Machine Vibration Measurement

Question 1

What mathematical functions can be used to express the properties of waves?

Answer 1

Trigonometric sine and cosine functions.

Question 2

What is the Fourier Transform used for?

Answer 2

To determine the frequency spectrum of a waveform.

Question 3

True or False: The same tools used for analyzing water waves can be used for mechanical vibrations.

Answer 3

True.

Question 4

What is an example of the Fourier Transform applied to music?

Answer 4

The spectrum analyzer display on high-quality audio reproduction equipment.

Question 5

How can the technology used in audio spectrum analyzers be applied to machine vibration?

Answer 5

It can indicate sources of vibration by analyzing different frequencies generated by machine components.

Question 6

What is the formula for Newton’s Second Law of motion?

Answer 6

F = ma

Describes how the acceleration of an object is directly proportional to the applied force and inversely proportional to its mass.

Question 7

What does differentiation in calculus involve?

Answer 7

A variable is proportional to the rate-of-change of two others

Results in a division of units and is relevant for converting position to velocity and velocity to acceleration.

Question 8

What does integration in calculus involve?

Answer 8

A variable is proportional to the accumulation of the product of two others

Results in a multiplication of units and is relevant for converting acceleration to velocity and velocity to position.

Question 9

What are Fourier transforms used for?

Answer 9

Decomposing vibrational wave signals into constituent harmonic frequencies

Any repetitive waveform can be expressed as a series of sinusoidal waves of different frequencies, amplitudes, and phase shifts.

Question 10

What is the relationship between acceleration and force according to Newton’s Second Law?

Answer 10

Acceleration is directly proportional to force and inversely proportional to mass

This principle is relevant for calculating the force developed on a machine part from vibration.

Question 11

Fill in the blank: Differentiation results in a ______ of units.

Answer 11

division

Question 12

Fill in the blank: Integration results in a ______ of units.

Answer 12

multiplication

Question 13

True or False: Fourier transforms can only be applied to linear waveforms.

Answer 13

False

Any repetitive waveform is mathematically equivalent to sinusoidal waves.

Question 14

What kind of waves does Fourier transform analyze?

Answer 14

Sinusoidal waves

These waves can have different frequencies, amplitudes, and phase shifts.

Question 15

What is the significance of harmonics in Fourier transforms?

Answer 15

They are integer multiples of the frequencies of sinusoids

Harmonics help identify which parts of a machine are vibrating the most.

Question 16

What principle enhances problem-solving abilities in engineering?

Answer 16

Mastering the applications of fundamental principles

A wide variety of topics should be considered for better learning.

Question 17

What mathematical function is used to express vibration physics?

Answer 17

Cosine function

The cosine function alternates between extreme values of +1 and -1.

Question 18

What does ‘I’ represent in the vibration formula?

Answer 18

Displacement as measured by sensor at time t

Displacement is a key variable in understanding vibration.

Question 19

What is represented by ‘D’ in the vibration equation?

Answer 19

Peak displacement amplitude

D is a coefficient relating the cosine function to peak displacement.

Question 20

What does ‘w’ stand for in the formula for vibration?

Answer 20

Angular velocity

Angular velocity is typically expressed in units of radians per second.

Question 21

What is the meaning of ‘b’ in the vibration equation?

Answer 21

Bias air gap measured with no vibration

It accounts for any offset in the measurement.

Question 22

What does ‘t’ represent in the context of vibration physics?

Answer 22

Time (seconds)

Time is critical in describing the motion and changes in displacement.

Question 23

What is the unit of the product ‘wt’ in the vibration formula?

Answer 23

Radians

‘w’ is in radians per second and ‘t’ is in seconds.

Question 24

At time-0, what is the value of the cosine function?

Answer 24

+1

This occurs when the mass is aligned with the sensor.

Question 25

What is the angular velocity for a wheel spinning at 1720 RPM?

Answer 25

Approximately 180.1 radians per second ## Footnote This value is significant for calculating angle and displacement.

Question 26

What does 'v' represent in the context of motion?

Answer 26

Velocity of an object ## Footnote Velocity is the rate at which displacement changes over time.

Question 27

In physics, how is velocity mathematically defined?

Answer 27

As the time-derivative of displacement ## Footnote This relationship can be expressed using calculus notation.

Question 28

What does 'z' denote in the context of motion?

Answer 28

Displacement (position) of an object ## Footnote Displacement is crucial for understanding motion and velocity.

Question 29

What is the relationship between displacement and time in the context of velocity?

Answer 29

Velocity is the rate at which displacement changes over time ## Footnote This relationship is fundamental in physics.

Question 30

Fill in the blank: The angle between the off-center mass and the sensor is calculated using _______.

Answer 30

T = Dcos(wt) + b ## Footnote This equation incorporates displacement, angular velocity, and bias.

Question 31

True or False: The cosine function's argument is the product of angular velocity and time.

Answer 31

True ## Footnote This product determines the angle used in the cosine function.

Question 32

What is the equation for velocity derived from displacement?

Answer 32

V = -wD sin(wt) ## Footnote Where D is peak displacement, w is angular frequency, and t is time.

Question 33

What is the relationship between peak displacement and velocity?

Answer 33

The velocity of the wheel's wobble increases linearly with speed (w) ## Footnote This is because an increase in rotational speed means the wheel displaces the same vibrating distance in less time.

Question 34

What is the equation for acceleration derived from velocity?

Answer 34

a = -wD cos(wt) ## Footnote Acceleration is the time-derivative of velocity.

Question 35

How does acceleration relate to peak displacement and speed?

Answer 35

The acceleration of the wheel's wobble increases with the square of the speed (w) ## Footnote This indicates that higher speeds lead to greater acceleration for the same peak displacement.

Question 36

What principle from Newton's Second Law relates acceleration to lateral force?

Answer 36

P = ma ## Footnote Where P is the lateral force, m is the mass of the wheel, and a is the lateral acceleration.

Question 37

True or False: The derivative of a constant is always zero.

Answer 37

True ## Footnote This principle simplifies calculations when differentiating equations.

Question 38

Fill in the blank: The derivative of a cosine function is _______.

Answer 38

-sin(wt) ## Footnote This is a fundamental rule from calculus.

Question 39

What happens to the rate of change of a waveform as its frequency increases?

Answer 39

The rate of change must increase as well ## Footnote This is because higher frequency means less time for the waveform to travel the same amplitude.

Question 40

What does the notation 'd/dt' represent in calculus?

Answer 40

The derivative with respect to time ## Footnote This notation is used to indicate the process of differentiation.

Question 41

What is the significance of a constant multiplier in a function's argument during differentiation?

Answer 41

The multiplier is applied to the entire derivative ## Footnote This applies to both sine and cosine functions.

Question 43

What is the formula for vibrational force experienced by a wheel as rotational speed increases?

Answer 43

F = ma = -mw'D cos wt

Question 44

How does vibration affect high-speed rotating machinery?

Answer 44

Vibration can be terribly destructive due to enormous forces generated by small lateral displacements.

Question 45

What happens to the vibrational force when rotational speed is doubled?

Answer 45

The force quadruples.

Question 46

What happens to the vibrational force when rotational speed is tripled?

Answer 46

The force increases by 9 times.

Question 47

What assumption is made about displacement (D) in calculating vibrational forces?

Answer 47

It is assumed to be constant.

Question 48

What effect does centrifugal force have on an imbalanced rotating shaft?

Answer 48

It bends the rotating shaft, placing the mass farther from the centerline.

Question 49

In the United States, how is vibrational displacement (D) measured?

Answer 49

In mils, with one mil being 0.001 inch.

Question 50

How is vibrational velocity measured?

Answer 50

In inches per second.

Question 51

What unit is often used to represent acceleration in vibrational physics?

Answer 51

The unit of G, representing multiples of Earth's gravitational acceleration.

Question 52

What is the peak displacement (D) of a rotating machine vibrating in a sinusoidal manner at 1720 RPM?

Answer 52

2 mils (0.002 inch).

Question 53

What is the frequency of rotation for a machine operating at 1720 RPM?

Answer 53

28.667 Hz.

Question 54

What is the angular frequency (ω) in radians per second for a machine rotating at 1720 RPM?

Answer 54

180.1 rad/s.

Question 55

What is the period (T) of a machine vibrating at a frequency of 28.667 Hz?

Answer 55

34.88 ms.