29.9 PRACTICE QUESTIONS

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V14. A machine contains a 60Hz source of vibration that disturbs other machines in the same room. Find the spring coefficient, and natural frequency of an elastomer (with a damping coefficient of .05) that will isolate the vibration source.

V15. There is a large machine that weighs 1000 Kg, and has three legs. We will mount some elastomer under each leg. The graph below shows the characteristics of the isolator. From the graph determine a spring constant (hint: a slope), and determine the natural frequency, and damping ration of the mount.

V16. A piece of electronic equipment is to be isolated from a mounting panel which is vibrating at 8Hz. If 90% isolation is specified what static deflection would you expect? (ans. 0.042 m)

V17. A piece of mechanical equipment contains a 60Hz electric motor driving a reciprocating mechanism which generates motion excitation at 12Hz. The equipment has a total mass of 450 Kg and is mounted on isolators. To establish some criteria regarding the actual isolation an accelerometer is mounted along the vertical axis of the machine. First, the static deflection is measured and found to be 11mm. When the machine is switched off after operation, the output from the accelerometer is captured as a trace, on a storage oscilloscope. The response ratio between two adjacent positive maxima on the trace (i.e., one cycle separation) is 1.65,

a) find the damped natural frequency of the equipment. (ans. 29.77 rad/s)

b) determine the percentage isolation (interpolating fig 9.8is adequate if you note your entry figures). (ans. 72%)

c) if the equipment was operated on a 50Hz supply (motor speed reduced by 17%), explain briefly what changes you would expect in the above results. (ans. <60%)

V18. It is required to provide 80% isolation for a machine using isolation pads conforming to density C specifications in the figure below (note the damping ratio is 0.05). Mounted equipment will generate dynamic forces at 2600 rpm.

a) What would be the minimum mass of the machine if the total isolator pad area is limited to 750 cm2?

b) for the same pad area and a machine mass of 275 kg, what isolation efficiency would be afforded if density A material were used?

V19. Using the data shown in the figure above a total of 4 isolators were selected to isolate a small piece of equipment having a mass of 10kg. The dominant forcing frequency is 60Hz and damping can be neglected.

a) determine the isolation afforded by Isolator B.

b) What would be the result if Isolator C were substituted?

V20. Using the figure above, select an isolator to assure 96% isolation efficiency at a forcing frequency of 112Hz (assume no damping). What would be the static deflection of the isolator selected for a load/isolator of 15N (ans. 0.075cm)

V21. For a system weight/per isolator of 35N, use the figure above determine what isolation efficiency would be obtained if isolator D were used. System forcing frequency is 80Hz. Damping ratio = 0. (ans. 95%)

V22. Cork isolation pads (damping ratio 0.05 use the figure above) are used at the 4 corners of the base of a small machine that weighs 2 KN and generates a forcing frequency of 105Hz. If the pads are 10cm wide by 20cm long, what isolation efficiency would be afforded if they are made of a slab of density C? (ans. 95%)

V23. Four springs, each having a spring constant of 2400 N/m are placed at the 4 corners of a centrally loaded baseplate. What is the system weight if the static deflection is 10mm? (ans. 96N)

V24. An accelerometer is attached to a piece of equipment and connected via a preamplifier to an oscilloscope. The equipment is mounted on vibration isolators. Assuming a single d.o.f. model which we can see is underdamped, what is the system damping ratio if the measured ratio of the amplitude response at two consecutive positive peaks on the decaying harmonic waveform is 1.8? (ans. )

V25. In the design of an accelerometer the requirement is for the maximum measurement error to be limited to 6% at 1/3 of the resonant frequency (i.e. magnification factor 1.06). What would be the maximum damping ratio to meet these requirements? (ans. 0.48)

V26. Using an oscilloscope the ratio of two subsequent maxima was found to be 1.6. Determine the damping ratio. (ans. 0.075) -- EXPLAIN

V27. A motor (mass M1) and controller (mass M2) are mounted on a heavy base (mass M3). Isolators are used to mount all masses as shown below. Dynamic forces generated by the motor are forcing the system so it will be necessary to determine the forces on M2 and the support structure below M3.

a) show the lumped parameter model which represents the system.

b) using the Force/Current Mobility Analogue show the equivalent electrical circuit.

V28. (A long problem) Find the force transmitted by the unbalanced load in the washing machine, to the floor. The rotating mass is 1kg at a distance of 20cm from center, turing at a speed of 30rpm. The assembly consists of the upper drum and motor assembly with mass M2, is rested on a spring, that in turn rests on a large mass. This mass is suspended on a solid floor using a spring/damper combination.

a) Develop the transfer function for the force applied by the eccentric mass to the ground.

b) Determine the input forcing function (from the eccentric mass)

c) Develop the time based reaction using Laplace transforms.

d) Use Fourier transforms to find the effect of the system in steady state.

e) Draw Bode plots for the system.

V29. A machine stands on 6 legs in a corner of a room. in total the machine weighs 10,000 kg, and a vibrational force of 50N is applied at 120Hz by a rotating mass, and a force of 2 N is applied at 60Hz by an AC motor. It has been decided that an isolator will be added to reduce the vibration passed to the floor. 6 isolators will be attached to the legs of the machine. The isolators will be spring damper pairs connected in parallel. (Note: assume the floor movement is negligible). The spring constant is 100KN/m, and the damper is 200KNs/m.

a) Draw a Free Body Diagram of the system.

b) Develop a transfer function for the force input to the machine mass, to the force output applied to the floor.

c) Find the isolation for the two vibrations using the results in b).

d) Find a Laplace input function for the vibration, and determine what the Laplace output function will be.

e) Determine the time based response of the function in d).

f) Draw a Bode Plot for the transfer function in b).

g) Use the Bode plot in f) to find the steady state forces applied to the floor.

h) Use the Bode plot in f) to find the isolation of the vibrations.

i) Design an elastomeric isolator (instead of the spring-damper) to get 90% isolation for the 60Hz force.

V30. Given the car wheel modelled below, relate a change in the height of the wheel to a change in the height of the car. The final result should be a Laplace transfer function of `ycar/yroad'.

V31. Find the time response `x(t)' of a system with a transfer function G(s) that is excited by the force `F(t)'.

V32. A large machine weighs 1000kg and vibrates at 20Hz, design an inertial damper.

V33. A 10kg machine is set on isolation pads and vibrates at 60Hz, what should the natural frequency of an elastomer isolation pad be? If there are 3 pads, what should their spring constant be?

V34. A machine stands on 6 legs in a corner of a room. in total the machine weighs 10,000 kg, and a vibrational force of 50N is applied at 120Hz by a rotating mass, and a force of 2 N is applied at 60Hz by an AC motor. It has been decided that an isolator will be added to reduce the vibration passed to the floor. 6 isolators will be attached to the legs of the machine. The isolators will be spring damper pairs connected in parallel. (Note: assume the floor movement is negligible). The spring constant is 100KN/m, and the damper is 200KNs/m.

a) Draw a Free Body Diagram of the system.

b) Develop a transfer function for the force input to the machine mass, to the force output applied to the floor.

c) Find the isolation for the two vibrations using the results in b).

d) Find a Laplace input function for the vibration, and determine what the Laplace output function will be.

e) Determine the time based response of the function in d).

f) Draw a Bode Plot for the transfer function in b).

g) Use the Bode plot in f) to find the steady state forces applied to the floor.

h) Use the Bode plot in f) to find the isolation of the vibrations.

i) Design an elastomeric isolator (instead of the spring-damper) to get 90% isolation for the 60Hz force.