Kinematics, Dynamics, and Vibrations FE Review...
Transcript of Kinematics, Dynamics, and Vibrations FE Review...
Kinematics, Dynamics, and Vibrations
FE Review Session
Dr. David Herrin March 27, 2012
Example 1 A 10 g ball is released vertically from a height of 10 m. The ball strikes a horizontal surface and bounces back. The coefficient of restitution between the surface and the ball is 0.75. The height that the ball will reach after bouncing is most nearly
00
0202
1
mghv
mghmv
PEKE
=
=
=
mh
mghmv
vve
r
rr
r
6.521 2
0
=
=
=
Example 2 A 5 g mass is to be placed on a 50 cm diameter horizontal table that is rotating at 50 rpm. The mass must not slide away from its position. The coefficient of friction between the mass and the table is 0.2. What is the maximum distance that the mass can be placed from the axis of rotation?
5 g
ω
R
ω
R 5 g
mgR
RmNmaF r
072.2
2
==
−=−
=∑
ωµ
ωµ
Example 3 A truck of 4000 kg mass is traveling on a horizontal road at a speed of 95 km/h. At an instant of time its brakes are applied, locking the wheels. The dynamic coefficient of friction between the wheels and the road is 0.42. The stopping distance of the truck is most nearly (A) 55 m (B) 70 m (C) 85 m (D) 100 m
mN
mvx
xNmv
WKEKE
k
k
5.84121
21
2
2
21
==Δ
Δ=
+=
µ
µ
Example 4 A translating and rotating ring of mass 1 kg, angular speed of 500 rpm, and translational speed of 1 m/s is placed on a horizontal surface. The coefficient of friction between the ring and the surface is 0.35. For an outside radius of 3 cm, the time at which skidding stops and rolling begins is most nearly
ω
v0
a
N µN
mg
maNmaFx
=
=∑
µ mgNFy=
=∑ 0
αµ
α2mRNR
IM=
=∑
tatvv o
αωω −=
+=
0
Skidding Stops when ωRv =
Example 5 A stationary uniform rod of length 1 m is struck at its tip by a 3 kg rigid ball moving horizontally with velocity of 8 m/s as shown. The mass of the rod is 7 kg, and the coefficient of restitution between the rod and the ball is 0.75. The velocity of the ball after impact is most nearly
v
1 m
v
ωr
v’
ω’r Before After
r
r
r
vvvve
LmLmvmvL
−
−=
+=
''
'31' 2ω
smv /88.1'=
Example 6 The D’Alembert force is
(A) Force of gravity in France. (B) Force due to inertia. (C) Resisting force due to static friction (D) Atomic force discovered by D’Alembert.
Example 7 A 5 kg pendulum is swung on a 7 m long massless cord from rest at 5° from center. The time required for the pendulum to return to rest is most nearly
Lg
n =ωLgf
π21
=Hzf 1885.0=
fT 1=
Example 8 A 2 kg mass is suspended by a linear undamped spring with a spring constant 3.2 kN/m. The mass is given an initial velocity of 10 m/s from the equilibrium position.
sradmk
n /40==ω
Example 8.1 Calculate the natural frequency of the mass.
nf ωπ21
=
Example 8.2 How long does it take for the mass to complete one complete cycle.
sf
T 156.01==
Example 8.3 What is the maximum deflection of the spring from the equilibrium position?
( ) ( ) ( )tvtxtx nn
n ωω
ω sincos 00 +=
Example 9 A 115 kg motor turns at 1800 rpm, and it is mounted on a pad having a stiffness of 500 kN/m. Due to an unbalanced condition, a periodic force of 85 N is applied in a vertical direction, once per revolution. Neglecting damping and horizontal movement, the amplitude of vibration is
mmD
m
C
mEkF
srad
sradmk
pst
n
f
n
f
opst
f
n
024.0
14.0
21
1
47.1
/5.188
/66
22
==
=
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−
=
−==
=
==
βδ
ω
ω
ω
ωβ
δ
ω
ω
Example 10 A uniform disk of 10 kg mass and 0.5 m diameter rolls without slipping on a flat horizontal surface, as shown. When its horizontal velocity is 50 km/h, the total kinetic energy of the disk is most nearly
22
21
21
ωoo ImvKE +=
Example 11 A homogeneous disk of 5 cm radius and 10 kg mass rotates on an axle AB of length 0.5 m and rotates about a fixed point A. The disk is constrained to roll on a horizontal floor. Given an angular velocity of 30 rad/s in the x direction and -3 rad/s in the y direction, the kinetic energy of the disk is
222
21
21
21
zzyyxx IIIKE ωωω ++=
Example 12 The natural frequency of the system is designed to be ωn = 10 rad/s. The spring constant k2 is half of k1, and the mass is 1 kg. The mass associated with the other components may be assumed to be negligible. For the given natural frequency, the spring constant k1 is most nearly
0=+ xkxm eq
21
1211
kkkeq+=
mkeq
n =ω
Example 13 A two-bar linkage rotates about the pivot point O as shown. The length of members AB and OA are 2.0 m and 2.5 m, respectively. The angular velocity and acceleration of member OA is ωOA = 0.8 rad/s CCW and αOA = 0 rad/s2. The angular velocity of member AB is ωAB = 1.2 rad/s CW, and the acceleration of member AB is αAB = 3 rad/s2 CCW. When the bars are in the position shown, the magnitude of the acceleration of point B is most nearly
( )( )
ABABAB
ABABABABABAB
AOOAOAOAOAA
avrraa
rra
//
//
/
2
+×
+×+××+=
×+××=
ω
αωω
αωω
Gears – Quick Review
Spur Gears" Helical Gears"
Bevel Gears"
Types of Gears
Worm Gear"
Rack and Pinion"
Concepts Review
Terminology
Fundamental Law of Gearing
N1 N2 θ2
θ1 1
2
1
2
NN
rrN ==
2211 θ=θ rr
Gear Ratio Fixed Shafts
2
1
2
1
1
2
1
2
ωω
=θθ
===NN
rrN
Gear Compatibility USCS
rNp2
= Units???"
SI
Nrm 2
=
Idlers
N3
N2 θ1
θ3
3
1
2
1
3
2
1
2
2
3
1
3
NN
NN
NN
==ωω
ωω
=ωω
Gear Ratio Fixed Shafts
2
1
2
1
1
2
1
2
ωω
=θθ
===NN
rrN
θ2
N1
Double Reductions Gear Ratio
Fixed Shafts
⎟⎟⎠
⎞⎜⎜⎝
⎛−==
2
1123 NNnnn
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
4
334 NNnn
42
31
1
4
NNNN
nn
=
Double Reductions Gear Ratio
Fixed Shafts
thdriven tee ofproduct teethdriving ofproduct
input
output =nn
42
31
1
4
NNNN
nn
=
Example 1 - Speed Reducer
502 =N
minrev4032 == nn
Example 2 The compound gear train shown is attached to a motor that drives gear A at ω in clockwise as viewed from below. What is the expression for the angular velocity of gear H in terms of the number of teeth on the gears? What is the direction of rotation of gear H as viewed from below.
Gear Forces
Wt
Wr
t
r
WW
=φtan
mNT
dTWt
22==
SI Units: N, mm, kW
ωω mNH
dHWt
22==
• Grashof Mechanism – One link can perform a full rotation relative to another link
• Grashof Criterion
ba LLLL +<+ minmax
Classification of 4-Bar Linkages
min1 LL =
Crank-Rocker Mechanism
min0 LL =
Drag Link Mechanism
min2 LL =
Double-Rocker Mechanism
ba LLLL +=+ minmax
Also called Crossover-Position Mechanism Change-Point Mechanism
ba LLLL +>+ minmax
• No link can rotate through 360° • Double Rocker Mechanism of the 2nd Kind • Triple-Rocker Mechanism
Non-Grashof Mechanism
Crank-Slider Mechanism
Quick Return Mechanism
Geneva Mechanism