kinematics, dynamics and vibrations dynamics... · 2010. 3. 7. · kinematics, dynamics and...
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Kinematics, Dynamics and Vibrations
Dr. Mustafa ArafaMechanical Engineering Department
American University in [email protected]
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Kinematics, dynamics and vibration
• Kinematics: study of motion (displacement,
velocity, acceleration, time) without reference to
the cause of motion (i.e. regardless of forces).
• Dynamics: study of forces acting on a body, and
resulting motion.
• Vibration: Oscillatory motion of bodies &
associated forces.
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Outline
A. Kinematics of mechanisms
B. Dynamics
C. Vibration: natural frequency and resonance
D. Balancing
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Kinematics of mechanisms
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Four-bar mechanism
Single Degree of Freedom
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Slider-crank mechanism
Single Degree of Freedom
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Position analysis
Given: a,b,c,d, the ground position, q2.
Find: q3 and q4
a
dq2
b
c
q3
q4
A
B
O2 O4
b
c
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Graphical solution
• Draw an arc of radius b, centered at A
• Draw an arc of radius c, centered at O4
• The intersections are the two possible positions for the linkage, open and crossed
a
dq2
b
c
q3
q4
b
c
A
O2 O4
B1
B2
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Analytical solution
2
2
sin
cos
q
q
aA
aA
y
x
Obtain coordinates of point A:
222
222
yx
yyxx
BdBc
ABABb
Obtain coordinates of point B:
These are 2 equations in 2 unknowns: Bx and By
See “position analysis” on page 242
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Dynamics
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Types of motion
Rectilinear Curvilinear
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OverviewKinematics: equations for constant velocity and acceleration
Kinetics: Newton’s second law of motion: d
F mvdt
For constant mass: F ma
Kinetic energy: 212
T mv
Potential energy: U mgh
212
U kx
Gravity
Elastic
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Friction
P
F
W
N
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Basic equations
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Projectile
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Projectile
v
v
v
a
a
a
x
y
g
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17
Plane Motion of a Rigid Body
x x y y G gF ma F ma M I
O OM I
For rotation about a fixed axis:
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18
Example
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.
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19
Solution
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:
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VibrationsNatural frequency and resonance
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Overview
mx cx kx f m
k c
x
fEquation of motion:
n k m Natural frequency:
2 f
rad/s Hz
/ 2 nc m
Damping
ratio
Damping
coeff.
critical
critical
critical
1 overdamping
1 critical damping
1 underdamping
c c
c c
c c
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Free vibrations with no damping
0mx kx Equation of motion:
mx
1 2cos sinn nx C t C t
Solution:
C1 and C2 are constants to be determined from
initial conditions x0 and v0
0 0cos sinn n nx x t v t
k
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Model of a vibrating system
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Spring and mass
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Spring, mass and damper
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Forced vibration
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Balancing of machinery
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Static unbalance
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Couple unbalance
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Dynamic unbalance