Student Code Number:_____________

Ph.D. Qualifying Exam

Dynamics and Vibrations

January, 2010

T. Pence and B. F. Feeny

Directions: Work all four problems. Note that the problems are EVENLY WEIGHTED.

You may use two books and two pages of notes for reference.

3. A trailer is pulled down a wavy road at a speed v. The road profile is approximated as sinusoidal with a wavelength of L. Modeling the tires as rigid and the trailer as a single degree of freedom mass spring system with base excitation, schematically depicted below, where the deflection of the base is

€

y(t) =Y sin(ωt), the absolute vertical deflection of the mass is x, and the deflection of the mass relative to the road surface is z = x – y. The damper is designed for

€

ζ = 12

.

(a) Write ω in terms of v and L. (b) Design the suspension spring k such that at a frequency

€

ω or higher, the trailer

deflection x has magnitude

€

XY

<12

. Use

€

ζ =12

for this calculation.

(c) The driver finds that the driving on these roads, the shocks often need replacement. Write an expression for the magnitude

€

Fd of the steady-state dynamic force in the

dashpot as a function of r

€

r =ωωn

, where

€

ωn is the undamped natural frequency of the

trailer.

(d) Plot the nondimensional force magnitude, i.e.

€

FdcYωn

, and show the features of the plot

as

€

r→ 0, r = 1, and

€

r→∞.

4. Some tests are performed on the three-mass structure depicted below. The masses are known, but the stiffnesses are unknown. The equations of motion of the system with the displacement coordinates shown is

€

M ˙ ̇ x + C ˙ x + K x = f , where

€

x =

x1x2x3

"

#

$ $ $

%

&

' ' ' and

€

f =

f1(t)f2(t)f3(t)

"

#

$ $ $

%

&

' ' '

are the displacement vector and applied force vector, and where

€

M ,

€

C , and

€

K are the mass, damping and stiffness matrices. Damping is modeled as “proportional” (or “Rayleigh”), and in the form

€

C =αM +βK , where

€

α is unknown and

€

β = 0 . Information on the mode shapes and corresponding undamped modal frequencies (in rad/sec) is partially known as

€

u1 =

a2.44952

"

#

$ $ $

%

&

' ' ' ,

€

u2 = γ

−101

$

%

& & &

'

(

) ) ) ,

€

u3 =

b−2.44955

2

#

$

% % %

&

'

( ( ( ,

€

ω12 = 6.2020,

€

ω22 =16,

€

ω32 = 25.7980

The mode shapes are normalized (units are kg-1/2), except that γ is an unknown normalization constant, and vector elements a and b are unknown. Also, the stiffness matrix K is unknown, but it is known that

€

M =

2m 0 00 m 00 0 m

"

#

$ $ $

%

&

' ' '

, where m = 1/12 kg.

(a) Determine the unknown element a of the first modal vector. (b) Determine normalization constant γ for mode 2. (c) The second mode is excited into a free vibration. The response of one of the coordinates is measured and plotted on the next page. Estimate the modal damping of this mode, i.e.

€

ζ 2 , and then obtain the unknown proportional damping coefficient α. (Recall that undamped modal frequencies squared are given above, and

€

β = 0 .) You may make simplified calculations by assuming

€

ζ 2 is small. (But do not assume

€

ζ 2 = 0!) (d) Now an impulse is applied such that

€

f = (0 F0δ(t) 0)T in where F0 = 2 Ns, when the system starts at rest (zero initial conditions). Find the response

€

x 3(t) of the third mass for the undamped case (for simplicity).