free & forced vibration.docx

Kuwait University

College of Engineering & Petroleum

Mechanical Engineering Department

Lab C1

ME-474

Experiment # 1

Free and Forced Vibration with Damping for a 1-DOF System

Conducted On: September 7th , 2015

By

Asilah Alshatti 209117171

Salma Al-Fahhad 2111113048

Taiba Al-Baloul 210111776

April 18, 2023

On my honor, I pledge that this work of mine does not violate the University provisions

on academic misconduct. By signing below, I certify that I understand the University

Policies on academic misconduct and that when an act of academic misconduct is

committed, all parties involved are in violation.

Signature:

Abstract

Free undamped and damped vibration:

This experiment was conducted in order to study undamped free vibration, damped

vibration, and forced vibration. For the purpose of studying the undamped free vibration,

a setup with no damping current was used. For different masses, the motion of the

vibrating mass was recorded on a strip of teledeltos paper. Afterwards, the natural

frequencies corresponding to the masses were calculated, and the squared values of the

natural frequencies were plotted against the reciprocal of the masses. The slope of the

graph represents the value of the experimental spring constant, which equals (54.19 N/m).

This value was compared to the theoretical value of the spring constant, which equals (54

N/m). The error between the theoretical value and the experimental value of the spring

constant was found to be (% ¿.

In order to study the damped vibration, the same setup was used with an electromagnet

to provide a damping current. For different damping currents, the damping ratios were

calculated, and then plotted against the damping current. In addition, the values of the

damping coefficients were calculated, and then plotted against the damping currents. It

was concluded that the increase in the mass results in a decrease in the damping ratios,

but provides a higher damping coefficient.

Forced vibration:

In order to study the forced vibration, a motor was used to oscillate the setup-base, that

resulted in a forced vibration of the mass which is called an oscillation motion. The

purpose of the experiment was to get the theoretical and the experimental values of the

transmissibility ratio. The theoretical and the experimental transmissibility ratios were

then plotted against the frequency ratio. After that, the half-power-point method was used

in order to plot the transmissibility ratio against the frequency ratio for the case of

applying a damping current of (0.7 A) to the system. The damping ratio corresponding to

this damping current was found to be equal to ( ) while the experimental damping ratio

was found to be equal to ( ).It was concluded that the increase in the damping current

results in a decrease in the transmissibility ratio. In addition, it was found that the

theoretical values of the transmissibility ratios were greater than the experimental values.

i

Table of Contents

Abstract.................................................................................................................................i

Table of Contents.................................................................................................................ii

List of Figures and Tables..................................................................................................iii

Introduction:.........................................................................................................................5

Tabulated Data:..................................................................................................................14

ii

List of Figures and Tables

Figure 1: The experimental setup for 1-DOF.....................................................................11

Table 1: Measured data resulting from the variation of time period with vibrating mass.. .1

Table 2: Measured data of the variation of damping ratio with damping current (m=1.35

kg)........................................................................................................................................1

Table 3: Measured data of the variation of damping ratio with damping current (m=4.85

kg)........................................................................................................................................1

Table 4: Measured data of the damping ratio for a value of damping current (I=0.7 A).....1




Table 8: Calculated data of the variation of time period with vibrating mass.....................1

Table 9: Calculated data of the variation of damping ratio with damping current (m=1.35

kg)........................................................................................................................................1

Table 10: Calculated data of the variation of damping ratio with damping current (m=4.85

kg)........................................................................................................................................1

Table 11: Calculated data of the damping ratio for a value of damping current (I=0.7 A). 1




iii

Nomenclature

A, B constants determined by initial conditions.

c damping coefficient.

F(t) external impressed time dependent force.

k spring stiffness.

m vibrating mass.

r frequency ratio.

t time.

wn natural frequency of the system.

wd damped natural frequency of the system.

x displacement in x-direction at time t.

y displacement of the support in x-direction at time t.

x velocity in x-direction at time t .

x acceleration in x-direction at time t .

X steady state amplitude of forced vibration in x-direction.

Y amplitude of frame oscillation in x-direction.

x damping ratio.

4

XY

amplitude ratio (Amplification Factor or Transmissibility )

( xn

xn+1) ratio of successive maxima

(−xnxn+ 1

2) ratio of successive maxima to minima

Introduction:

Vibration dampers are used in many application where dissipation of energy is

required. Mostly, mechanical energy is converted into heat. When the dissipation is due

to internal friction or hysteresis characteristics of to the molecular structure, it is called

material damping. When the dissipation is generated by friction, snapping, rubbing,

slapping or impacting at the joints and interfaces of structural assemblies, it is called

structural damping. Dampers are used for increasing life, reducing noise, and preventing

premature failures. Dampers have become more popular recently for vibration control of

structures, because of their safe, effective and economical design. In addition, another

application is in cars where the damper protects components from premature wear,

therefore increasing life. It also isolates vibration and oscillation noise so that it is not

transmitted to the vehicle structure. Many other uses of dampers exist in the industry

where machinery is involved. Vibration is divided into:

(1)Free Vibration:

The free vibration results from an initial impact energy that is changes continually

from potential energy to kinetic energy. If a mechanical system was displaced by an

initial impact energy from its equilibrium position and then released, the restoring force

will return the system towards its equilibrium position. This initial impact energy can be

an initial velocity, or an initial displacement through an applied force. In this type of

vibration, the system vibrates at its natural frequency. However, due to various reasons,

some mechanical energy will be dissipated during each cycle of vibration, and this effect

is called damping.

In the first part of the experiment, the system is allowed to vibrate freely, without

damping, in order to study the effect of varying the mass on the natural frequency. In the

second part of the experiment, the mass is fixed while the current damping is increased in

order to study its effect on the system. After that, the readings are collected and used to

determine the damping coefficient.

5

The logarithmic decrement is the natural log of the ratio of the amplitudes of any two

successive peaks. This method states that the displacement of an underdamped system is

a sinusoidal oscillation with decaying amplitude. This method method is used in order to

find the damping ratio of the system.

(2)Forced Vibration:

When an external excitation input is added to the system, the resulting vibration is

called forced vibration. In the third part of the experiment, the setup-base is oscillated

using a motor that oscillates at a set angular speed in order to cause an oscillation motion.

In this case, the transmissibility ratio (X/Y), which is the ratio of the amplitude of the

response X to that of the base Y, is obtained.

To estimate the damping ratio from the frequency domain, the half-power bandwidth

method is used. The half-power point damping method is defined as the ratio of the

frequency range between the two half power points to the natural frequency at this mode.

In this method, the amplitude of the frequency response function of the system is

obtained first. Corresponding to each natural frequency, there is a peak in the amplitude

of the frequency response function. The half-power points are the frequencies at which

the value of response amplitude drops 3 decibels, or 70.7%, in relation to the amplitude

of the response at the center. The more the damping, the more the frequency range

between these two points.

6

Objectives:

The objectives of this experiment were:

1) To investigate the variation of the undamped natural frequency with the vibrating

mass.

2) To study damped free vibration and to determine the damping ratio.

3) To study the forced vibration excited by an oscillating support.

7

Theoretical Background:

Mathematical Model:

In an ideal, 1-DOF system a rigid mass, m, is connected to a fixed, rigid support by a

light spring of stiffness k and a viscous damper with damping coefficient c, and is

constrained to move in the x-direction only. An external force F(t) acts on the mass.

The resulting equation of motion of the system is:

Case 1: Free Vibration

If a system, after an initial disturbance, is left to vibrate on its own, the ensuing

vibration is known as free vibration. When all externally applied forces F(t) are zero

Equation 1.1 becomes:

In the case of un-damped free vibration,

with a circular frequency, wn

where , wn is the natural frequency of the system and Equation 1.4 is a second order

homogeneous, linear differential equation. It has a solution of the form x = A sin wn t, or

x = A cos wn t. Introducing the damping ratio

8

m x..

+c x.

+kx=F ( t ) (1 .1 )

m x..

+c x.

+kx=0 (1 .2 )

m x..

+kx=0 (1. 3)

x. .

+ωn2x=0 (1 . 4 )

The Equation 1.2 becomes:

The solution of Equation 1.5 depends on the value of x thus:

for x <1

for x =1

and for x >1

where A and B are constants determined by the initial conditions. If x = X0 and

for x <1

for x =1

and for x >1

Note that for x <1 the free motion is oscillatory with damped natural frequency

9

ξ= c2√m×k

(1 .4a )

x=e−ξωn t¿¿

x. .

+2ξωn x.

+ωn2x=0 (1 .5 )

x=e−ωn t {A+Bt } (1 . 7)

x=e−ξωn t¿¿

xX0

=e−ξωn t ¿¿

xX0

=e−ωn t¿¿

xX0

=e−ξωn t ¿¿

and the ratio of successive maxima for x <1, is given by:

and the ratio of successive maxima to minima for x <1, is given by:

where , d is called the logarithmic decrement. One can find the minimum velocity (that is

the most negative velocity) by differentiating Equations 1.10, 1.11 and 1.12 as:

for x <1

for x =1

for x >1

Case 2: Forced Vibrations

If the support is oscillated such that its motion in the x-direction is y = Y sin w t the

equation of motion for the system becomes:

The steady-state solution of Equation 1.18 is

10

ωd=ωn( √1−ξ2 ) (1 . 12)

δ= ln{−XnXn+ 1

2

}= πξ

√1−ξ2 (1.14 )

δ=ln{XnXn+1

}= 2πξ

√1−ξ2 (1. 13)

xmin

.

=−ωn X0{exp(−ξωn (sin−1√1−ξ2 )) /√1−ξ2} (1. 15 )

xmin

.

=−ωn X0 exp {−1}=−0 .368ωnX 0 (1 . 16)

xmin

.

=−ωn X0 exp [{−ξωn (sin−1√ξ2−1)}/√ξ2−1 ] (1 .17 )

x. .

+2ξωn x.

+ωn2x=2ξωnY cosωt+ω

n2Y sinωt (1 .18)

where the amplitude of the forced vibration, X, is given by the amplification factor:

where ,

Experimental Details (set up and procedure):

Experimental set up:

1) Vibrating Mass: Constrained in an externally pressurized air bearing guide (2).

2) Air-Bearing Guide.

3) Frame.

4) Springs.

5) Copper Plate: It moves in the air gap of an electromagnet (6) attached to the frame.

6) Electromagnet: It gives eddy current damping.

7) Stylus: It is fixed to the mass and records its motion on a strip of teledeltos paper.

8) Teledeltos Paper.

9) Paper Drive.

10) Base Plate.

11) Driving Motor.

12) Gear Box: It is 1 : 20 speed reduction gear. Correct your RPM accordingly.

13) Eccentric: For providing oscillations to the frame (3).

14) Air Pressure Regulator and Filter: For air bearing.

15) Out-of-Balance Force: It may be fitted to the vibrating mass to force the system via

motor.

11

XY= √(1+4ξ2r2 )

√[ (1−r2 )2+4ξ2r2 ] (1 .19)

x=X cos (ωt+φ )

r= ωωn

and tanφ= 2ξr

1−r2 (1 .20 )

16) Control Unit.

Figure 1: The experimental setup for 1-DOF

Experimental procedure:

Part 1: Variation of undamped natural frequency with vibrating mass:

1) Prepare the setup for free un-damped vibrations by setting the eccentric to zero radius

and zero damping current. Adjust the air bearing pressure so that the mass floats. Set

up the recorder and time it for mm/sec of paper speed.

2) For a given value of vibrating mass make three measurements of the time taken for 10

n cycles of free vibration, where n is chosen to give a time of between 35 and 50

seconds. From the average time for one cycle, Tn, the un-damped natural frequency,

wn, can be computed from wn = 2p/Tn.

3) Repeat for different values of the vibrating mass, m, chosen to give approximately

equal intervals for 1/m.

4) Plot a graph of wn2 against m-1 which should be a straight line with slope equal to

the spring stiffness, k. Be consistent in units (if the unit of m is kg and Tn is sec, then

the unit of wn is rad/sec and of k is N/m).

12

Part 2: Damped free vibration and determination of the damping ratio

1) Set a given value of the damping current and hold it constant throughout the test.

Remember the safety precaution.

2) Measurement of the damping ratio,x, is from a record of the free vibration in which

the vibration mass is displaced to one of the stops limiting its motion and released

from rest. The equilibrium position of the mass should be recorded before and after

the free vibration and the initial position of the mass (x = X0) should be recorded

before the mass is released.

3) For x < 0.05: determine the mean value of the ratio of successive maxima, and

calculate the value of x from Equation 1.13.

4) For 0.05<=x<0.1: determine the mean value of the ratio of successive maxima and

obtain the value of x

5) For 0.1<=x ,0.3: determine the mean value of the ratio of successive maxima and

minima and obtain the value of x

6) For 0.3<=x <0.6: determine the ratio of the amplitude of the first minimum to the

initial displacement X 12

/X 0 and obtain ξ

7) Plot a graph of x against damping current to use as a calibration curve to guide in

setting x in other experiments. Note that the graph will apply for particular values of

m and k, but it can be adjusted for other values using V1.4a.

Part 3: Forced vibration excited by oscillating the support

1) Using the results of Part 2, set the damping current to give the required value of x

and hold it constant throughout the test.

2) Set the eccentric to give a suitable amplitude of frame oscillation, Y, and measure it

using a ruler. If Y is made too large with low values of x the motion of the vibrating

mass will be limited by the stops for that part of the forced vibration response in the

region of resonance.

3) Switch on the frame drive (selector D in Figure (1)).

4) Select the appropriate forcing frequency by controlling knob C of Figure (1). Count

the RPS using the photoelectric probe assembly.

13

5) Set up the recorder as before to record several cycles of steady-state forced vibration

for values of the forcing frequency, w, from approximately 0.5wn to 2.5wn, with close

spacing of the readings in the region of resonance.

6) Measure the amplitude of vibration of the vibrating mass, X, from the recorded traces

and plot X/Y against w /wn. Compare this graph with the theoretical response given

by Equation 1.19, evaluated for the appropriate value of x.

Tabulated Data:

Measured:

(A) Free Vibration

Table (1) below displays the data obtained for a free, undamped vibration of a single

degree of freedom system. The first column represents the serial number of the runs,

while the second column represents the mass of the cradle in (kg). The third column lists

the reciprocal of the mass in (kg-1), while the fourth column lists the number of cycles

taken for each run. The fifth column displays the length of the record in (mm).The sixth

column represents the average time period in (sec).

Table 1: Measured data from the variation of the time period with vibrating mass.

SerialNo.

Mass[m, kg]

1/m[kg-1]

No. of Cycles[n]

Length of the Record[L, mm]

Average Time Period

[T,sec]

1 1.35 0.74 2 94.5 1.00

2 1.85 0.54 2 111 1.17

14

3 2.35 0.43 1.5 94.5 1.33

4 2.85 0.35 1 70 1.48

5 3.35 0.30 1.5 114 1.61

6 3.85 0.26 1.5 121.5 1.71

7 4.35 0.23 1 86 1.82

8 4.85 0.21 1.5 136 1.92

Tables (2) and (3) below displays the data obtained for an undamped vibration of a

single degree of freedom system. The mass is fixed at (1.35 kg) in table (2) and at (4.85

kg) in table (3). The first column represents the serial number of the runs taken, while the

second column represents the damping current in (A). The third column lists the

amplitude of vibration in (mm) for the first maximum or minimum amplitude, while

columns four and five list the second and third maximum or minimum amplitude of

vibration in (mm), respectively.

Table 2: Measured data of the variation of damping ratio with damping

current (m=1.35 kg).

SerialNo.

Damping Current

[I, A]

xn

[mm]xn+1

[mm]xn+0.5

[mm]

1 0.0 21.5 20.0 19.0

2 0.1 20.5 19.0 18.5

3 0.2 19.5 18.5 18.0

4 0.3 19.5 17.0 17.5

5 0.4 19.0 15.0 17.0

6 0.5 18.0 14.0 15.0

7 0.6 17.0 12.0 14.0

8 0.7 16.5 10.5 13.0

9 0.8 16.5 10.0 12.0

10 0.9 14.5 8.0 10.0

11 1.0 14.0 6.0 8.5

12 1.1 13.0 4.5 7.5

13 1.2 12.0 4.0 6.5

14 1.3 11.0 3.0 5.0

15

Table 3: Measured data of the variation of damping ratio with damping

current (m=4.85 kg).

SerialNo.

Damping Current

[I, A]

xn

[mm]xn+1

[mm]xn+0.5

[mm]

1 0.0 24.0 20.0 23.0

2 0.1 23.5 19.0 21.5

3 0.2 22.5 18.0 20.5

4 0.3 21.5 17.0 18.5

5 0.4 21.0 16.0 17.0

6 0.5 19.0 14.0 15.0

7 0.6 17.5 12.5 12.5

8 0.7 17.0 10.5 10.0

9 0.8 14.5 9.0 8.0

10 0.9 13.0 7.0 6.0

11 1.0 11.0 5.5 4.5

12 1.1 9.0 4.5 3.0

13 1.2 8.0 3.5 2.5

14 1.3 6.5 3.0 1.0

(B) Forced Vibration

Tables (4), (5), (6), and (7) display the data obtained for a damped, forced vibration of

a single degree of freedom system. The damping current is fixed to (0.7 A) in table (4), to

(0.9 A) in table (5), to (1.1 A) in table (6), and to (1.3 A) in table (7). The mass is fixed at

(3.35 kg) for tables (4) through (7). The first column represents the serial number of runs

taken, while the second column represents the desired frequency ratio. The third column

lists the desired motor speed in (rpm), while the fourth column lists the actual motor

speed in (rpm). The fifth column displays the doubled amplitude of vibration in (mm)

while the sixth column displays the amplitude of vibration is listed in (mm).

Table 4: Measured data of the damping ratio for a value of damping current (I=0.7 A).

Serial

No.

Desired Frequency Ratio

[r]

Desired Motor Speed

[n, rpm]

Actual Motor Speed

[n, rpm]

2X[mm]

X[mm]

1

2

3

4

16

5

6

7

8

9

10

11

12

13


Serial

No.


[r]

Desired Motor Speed

[n, rpm]

Actual Motor Speed

[n, rpm]

2X[mm]

X[mm]

1

2

3

4

5

6

7

8

9

10

11

12

13


Seria Desired Frequency Desired Motor Actual Motor 2X X

17

lNo.

Ratio[r]

Speed[n, rpm]

Speed[n, rpm]

[mm] [mm]

1

2

3

4

5

6

7

8

9

10

11

12

13


Serial

No.


[r]

Desired Motor Speed

[n, rpm]

Actual Motor Speed

[n, rpm]

2X[mm]

X[mm]

1

2

3

4

5

6

7

8

9

10

11

12

13

18

Calculated:

(A) Free Vibration

Table (8) below displays the data obtained for a free, undamped vibration of a single

degree of freedom system. The first column represents the serial number of the runs,

while the second column represents the mass of the cradle in (kg). The third column lists

the reciprocal of the mass in (kg-1), while the fourth column lists the number of cycles

taken for each run. The fifth column displays the length of the record in (mm).The sixth

column represents the average time period in (sec).The seventh column shows the square

of the natural frequency of the system in (rad2/s2).

Table 8: Calculated data of the variation of time period with vibrating mass.

Serial

No.

Mass[m, kg]

1/m[kg-1]

No. of Cycles

[n]

Length of the Record[L, mm]

Average Time Period[T, sec]

Natural Frequency

[wn2, rad2/s2]

1 1.35 0.74 2.00 94.50 1.00 39.60

2 1.85 0.54 2.00 111.00 1.17 28.70

3 2.35 0.43 1.50 94.50 1.33 22.27

4 2.85 0.35 1.00 70.00 1.48 18.04

5 3.35 0.30 1.50 114.00 1.61 15.30

6 3.85 0.26 1.50 121.50 1.71 13.47

7 4.35 0.23 1.00 86.00 1.82 11.95

8 4.85 0.21 1.50 136.00 1.92 10.75

Tables (9) and (10) displays the data obtained for an undamped vibration of a single

degree of freedom system. The mass is fixed at (1.35 kg) in table (2) and at (4.85 kg) in

table (3). The first column represents the serial number of the runs taken, while the

second column represents the damping current in (A). The third column lists the

amplitude of vibration in (mm) for the first maximum or minimum amplitude, while

19

columns four and five list the second and third maximum or minimum amplitude of

vibration in (mm), respectively. The sixth and seventh columns list the logarithmic

decrement for successive maxima and successive maxima to minima, respectively. The

eighth column represents the damping ratio of the system, while the ninth column lists

the damping coefficient of the system in (N-s/m).

Table 9: Calculated data of the variation of damping ratio with damping current (m=1.35 kg).

SerialNo.

Damping Current

[I, A]

xn

[mm]xn+1

[mm]xn+0.5

[mm]ln (

xnxn+1

)Damping

Ratio[ζ]

Damping Coefficient[c, N-s/m]

1 0.00 21.50 20.00 19.00 1.08 0.07 0.01

2 0.10 20.50 19.00 18.50 1.08 0.08 0.01

3 0.20 19.50 18.50 18.00 1.05 0.05 0.01

4 0.30 19.50 17.00 17.50 1.15 0.14 0.02

5 0.40 19.00 15.00 17.00 1.27 0.24 0.04

6 0.50 18.00 14.00 15.00 1.29 0.25 0.04

7 0.60 17.00 12.00 14.00 1.42 0.35 0.06

8 0.70 16.50 10.50 13.00 1.57 0.45 0.07

9 0.80 16.50 10.00 12.00 1.65 0.50 0.08

10 0.90 14.50 8.00 10.00 1.81 0.59 0.09

11 1.00 14.00 6.00 8.50 2.33 0.85 0.13

12 1.10 13.00 4.50 7.50 2.89 1.06 0.17

13 1.20 12.00 4.00 6.50 3.00 1.10 0.17

14 1.30 11.00 3.00 5.00 3.67 1.30 0.20

20

Table 10: Calculated data of the variation of damping ratio with damping current (m=4.85 kg).

SerialNo.

Damping Current

[I, A]

xn

[mm]xn+1

[mm]xn+0.5

[mm]ln (

xnxn+1

)Damping

Ratio[ζ]


1 0.00 24.00 20.00 23.00 1.20 0.18 0.03

2 0.10 23.50 19.00 21.50 1.24 0.21 0.03

3 0.20 22.50 18.00 20.50 1.25 0.22 0.04

4 0.30 21.50 17.00 18.50 1.26 0.23 0.04

5 0.40 21.00 16.00 17.00 1.31 0.27 0.04

6 0.50 19.00 14.00 15.00 1.36 0.31 0.05

7 0.60 17.50 12.50 12.50 1.40 0.34 0.05

8 0.70 17.00 10.50 10.00 1.62 0.48 0.08

9 0.80 14.50 9.00 8.00 1.61 0.48 0.08

10 0.90 13.00 7.00 6.00 1.86 0.62 0.10

11 1.00 11.00 5.50 4.50 2.00 0.69 0.11

12 1.10 9.00 4.50 3.00 2.00 0.69 0.11

13 1.20 8.00 3.50 2.50 2.29 0.83 0.13

14 1.30 6.50 3.00 1.00 2.17 0.77 0.12

(B) Forced Vibration

Tables (11), (12), (13), (14) list the data obtained for a damped, forced vibration of a

single degree of freedom system. The damping current is fixed to (0.7 A) in table (11), to

(0.9 A) in table (12), to (1.1 A) in table (13), and to (1.3 A) in table (14). The mass is

fixed at (3.35 kg) for tables (11) through (14). The first column represents the serial

number of runs taken, while the second column represents the desired frequency ratio.

The third column displays the desired motor speed in (rpm), while the fourth column

displays the actual motor speed in (rpm). The fifth column lists the doubled amplitude of

vibration in (mm), while the sixth column lists the amplitude of vibration in (mm). The

seventh column shows the actual frequency ratio. The eighth and ninth columns

represents the experimental and theoretical amplitude ratios, respectively. The first row

of the table displays the damping coefficient in (N-s/m) and the damping ratio.

21

Table 11: Calculated data of the damping ratio for a value of damping current (I=0.7 A).


Damping Ratio[ζ]

SerialNo.

Desired Frequency

Ratio[r]

Desired Motor Speed

[n, rpm]

Actual Motor Speed

[n, rpm]

2X[mm]

X[mm]

Actual Frequency

Ratio[r]

XY

XY theo.

1

2

3

4

5

6

7

8

9

10

11

12

13



Damping Ratio[ζ]

SerialNo.

Desired Frequency

Ratio[r]

Desired Motor Speed

[n, rpm]

Actual Motor Speed

[n, rpm]

2X[mm]

X[mm]

Actual Frequency

Ratio[r]

XY

XY theo.

1

2

3

4

5

6

7

8

9

10

11

12

13

22



Damping Ratio[ζ]

SerialNo.

Desired Frequency

Ratio[r]

Desired Motor Speed

[n, rpm]

Actual Motor Speed

[n, rpm]

2X[mm]

X[mm]

Actual Frequency

Ratio[r]

XY

XY theo.

1

2

3

4

5

6

7

8

9

10

11

12

13



Damping Ratio[ζ]

SerialNo.

Desired Frequency

Ratio[r]

Desired Motor Speed

[n, rpm]

Actual Motor Speed

[n, rpm]

2X[mm]

X[mm]

Actual Frequency

Ratio[r]

XY

XY theo.

1

2

3

4

5

6

7

8

9

10

11

12

13

23