage difference of 46.936%.Tthe damping ratio, obtained for this experiment would be

. Lastly the damped natural frequency d for this experiment is 96.66 rad/s.7.0 REFERENCES

1.

Rao, S.S. (1986),Mechanical Vibrations, Cambridge, MA: Addison-Wesley.

2. Laboratory worksheet

3. Degree of freedom: http://en.wikipedia.org/wiki/Degrees_of_freedom_(mechanics)

4.

Vibration ;http://en.wikipedia.org/wiki/Vibration

5. Linear Variable Differential Transformer:

http://en.wikipedia.org/wiki/Linear_variable_differential_transformer
http://en.wikipedia.org/wiki/Degrees_of_freedom_(mechanics)http://en.wikipedia.org/wiki/Degrees_of_freedom_(mechanics)http://en.wikipedia.org/wiki/Vibrationhttp://en.wikipedia.org/wiki/Vibrationhttp://en.wikipedia.org/wiki/Vibrationhttp://en.wikipedia.org/wiki/Linear_variable_differential_transformerhttp://en.wikipedia.org/wiki/Linear_variable_differential_transformerhttp://en.wikipedia.org/wiki/Linear_variable_differential_transformerhttp://en.wikipedia.org/wiki/Vibrationhttp://en.wikipedia.org/wiki/Degrees_of_freedom_(mechanics)

one degree of freedom 2014-2015

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