NATURAL FREQUENCY AND DAMPING OF SINGLE DEGREE OF FREEDOM

SYSTEMS

Laboratory Report - 01

Name :- M.K.S.Liayanarachchi Curtin ID :- 18373350 SLIIT ID :- EN 13522148 Date of submission :- 24th April 2015

Fundamentals of Mechanical Vibrations

(MCEN3005)

[i]

Executive Summary The objectives of this experiment is to validate two theoretical expression,

1. The mass of the spring can be neglected when calculating natural frequency oscillation,

if the mass of the spring is small when compared with the oscillating mass.

2. The damped natural oscillation frequency is virtually same as undamped natural

oscillation frequency, when the damping ratio is small.

This experiment can be divided into three section. The stiffness of the spring was calculated in

the first part of the experiment. This enabled the use of an accurate value for stiffness of spring

to be used in the latter part of the experiment. Second part of this experiment is to validate the

accuracy of the first expression regarding the mass of the spring and frequency of oscillation.

This was done by directly measuring the period of oscillation of spring mass system for

different applied loads. In third section of this experiment a damper was used to validate the

second expression. The damping ratio of the system was calculated during this section.

The summary of results obtain during first two section of the experiment is given bellow,

Table 1 Summary of results

Theoretical stiffness of spring = 3295.15 Nm-1 Experimental stiffness of spring = 3333.33 Nm-1

Total mass of weights on

hanger mi (Kg)

Experimental Period of oscillations, T (sec)

Theoretical Period of oscillations without considering

spring mass, T (sec)

Theoretical Period of oscillations with considering

spring mass, T (sec)

1.02 0.3018 0.121088 0.121232

1.53 0.324 0.143884 0.144005

2.04 0.378286 0.163532 0.163638

2.24 0.370923 0.170621 0.170723

2.55 0.418167 0.181061 0.181157

3.06 0.4576 0.197036 0.197124

Following conclusion were derived for the experiment.

1. The stiffness of the spring can be accurately calculated using equation (18)

2. The mass of the spring can be neglected when calculating natural frequency oscillation,

if the ratio of ms/mi is less than 0.06.

3. If the damping ratio is less than 0.2, the natural frequency of damped oscillation is

virtually identical to undamped natural frequency.

[ii]

Table of Contents Executive Summary ................................................................................................................................................. i

List of Figures ......................................................................................................................................................... iii

List of tables ........................................................................................................................................................... iv

1.0 Introduction and Aims ...................................................................................................................................... 1

2.0 Theory ............................................................................................................................................................... 3

2.1 Calculating equivalent mass for a one degree spring mass system when the mass of the spring is not

negligible. ........................................................................................................................................................... 3

2.2 Viscous Damping and its effects .................................................................................................................. 4

3.0 Procedure ......................................................................................................................................................... 6

4.0 Results ............................................................................................................................................................. 7`

4.1 Measuring the Stiffness of a spring .............................................................................................................. 7

4.2 Natural Frequency of Oscillation (without and with lumped mass correction) ............................................ 7

5.0 Calculations ....................................................................................................................................................... 8

5.1 Theoretical calculation of stiffness of the spring .......................................................................................... 8

5.2 Experimentally calculating stiffness of the spring ....................................................................................... 9

5.3 Theoretically predicting the period of oscillations with and without considering the spring mass ............ 10

5.4 Graph plotted between time periods versus (mi +mh)^0.5 .......................................................................... 11

5.5 Graph plotted between time periods versus (mi +mh+ ms/3)^0.5 ............................................................... 12

5.6 Graph plotted between time periods versus square root mass for both heavy and light spring .................. 13

6.0 Discussion ....................................................................................................................................................... 14

7.0 Conclusions ..................................................................................................................................................... 16

[iii]

List of Figures

Figure 1 Simple one degree freedom spring mass system ..................................................................... 1

Figure 2 Simple one degrees freedom spring mass system with a heavy spring ................................... 3

Figure 3 Distance to viscous damper and spring for the pivot O ............................................................ 4

Figure 4 Plotted graph between forces vs. deflection ............................................................................ 9

Figure 5 Plotted graph between time period and total mass ............................................................... 11

Figure 6 Plotted graph between time period and total mass ............................................................... 12

[iv]

List of tables

Table 1 Summary of results ..................................................................................................................... i

Table 2 Recorded data on applied load and spring deflection ............................................................... 7

Table 3 Recorded data on total mass and period of oscillation ............................................................. 7

Table 4 Comparison on result obtained on period of oscillation ......................................................... 10

[1]

1.0 Introduction and Aims This experiment is designed to confirm the accuracy two of the theoretical expressions. First

equation is used when calculating oscillation frequency simple spring-mass oscillator where

spring is not massless. The second equation that would be verified during this laboratory is that

damped natural oscillation frequency is virtually the same as the undamped natural oscillation

frequency when damping ratios less than 0.1 or when there is no deliberately designed damper.

The fundamentals of mechanical Vibrations is an important aspect of mechanical engineering.

Mechanical vibration is strongly connected to classical mechanics, solid mechanics and fluid

dynamics. In most of engineering applications mechanical vibration is significant and proper

calculation must be conducted to understand the dynamic behavior of a system. Vibration may

results in the failure of machines or their critical components. The effects of vibration on

mechanical systems depends on frequency, magnitude of displacement, acceleration and total

duration of the vibration.

Galileo discover the relationship between the length of a pendulum and its frequency. He also

observed the resonance occurring in two bodies with same natural frequency. Just like the mass

pendulum, frequency of oscillation in a one degree freedom simple spring mass system can be

easily calculated using the equation given below.

=1

2

(01)

Spring (K)

X

Mass (M)

Figure 1 Simple one degree freedom spring mass system

[2]

Natural frequency of oscillation in a one degree freedom simple spring mass system can be

calculated by applying newtons second law to the spring mass system and deriving a value for

. This is done using the lamp parameter assumption. There it is assumed that the mass of the

spring is negligible when compared with the oscillating mass. But there may be complicated

systems this assumption is invalid. It could be shown that for those types of systems, the total

mass of the system can be calculated by adding one third of the mass of the spring to the

oscillating mass.

=

1

2

+3

(02)

Generally amplitude of natural vibration decays with time. This is due to the effects of

damping. The decaying of amplitude happens since the initial energy contained in the system

has been converted in to form other than kinetic or strain energy. A part of the initial energy

has been converted into other form- usually heat by the damper. Damping mechanism can take

several forms. Mainly,

1) Coulomb Damping

2) Hysteretic Damping

3) Viscous Damping

Viscous damping can be found in many mechanical systems. Viscous drag or viscous damping

force can be represented as,

= (03)

Natural frequency of a damped system, compared to the undammed natural frequency, is

reduced by a factor.

= 1

2 (04)

But when damping ratio is small, the ratio between damped and undamped natural frequencies

is almost equal to 0ne. A typical range for the damping ratio for an engineering application is

around 0.002-0.05. Values higher than 0.2 is archived by deliberately designing dampers in the

system.

[3]

2.0 Theory 2.1 Calculating equivalent mass for a one degree spring mass system when the mass of the

spring is not negligible.

Kinetic energy of the body only =1

22

(05)

Kinetic energy of the element of the spring =1

2(

) 2

(06)

From triangles , 2 = (

)

2

(07)

kinetic energy of the whole spring =1

2

2

2 2

0

(08)

kinetic energy of the whole spring =1

2

3

(2) (09)

By equating the maximum strain energy and kinetic energy, natural frequency of oscillation can be

calculated,

=

+3

(10)

Heavy Spring (K)

X

Mass (M)

c

C

dc

v

V

Figure 2 Simple one degrees freedom spring mass system with a heavy spring

[4]

2.2 Viscous Damping and its effects

Figure 3 Distance to viscous damper and spring for the pivot O

For small angel ,

Vertical deflection at the point which the spring is attached, () =

Equivalent torsional stiffness (

) = k2

Similarly, the tortional damping coeffient (

) = q2

Applying newtons II law about the pivot at O: 2 2 =

(11)

+

2

+

2

(12)

Comparing this with the standard form

2 =24

42 =

2

(13)

Frequency of damped vibration would be

= 1

2=

1

2

2

(1

24

42) (14)

[5]

From the logarithmic decrement,

=

1

(

22

) =2

1

2 1

(15)

By substitution

=

()

2

[

2

(22

)] 2

+ 1

(16)

And

= 1 24

42= 1 2 (17)

[6]

3.0 Procedure The objective of this experiment is to measure effects of spring mass and damping on simple

one-degree spring mass systems. Stiffness of the spring is used in calculating the natural

frequency. So first part of this laboratory is to measure the stiffness of the spring. This can be

determined by measuring its stretch due to a known load. To improve the accuracy of

calculating the mass of the spring a graph was used to generate the average value. During the

experiment deflection of the spring was measured for appropriate mass increment. Then a

graph was drawn using load as the abscissa and deflection as Ordinate. The slope of that graph

is equal to 1/k (m/N). So the spring stiffness would be equal to 1/slope (N/m).

After calculating spring stiffness, the mass of the spring and hanger was measured. The spring

and hanger was the reinstalled to the apparatus. To calculate the natural frequency of

oscillation, a vertical oscillation with an amplitude about 10mm was induced to the apparatus

and recorded the time it takes to complete a specific number of cycles. After tabulating the data

two graphs were drawn. The first graph is drawn between T versus square root of (mi + mh).

The second graph was drawn between T versus square root of (mi + mh + ms/3). These graphs

would enable comparison between natural frequencies calculate with considering the mass of

the spring and neglecting the mass of the spring.

After measuring the effects of spring mass for natural frequency, another part of this

experiment must be carried out to calculate the effects of damping on natural frequency. First

measure a (distance to damper form pivot) and c (distance to spring for pivot) to the nearest

mm. Then drain the oil from dashpot and measure the undamped natural frequency by

measuring the time it takes to complete 40 oscillations. After that calculate Io from equation

(??). Next refill the dashpot and enable the chart recorder function. Displace the free end of

oscillating arm about 35mm and release it. Record a time trace of amplitude y versus time

which contains about 20 cycles. From these data calculate damping coefficient q. after that

determine the frequency of damped free oscillation with the use of digital counter.

[7]

4.0 Results

Mass of spring =8.82g

Mass of hanger =218.16g

Spring length =31.38g

Wire diameter =1.30mm

No of coils =25

Outer coil diameter =10.96mm

Inner coil diameter =8.36mm

4.1 Measuring the Stiffness of a spring

Table 2 Recorded data on applied load and spring deflection

i Total suspended

mass, mi (kg)

Total force,

Fi=mig (N)

Scale reading,

i (mm) Deflection,

i-o (mm) Increment in

Deflection (mm)

0 0.218 2.14 0 0 0

1 0.728 7.14 2 2 2

2 1.238 12.14 8 8 6

3 1.747 17.14 14 14 6

4 2.257 22.14 23 23 7

5 2.767 27.14 30 30 7

6 3.276 32.14 37 37 7

4.2 Natural Frequency of Oscillation (without and with lumped mass correction)

Table 3 Recorded data on total mass and period of oscillation

Total mass of weights on

hanger mi (Kg)

Number of cycles, N

Mean time for N oscillations, t

(sec)

Period of oscillations,

T (sec)

+

(Kg)-0.5 + +

3

.(Kg)-0.5

1.02 100 30.18 0.3018 1.112654 1.113975

1.53 80 25.92 0.324 1.322120 1.323231

2.04 70 26.48 0.378286 1.502664 1.503642

2.24 65 24.11 0.370923 1.567801 1.568738

2.55 60 25.09 0.418167 1.663731 1.664614

3.06 50 22.88 0.4576 1.810525 1.811337

[8]

5.0 Calculations

5.1 Theoretical calculation of stiffness of the spring

From strength of material text will show that,

=

83

4 (18)

Where, = stretch

P = axial load

D = mean coil diameter

n = number of coils

G = shear modulus of wire

d = wire diameter

The stiffness of the spring can be calculated from following equation

=

83

4=

1

(18)

=4

83

=(208109)(0.0013)4

8(9.66)325

= 3295.151 1

[9]

5.2 Experimentally calculating stiffness of the spring

Figure 4 Plotted graph between forces vs. deflection

Calculating the stiffness of the spring using the slope of the graph,

() =1

= 0.0003

= 3333.33 1

y = 0.0003x

-0.001

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

-2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

De

fle

ctio

n (

m)

Force (N)

Deflection vs. Load

[10]

5.3 Theoretically predicting the period of oscillations with and without considering the spring

mass

() = (2

) +

() = (2

333.333) +

() = (0.1088280163) +

, () = (2

) + +

3

() = (2

3333.333) + +

3

() = (0.1088280163) + +3

Table 4 Comparison on result obtained on period of oscillation

Total mass of weights on

hanger mi (Kg)

Experimental Period of oscillations, T (sec)

Theoretical Period of oscillations without considering spring mass,

T (sec)

Theoretical Period of oscillations with considering

spring mass, T (sec)

1.02 0.3018 0.121088 0.121232

1.53 0.324 0.143884 0.144005

2.04 0.378286 0.163532 0.163638

2.24 0.370923 0.170621 0.170723

2.55 0.418167 0.181061 0.181157

3.06 0.4576 0.197036 0.197124

[11]

5.4 Graph plotted between time periods versus (mi +mh)^0.5

Figure 5 Plotted graph between time period and total mass

y = 3.9934x

-0.5

0

0.5

1

1.5

2

-0.1 0 0.1 0.2 0.3 0.4 0.5

Time Period vs. +

Tim

e P

erio

d (

s)

+ (1

2)

[12]

5.5 Graph plotted between time periods versus (mi +mh+ ms/3)^0.5

Figure 6 Plotted graph between time period and total mass

y = 3.996x

-0.5

0

0.5

1

1.5

2

-0.1 0 0.1 0.2 0.3 0.4 0.5

.

Time Period vs.

+ +

3

1

2

Tim

e P

erio

d (

s)

+ +

3(

1

2)

[13]

5.6 Graph plotted between time periods versus square root mass for both heavy and light

spring

Figure 7plotted graph of time period vs total mass

This graph indicated that there is no significant difference for the time period of heavy spring

and light spring for this experiment.

0.25

0.3

0.35

0.4

0.45

0.5

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

TIM

E P

ERIO

D (

S)

M

Time Period vs. M Heavy Spring Light Spring

[14]

6.0 Discussion First part of this experiment is about calculating the spring stiffness. The spring stiffness was

calculated using a graph. The graph was drawn using load as the abscissa and deflection as

Ordinate. The slope of that graph is equal to 1/k (m/N). So the spring stiffness would be equal

to 1/slope (N/m). The theoretical shape of the graph is y=mx, so that the origin (0, 0) is a point

in graph that is known without any uncertainty.

Spring stiffness can be also calculated using the relationship between load and stretch for a

close-coiled helical spring. The spring stiffness is equal P/. Measurements on mean coil

diameter, wire diameter, shear modulus of the spring material and number of coils are needed

calculate spring stiffness using this formula. The difference between spring stiffness is due to

errors in measurement, non-uniformity of the spring material, errors in applied load.

Data point that were collected to draw the graph of force verse deflection is not in straight line,

this due to errors on measuring and non-linearity of the spring. There is a small difference

between spring stiffness calculated from the equation and stiffness calculated form the graph.

The difference between experiment and theoretical values of spring stiffness is about 38Nm-1

and this is due errors in measuring the diameter of spring, difference actual youngs modulus

and effects of helix angle of the spring

When measuring period of oscillation in the system is done by measuring the mean time for a

large number of oscillations and dividing it by number of cycles. When selecting the number

of cycles it must not be very small so that the error when measuring the time is high and number

of cycle should not be very large so that the vibration amplitude decays significantly. Doing

the experiment for the same weight three or more times and taking the average of the period of

oscillation lowers the uncertainty errors. When taking the average reject any obvious outlying

measurements to improve the accuracy of the period.

Since the mass of the spring that were used in this experiment is negligible when compared

with the oscillating mass. There are no significant difference on natural oscillation frequency

between with and without the spring mass. The natural frequency of oscillation is predicted

theatrically will be more accurate when the mass of the spring is considered. When the ratio

between ms/mi is equal to 0.06 the different between natural frequencies obtained considering

with and without the spring mass is less than 1%. So if the ratio between ms/mi is less than

0.06, then the mass of the spring can easily be neglected.

[15]

If the mass of the spring used in this experiment is large such as around 2kg, the effect of the

mass of the spring could be observed easily. But due to resource limitation this spring was the

heaviest spring that could be used with the apparatus.

There is more than 50% difference between theoretical periods of oscillation and experimental

period of oscillation is mainly due to error of measuring the mean time for N number of

oscillations.

Damping is important in engineering applications since it prevents systems oscillating at high

amplitudes at resonance. Natural frequency of oscillation changes when there is significant

amount of damping present in the system. But this effect is negligible if the damping ratio is

less than 0.2.

During the practical the oil from the dashpot is removed to get the undamped natural frequency

of the system. This method is more accurate than removing the damper from the apparatus

when calculating natural frequency. This is due to the fact if the dashpot was removed from

the system, the mass of the system will change thus changes the natural frequency.

[16]

7.0 Conclusions From looking at the results obtained from the first part of the experiment following conclusion

can be derived. The experimental and theoretical values for the spring stiffness is almost

identical, so a conclusion could be derived that the equation that were used to predict the spring

stiffness is accurate. And also another conclusion can be derived about the relationship between

applied load and deflection. From the data obtained it is safe to assume that there is a linear

relationship between applied load and deflection.

Form the second part of the experiment following conclusions can be derived. By looking at

the collected data from the experiment it is safe to assume that if the mass of the spring is small

when compared with the oscillating mass, then the mass of the spring can be neglected when

calculating the natural frequency of oscillation. So if the ratio between ms/mi is less than 0.06,

then the mass of the spring can easily be neglected when calculating the natural frequency.

Another conclusion that can be derived for this part of the experiment is that there is a linear

relationship between the period of oscillation and square root of the total applied load on the

spring.