Sheet4 Vibration Ahmedawad

HarrrEonically Excited !'ibration of Singtre Degree of F'reedom
Systems
4.1. A spring-mass-damper system, of stiffiress k, mass m and coefficient of viscous
damping c, is excit"a Uy a harmonic force of amplitude Fo and frequency alZx'
a) Derive an expression for the energy dissipated by darnping during one cycle of
steady state vibration in terms of k, rt, c, Fs and tD'
b) Derive an expression 1br the energy input to the system by '.-hc excitation
during one cycle o1'steady state vibration in tertls of k, m, c, F1v zLtrd o)'
c) what do yo,-i d".lu"e fiorn the expressions deri'red in (a) and (b)'/
4.2. A spring-mass-damper sysiem, of stiffness 800 N/mm, mass 80 kg and coefficient
of viscous damping 1.6 N.s/mm, is excited by a harmonic force given by,
F : 1 200 sin Zrcft NI,
where f is the exciting frequency in Hz and t is time in seconds'
a) Write down the equation of motion, at steady state, for each of the following
vaiues of the exciting frequencY:
i- f :12.732H2.
ii- f : 15.9 i5 Hziii-
f:i9.099H2.
b) For steady state vibration, determine the maximum anrplitude and the exciting
. frequency. which causes this amplitude.
4.3. A machine of mass 80 kg is mounied on an elastic support, which pei-mits vertical
displacement only. The support has stiffness 800 N/mrn and coefTjcient ofequivalent
viscous damping i.O X.rl**. The machine iras an unbalanced rotor, of
unbalance 120 kg.mm, rotating at a speed hI.
a) For steady state vibration, determine the amplitude and the phase angle
between the systern vibration and the rotor unbalance for each of the following
values of the rotor sPeed:
i- N :763.9 rPm'
ii- N:954.9 rpm.
iii- N: 1146 rpnt.
b) For steady state vibration, determine the maximum amplitude and the rotor
speed, which causes this aniplitude.
4.4. A machine of mass 65 kg is rnounted on an elastic support, which permits vertical
dispiacement only. The support has stif'fness 720 N/mm and coefficient of
equivaient viscous damping ijos N.s/rn. The machine has unbalanced rotor' when
this rotor was rotating at 804 ,p*, the amplitude of steady state vibration of the
machine was 1 mm. Determine:
a) The unbalance of the rotor'
b) The rotor speed, which produces maximurn amplitude.
,./b') The rotor speed, above which the increase of the suppott dampins increases
'; the force transmitted through the support'
..;/
4.5. A machine of mass 50 kg is rnounted on an elastic support, which perrnits vertical
displacement only. The machine has an unbalanced rotor. The support has stiffiness
500 N/mm. 'When the rotor was rotating atdifferent uniform speeds, the maximum
^*pii*Je-of steady state vibration of the machine was 3 mm, and this amplitude
occurred at964.6 rpm. Determine the unbalance of the rotor'
If the rotor is rotating al l33l 1pm, determine, for steady state vibration:
a) 'fhe amPlitude of the machine'
b) The phase angle between the machine vibration and the rotor unbalance'
c) The amplit Ol of the force transmitted through the support.
d) The phase angle between the force transmitted through the support and the
rotor unbaiancc.
e) The energy dissipated in damping during one cyctre'

The system shown in Fig'
given by,
T:800 cos 70t N.nt,
where t is time in seconds. For steady state vibration, determine:
a) The equation of motion of the roci'
ti The dynamic force in spring k1 as function of time'
"i The dynamic force in spring k2 as function of time'
d) The dynamic force in the damper as function of time'
The sy,stem shown in Fig. 2 is excited by a vertical harmonic force, acting on mass
m2, with amplitude 3 N and frequency 0'7958 Hz' For steady state vibration'
determine:
a) The angular amplitude of the puiley'
b) The amplitude of the center of the pulley'
c) The amPlitude of mass m2'
d) The phase angle betweenlhe systern motion and the exciting force'
"j The maximum dynamic force in the spring'
0 The maxifilum dynarnic force in the damper'
In the system shown in Fi