J.H. Janssen 329 
Moreover, only bending waves are tolerated inour model; i.e., it is supposed 
that Eq. (12) suffices to describe the wave phenomena in plates and beams. This 
is not necessarily a serious restriction for at least two reasons: (1) for very 
general mechanical structures similar statements can be made as those derived 
below) [10]; (2) results of measurements agree reasonably with the very simple 
theory. 
Let us consider a specific example of a mechanical structure, a beam. It is 
well known that a beam can vibrate laterally at an infinite number of natural 
frequencies. For each frequency there is a definite shape in which the beam 
will deflect while vibrating harmonically; this shape iscalled a normal mode 
of vibration of the beam. Such a normal mode will be designated by @, ,the sign ” 
indicating the maximum value (amplitude) of the displacement of the beam at 
any point x, the instantaneous value ‘u varying sinusoidally with time t; v is the 
number of the mode. 
The problem now is to describe the forced vibrations of this beam when 
excited by a general harmonic force. The directions of the force(s) and of the 
displacements are supposed to be parallel. In most instances, the force will be 
acting near a certain point A, e.g., a resilient mount; this point will be given 
some mathematical preference below. If p is the force as a function of the coor- 
dinates on the surface S$ of the beam, the total force F is given by 
F = fpdS = Fe! (16) 
The velocity of the beam caused at the point C by the action of the force at A 
can be written as the product of this force and the sum of an infinite series of 
"mechanical admittance" or mobility terms: 
A A 
(=F ur(C)/a,(A) (17) 
a > alAYO LAD 
The meaning of the various symbols is explained below. As is usual, the real 
part of the complex v(C) is the measurable quantity, "velocity at C." 
The numerator of Eq. (17) contains only the ratio of the values at C and A 
of the vth natural mode. This ratio is equal to ¢,(C)/¢,(A), where ¢, is the 
characteristic function describing a natural mode. With the aid of tables of these 
functions [11] this ratio can be determined. 
In the denominator of Eq. (17), the so-called "quadratic amplitude transform 
factors" q,(A) are given by 
R32 
up 
02(A) 
bo. 
qv(A) = (18) 
This name suggests the same action so weil known from electric transformers. 
These impedances are transformed in the ratio 1 to n”, while currents or volt- 
ages are transformed in the ratio 1 to n. In q,(A), a more general seesaw action 
is described: the mechanical impedance felt by an exciting force at one end of a 
seesaw due to a load at the other end is proportional to the square of the ratio 
of the distance load-pivot to the distance force-pivot. The bar over 4? indicates 
the mean value over the homogeneous beam of @ squared. In [11], the charac- 
teristic functions are normalized in such a way that 
