Students are allowed to consult their colleagues to gure out how to approach
the solution. The students must write their own solutions though. Copying
someone else's text of the solution is illegal.
The solution is due by 5 PM on Frida
An industrial press is mounted on a rubber pad to isolate it from its foundation. If the rubber pad
is compressed 5 mm by the self weight of the press, nd the natural frequency of the system.
A helical spring, when xed at one e
6.71 Consider the eigenvalue problem
[k] night? = 6
[m]=|:cfw_2) and [k] [j '1]
Find the natural frequencies and mode shapes of the system:
a. by solving the equation
[mr'm Fun? = 6'
b. by solving the equation
[-w2[k]"[m] + [111x = 6
6.16.5 Derive the equations of motion. using Newton's second law of motion. for each of the sys-
tems shown in Figs. 6.18 to 6.22.
6.24 Find the flexibility and stiffness inuence coelTIcients of the system shown in Fig. 6.29.
How do we run this course?
Lecturing much reduced
Peer-instruction (yes, that means you)
A clicker is required for this class
Its not optional, using it will be a percentage of your
Clickers are used to help both YOU and ME figure out
2.8 An automobile having a mass of 2,000 kg deects its suspension springs 0.02 rn under static
conditions. Determine the natural frequency of the automobile in the vertical direction by
assuming damping to be negligible.
2.9 Find the natural fr
3.8 A mass m is suspended from a spring of stiffness 4000 N/m and is subjected to a harmonic
force having an amplitude of 100 N and a frequency of 5 Hz. The amplitude of the forced
motion of the mass is observed to be 20 mm. Find the value of m
The equation of motion of free Vibration
E-EX 2 0
+7ix : 0
-X : 0
Z7UmL/sj 1 am acmlcvomdu od' 11M
for murals horiunit'al (Cd
of WWIMIA. D'sj / how larje
is flat Asrlacfmmf Wuh?
Rafael bf ~Hn
the characterisric equation a3 + mi: 0, whose solutions are ot=iim,. Then,
using complex numbers we have 3 ~
._ ' 4&5? r
xi!) = Ae" + Betm" xctl hx c J .4)
where A and fare two complex constants determined by the initial condi-
Nn cur-I1 rl'ldr 1
6. Consider the illustrated system. A rigid ring with mass m is connected via four massless. inextensible
wires to a central hub (radius r). Each wire has a tension T. If the motion of the hub is given by
90) = 90 cos wt (60 small), determine the angular
I64 Applied Struccural and mechanical vibratlns: Theoaffid methods
0 2 4 G 8 10
FigureSJ Response to step function. AF E- 'f
where in this case we have a = 35(1) Inn). and 31 +152 = mi . Ta king
' 1. If x = aleiw' + agem, x(0) = 2, and xtol = 3. what are a and at: equal to?
2. If x = b1c05(wt) + b; sin(wt), with the same initial conditions as in problem 1, what are In and in
3. A sinusoid, x(t) = a cos(7t + (it). has initial conditions
4. Find the natural frequencies and eigenveetors for the system shown in Figure 4.1. ml = 10 kg. m; =
20 ngc. = 100 N/m, 1:; = 100 mm, and k3 = so N/m.
:-v-xl :l- x:
kl . 0.01.]. k: . ojojolo h
27. Find the natural frequencies and th
m: 2000 kg . 85L. - -
w: (9/sst)% = 9'31)% :22. 1472 rad/sec
| Leif it be measured 53mm
'Hwe Pasibon of mass at which
re SPHan axe uthrcfcked.
Ezuai'uon 03C mafn'on cfw_5
when: 656 (R. + k;) = w sin 9 .
Thu: 533- (Et) becomes 111: +
Find the total response of a singledegreeof-freedom system with m = IO kg, 6 = 20 Ns/m,
k = 4000 N/Ill, x0 = 0.01 m, and in = 0 under the following conditions:
a. An external force F (r) = F0 cos to facts on the system with F0 = 100 N and m = 10
Free Vibration with Coulomb Damping
In many mechanical systems, Coulomb or dryfriction dampers are used because of their
mechanical simplicity and convenience [2.9]. Also, in vibrating structures, whenever the
components slide relative to each other, dryf
The equation of motion of free Vibration
is F K disrlaCWWt' mafbi'l'm 3.
Fahd bf ~an Mtderodim
meow/m3 :5 A0 second-s.
A Hug g.
Inn I'm m
i m3 : o lkt+wim=o
T d, [X[u~%_b;
[MOI/J GE'T WET l"
: [OEM'EL] 1
1X|=g X "
SE101C/ MAE130C Spring 2010
MECHANICS III: VIBRATIONS
Dr. Van Den Einde Page 1
SE101C/MAE130C is designed to teach free and forced vibrations of undamped and damped one-degree of freedom systems; vibration isolation, analysis of discrete M