Eqn of Motion

1
Forced Vibrations - Summary
Harmonic forcing only
Eqn of Motion
Or
Solution
2
The Steady State Solution
where
3
No damping case
when no damping
?
Two solutions
(Use when )
And
(Use when )
Using the two possible solutions in the above way
keeps X positive for all w
(Alternative is to use the first equation for all
w and allow X to change sign at w gt wn but this
can get a little confusing. What is actually
happening is a phase change of 180 degrees
which we are generally not that interested in.)
4
Rotor Excitation
Equation of motion
Like before, but the forcing function is
And the driven mass is M (not m), so
Solution as before, e.g.
Note M is the TOTAL mass including m
if define
5
Prob 8/69
Total mass of the device 10 kg
i.e. M 10 kg
Determine the two possible values of the
equivalent spring stiffness k for the mounting to
permit the amplitude of the force transmitted to
the fixed mounting due to the imbalance to be
1500 N at a speed of 1800 rpm
Finished?
No!
This is the force driving the device.
If the device is connected to the fixed mounting
through a spring, the force transmitted to the
mounting is
which has a maximum value of
where X is the magnitude of the displacement
6
Prob 8/69
From before
Or, with no damping
OR
Thus
AND
(Aside what actually happens is the transmitted
force is in phase with forcing force for
and out of phase by 180o for )
We want the max force transmitted to be 1500 N,
so there are two acceptable solutions
or
7
Prob 8/69
or
?
or
If the machine is to be run at 1800 rpm (188.5
rads/s), then we need
or
With M 10 kg and
We get
or
i.e. we can choose a stiff spring and run the
machine BELOW the natural freq
or we can choose a soft spring and run the
machine ABOVE the natural freq
8
Force Transmission to Base
In the last problem, considered force
transmission to the ground (kx)
With vibrating machinery this can be an important
consideration, so we mount such equipment on
isolating mounts
What is the force transmitted if the mount has
damping?
It is the sum of the spring and dashpot forces.
These are always 90o out of phase, so can
represent the time-varying force transmitted as
In general, the maximum value over all q of
is
(prove!)
So the maximum force transmitted is
9
Force Transmission to Base
or
From before
So the force transmission ratio is
10
Force Transmission to Base
A little different from the previous plot we had
for
Dont worry (be happy)
especially here
11
Force Transmission to Base
For a force transmission ratio lt 1 (i.e.
vibration isolation), need
Since the operating frequency of the machine is
probably fixed, this means we need to choose a
soft enough spring so the above is true
Also, in the isolation region ( )
a lower value of z lowers
However, the machine has to pass the resonant
frequency on its way to its operating speed from
start-up, so z cannot be too low!
If the force transmission is not satisfactory
with standard mount materials, another option is
to add weight to the machine, or mount it
directly on another mass which is then placed on
the mounting pads. This will also bring down wn.
12
Harmonic Base Excitation
Base moves. How does the mass move?
e.g. earthquakes, hilly road acting on car etc.
Forces in the spring and dashpot are proportional
to the RELATIVE displacement and velocity
respectively
13
Harmonic Base Excitation
Equation of motion
?
Or
Or
Two terms on RHS so a little different from
before, but RHS can be re-written
(dont worry about the a - we could get it if we
wanted)
which is the same as before (apart from the a) if
we let
14
Harmonic Base Excitation
?
From before
So the displacement transmission ratio is
15
Harmonic Base Excitation
Displacement transmission ratio
Notice this is exactly the same expression as we
got for the force transmission ratio of a
rotating machine!! Even though this is a
different problem!
Exactly same picture except y-axis is X/Y instead
of FT/F0 this time
Not surprisingly the design criteria for limiting
the displacement transmitted to the mass here are
the same as for limiting the force transmitted in
the previous problem.
i.e. a soft enough spring, limited damping, and
add mass if needed
16
Harmonic Base Excitation
No isolation
soft spring
17
Example
Car weighing 3000 lbs drives over a road with
sinusoidal profile, as shown
Design the suspension (i.e. pick k and c) so
that
1. The vibration amplitude of the car is lt 14 at
all speeds, and
2. The vibration amplitude of the car is lt 4 at
55 mph
18
Example
Equation of road profile (base motion)
At speed v, distance is svt, and Y 8 2/3ft
So the forcing frequency of the base motion is
Vibration amplitude lt 14 at all speeds
at all speeds
Use the graph
gives about the best soln to this criterion
19
2. Vibration amplitude lt 4 at 55 mph
Example
at 55 mph
With z 0.375
when
(approx)
So wn must be
Now solve for required k
20
Example
Finally solve for c
So
At what speed (approx) will the worst vibrations
occur?
21
Prof. McCarthys first car
The ultra-cool Morris Minor
Even the name evokes excitement
On Irish roads, had natural frequency at about 45
mph a WHOLE lot of shaking going on
Excellent excuse if stopped doing 55 mph in a 45
mph zone
22
Harmonic Base Excitation Motion Relative to Base
Sometimes the motion relative to the base is of
interest
Introducing the relative displacement z x y,
the equation of motion
becomes, in terms of the relative displacement
Or
This case is dealt with on page 625 of the book
with slight difference in nomenclature
Base motion
instead of
instead of
Relative motion variable
Writing the solution on page 625 in the current
nomenclature
Or
23
Prob 8/61
The motion of the outer cart is varying
sinusoidally as shown.
For what range of w is the amplitude of the
motion of the mass m, relative to the cart less
than 2b?
?
when no damping
As before there are two solutions
(Use when )
And
(Use when )
24
Prob 8/61
We want
?
for
?
for
25
Recitations as usual tomorrow (Wednesday)
No lecture this Thursday (May 1st)
HW13 due Friday 1200. Hand in to Stephanie on
7th floor.
No recitations next week
Review lecture next Tuesday (May 6th). Exam Wed
May 7th, 9am.
Good luck on the final, best wishes for your
future career, and be sure to visit Ireland
sometime!