ME351/Jan11/KVG/VI M1&M2
Machine Dynamics Dynamics and Vibration- Forced Forced Vibration Assignment
1
From a forced vibration response analysis of a motor on foundation (treating as single degree of freedom system), it is found that maximum amplitude occurs at 600 rpm. From free vibration test it is found that damped natural frequency of the system is 11 Hz. The mass of the system is found to be 100 kg. Estimate the stiffness and equivalent viscous damping present in t he system
2
A machine part of mass 1.95 kg vibrates in a viscous medium. Determine the damping coefficient when a harmonic exciting force of 24.46 N results in a resonant amplitude of 1.27 cm with a period of 0.20 s. If the system is excited by a harmonic force of frequency 4 cps, what will be the percentage increase in the amplitude of forced vibration when the viscous medium is removed?
3
A weight attached to a spring of stiffness 525 N/m has a viscous damping device. When the weight is displaced and released, the period of vibration is found to be 1.80 s, and the ratio of consecutive amplitudes is 4.2 to 1.0. Determine the amplitude and phase when a force F = 2cos3t acts on the system.
4
A springmass is excited by a force F 0sint. At resonance resonance the amplitude is measured to be 0.58 cm. At 0.8 resonant frequency, the amplitude is measured to be 0.46 cm. Determine the damping factor of the system.
5
For the system shown in Fig. 1 set up the equation of motion and solve for the steady state amplitude and
x1
c m
phase angle. 6
x2
k Fig. 1
A counter rotating eccentric mass exciter shown in Fig.2 is used to determine the dynamic characteristics of a structure of mass 181.4 kg. At a speed of 900 rpm, a stroboscope shows the eccentric masses to be at t he
m
top at the instant the structure is moving upward through its static equilibrium position, and the corresponding amplitude is 21.6 mm. If the c
k
unbalance of each wheel of the exciter is 0.0921 kg m, determine a) the natural frequency of the structure, (b) the damping factor of the structure, (c) the amplitude at 1200 rpm, and (d) the angular position of the
Fig. 2
eccentrics at the instant the structure is moving upward through its equilibrium position.
7
A balanced wheel supported on springs, as shown in Fig.3, is rotating at 1200 rpm, If a bolt weighing 15 g and located 5 cm from center suddenly comes loose and flies Fig. 3
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off, determine the build up of vibration if the natural frequency of the system is 18 Hz with damping of = 0.01 8
A solid disk of weight 45 N is keyed to the center of a 16 mm. steel shaft 600 mm between bearings. Determine the lowest critical speed. (Assume simply supported).
9
Machinery kept on isolators is vibrating violently in horizontal direction during starting and stopping. No
Fig 4
system details are available. To investigate the problem,
Horizontal excitation
a counter rotating double mass exciter is used to produce a forced vibration in the system (as shown in Figure 4).
Exciter
Excitation is in the horizontal direction. By varying the speed of the exciter, a resonant amplitude was measured
Rubber isolator designed to give stiffness in horizontal direction
as 6 mm. When the exciter speed was increased further (much beyond the natural frequency) the amplitude of
vibration appeared to approach a value of 0.12 mm. Estimate the damping factor of the system. Suggest a method (giving numerical values) to bring down the resonant amplitude to 1 mm. 10 The rotor of a turbine 13.6 kg in mass is supported at the midspan of a shaft with bearings 0.4064 m apart. The rotor is known to have an unbalance of 0.2879 kg cm. Deter mine the forces exerted on the bearings at a speed of 6000 rpm if the diameter of the steel shaft is 2.54 cm. Compare this result with that of the same rotor mounted on a steel shaft of diameter 1.905 cm (assume the shaft to be simply supported at the bearings) 11 For turbines operating above the critical speed, stops are provided to limit the amplitude as it runs through the critical speed. In the turbine of Prob. 8, if the clearance between the 2.54 cm shaft and the stops is 0.0508 cm, and if the eccentricity is 0.0212 cm, determine the time required for the shaft to hit the stops. Assume that the critical speed is reached with zero amplitude.
x
12 Fig.5 represents a simplified diagram of a
v m
springsupported vehicle traveling over a rough road. Determine the equation for the amplitude of m as a function of the speed v and determine the
Fig. 5
y L
most unfavorable speed. 13 The spring of an automobile trailer are compressed 10.16 cm under its weight. Find the critical speed when the trailer is traveling over a road with a profile approximated by a sine wave of
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amplitude 7.62 cm and wave length of 14.64 m. What will be the amplitude of vibration at 70 kmph? (Neglect damping) 14 A machine having a mass of 2250 kg is mounted on a foundation. It is required to verify the operating speed range that can be used, a small double
m
mass exciter is fixed to the machine and amplitude of vibrations were measured at different operating speeds of the exciter (engineering model of the system is given in figure 6). At 690 rpm amplitude of amplitude was
c
k
noted as 0.3 mm, Resonant amplitude noted was 5 mm and the when the frequency of excitation was very high the amplitude tends to a constant value of 1 mm. Estimate the stiffness and damping present in the foundation.
Fig. 6
Comment on suitability of running the machine at 1440 rpm. 15 An aircraft radio weighing 106.75 N is to be isolated from engine vibrations ranging in frequencies from 1600 cpm to 2200 cpm. What static defection must the isolators have for 85% isolation. 16 A refrigerator unit weighing 300 N is to be supported by 3 springs of stiffness k N/mm each. If the unit operates at 720 rpm, what should be the value of the spring constant k if only 10 % of the shaking force of the unit is to be transmitted to the supporting structure? 17 An industrial machine of mass 453.4 kg is supported on isolation pad with a static deflection of 0.508 cm. If the machine has a rotating unbalance of 0.2303 kg m, determine (a) the force transmitted to the floor at 1200 rpm and the dynamical amplitude at this speed. (Assume damping to be negligible). 18 If the machine of problem 14 is mounted on a large concrete block of mass 1136 kg and stiffness of the springs or pads under the block is increased so that the static deflection is still 0.508 cm, what will be dynamical amplitude? 19 An electric motor of mass 68 kg. is mounted on an isolator block of mass 1200 kg and the natural frequency of the total assembly is 160 cpm. With a damping factor of = 0.10. If there is an unbalance in the motor that results in a harmonic force of F = 100sin31.4t, determine the amplitude of vibration of the block and force transmitted to the floor 20 A sensitive instrument with mass 113 kg. Is to be installed at a location where the acceleration 2
is 15.24 cm/s at a frequency of 20Hz. It is proposed to mount the instrument on a rubber pad with the following properties. K =2802 N/cm and = 0.10 What acceleration is transmitted to the instrument?
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21 If the instrument in problem 20. Can only tolerate an acceleration of 2.03 cm/s 2, suggest a solution assuming that the same rubber pad is the only isolator available. Give numerical values to substantiate your solution. 22 A commercial type vibration pickup has a natural frequency of 4.75 cps and a damping factor
= 0.65 what is the lowest frequency that can be measured with a) 1% error and b) 2% error. 23 An undamped vibration pickup having a natural frequency of 1Hz is used to measure a harmonic vibration of 4 Hz. If the amplitude indicated by the pickup (relative amplitude between the pickup mass and the frame) is 0.052 cm, What is the correct amplitude. 24 A manufacturer of vibration measuring instruments gives the following specification for one of its vibration pickups. Frequency range: velocity response
flat from 10 Hz to 1000 HZ
Sensitivity: 0.096 v/cm/s, both volts and velocity in rms values. Amplitude range almost no lower limit to maximum stroke between stops of 0.60 in a) this instrument was used to measure the vibration of a machine with a known frequency of 30 Hz. If a reading of 0.024 v is indicated, determine the rms amplitude. B) could this instrument be used to measure the vibration of machine with the known frequency of 12 Hz. And double amplitude of 0.80 cm ? Give reasons 25 A vibration pickup has a sensitivity of 40 mv/cm/s between f = 10Hz to 2000Hz. If 1g acceleration is maintained over this frequency range what will be the output voltage at a) 10 Hz b) 2000 Hz? 26 A vibration pickup has a sensitivity of 20 mv/cm/s. Assuming that 3 mv (rms) is the accuracy limit of the instrument, determine the upper frequency limit of the instrument for 1g excitation. What voltage would be generated at 200 Hz? 27 The sensitivity of a certain crystal accelerometer is given as 18 pc/g , with its capacitance 450 pF. It is used with a vacuum tube vote meter with connecting cable 5 m long with a capacitance of 50 pF/m. Determine its voltage output per g. 28 Compute the critical speed of a rotor having a disc of 100 N weight at midspan of a horizontal shaft having 10 mm in diameter, 300 mm long supported on spherical bearings at its ends. The rotor is manufactured for a quality of balance G16 (ie e =16 mm/s, e is the linear velocity of cg of the rotor) at a speed of 600 rpm. Compute the maximum dynamic stress on the shaft for a speed of 1200 rpm and 2616 rpm.
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