3 Fundamental Equations Employed in Ultrasonic
3.5 TRANSVERSE-WAVE EQUATION
3.6.1 c lamped –f Ree u nifoRm B aR
A clamped–free bar, such as a cantilever, is rigidly mounted at one end so that it is perfectly restrained from moving at that end but unrestrained and free to vibrate at the other end. Under these conditions, at the clamped end (x = 0), the displacement ξ is zero and the slope ∂ξ/∂x is zero for all values of t. At the free end (x = ℓ), no external force is applied and both the bending moment and the shearing force are zero in a plane through the end face of the bar. Therefore, at the free end (x = ℓ), ∂2ξ/∂x2 = 0 and ∂3ξ/∂x3 = 0.
Applying the boundary conditions at x = 0 to Equation 3.56, we obtain ξ = =0 A C+ or A= −C
and
∂
∂ξ= = + = −
x 0 B D or B D
from which
Applying the boundary conditions at x = ℓ to Equation 3.57 gives
∂ which can be true only at discrete frequencies. These discrete, or allowed, frequen-cies are determined by dividing Equation 3.58a by Equation 3.59a, as follows:
cosh( / ) cos( / )
Fundamental Equations Employed in Ultrasonic Design and Applications 123
From the identities
cosh2θ=cosh2θ+sinh2θ cos2θ=cos2θ−sin2θ Equation 3.60a may be rewritten
cosh2 sinh2 cos sin2
2 2 2 2
ω ω ω ω
v+ v 2 v− v = −1 (3.61)
Dividing both sides of Equation 3.61 by cosh2(ωℓ/2v) cos2(ωℓ/2v) gives
1 2 1
The intersections of the curves corresponding to cot ωℓ/2v and ±tanh ωℓ/2v plotted against (ωℓ/2v) give the frequencies corresponding to the allowed modes of vibration of the clamped–free uniform bar. These curves are plotted in Figure 3.9.
−2.0
FIGURE 3.9 Graphical determination of allowed frequencies for flexural resonances in fixed–free uniform bars.
The intersections occur at ω π
2 4 1 194 2 988 5 7
v= ( . , . , , , ) … (3.63)
The numerals 5, 7, … follow from the fact that the hyperbolic tangent is nearly unity for all angles greater than π.
Since
v= ω κc or v2=ω κc =2π κfc it follows from Equation 3.63 that
f c
= π κ …
8 2 1 194 2 988 5 72 2 2 2
( . , . , , , ) (3.64)
are allowed resonant frequencies of a clamped–free bar vibrating in flexure.
The nodes of the clamped–free bar (corresponding to ξ = 0) are not evenly dis-tributed along the bar. Since the hypothesis states that the bar is clamped at one end (x = 0), the clamped end is a nodal position. Here, both ξ = 0 and ∂ξ/∂x = 0. This is the only node occurring at the fundamental frequency. One additional node occurs near the free end at the first overtone. This node is characterized by ξ = 0 but ∂ξ/∂x ≠ 0 and ∂2ξ/∂x2≠ 0. Therefore, two different types of nodes exist for this mode. The sec-ond overtone includes nodes of these two types, one at the clamped end and the other near the free end, plus a third node characterized by ξ = 0 and ∂2ξ/∂x2= 0.
Each successive overtone adds an additional node characterized by the conditions ξ = 0 and ∂2ξ/∂x2= 0, while also retaining the two types of nodes near the extremities of the lower modes.
These intermediate nodes associated with the second overtone and higher modes (f3, f4, f5,…,fn) can be determined by the equations
Fundamental Equations Employed in Ultrasonic Design and Applications 125
Dividing Equation 3.57b by Equation 3.58b and simplifying gives 2coshωvsinω 2sinhω cosω
Equation 3.65 is solved graphically as was Equation 3.62 for discrete values of x cor-responding to y = 0. The quantity ωx/v is greater than π and lies only in the quadrants where tan ωx/v is positive, that is, where
ωx π π π
corresponding to the second, third, fourth, and so on, overtones. For example, the nodal positions for the second overtone are as follows:
First nodal position: x = 0 Second nodal position:
The third nodal position of the second overtone is not accurately determined by Equation 3.65 because the condition that ∂2ξ/∂x2 = 0, on which Equation 3.65 is based, does not exist at the node nearest the free end. Rather ∂2ξ/∂x2 = 0 occurs at the free end. Neither is ∂ξ/∂x = 0. However, ξ must be zero for a node to exist. Equations 3.63a and 3.66 may be used to obtain a first approximation to the location of the third node of the second overtone, and this approximate value is x = 0.9 ℓ, which would be the true value if ∂2ξ/∂x2 = 0 at that position. The actual location of the third node is at x = 0.868 ℓ.
Actual nodal positions, corresponding frequencies, and phase velocities are given in Table 3.1. The modes of transverse vibrations of a clamped–free bar are shown in Figure 3.10.
3.6.2 fRee–fRee BaR (BaR fRee at Both endS)
Again referring to the general solution of the wave equation for a bar in transverse vibration (Equation 3.56), the boundary conditions for a free–free bar are ∂2ξ/∂x2= 0 and ∂3ξ/∂x3= 0 at both x = 0 and x = ℓ.
Applying these conditions at x = 0,
∂
∂ = + + − −
2 2
2
2 0 0 0
ξ ω φ ω
x t
v A B C D
cos( ) ( cosh sinh cos siin )0 =0
TABLE 3.1
Locations of Nodes and Corresponding Phase Velocities of a Clamped–Free Uniform Bar in Transverse Vibration
Frequency Phase Velocity Nodal Positions
fl v1 x = 0
f2(6.267f1) 2.50v1 x = 0, 0.774 ℓ f3(17.55f1) 4.18v1 x = 0, 0.58 ℓ, 0.868 ℓ f4(34.39f1) 5.87v1 x = 0, 0.356 ℓ, 0.644 ℓ, 0.905 ℓ
0
0
0
0
f1
f2
f3
f4 0.774 ℓ
0.5 ℓ 0.868 ℓ
0.356 ℓ 0.644 ℓ 0.905 ℓ ℓ
ℓ ℓ ℓ
FIGURE 3.10 The modes of transverse vibration of a clamped–free uniform bar.
Fundamental Equations Employed in Ultrasonic Design and Applications 127
is the equation of motion of a free–free bar in flexure.
At x = ℓ, Using the procedure in Section 3.6.1 for the clamped–free bar, we obtain
cosh ω cos ω
v v = 1
tan ω tanh ω
2v = ± 2v (3.70)
from which the allowed frequencies are obtained. The allowed frequencies corre-sponding to the intersection of tan(ωℓ/2v) with ±tanh (ωℓ/2v) plotted against ωℓ/2v are given by
As with the clamped–free bar, the overtones are not harmonics. The nodes are of two types. Those nearest the ends are characterized by ξ = 0, but ∂ξ/∂x ≠ 0 and
∂2ξ/∂x2 ≠ 0. The nodes between those nearest the ends are characterized by ξ = 0 and ∂2ξ/∂x2= 0, and the same procedure as used to determine the locations of the
nodes of a clamped–free bar is used here. In comparison with the clamped–free bar, two nodes instead of one are associated with the fundamental frequency. The first overtone produces three, and so on. Table 3.2 gives the relative frequencies, phase velocities, and positions of nodes of the fundamental and the first three overtones.
The corresponding modes are plotted in Figure 3.11. As one would expect, the positions of the nodes are symmetrical about the center of the bar for all modes. The shapes of the displacement curves for all modes corresponding to odd- numbered frequencies, including the fundamental f1, are symmetrical about the center.
However, the modes corresponding to even-numbered frequencies are asymmetri-cal, that is, not symmetrical about the center. The latter modes exhibit true nodes at the center.
TABLE 3.2
Vibration Characteristics of a Free–Free Uniform Bar in Flexure
Frequency Phase Velocity Nodal Positions
f1 v1 x = 0.224ℓ, 0.776ℓ
f2= (2.756f1) 1.66v1 x = 0.132ℓ, 0.5ℓ, 0.868ℓ
f3= (5.404f1) 2.32v1 x = 0.0942ℓ, 0.356ℓ, 0.644ℓ, 0.906ℓ f4= (8.933f1) 2.99v1 x = 0.073ℓ, 0.277ℓ, 0.5ℓ, 0.723ℓ, 0.927ℓ
0.224 ℓ 0.776 ℓ
0.132 ℓ 0.5 ℓ 0.868 ℓ
0.094 ℓ 0.356 ℓ 0.644 ℓ 0.906 ℓ
0.073 ℓ 0.277 ℓ 0.5 ℓ 0.723 ℓ 0.927 ℓ
f1
f2
f3
f4 ℓ
ℓ
ℓ ℓ
FIGURE 3.11 The modes of transverse vibrations of a free–free uniform bar.
Fundamental Equations Employed in Ultrasonic Design and Applications 129
3.6.3 clamped–clamped BaR (BaR clamped at Both endS)
A clamped–clamped bar, that is, a bar which is rigidly clamped at each end, will vibrate at the same frequencies as the free–free bar of equal dimensions. Here, again, two types of nodes occur. The nodes at the ends are characterized by ξ = 0 and
∂ξ/∂x = 0. Those in between are characterized by ξ = 0 and ∂2ξ/∂x2= 0. The nodal positions do not coincide with those of the free–free bar but are as shown in Table 3.3.
3.6.4 effect of GeometRy on tRanSVeRSe ViBRationS of BaRS
The effect of geometry on the vibratory characteristics of a bar depends upon the corresponding distribution of the mass of the bar.
The fundamental resonant frequencies of various tapered bars, which are rigidly mounted at their large end, are listed in Table 3.4. In Table 3.4, a is the radius of the cone at the base, in meter, Y is Young’s modulus, ρ is the density (kg/m3), and b is the thickness of the bar at the base in the direction of vibration, in meter.
Further discussion of wave equations, vibrations and waves in rods and bars are given in references [1,4].
TABLE 3.3
Locations of Nodes and Corresponding Phase Velocities of a Fixed–Fixed Uniform Bar Resonant in Transverse Vibration
Frequency Phase Velocity Nodal Positions
fl v1 0, ℓ
f2= (2.756f1) 1.66v1 0, 0.5ℓ, ℓ
f3= (5.404f1) 2.32v1 0, 0.357ℓ, 0.643ℓ, ℓ f4= (8.933f1) 2.99v1 0, 0.277ℓ, 0.5ℓ, 0.723ℓ, ℓ f5= (13.3446f1) 3.653v1 0, 0.227ℓ, 0.409ℓ, 0.591ℓ, 0.773ℓ, ℓ
TABLE 3.4
Resonant Frequencies of Tapered Bars Rigidly Mounted at the Large End
Geometry Direction of Vibration Frequency (Hz)
Wedge-shaped bar Normal to parallel sides 1 14
12
2
. 2
Yb
ρ
Wedge-shaped bar Parallel to parallel sides 0 85
12
2
. 2
Yb
ρ
Conical bar 1 39
4
2
. 2
Ya
ρ
3.7 PLATE WAVES 3.7.1 GeneRal
The form of the wave equation for plates is
− ∂
∂ =
4 2
2 2
2
3 1 0
ξ ρ( σ ) ξ
Yh t (3.73)
where
ξ is the displacement amplitude
ρ is the density of the material of the plate Y is the modulus of elasticity (Young’s modulus) σ is the Poisson’s ratio
h is the half-thickness of the plate t is time
The equation for the bar vibrating in flexure, Equation 3.50 (Section 3.5), may be derived from Equation 3.73 by applying the boundary conditions of a thin bar in flexure. The thin bar in flexure is not constrained laterally by adjacent material.
The stresses are assumed to be directed in planes that are parallel to the direction of bending. The directions of stress are (1) parallel to the length of the bar and (2) parallel to the direction of bending.
The stresses developed in plates in flexure are more complicated than those for the flexural bar because each incremental element in a plate under stress may be subjected to constraints in all directions. Hence, the term involving Poisson’s ratio, which is incorporated in Equation 3.73.
Solutions to the plate equation allow an infinite number of vibrational conditions.
For instance, the simple harmonic solution given by Morse [5] is
ξ( , )r φ =sincos (mΦ)[AJm( )γr +BIm( )]γr (3.74) where
Jm is a cylindrical Bessel function Im is a hyperbolic Bessel function γ is the wavelength constant for a plate
For the circular plate of uniform thickness and radius a clamped at the edge, the boundary conditions are that both ξ and ∂ξ/∂r are zero at r = a. To satisfy these conditions, from Equation 3.74
ξ( , )aφ =sincos(mΦ)[AJm( )γa +BIm( )]γa =0
B A J a
I a
m m
= – ( )
( ) γ
γ (3.75)
Fundamental Equations Employed in Ultrasonic Design and Applications 131
As with the flexural bar, these conditions can be satisfied only at discrete frequen-cies. Again referring to Morse [5], these frequencies are fixed by the values
β β β
where γmn= (π/a)βmn. Therefore, the allowed frequencies for the normal plate clamped at its circumference are
where the number of nodal circles is n, including the clamped edge as a nodal circle, and the number of nodal diameters is m.
From the values for βmn given in Equation 3.77, the allowed frequencies are
f h
From the βmn values in Equation 3.77 and from Equation 3.79, one may easily see that the overtones of the clamped plate are not harmonics. It can be shown simi-larly that this is generally true for all other boundary conditions for vibrating plates.
Therefore, the velocity of propagation of plate waves is dependent not only upon the elastic properties and the density but also upon the frequency of the oscillations.