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Theory and problems of mechanical vibrations

S. Graham Kelly

Chapter 8

VIBRATION CONTMOL.$\ldots \ldots \ldots \ldots \ldots \ldots \ldots$ - all with Video Answers

Educators

Chapter Questions

02:57

Problem 1

What is the maximum stiffness of an undamped isolator to provide 81 percent isolation for a $200-\mathrm{kg}$ fan operating at $1000 \mathrm{r} / \mathrm{min}$ ?

James Kiss
James Kiss
Numerade Educator
01:44

Problem 2

What is the minimum static deflection of an undamped isolator to provide 75 percent isolation to a pump that operates at speeds between 1500 and $2000 \mathrm{r} / \mathrm{min}$ ?

James Kiss
James Kiss
Numerade Educator
02:07

Problem 3

A $150-\mathrm{kg}$ sewing machine operates at $1200 \mathrm{r} / \mathrm{min}$ and has a rotating unbalance of $0.45 \mathrm{~kg}-\mathrm{m}$. What is the maximum stiffness of an undamped isolator such that the force transmitted to the machine's foundation is less than 2000 N?

Anand Jangid
Anand Jangid
Numerade Educator
02:38

Problem 4

A $20-\mathrm{kg}$ laboratory experiment is to be mounted to a table that is bolted to the floor in a laboratory. Measurements indicate that due to the operation of a nearby pump that operates at $2000 \mathrm{r} / \mathrm{min}$, the table has a steady-state displacement of $0.25 \mathrm{~mm}$. What is the maximum stiffness of an undamped isolator, placed between the experiment and the table such that the experiment's acceleration amplitude is less than $4 \mathrm{~m} / \mathrm{s}^2$ ?

Narayan Hari
Narayan Hari
Numerade Educator
01:35

Problem 5

A $100-\mathrm{kg}$ turbine operates at $2000 \mathrm{r} / \mathrm{min}$. What percent isolation is achieved if the turbine is mounted on four identical springs in parallel, each of stiffness $3 \times 10^5 \mathrm{~N} / \mathrm{m}$ ?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
06:05

Problem 6

What can be done to the turbine of Problem 8.5 to achieve 81 percent isolation if the same mounting system is used?

Km Neeraj
Km Neeraj
Numerade Educator
01:06

Problem 7

List one negative and two beneficial effects of adding damping to an isolator.

Suzanne W.
Suzanne W.
Numerade Educator
02:37

Problem 8

An isolator of damping ratio $\zeta$ is to be designed to achieve a transmissibility ratio $T<1$. Derive an expression, in terms of $\zeta$ and $T$, for the smallest frequency ratio to achieve appropriate isolation.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:53

Problem 9

Solve Problem 8.1 as if the isolator had a damping ratio of 0.1 ,

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
02:53

Problem 10

Solve Problem 8.3 as if the isolator had a damping ratio of 0.08 .

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
01:04

Problem 11

What are the maximum start-up amplitude and the steady-state amplitude of the system of Problem 8.10?

Hunza Gilgit
Hunza Gilgit
Numerade Educator
02:07

Problem 12

Design an isolator by specifying $k$ and $\zeta$ for the system of Problem 8.3 such that the maximum start-up amplitude is $30 \mathrm{~mm}$ and the maximum transmitted force is $3000 \mathrm{~N}$.

Anand Jangid
Anand Jangid
Numerade Educator
02:36

Problem 13

Using the isolator designed in Problem 8.12 , what, if any, mass should be added to the machine to limit its steady-state amplitude to $3 \mathrm{~mm}$ ?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:19

Problem 14

Solve Problem 8.4 as if the isolator had a damping ratio of 0.13 .

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
02:36

Problem 15

For the isolator designed in Problem 8.14, what is the steady-state amplitude of the experiment, and what is the maximum deformation in the isolator?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
View

Problem 16

A 200-kg turbine operates at speeds between 1000 and $2000 \mathrm{r} / \mathrm{min}$. The turbine has a rotating unbalance of $0.25 \mathrm{~kg}-\mathrm{m}$. What is the required stiffness of andamped isolator such that the maximum force transmitted to the turbine's foundation is $1000 \mathrm{~N}$ ?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
11:24

Problem 17

Repeat Problem 8.16 as if the isolator had a damping ratio of 0.1 .

Susan Hallstrom
Susan Hallstrom
Numerade Educator
01:23

Problem 18

Repeat Problem 8.17 as if the upper operating speed were $2500 \mathrm{r} / \mathrm{min}$.

Shahab Ullah
Shahab Ullah
Numerade Educator
02:02

Problem 19

Measurement indicates that the peak components of the table vibration of Problem 8.4 are a $0.25-\mathrm{mm}$ component at $100 \mathrm{rad} / \mathrm{s}$ and a $0.4-\mathrm{mm}$ component at $150 \mathrm{rad} / \mathrm{s}$. An available isolator has a stiffness of $8 \times 10^4 \mathrm{~N} / \mathrm{m}$ and a damping ratio of 0.1 . Will the acceleration felt by the apparatus exceed $6 \mathrm{~m} / \mathrm{s}^2$ when this isolator is installed?

James Kiss
James Kiss
Numerade Educator
01:03

Problem 20

Repeat Problem 8.1 as if the isolator damping is assumed to be hysteretic with a hysteretic damping coefficient of 0.2 .

Manish Jain
Manish Jain
Numerade Educator

Problem 21

If the model is mounted on an isolator of stiffness $5.4 \times 10^5 \mathrm{~N} / \mathrm{m}$ and damping ratio 0.1 , what is the maximum transmitted force and maximum model displacement for a pulse of magnitude $2500 \mathrm{~N}$ and duration $0.12 \mathrm{~s}$ ?

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03:06

Problem 22

What is the maximum stiffness of an isolator such that the maximum transmitted force is less than $2000 \mathrm{~N}$ for a pulse of magnitude $2500 \mathrm{~N}$ and duration $0.12 \mathrm{~s} ?$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 23

What is the maximum displacement of the model with the isolator of Problem 8.22 installed?

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02:07

Problem 24

If the model is mounted on an isolator of stiffness $5.4 \times 10^5 \mathrm{~N} / \mathrm{m}$ and damping ratio 0.14 , what is the maximum transmitted force if the pulse has a magnitude of $3000 \mathrm{~N}$ and a duration of $0.01 \mathrm{~s}$ ?

Anand Jangid
Anand Jangid
Numerade Educator
01:29

Problem 25

What is the model's maximum displacement for the situation described in Problem 8.24?

James Kiss
James Kiss
Numerade Educator
11:11

Problem 26

The $120-\mathrm{kg}$ hammer of a $300-\mathrm{kg}$ forge hammer is dropped from $1.3 \mathrm{~m}$. Design an isolator for the hammer such that the maximum transmitted force is less than $15,000 \mathrm{~N}$ and the maximum displacement is a minimum.

Keshav Singh
Keshav Singh
Numerade Educator
01:21

Problem 27

A $200-\mathrm{kg}$ machine is attached to a spring of stiffness $4 \times 10^5 \mathrm{~N} / \mathrm{m}$. During operation the machine is subjected to a harmonic excitation of magnitude $500 \mathrm{~N}$ and frequency $50 \mathrm{rad} / \mathrm{s}$. Design an undamped vibration absorber such that the steady-state amplitude of the primary mass is zero and the steady-state amplitude of the absorber mass is less than $2 \mathrm{~mm}$.

James Kiss
James Kiss
Numerade Educator
01:18

Problem 28

What are the natural frequencies of the system of Problem 8.27 with the absorber in place?

James Kiss
James Kiss
Numerade Educator
02:50

Problem 29

A piping system experiences resonance when the pump supplying power to the system operates at $500 \mathrm{r} / \mathrm{min}$. When a $5-\mathrm{kg}$ absorber tuned to $500 \mathrm{r} / \mathrm{min}$ is added to the pipe, the system's new natural frequencies are measured as 380 and $624 \mathrm{r} / \mathrm{min}$. What is the natural frequency of the piping system and its equivalent mass?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:10

Problem 30

Redesign the absorber used in Problem 8.29 such that the system's natural frequencies are less than $350 \mathrm{r} / \mathrm{min}$ and greater than $650 \mathrm{r} / \mathrm{min}$.

James Kiss
James Kiss
Numerade Educator

Problem 31

A 100-kg machine is placed at the midspan of a simply supported beam of length $3 \mathrm{~m}$, elastic modulus $200 \times 10^9 \mathrm{~N} / \mathrm{m}^2$, and moment of inertia $1.3 \times 10^{-6} \mathrm{~m}^4$. During operation the machine is subjected to a harmonic excitation of magnitude $5000 \mathrm{~N}$ at speeds between 600 and $700 \mathrm{r} / \mathrm{min}$. Design an undamped vibration absorber such that the machine's steady-state amplitude is less than $3 \mathrm{~mm}$ at all operating speeds.

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02:54

Problem 32

If an optimally designed damped vibration absorber is used on the system of Problem 8.31 with a mass ratio of 0.25 , what is the machine's steady-state amplitude at $600 \mathrm{r} / \mathrm{min}$ ?

Supratim Pal
Supratim Pal
Numerade Educator

Problem 33

A $300-\mathrm{kg}$ machine is placed at the end of a cantilever beam of length $1.8 \mathrm{~m}$, elastic modulus $200 \times 10^9 \mathrm{~N} / \mathrm{m}^2$, and moment of inertia $1.8 \times 10^{-5} \mathrm{~m}^4$. When the machine operates at $1000 \mathrm{r} / \mathrm{min}$, it has a steady-state amplitude of $0.8 \mathrm{~mm}$. What is the machine's steady-state amplitude when a $30-\mathrm{kg}$ absorber of damping coefficient $650 \mathrm{~N}-\mathrm{s} / \mathrm{m}$ and stiffness $1.5 \times 10^5 \mathrm{~N} / \mathrm{m}$ is added to the end of beam?

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01:52

Problem 34

An engine has a moment of inertia of $3.5 \mathrm{~kg}-\mathrm{m}^2$ and a natural frequency of $100 \mathrm{~Hz}$. Design a Houdaille damper such that the engine's maximum magnification factor is 4.8 .

Supratim Pal
Supratim Pal
Numerade Educator
02:56

Problem 35

During operation the engine of Problem 8.34 is subjected to a harmonic torque of magnitude $100 \mathrm{~N}-\mathrm{m}$ at a frequency of $110 \mathrm{~Hz}$. What is the engine's steady-state amplitude when the Houdaille damper designed in Problem 8.34 is used?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:32

Problem 36

A 40-kg rotor has an eccentricity of $1.2 \mathrm{~cm}$. It is mounted on a shaft and bearing system whose stiffness is $3.2 \times 10^5 \mathrm{~N} / \mathrm{m}$ and has a damping ratio of 0.07 . What is the amplitude of whirling when the rotor operates at $1000 \mathrm{r} / \mathrm{min}$ ?

James Kiss
James Kiss
Numerade Educator
03:11

Problem 37

An engine flywheel has an eccentricity of $1.2 \mathrm{~cm}$ and mass of $40 \mathrm{~kg}$. Assuming a damping ratio of 0.05 , what is the necessary stiffness of its bearings to limit its whirl amplitude to $1.2 \mathrm{~mm}$ at all speeds between 1000 and $2000 \mathrm{r} / \mathrm{min}$ ?

James Kiss
James Kiss
Numerade Educator
01:41

Problem 38

What is the maximum stiffness of an undamped isolator to provide 81 percent isolation to a $350-\mathrm{kg}$ sewing machine when it operates at $2100 \mathrm{r} / \mathrm{min}$ ?

James Kiss
James Kiss
Numerade Educator
01:41

Problem 39

What is the maximum stiffness of an undamped isolator to provide 70 percent isolation to a $200-\mathrm{kg}$ pump that operates at speeds between 1000 and $1500 \mathrm{r} / \mathrm{min}$ ?

James Kiss
James Kiss
Numerade Educator
01:04

Problem 40

Repeat Problem 8.38 for an isolator with a damping ratio of 0.1 .

David Collins
David Collins
Numerade Educator
01:20

Problem 41

Repeat Problem 8.39 for an isolator with a damping ratio of 0.08 .

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:09

Problem 42

A $50-\mathrm{kg}$ compressor operates at $200 \mathrm{rad} / \mathrm{s}$. The only available isolator has a stiffness of $1.3 \times 10^{\circ} \mathrm{N} / \mathrm{m}$ and a damping ratio of 0.1 . What is the minimum mass that must be added to the compressor to provide 68 percent isolation?

James Kiss
James Kiss
Numerade Educator
02:07

Problem 43

During operation at $1000 \mathrm{r} / \mathrm{min}$, a $200-\mathrm{kg}$ tumbler produces a harmonic force of magnitude $5000 \mathrm{~N}$. What is the minimum static deflection of an isolator of damping ratio 0.12 such that the transmitted force is less than $2000 \mathrm{~N}$ ?

Anand Jangid
Anand Jangid
Numerade Educator
02:36

Problem 44

What is the steady-state amplitude of the system of Problem 8.43 when the isolator with the minimum static deflection is installed?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:02

Problem 45

A $15-\mathrm{kg}$ flow meter is mounted on a table in a laboratory. Measurements indicate that the dominant frequency of surrounding vibrations is $250 \mathrm{rad} / \mathrm{s}$. The amplitude at this frequency is $0.8 \mathrm{~mm}$. What is the maximum stiffness of an isolator of damping ratio 0.1 such that the acceleration transmitted to the flow meter is $5 \mathrm{~m} / \mathrm{s}^2$ ?

James Kiss
James Kiss
Numerade Educator
01:12

Problem 46

A $60-\mathrm{kg}$ engine operates at $2000 \mathrm{r} / \mathrm{min}$ and has a rotating unbalance of $0.2 \mathrm{~kg}-\mathrm{m}$. Can an isolator of damping ratio 0.1 be designed to limit the transmitted force to $1000 \mathrm{~N}$ and the steady-state amplitude to $3 \mathrm{~mm}$ ?

James Kiss
James Kiss
Numerade Educator
01:36

Problem 47

What is the minimum mass that can be added to the engine of Problem 8.46 such that a steady-state amplitude of $3 \mathrm{~mm}$ can be attained when a transmitted force of $1000 \mathrm{~N}$ is attained using an isolator of damping ratio 0.1 ?

Narayan Hari
Narayan Hari
Numerade Educator

Problem 48

What is the maximum stiffness of an isolator of damping ratio 0.1 that limits the transmitted force to $1000 \mathrm{~N}$ when 12.8 is added to the engine of Problem 8.47 ?

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02:01

Problem 49

What is the maximum start-up amplitude of the system of Problem 8.48.

Karan Soni
Karan Soni
Numerade Educator
01:57

Problem 50

Repeat Problem 8.4 as if the isolator had hysteretic damping with a damping coefficient of 0.15 .

Narayan Hari
Narayan Hari
Numerade Educator
View

Problem 51

What is the maximum stiffness of an isolator of damping ratio 0.1 such that the acceleration felt by the apparatus of Problem 8.19 is less than $6 \mathrm{~m} / \mathrm{s}^2$ ?

Victor Salazar
Victor Salazar
Numerade Educator

Problem 52

A $50-\mathrm{kg}$ machine is mounted on four parallel springs, each of stiffness $3 \times 10^5 \mathrm{~N} / \mathrm{m}$. What is the maximum transmitted force when the machine is subject to an excitation of the form of Fig. 8-12 with $F_0=1200 \mathrm{~N}$ and $t_0=0.05 \mathrm{~s}$ ?

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01:01

Problem 53

What is the maximum displacement of the machine of Problem 8.52?

Narayan Hari
Narayan Hari
Numerade Educator
02:07

Problem 54

What is the maximum stiffness of an isolator of damping ratio 0.1 such that the maximum transmitted force for a $100-\mathrm{kg}$ machine is $1125 \mathrm{~N}$ when it is subjected to the excitation of Fig. 8-12. with $F_{\mathrm{o}}=1500 \mathrm{~N}$ and $t_{\mathrm{a}}=0.04 \mathrm{~s}$ ?

Anand Jangid
Anand Jangid
Numerade Educator
02:07

Problem 55

What is the minimum stiffness of an isolator of damping ratio 0.1 such that the maximum displacement of a $150-\mathrm{kg}$ machine is $2.2 \mathrm{~mm}$ when it is subjected to an excitation of the form of Fig. 8-12 with $F_0=2000 \mathrm{~N}$ and $t_0=0.06 \mathrm{~s}$ ?

Anand Jangid
Anand Jangid
Numerade Educator
02:07

Problem 56

What is the range of stiffness of an isolator of damping ratio 0.1 such that when a 200-kg machine is subjected to an excitation of the form of Fig. 8-12 with $F_6=2000 \mathrm{~N}$ and $t_0=0.05 \mathrm{~s}$, the maximum transmitted force is $1500 \mathrm{~kg}$ and the maximum displacement is $6 \mathrm{~mm}$ ?

Anand Jangid
Anand Jangid
Numerade Educator
02:07

Problem 57

A $200-\mathrm{kg}$ machine rests on springs whose equivalent stiffness is $1 \times 10^5 \mathrm{~N} / \mathrm{m}$ and damping coefficient $1500 \mathrm{~N}-\mathrm{s} / \mathrm{m}$. During operation the machine is subjected to an impulse of magnitude $75 \mathrm{~N}$-s. What is the maximum force transmitted to the machine's foundation due to the impulse?

Anand Jangid
Anand Jangid
Numerade Educator
03:01

Problem 58

What is the maximum displacement of the machine of Problem 8.57?

Eric Mockensturm
Eric Mockensturm
Numerade Educator

Problem 59

During operation a $65 \mathrm{~kg}$ machine is subjected to an impulse of magnitude $100 \mathrm{~N}-\mathrm{s}$. Specify the stiffness and damping coefficient of an isolator such that the transmitted force is $4000 \mathrm{~N}$ and the machine's maximum displacement is minimized.

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01:11

Problem 60

A $50-\mathrm{kg}$ machine is mounted on a table of stiffness $1 \times 10^5 \mathrm{~N} / \mathrm{m}$. During operation it is subjected to a harmonic excitation of magnitude $1200 \mathrm{~N}$ at $45 \mathrm{rad} / \mathrm{s}$. What is the required stiffness of a $4-\mathrm{kg}$ absorber to eliminate steady-state vibrations of the machine during operation?

Penny Riley
Penny Riley
Numerade Educator
01:21

Problem 61

What is the steady-state amplitude of the absorber mass for the system of Problem 8.60?

James Kiss
James Kiss
Numerade Educator
01:18

Problem 62

What are the natural frequencies for the system of Problem 8.60 with the absorber in place?

James Kiss
James Kiss
Numerade Educator
02:36

Problem 63

For what range of frequencies near $45 \mathrm{rad} / \mathrm{s}$ is the steady-state amplitude of the machine of Problem 8.60 less than $5 \mathrm{~mm}$ when the absorber is in place?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
View

Problem 64

When a $10-\mathrm{kg}$ undamped absorber tuned to $100 \mathrm{rad} / \mathrm{s}$ is added to a 1-degree-of-freedom structure of stiffness $5 \times 10^6 \mathrm{~N} / \mathrm{m}$, the lowest natural frequency of the structure is $85.44 \mathrm{rad} / \mathrm{s}$. What is the higher natural frequency of the structure?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
01:21

Problem 65

A $15-\mathrm{kg}$ undamped absorber tuned to $250 \mathrm{rad} / \mathrm{s}$ is added to a $150-\mathrm{kg}$ machine mounted on a foundation of stiffness $1 \times 10^7 \mathrm{~N} / \mathrm{m}$. At $250 \mathrm{rad} / \mathrm{s}$, the amplitude of the absorber mass is $3.9 \mathrm{~mm}$. What is the amplitude of the machine at $275 \mathrm{rad} / \mathrm{s}$ ?

James Kiss
James Kiss
Numerade Educator

Problem 66

A 50-kg machine is placed at the midspan of a $1.5-\mathrm{m}$ simply supported beam of elastic modulus $210 \times 10^9 \mathrm{~N} / \mathrm{m}^2$ and moment of inertia $1.5 \times 10^{-6} \mathrm{~m}^4$. When running at $3000 \mathrm{r} / \mathrm{min}$, the machine's steady-state amplitude is measured as $1.2 \mathrm{~cm}$. Design an undamped absorber such that the steady-state amplitude is less than $2 \mathrm{~mm}$ at all speeds between 2900 and $3100 \mathrm{r} / \mathrm{min}$.

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01:21

Problem 67

If an optimally designed $15-\mathrm{kg}$ damped vibration absorber is used for the system of Problem 8.66, what is the steady-state amplitude of the machine when operating at $3000 \mathrm{r} / \mathrm{min}$ ?

James Kiss
James Kiss
Numerade Educator
02:04

Problem 68

A 20-kg machine is mounted on a foundation of stiffness $1.3 \times 10^6 \mathrm{~N} / \mathrm{m}$. What are the stiffness and damping coefficient of an optimally designed $4-\mathrm{kg}$ damped vibration absorber?

James Kiss
James Kiss
Numerade Educator

Problem 69

With the absorber designed in Problem 8.68 in place, what is the steady-state amplitude of the machine when operating at $85 \mathrm{rad} / \mathrm{s}$ if the machine has a rotating unbalance of $0.5 \mathrm{~kg}-\mathrm{m}$ ?

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02:07

Problem 70

A $110-\mathrm{kg}$ machine is subjected to an excitation of magnitude $1500 \mathrm{~N}$. The machine is mounted on a foundation of stiffness $3 \times 10^6 \mathrm{~N} / \mathrm{m}$. What are the mass and damping coefficient of an optimally designed vibration damper such that the maximum amplitude is $3 \mathrm{~mm}$ ?

Anand Jangid
Anand Jangid
Numerade Educator
01:59

Problem 71

What is the steady-state amplitude at $180 \mathrm{rad} / \mathrm{s}$ of the machine of Problem 8.70 when the optimally designed vibration damper is added?

Penny Riley
Penny Riley
Numerade Educator
02:33

Problem 72

An engine has a mass moment of inertia of $3.5 \mathrm{~kg}-\mathrm{m}^2$ and is mounted on a shaft of stiffness $1.45 \times 10^5 \mathrm{~N}-\mathrm{m} / \mathrm{rad}$. If the applied moment has a magnitude of $1000 \mathrm{~N}-\mathrm{m}$, what is the engine's steady-state amplitude at $2000 \mathrm{r} / \mathrm{min}$ when an optimally designed Houdaille damper of mass moment of inertia $1.1 \mathrm{~kg}-\mathrm{m}^2$ is added?

James Kiss
James Kiss
Numerade Educator
02:32

Problem 73

The center of gravity of a $12-\mathrm{kg}$ rotor is $1.2 \mathrm{~cm}$ from its geometric center. The rotor is mounted on a shaft and spring-loaded bearings of stiffness $1.4 \times 10^5 \mathrm{~N} / \mathrm{m}$. Assuming a damping ratio of 0.05 , what is the amplitude of whirling when the rotor operates at $1500 \mathrm{r} / \mathrm{min}$ ?

James Kiss
James Kiss
Numerade Educator