final exam dynamic paper

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CONFIDENTIAL

EM/APR 2007/KJM457/421

UNIVERSITI TEKNOLO GI M ARA

FINAL EXAMINATION

COURSE

COURSE CODE

EXAMINATION

TIME

: VECTOR DYNAMICS AND VIBRATION

: KJM457/421

: APRIL 2007

: 3 HOURS

INSTRUCTIONS TO CANDIDATES

1 . This question paper consists of two (2) parts: PAR T A (3 Questions)

PART B (3 Questions)

2.

Answer ALL questions in PART A and any two (2) question in PART B.

3. Answer ALL questions in the Answer Booklet. Start each answer on a new page.

4 . Do not bring any material into the examination room unless permission is given by the

invigilator.

5. Please check to make sure that this examination pack consists of:

i) the Question Paper

ii) an Answ er Booklet - provided by the Faculty

iii) a one - page Appe ndix

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO

This examination paper consists of 8 printed pages

Hak Cipta Universiti Teknologi MARA

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PART A 40 Marks)

Q U E S T I O N 1

The assembly which consists of a sphere of mass 0.5 kg, attached to a hydraulic arm A Bof

negligible mass spins about its axis at a constant rate of C 2= 4 rad/s. At the same instant,

the hydraulic arm AB moving outward at a constant rate of b = 2 m/s and rotates at a

constant rate of , = 2 rad/s about its V-axis shaft. For the instant represented in Figure

Q1 , determine

a) the angular momentum of the assembly with respect to point A , and

b) the percentage of angular kinetic energy of the assembly.

(10 m arks)

(5 marks)

Given:

r = 30m m

/ = 120 mm

Figure Q1


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Q U E S T I O N 2

The plate which has a moving belt is rigidly fitted to the rotating arm ABC. At the instant

shown, the arm rotates at the constant ratea = 5 rad/s while the belt moves at the constant

rate/? = 120 m m/s relative to the plate in the direction as shown in Figure Q2.

a) Determ ine, at this instant, the absolute velocity and acceleration of point E on the link of

the belt.

(10 marks)

b) Does the a cceleration of point on the link of the belt have the Coriolis s terms?

Discuss or explain about your answer.

(2 marks)

c) If we were to determine the acceleration of point H on the link of the belt, will you have

the Coriolis s terms in this acce leration? Discuss or exp lain about your answer. If your

answer is yes, also compu te its value.

(3 marks)

a

= 240 mm

b=

100 mm

c=80 mm

r

= 60 mm

Figure Q2


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EM/APR 2007/KJM457/421

Q U E S T I O N 3

A machine element having a mass of 600 kg is supported by two springs and one damper

as shown. Each spring has a constant of 50 kN/m and the coefficient of damping is 1500

N-s/m.

a) If a periodic force of maximum magnitude eq ual to 200 N is applied to the element at a

frequency of 3.2 Hz, determine the am plitude of vibration at steady state.

(5 marks)

b) If the damping element is now being rem oved, determine the critical mass of this

machine element for which the amplitude ratio is most excessive at this same operating

frequency of 3.2 Hz.

(5 marks)

Figure Q3


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PART B 60 Marks)

QUESTION 1

At the instant shown in Figure Q 1 , collar P is sliding outward at the rate of 1.5 m/s and is

decreasing at 20 m/s

2

relative to arm BC which is rotating at the constant rate 6 = 1 rad/s

relative to arm ABwhich itself is rotating at the constant rate of 4.5 rad/s relative to arm OA.

Knowing that at the same instant, arm OA is rotating at 3 rad/s and is increasing at 5 rad/s

2

,

determine at this instant,

a) the absolute angular velocity and acceleration of arm OA,arm A B,and arm BC,and

(10 marks)

b) the velocity and acceleration of collarP.

(15 marks)

c) the kinetic energy of collar P if the mass of the collar is 0.45 kg. Does this kinetic energy

change with time? Explain.

(5 marks)

Given:

BP= 100 cm

9=30

Figure Q1

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QUESTION 2

The 2-kg plate which has a moving belt is rigidly fitted to the rotating arm ABC of negligible

mass. At the instant shown, the arm rotates at the constant rateco = 5 rad/s while the belt

moves at the constant ratep = 120 mm/s relative to the plate in the direction as shown in

Figure Q2.

a) Determ ine, at this instant, the absolute velocity and acceleration of point

D

on the link of

the belt.

(8 marks)

b) Do pointsDand Hon the link of the belt have the same velocity? Explain.

(2 marks)

c) Determ ine the m ass mom ent of inertia of the plate with respect to the centroidalx axis

i.e. through the centroid C) parallel to the Xaxis and also all products of inertia with

respect to the centroidal planes i.e.planes through the centroid C).

(8 marks)

d) Determine the dynamic reaction at the support at A. Neglect the mass of the moving

belt.

(10 marks)

e) Does the dynamic reaction at the support atA depend on the speed of rotation of arm

ABC and/or the mass of the plate? Explain.

(2 marks)

a

a=240 mm

b=1 0 0 mm

c=80 mm

r=60mm

Figure Q2


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QUESTION 3

a) The square plate ABCD is confined within the slots at A and 6 as shown in

Figure Q3(a). At the instant whenG= 30, slot A has a velocity of 8 m/s to the right.

Determine at this instant the angular velocity of the plate ABCD and the velocity of

slot B. Discuss your selection of the location of the origin O of the global coordinate

system

OXYZ.

(10 marks)

iii) Do you think that we can use the sam e angular velocity found in part (i) to calculate

the velocity of point C? Discuss your answer. If so, then calculate the velocity of

point C with respect to pointA .

(5 marks)

Y

0.3 m

Qm/s

Figure Q3 a)


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b) The 2.1-kg slender rodACE is supported at A as shown in Figure Q3(b). A spring and a

dashpot are attached to the rod at point C and its free end Eand together they hold the

assembly at rest in the position shown. Use / = 600 mm, k = 0.25 kN/m, and

b = 1 0 N s /m .

i) Determine the period of vibration of the assembly when end E of the rod is given a

small displacement and released.

(10 marks)

ii) Classify the type of vibration induced i.e. free or forced, dam ped or unda mped , etc.

(1 m ark)

iii) If a constant torque of magn itude 10 N-m is applied to the rod, determ ine the system

response and calculate the percent maximum overshoot.

(4 marks)

D

Figure Q3 b)

END OF QUESTION PAPER


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APPENDIX

EM/APR 2007/KJM457/421

Kinematics

MainFormulasin ynamics

R

= Ru

R=Ru+QAR

R=Ru

+

2QA(Ru)

+

aAR

+

QA(QAR)

Moment

of

Inertia

Iv

=

I

x

u\ +I

r

uj +I

z

uj -

Angular Momentum

where

H

o

= R

G

A mv

G

+ H

KG =

H

x

i

H

y

j

+

H

z

k

+Izxu

z

u

x

)

H

H

H

X

y

z

Ix

- I yx I y

- I zx 1zy

-Ixv -I

y

i xy

lyz

Iz

Moments

where

M_

G

=

M

x

I +

My

j

M

z

k

M

x

= / ;

My j

M

z

= U

Kinetic Energy

x

\Iy

^ - { 1 2 -

:(X

J

-(Ix-

Iz

ix

Oy O

z

)a>

z

G)

x

o

x

o

y

IzCO

2

z

) - IxyO)

x

O)y+ Iy~,0)

y

C0

2

+ I zxG

Z

G

X


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final exam dynamic paper

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