## 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.

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.

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 = F0eiωt 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=x0eiωt, velocity v=dx/dt=v0eiωt, (v0=iωx0), and acceleration a=dv/dt=a0eiωt, (a0=iωv0=-ω2x0) 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 xo 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 eiωt 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 θz, 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 = z1 + z2 + z3 + z4.

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 /z1+1/z2+1 /z3).

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=iωx=iωF/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= (v1+v2+v3+)/F=z1+z2+z3+,

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

zi = vi/Fi .                                                       Definition 1

Given the driving force across a mass mi

Fi = Foi exp(iwt).                                          Equation 1

By dividing Definition 1-6 Fi= miai= mi d2ri/dt2 (Newton's Third Law) by mi we have

Fi/mi = ai = d2ri/dt2

= (Foi/mi) exp(iwt)

= aoiexp(iwt).                                                  Equation 2

Where aoi = Foi/mi.

From integration of ai = aoi exp(iwt) Equation 2, we have veloccity vi

vi = dri/dt. = (-iaoi/w) exp(iwt) =voi exp(iwt).                        Equation 3

Where v0i = -iaoi/w = -iFoi/(miw).

Using vi = voi exp(iwt) Equation 3 and Fi = Foi exp(iwt) Equation 2 in z = v/F Definition 1 gives

zmi = (voi exp(iwt))/( Foi exp(iwt))

zmi =(-iFoi / miw) exp(iwt))/( Foi exp(iwt))                               Equation 4

zmi = -i/ miw .

Given an oscillating force through a bond ki is

Fi = Fio exp(iwt).                                         Equation 5

Using Fi = Fio exp(iwt) Equation 5 and Fi = k xi Assumption 3-1 (Hooke's Law) gives

k xi = Fio exp(iwt) xi = (Fio/k) exp(iwt).                                  Equation 6

Differentiating xi = (Foi/k) exp(iwt). Equation 5 we have

vi = (iwFi0/k)exp(iwt)

=(iw/k) Fio exp(iwt)

= (iw/k) Fi .                                              Equation 7

Using vi = (iw/k) Fi Equation 7 and Fi = Fio exp(iwt) Equation 5 zi = vi/Fi Definition

1 gives in zi = vi/Fi Law 3-1 gives

zki = vi/Fi

zki = (iw/k) Fi/Fi

zki = iw/ki.                                              Equation 8 go to the top