Download Static Machine Forces - University of Dayton

Section 8
Energy Methods
 Energy is the capacity of
doing work.
 Energy methods are useful in analyzing
machines that store energy.
 counterweights
 springs
 flywheels
Energy Methods from Newton
dv
F  ma  m
dt
dv  ds 
dv ds
dv
m  m
 mv
dt  ds 
ds dt
ds
F ds  m v dv
1 2
F s  mv
2
where:
s = displacement
v = velocity
a = acceleration
Linear Work (U12)
 Mechanical energy imposed onto an object
between two states (locations).
 Translating Objects:
U12   Fs ds  Fs s
Fs = force in the direction of motion
s = displacement
Problem 8-1:
A cart is pushed 6 m along the slope by the
300 N force. Determine the work done on the
cart.
300
300 N
100
Rotational Work (U12)
 Mechanical energy imposed onto an object
between two states (angles).
 Rotating Objects
U12   Ts dq  Ts Dq
Ts = torque about the axis of rotation
Dq = displacement
Problem 8-8:
The flywheel on a punch press delivers
3000 ft-lbs of energy to punch a hole. The
flywheel turns two revolutions, for every
punch stroke. Determine the average torque
that needs to be supplied to the punch press.
Potential Energy (PE)
 Amount of energy stored in a stationary
object.
 Potential Energy due to gravity:
PE = Wh
W = weight of the object
h = elevation of the object
Problem 8-10:
An 1200 lb elevator moves from the 2nd floor
to the 5th floor, a distance of 36 ft. Determine
the increase of potential energy of the
elevator.
Potential Energy (PE)
 Energy Stored in a Spring
PE = ½ k x2
k = spring constant (rate)
x = deformation of the spring
Problem 8-18:
A coil spring is used on a bumper, and has a
spring constant of 200 lb/in. Determine the
amount of stored energy, as it is compressed
the first 3 inches. Also, determine the
additional stored energy as it is compressed
two additional inches from 3 to 5 inches.
Kenetic Energy (KE)
 Energy stored in a moving object.
 Translating Objects:
KE = ½ mv2
m = mass of the object
v = linear velocity of the object
Problem 8-20:
Determine the kinetic energy of a truck
weighing 9000 lbs and traveling at 35 mph.
How high would the equivalent amount of
energy lift the truck?
Kenetic Energy (KE)
 Rotating Objects
KE = ½ Iw2
Ts = torque about the axis of rotation
Dq = angular displacement
Problem 8-23:
Determine the kinetic energy of a 2 m slender
rod, with a mass of 10 kg. It is rotated at 20
rpm as shown.
1m
2m
Conservation of Energy
Energy is neither created nor destroyed. It is
converted from form to form.
DPE + DKE = U12
First Law of Thermodynamics
General Energy Equation
Problem 8-26
A dumb waiter and its load have a combined weight
of 600 lb. It is configured with an 800 lb
counterweight as shown. Determine the energy
(work) required from the motor to raise the dumb
waiter 12 ft.
dumb waiter
counterweight
Problem 8-30
A lift that is used to dump the contents of a 55 gallon
drum is shown below. The maximum density of the
contents is12 lbs per gallon. Determine the work
required to lift the container 6 ft. The coefficient of
friction between the collars and the guide rod is 0.1
3 ft
6 ft
Power (P)
 Rate of doing work
dU DU
P

dt
Dt
 Units:
1 hp = 550 ft lb/s
1 Nm = 1 Joule (J)
1 Watt (W) = 1 Nm/s = 1 J/s
1 hp = 746 W
Power
 Translating Objects:
d ( F s)
ds
P
F
 Fv
dt
dt
 Rotating Objects:
d (Tq )
dq
P
T
 Tw
dt
dt
Problem 8-34
A belt sander has a belt speed of 400 fpm. The
coefficient of friction of new, 100 grit paper is
0.5. Determine the power required to operate
the belt when a user
presses down with
30 lb while sanding.
Problem 8-38
A flexible coupling is
rated for 5 hp, when used
on a 1200 rpm shaft.
Determine the allowable
torque for the coupling
Problem 8-42
A jack hammer uses a rotary hydraulic motor with a
cylindrical cam. The cam has a ramp, which causes
the follower to compress a spring, and a step to
release the follower. The spring has a rate of 2000
lb/in and the cam lift is 2 in. At the end of the stroke,
the spring is compressed 0.5 inches. The hammer
weighs 30 lbs. Determine the impact velocity of the
hammer. Also, estimate the hydraulic power needed
to drive the jackhammer if it delivers 5 strokes per
second.
Problem 8-42 (con’t)
Hydraulic
Motor
Splined shaft
Cam follower
Stationary cam
Compression spring
Hammer
Impactor bit
Efficiency (h)
Energy conversion is not an ideal process.
Some energy is lost with friction and
generates heat.
power output work output
h

power input
work input
Problem 8-49
Crushed stone is being moved from a quarry to a
loading dock at a rate of 500 tons/hr. An electric
generator is attached to the system in order to
maintain constant belt speed. Knowing that the
efficiency of the belt/generator system is 70%,
determine the average killowatts developed by the
generator. The belt speed is 10 ft/sec.
240 ft
3 mi