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23. MEC 706: Sound and Vibration Control

The notes are based on the book “Industrial Noise and Vibration Control” by Irwin and Graf. Whenever figures, table, questions, etc. are referred to that are not in these notes, they may be found there. The notes generally follow the order of the text, except for the vibration component of the notes. Sets of questions follow the applicable note sections, these are from previous assignments, midterms, and finals. Where available answers are provided, but at this point the only promise about their value is that they will contain mistakes (hopefully many of these mistakes will be corrected in future revisions). Please take note of mistakes in the notes, and indicate them to me later. I will upgrade the notes for the benefit of future years. Please keep track of additional topics you think would be of value, and I will endeavor to add those if possible. A set of log graph paper sheets are provided near the front of the notes. These can be used for some of the vibration, and sound problems.

A note of value is that the problems do tend to focus on industrial noise and vibration control, but the approaches discussed are directly applicable to other areas, such as vibration in an airframe.

The selection of topics, first vibration then sound might apparently seem distant at best. But, when considered, these phenomenon are natural complements. In fact the underlying mathematical concepts are identical, the major difference is that the transmission medium is varied. This rationale leads the review of vibrations concepts, to lay a familiar basis for the student to consider the nature of sound. The essential nature of these problems cannot be underplayed, as environmental issues are becoming an essential component of all engineering design. It is for this reason that the scientific properties of sound will be related to various existing legal statutes for noise control.

23.1 Theoretical Fundamentals
The student is expected to have a grasp of a number of basic topics,

Properties of logarithms

Laplace and Fourier transforms

Wave properties

A knowledge of statics and dynamics

A basic background in material properties

A previous course in vibration

23.2 Evaluation
The student will be evaluated throughout the year with two midterm tests, tentatively schedules for,

Midterm #1: February _____, 1995

Midterm #2: March _____, 1995.

In addition there will be a total of four assignments that will be assigned throughout the term as material is covered. These assignments descriptions will be given later.

23.3 RESOURCES

#### 23.3.1 Conversion Table

A conversion table is attached to the end of these notes for many common unitary translations that may be required.

23.4 Problems
Problem 23.32 a) A worst case analysis is being done on a four bladed propeller. Given the properties below, estimate the maximum force that will be applied to the engine mount if one of the blades falls off.

b) Describe how we would practically find the relationship between vibration at the engine mount, and a point in the cockpit.

c) Assuming the Bode Plots below was obtained in the test in part b), what would the transfer function be?

d) Given that the engine generates a vibration force of 100N at 6000 rad/sec, use the bode plots to estimate the vibration force in the cabin, with the missing propeller blade.

e) Using the conditions described in d), use a Fourier transform to find the vibration force in the cabin, with the missing propeller blade.

f) Given that the main source of sound in the cabin is a panel that shakes, and experimentally we determine the force to sound power to be related by the equation below. Find the sound power levels in the cabin.

g) Given that the cabin (2m by 2m by 2m) is finished with wood walls (alpha = .1), and 4 glass windows (40cm by 40cm each and alpha = .12). Find the reverberant sound pressure level in the cabin caused by the motor, and the missing propeller blade.

h) What is the total loudness in sones, based on the values found in g)?

i) What is the voice level that may be used from the curves below if the pilot and copilot sit 3 feet apart?

j) Outside, the aircraft engine is running, and noise POWER levels are as listed below. If the nearest neighborhood is 50m away, what are the sound pressure levels? What is the total sound pressure in each time period (add the A-weighted pressure values)?

k) What is the noise pollution level if a 10dB penalty is in place from 10pm to 7am?

l) A hangar is to be built to house the aircraft, and reduce the noise levels. (In effect this is a large hood.) If the hangar is to be built 100m by 100m by 20m, with a 1/4” plywood wall with insulation, what will the 1000Hz sound pressure level be near the hangar during the day?

m) As another alternative, a resonant muffler (an expansion chamber) is to be designed to reduce the 1000Hz sound. If the exhaust pipe has a 30cm radius, select the width and length for a Transmission Loss of 20dB. Assume c=300m/s.

Problem 23.33 When a force ‘F’ is applied to the spring-mass-damper system below, the result is a motion ‘x’. If the spring-damper pair should have a static spring coefficient of 500 KN/m, but at 100Hz we want an isolation of 99.9%. What is the required damper value. Clearly state any assumptions made.

Problem 23.34 The applied force ‘F’ is the input to the system, and the output is the displacement ‘x’.

Problem 23.35 Given the transfer function below,

Problem 23.36 a) draw the straight line approximation of the bode and phase shift plots.

Problem 23.37 b) determine the steady state output if the input is x(s) = 20 cos(9t+.3)

Problem 23.38 Given the following time response to a step input of F=5N for t >= 0, find the transfer function x/F.

Problem 23.39 Given the transfer function below, find the isolation at 2Hz.

Problem 23.40 Describe how an active vibration control system might work.

Problem 23.41 A 160lb sign is to be mounted onto the side of a building. One significant problem is that the sign vibrates at about 100 radians per second. Therefore isolators will be used between the sign and the side of the building. Using the specifications sheet that was supplied by a local sales office, select a suitable isolator.

Problem 23.42 Two identical sources have a combined amplitude (Lp) of 104dB. What would be the value of the root mean square pressure for just one of the sources?

Problem 23.43 The sound spectrum below is known to exist for an industrial machine. Keeping in mind the proximity of some of the peaks, what bandwidth would our sound instrumentation require to tell them apart?

Problem 23.44 There is a small room that contains an average 100 hp diesel engine turning at 7200 rpm. The room is 2.5m by 2.5m and has a ceiling height of 2m. The walls and ceiling of the room are covered by a sound absorption coating (with a coefficient of 0.92). There are two 50cm by 30cm heavy plate glass windows in the walls for safety, and the floor is unpainted concrete. (50%)

a) determine the sound power level of the sound source (in dB).

b) assuming uniform directivity, what would be the direct sound pressure level (in dB) would we hear standing 0.5m and 2m away?

c) what is the reverberant sound pressure level (in dB) we would hear standing 0.5 and 2m away?

d) assuming the noise from the engine is relatively uniform in intensity from 200 Hz to 4 KHz, how far apart would two people in the room have to stand to hear each other shouting based on reverberant sound only?

e) what would the longest period of time a worker could be in that room each day based on reverberant sound only?

Problem 23.45 Given the transfer function below,

a) draw the straight line approximation of the bode and phase shift plots.

b) determine the steady state output if the input is x(s) = 20 cos(100t+.3).

Problem 23.46 Given the following time response to a step input of F=5N for t >= 0, find the transfer function x/F.

Problem 23.47 A spring damper system supports a mass of 34N. If it has a spring constant of 20.6N/cm, what is the systems natural frequency?

Problem 23.48 A large machine weighs 1000kg and vibrates at 20Hz, design an inertial damper.

Problem 23.49 If we had two sound sources at 120 Hz and 142 Hz that we wanted to measure separately, what would be the bandwidth of the analyzer?

Problem 23.50 A design is specified so that it should have a maximum preferred noise criteria value of 35. When tested, the sound values measured at half octaves from 63Hz are, 41dB, 43dB, 46dB, 51dB, 48dB, 48dB, 35dB, 33dB

Problem 23.51 Expecting an increase in business, the Boris and Natasha Detective Agency has hired you to build a soundproof wall to divide their cork lined interrogation room. The dimensions and location of the new rooms are shown below. The loudest voice expected in either room will be 80dB, and a listener can hear voices at pressure levels above 20dB.Will a 2x4” stud wall with gypsum on both sides prevent a listener from hearing another interrogation? (15%)

Problem 23.52 You think that a bend in a hallway of your house reduces noise levels. This “plenum” has the dimensions below. Determine the Transmission Loss for 500 Hz.

Problem 23.53 The following data was recorded for a non-isotropic source. What would be the directivity factor (Q) in direction θ = 60° (use appropriate approximation for calculation of the average value).

θ(deg): 0, 60, 120, 180, 240, 300

Lp(dB): 77, 71, 69, 70, 69, 73

Problem 23.54 Given the transfer function below, find the isolation at 2Hz.

Problem 23.55 There is a small room that contains an average 100 hp diesel engine turning at 7200 rpm. The room is 2.5m by 2.5m and has a ceiling height of 2m. The walls and ceiling of the room are covered by a sound absorption coating (with a coefficient of 0.92). There are two 50cm by 30cm heavy plate glass windows in the walls for safety, and the floor is unpainted concrete.

a) determine the sound power level of the sound source (in dB).

b) assuming uniform directivity, what would be the direct sound pressure level (in dB) would we hear standing 0.5m and 2m away?

c) what is the reverberant sound pressure level (in dB) we would hear standing 0.5 and 2m away?

d) assuming the noise from the engine is relatively uniform in intensity from 200 Hz to 4 KHz, how far apart would two people in the room have to stand to hear each other shouting based on reverberant sound only?

e) what would the longest period of time a worker could be in that room each day based on reverberant sound only?

23.5 References
23.51 Irwin, J.D., and Graf, E.R., Industrial Noise and Vibration Control, Prentice Hall Publishers, 1979.