kinematics, dynamics and vibrations dynamics and...kinematics, dynamics and vibrations ......
TRANSCRIPT
Kinematics, Dynamics and Vibrations
Dr. Mustafa ArafaMechanical Engineering Department
American University in [email protected]
Outline
A. Kinematics of mechanisms
B. Dynamics of mechanisms
C. Rigid body dynamics
D. Natural frequency and resonance
E. Balancing of rotating &reciprocating equipment
F. Forced vibrations (e.g., isolation, force transmission, support motion)
References
• Shigley & Uicker: Theory of Machines and Mechanisms
• Norton: Design of Machinery
• Rao: Mechanical Vibrations
• Beer & Johnston: Vector Mechanics for Engineers
A.Kinematics of mechanisms
Four-bar linkage
Slider-crank mechanism
Scotch yoke mechanism
Four-bar mechanism
2
2
sin
cos
aA
aA
y
x
Obtain coordinates of point A:
222
222
yx
yyxx
BdBc
ABABb
Obtain coordinates of point B:
2 equations in 2 unknowns: Bx and By
Analytical position analysis
Write vector loop equation:
Using complex vectors:
01432 iiii
decebeae
01432 RRRR
Solve for 3 and 4
Velocity analysis
10
Vector loop equation
After solving for position, take the derivative:
01432 iiii
decebeae
0432432 iceibeiae
iii
0432432
iiiceibeiaei
Solve for 3 and 4
Acceleration analysis
01432 iiii
decebeae
0432432
iiiceibeiaei
04433224
243
232
22
iiiiiiceicebeibeaeiae
Write vector loop equation:
Take two derivatives:
B. Dynamics of mechanisms
• Free Body Diagrams
Force analysis
Links 2 and 3Link 2
232323232
1212121212
23212
23212
2
2
2
G
G
G
IFRFR
FRFRT
amFF
amFF
xyyx
xyyx
yyy
xxx
3
32233223
43434343
33243
33243
3
3
3
GPPPP
GP
GP
IFRFR
FRFR
FRFR
amFFF
amFFF
xyyx
xyyx
xyyx
yyyy
xxxx
Link 3 (F23=-F32)
Link 4
• F34=-F43
443434343
141414144
44314
44314
4
4
4
G
G
G
IFRFR
FRFRT
amFF
amFF
xyyx
xyyx
yyy
xxx
In one matrix equation
• We have 9 equations and 9 unknowns
44
4
4
3
3
3
2
2
2
12
14
14
43
43
32
32
12
12
14144334
43432323
32321212
4
4
4
3
3
3
2
2
2
00000
010100000
001010000
00000
000101000
000010100
10000
000001010
000000101
TI
am
am
FRFRI
Fam
Fam
I
am
am
T
F
F
F
F
F
F
F
F
RRRR
RRRR
RRRR
G
G
G
PPPPG
PG
PG
G
G
G
y
x
xyyx
yy
xx
y
x
y
x
y
x
y
x
y
x
xyxy
xyxy
xyxy
C. Rigid body dynamics
Basic equations
Particle dynamics
A thin circular rod is supported in a vertical plane by a bracket at A. A spring of stiffness k = 40 N/m is attached at A and fits loosely on the rod. The spring has an undeformed length equal to the arc of the circle AB. A 200-g collar C (not attached to the spring) can slide without friction. Knowing that the collar is released from rest when = 30⁰, determine the maximum height above point B reached by the collar. Determine also the maximum velocity.
20
Rigid body dynamics
At a forward speed of 30 ft/s, the truck brakes were applied, causing the wheels to stop rotating. It was observed that the truck skidded to a stop in 20 ft.
Determine the magnitude of the normal reaction and the friction force at each wheel as the truck skidded to a stop.
21
Rigid body dynamics
2 2
0 0
2
2
0 30 2 20
v v a x x
a
22.5 ft sa
x GxF ma
Free-body diagram:
22.5A BF F m
y GyF maA BW mg N N
G GM I 7 4 4 5B B A AN F F N
But ,A A B BF N F N
22.5A BN N m
A BN N mg 0.699
0.35 , 0.65A BN mg N mg
, ,A BN N Unknowns:
D. Natural frequency and resonance
Model of a vibrating system
Mass and spring
Mass, spring & damper
Forced vibration
2 DOF systems
Modal Analysis
E. Balancing of rotating & reciprocating equipment
Static Unbalance
• Acting through the center of mass of the rotor
• Can be corrected at a single location (plane)
• Can be detected without spinning the rotor
Couple Unbalance
• Rotor is statically balanced, but dynamically unbalanced
Dynamic Unbalance
• Combination of static & couple unbalance
F. Forced vibrations (e.g., isolation, force transmission, support motion)
Forced vibration
Equation of motion:
Steady-state solution:
Where:
Or:
Frequency response
Support motion
Relative motion:
Support motion
Absolute motion:
Vibration isolation
Force transmitted