## Synthesis 3.9
Mobility For Lumped Parameter Mass and Bond Systems
**The JOURNAL OF APPLIED PHYSICS VOLUME 9, JUNE 1938**
* Paper presented at the Symposium on Physics
in the Automotive Industry at Ann Arbor, Michigan,
March 14 and 15, 1938.
By F.A. FIRESTONE
Department of Physics, University of Michigan, at Ann Arbor, Michigan.
1 Introduction
**The
mobility method provides a similar set of distinctive tools, in a form suitable
for application to mechanical vibration problems, as follows:**
**
**
1. A set of conventionalized symbols with which the essential characteristics of
a mechanical system can be set forth in the form of a* *schematic
diagram.
2. The concept of the velocity across mechanical elements (velocity of one end
of the element relative to the other end) as contrasted with the velocity of
points in the system relative to ground; the advantage here is that the
relationship between the velocity across an element and the force through it,
depends only on the characteristics of the element itself and not on the
characteristics of the rest of the system.
3. The use of complex numbers to represent simple harmonic velocities and
forces, both the magnitude and phase of these quantities being represented by
the absolute value and angle of the complex numbers.
4. The concept of the "mobility" of an element (ease of motion), which
is defined as the complex ratio of the velocity across an element to the farce
through the element; simple rules are developed for computing the mobility of
series or parallel combination's of elements and the force can then be found
simply as the velocity divided by the mobility, or the velocity can be found as
the force multiplied by the mobility.
2 The
Mechanical-Electrical Analogy
**The
mechanical/electrical analogy is shown in Figure 1.**
MASS ----------------------------- CAPACITANCE
ELASTICITY ------------------------- INDUCTANCE
**
**
**
Figure 1. The electrical analogs of
the mass and spring.
**
**
**
**Thus by the mobility method we can compute the velocity across each
element in the system and the
velocity of each point in the system, due to the action of an impressed vibratory force or velocity. Not
only can the forced vibration be computed, but the natural frequencies of the
free vibration of systems having but small damping, can be found. While the
method is most easily applied to those systems ***in *which the vibrations of
all parts are parallel to a single line, it can be easily extended to cover many
more complicated cases.
**In
Figure 1 the condenser put analogous to the mass, and the inductance analogous
to the spring. Not only is the formal similarity of the equations complete as
before, but also velocity ***across
is *analogous
to voltage across, and force *through
is *analogous
o current *through. *This results in mechanical elements in series being
represented by electrical elements in series; similarly for elements
in parallel. The analogous electrical circuit is therefore of
a form very similar to the mechanical system itself, as illustrated by analogous
electrical circuit *of
*Figure 2 which represents the
mechanical system in the top of the Figure.
**
****
**
**Figure 2. A mechanical system
(top) with its analogous electrical circuit (bottom).
**
**
**
**We could now work out the
analogous electrical problem and transfer our answer into mechanical terms, but
with the analogy as close as is shown, we ask ourselves what
advantage is to be gained through working with the letters in the electrical
circuit when we could as well work with the letters in the mechanical system
itself, while carrying out the computation in the same manner. The mobility
method of computation is the final result of this
investigation and consists in applying the analogy merely in the development of
the method of computation; in making computations by the mobility method, no
reference is made to the analogy or to electrical systems.**
**
**
**
**
**Figure 3. The relationship
between the vectors representing displacement, velocity, and acceleration,
in simple harmonic motion.
**
**
**
3 The
Mobility Method of Vibration Computation
**For
our purposes we may consider mechanical systems as being constructed of only
three kinds of elements: springs, mechanical resistors, and masses. These
correspond to the three fundamental mechanical properties of matter in bulk:
elasticity, viscosity, and mass. A spring has a ***displacement
*across it proportional to the force through it (Hooke's law); a
mechanical resistor has a *velocity *across
it proportional to the force through it; and a mass has *acceleration *across
it proportional to the force through it (definition of force). The displacement
produced across a spring by unit force through it is a constant of its structure
called the *compliance 1/k *of* *the spring; the
acceleration produced across a mass by unit force acting on it is called the
mass of the mass. While any actual spring has some mass, and every mass has some
elasticity, for purposes of analysis of mechanical systems we consider our
mechanical element, as being "pure" and we take into account any
important additional properties of any structure by adding elements to the
schematic diagram. Regardless of the actual structure of our elements, we will
represent them in our schematic diagrams by the symbols shown in the Figures
which follow.
**If
a simple harmonic force F = F**_{0}e^{iωt} is sent through
our mechanical elements, it will produce a simple harmonic displacement across
the spring and a simple harmonic acceleration across the mass, in fact, if we
confine our attention to the oscillating components of the motion we may say
that each of these elements, will have a simple harmonic displacement x=x_{0}e^{iωt},
velocity v=dx/dt=v_{0}e^{iωt}, (v_{0}=iωx_{0}),
and acceleration a=dv/dt=a_{0}e^{iωt}, (a_{0}=iωv_{0}=-ω^{2}x_{0})
across it. Representing the vibratory motion as the real part of a vector
rotating in the complex plane we have the following relationships between the
displacement, velocity, and acceleration, across any element:
**
**
**
**
**
Here the angular frequency ω=2πf where f is the frequency of the
vibration in cycles per second.**
i=(-1)^{1/2 } The
displacement amplitude x_{o} is a complex constant whose norm or
absolute value |x| represents the actual displacement amplitude across the
element (maximum displacement in the vibration) and whose angle θ,
represents the epoch angle of the displacement (portion of the cycle of motion
in which the displacement started when t=0). The velocity amplitude, v and the
acceleration amplitude a are complex constants whose absolute values and angles
are the actual amplitudes and epoch angles of these quantities. The
relationships expressed in the above amplitude equations are set forth in Figure
3. The acceleration leads the velocity by 90 and the velocity leads the displacement by 90°. The multiplication
of each of these vectors by e^{iωt} causes the set of vectors to
rotate with angular speed in radians/sec. and the real part of each
vector, or its projection on the horizontal real axis, is the instantaneous
acceleration, velocity, and displacement,
respectively. If any one of the three quantities x, v, or a is known, the other
two can be found immediately from the
equations at the right ^{-}above. We will find it most convenient to compute v first,
even though we may be more interested in finding x or a.
**The
mobility z of an element (or group of elements) is by definition the ratio of
the velocity amplitude across the element to the force amplitude through the
element;
**
**Z
:= v/F
Definition 1 **
**The
mobility is a complex number, its absolute value |z| being the ease of motion,
the amount of velocity produced by unit force, and its angle θ**_{z},
being the angle by which the velocity leads the force.
Mobility of Series Components
The system below has a damper, two elastic elements and a mass in series.
**
**
**Figure
4. A series mechanical system. The mobility across a number of elements in
series is the sum of their separate mobilities; z = z**_{1} + z_{2}
+ z_{3} + z_{4}.
**
**
**The
mobility depends only on the structure of the element and on the frequency of
the impressed force or velocity, and is independent of the amount of farce or
velocity, which is impressed since these are proportional to each other. Far
convenience, we may give names to the real and imaginary parts of the mobility,
calling them the ***responsiveness *r and *excitability
x, *respectively. Thus mobility= responsiveness +i excitability; z = *r+ix. *If a system has some responsiveness in its mobility it absorbs
mechanical energy and turns it into heat or waves or some other form
of energy, that is, it really *responds
*to the vibration; while if the _{.}mobility of a system, is a pure
excitability, energy oscillates in and out of the system
but no energy is permanently extracted from the source, that
is, the system merely gets *excited*.
**
**
**Figure
5. A parallel mechanical system; z=1/(1 /z**_{1}+1/z_{2}+1 /z_{3}).
**The
mobility of each kind of element can be easily computed from knowledge of the
fundamental constant of the element and the frequency of the impressed
vibration. If an oscillating force of amplitude
F is sent through a, spring of compliance l/k, the displacement amplitude across
the spring will be x=F/k (from the definition of compliance). The velocity
amplitude across the spring will therefore
be v=i***ω*x=i*ω*F/k The mobility of a spring is therefore
*z = vlF = iωF/Fk = iω/k
*
*
*
**and
is a pure excitability proportional
to the frequency w. On the other hand,
if an oscillating force of amplitude F is sent through a mechanical resistor of
responsiveness r, it will produce a velocity across the resistor v=rF (from the
definition of the responsiveness of a resistor). The mobility of a resistor is therefore** **z=v/F=rF/F=r
****and is a pure responsiveness independent of the frequency.
**
**
**
**Figure 6. How to make a blunder in analyzing a mechanical system: the
three masses shown in the upper diagram are in
parallel since one
terminal of each is
the ground and their movable terminals are connected together; the two hooks on
opposite sides of each mass are not two terminals, they are the same terminal.
The schematic diagram of the structure at the top is shown in the lower part of
the Figure.
**
**
**
**Figure
6
The masses are ground. The upper system is in parallel not series.**
**
**
**If
an oscillating force of amplitude F is impressed on a mass m, it will produce an
acceleration across the mass of amplitude a=F/m (from the definition of force,
otherwise known as Newton's laws of motion). The velocity amplitude across the
mass will therefore be v=a/iw=F/iwm
and the mobility will be z = v/F =1/iwm
= - i/wm,
a pure negative excitability inversely proportional to the frequency.
Summarizing, the mobility of each of our three mechanical elements is simply:
**
**
**
**Element
Mobility
**
**
Spring
z-iw/k
**
**
Mass
z=-i/wm
**
**Resistor
z=r
**
**
**
**Elements
are said to
be connected in series when their
terminals are connected end to end (with not more than two
terminals to any junction point) as
shown in Figure
4. Elements are connected in parallel when their terminals are connected to two common
junction points as shown in
Figure 5. If there is any question as to whether in any specific example
the elements are connected in series
or in parallel,
one should
note that functionally a
series connection results in
the same force acting
through all the elements, while
the velocity across the combination is
the sum
of the velocities across the
individual elements; a
parallel connection results in the same
velocity across all
the elements while
the force through the
combination is the sum
of the forces through the individual
elements. The mobility of
a series combination of elements is
therefore**
** z=F/i=
***(v*_{1}+v_{2}+v_{3}+)/F=z_{1}+z_{2}+z_{3}+,
**and
is simply the
sum of the mobilities of the individual** **elements. The mobility of
a parallel combination of elements is
**
**,**
**and
is the
reciprocal of the sum of the reciprocals of the mobilities of the individual
elements. It should be remembered that. one terminal
of every mass is
the fixed point relative to which the velocity of the mass is measured,
otherwise one may
look at Figure 5 and conclude
erroneously that the masses shown there are in series because they are hooked
end to end; however, the two hooks on each mass are not the two
terminals of the mass, they are the same terminal, there is no relative velocity
between them, they move together, and the other terminal of each mass is
the floor of the laboratory. The schematic
diagram of these masses
is shown in the lower part of Figure 6, indicating that the masses are in
parallel and that the force or the parallel combination of elements is the sum
of the forces required by the individual elements. Since all masses have one
terminal in common (the frame of reference), it is not possible to connect a
number of them in series in any simple manner.
**
**The
mobility method of vibration computation consists in drawing the schematic
diagram of the mechanical system and applying the simple formulas of the last
two paragraphs. The system will usually be merely a series-parallel arrangement
of elements and with the aid of the above. formulas we can easily compute the
mobility of the combination through which the given oscillating force is
applied. The velocity amplitude across this
combination can then be found merely as v=Fz.**
4 The Mobility of an
object of mass m
Mobility z of an object of mass m is defined as the velocity v across the mass
divided by the force F through
**z**_{i}
= v_{i}/F_{i
.
}Definition 1
**Given
the driving force across a mass m**_{i}
**F**_{i}
= F_{oi} exp(iwt).
Equation 1
**
**
**By dividing Definition 1-6 F**_{i}= m_{i}a_{i}= mi d^{2}r_{i}/dt^{2}
(Newton's Third Law) by m_{i} we have
**
**
**F**_{i}/m_{i
}= a_{i} = d^{2}r_{i}/dt^{2}
**=
(F**_{oi}/m_{i}) exp(iwt)
**=
a**_{oi}exp(iwt).
Equation 2
**Where
a**_{oi} = F_{oi}/m_{i}.
**
**
**From
integration of a**_{i} = a_{oi} exp(iwt)
Equation 2, we have veloccity v_{i
}
**
**
**v**_{i}
= dr_{i}/dt. = (-ia_{oi}/w)
exp(iwt)
=v_{oi} exp(iwt).
Equation 3
**
**
**Where
v**_{0i} = -ia_{oi}/w
= -iF_{oi}/(m_{i}w).
**
**
**Using
v**_{i} = v_{oi} exp(iwt)
Equation 3 and F_{i} = F_{oi} exp(iwt)
Equation 2 in z = v/F Definition 1 gives
**z**_{mi}
= (v_{oi} exp(iwt))/(
F_{oi} exp(iwt))
**z**_{mi}
=(-iF_{oi }/ m_{i}w)
exp(iwt))/(
F_{oi} exp(iwt))
^{
}Equation 4
**z**_{mi}
= -i/ m_{i}w
.
5 The
Mobility of a Bond
**Given
an oscillating force through a bond k**_{i} is
**
F**_{i} = F_{io} exp(iwt).
Equation 5
**
**
**Using
F**_{i} = F_{io} exp(iwt) Equation 5 and F_{i} = k x_{i}
Assumption 3-1 (Hooke's Law) gives
**
**
**k
x**_{i} = F_{io} exp(iwt)
x_{i} = (F_{io}/k) exp(iwt).
Equation 6
**
**
**Differentiating
x**_{i} = (F_{oi}/k) exp(iwt).
Equation 5 we have
**
**
**vi
= (iwF**_{i0}/k)exp(iwt)
**=(iw/k)
F**_{io} exp(iwt)
** =
(iw/k)
F**_{i} .
Equation 7
**
**
**Using
vi = (iw/k) Fi Equation 7 and Fi = F**_{io} exp(iwt) Equation 5 z_{i}
= v_{i}/F_{i} Definition
**1****
gives in z**_{i} = v_{i}/F_{i} Law 3-1 gives
**z**_{ki}
= vi/Fi
**z**_{ki}
= (iw/k)
F_{i}/F_{i
}
**z**_{ki}
= iw/ki.
Equation 8
**
**
**
** |