L. Sun Assistant Professor, Transportation and Operation Research Program, Department of Civil Engineering, The Catholic University of America, Washington, DC 20064 e-mail:
[email protected]
1
Dynamic Response of Kirchhoff Plate on a Viscoelastic Foundation to Harmonic Circular Loads In this paper Fourier transform is used to derive the analytical solution of a Kirchhoff plate on a viscoelastic foundation subjected to harmonic circular loads. The solution is first given as a convolution of the Green’s function of the plate. Poles of the integrand in the integral representation of the solution are identified for different cases of the foundation damping and the load frequency. The theorem of residue is then utilized to evaluate the generalized integral of the frequency response function. A closed-form solution is obtained in terms of the Bessel and Hankel functions corresponding to the frequency response function of the plate under a harmonic circular load. The result is partially verified by comparing the static solution of a point source obtained in this paper to a well-known result. This analytical representation permits one to construct fast algorithms for parameter identification in pavement nondestructive test. 关DOI: 10.1115/1.1577598兴
Introduction
Nondestructive testing 共NDT兲 has been extensively used in pavement engineering since the 1980s, 关1– 4兴, to evaluate pavement structural parameters. The most commonly used NDT device for pavement structural evaluation are falling weight deflectometer and Dynaflect. Falling weight deflectometer applies an impulse load to pavement surfaces, while Dynaflect applies a steady-state vibrating harmonic load, 关3兴. Given that a typical physical model for rigid pavements 共e.g., cement concrete pavements兲 is a Kirchhoff plate resting on an elastic Winkler foundation, 关5,6兴, the mathematical problem involved here thus becomes to estimate the parameters of governing equation of the plate provided that the applied load is known 共i.e., the inverse problem兲. The structural evaluation is then achieved by identifying structural parameters based on pavement response to applied dynamic loads. Because of the complexity involved in the inverse problem, in current practice a widely used technique is to use the forward analysis of a plate under a static load. By comparing measured dynamic response and calculated static response using optimization techniques, pavement structural parameters are eventually determined while selecting a pavement structure whose calculated response is most closely to the measured maximum response in terms of certain objective functions, 关7兴. Clearly, pavement response under dynamic loads such as applied by Dynaflect is significantly different from pavement response under static loads. Finite element procedures have been developed to calculate numerically the response of a plate to dynamic loads, 关8 –10兴. However, in terms of efficiency, the computation using finite element methods is time-consuming. Computational efficiency can be improved if analytical solutions are available and used for numerical calculation. Achenbach et al. 关11兴 investigated the response of an infinite plate to harmonic plane waves. Freund and Achenbach 关12兴 and Oien 关13兴 investigated the response of a semi-infinite plate on an elastic half-space. In their study the displacement of
the plate is assumed to be harmonic. As a result, the time and spatial coordinates become separated and the governing partial differential equation turns out to be an ordinary differential equation. The solution is then obtained using the Bubnov-Galekin method and series expansion of the vibrational modes of the plate. Arnold et al. 关14兴 and Warburton 关15兴 conducted similar studies by means of integral transform methods. Using integral representation of the general solution of the plate provided by Bycroft 关16兴, Krenk and Schmidt 关17兴 studied the steady-state response of finite plate on an elastic half-space. In our previous work, 关18 –20兴, dynamic response of BernoulliEular beam 共one-dimensional situation兲 to specific loading condition has been studied. However, the analysis of a Kirchhoff plate on a viscoelastic foundation to a circular harmonic load applied by a Dynaflect has not been available in the literature. To this end, as a continuous effort in this paper the author extends previous work to deal with cases and the dynamic response of a ridge pavement structure 共two-dimensional situation兲 under a circular harmonic load. The availability of such analytical solutions will enable one to construct fast algorithm for parameter identification problem, a core issue in pavement NDT. This paper is organized as follows. In Section 2, the governing equation is established with the exploration of associated foundation models. In Section 3, the Green’s function of a Kirchhoff plate is derived analytically using integral transform method. In Section 4, by integrating the Green’s function with respect to time-spatial dimensions, we obtain the frequency response function corresponding to a harmonic circular load and concentrated load. In Section 5 we address several special cases such as the static solution and Winkler foundation, which is also used to verify the correctness of our result.
2 Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the Applied Mechanics Division, Jan. 25, 2000; final revision, Sept. 15, 2002. Associate Editor: V. K. Kinra. Discussion on the paper should be addressed to the Editor, Prof. Robert M. McMeeking, Chair, Department of Mechanics and Environmental Engineering, University of California–Santa Barbara, Santa Barbara, CA 93106 –5070, and will be accepted until four months after final publication in the paper itself in the ASME JOURNAL OF APPLIED MECHANICS.
Journal of Applied Mechanics
The Governing Equation of the Plate
Figure 1 depicts the coordinate system and significant dimensions. Three assumptions are commonly made to simplify the mathematical model of a Kirchhoff plate. These assumptions are 共1兲 the strain component z in the perpendicular direction of the plate is sufficiently small such that it can be ignored; 共2兲 the stress components zx , zy , and z are far less than the other stress
Copyright © 2003 by ASME
JULY 2003, Vol. 70 Õ 595
Fig. 1 A plate on a viscoelastic foundation subjected to a circular load
components, therefore, the deformation caused by zx , zy , and z can be negligible; and 共3兲 the displacement parallel to the horizontal direction of the plate is zero, 关6兴. Denote the displacement of the plate in the z-direction by W(x,y,t). Based on these assumptions and the fundamental equations of elastodynamics, the governing equation for the deflection of the Kirchhoff plate can be derived by considering the balance of all the forces acting on the element (x,x⫹dx;y,y⫹dy). These forces are the impressed force distribution F(x,y,t), the shearing force, the restoring force from the foundation q(x,y,t), and the inertial force h 2 W/ t 2 . The well-known result is
2 Dⵜ ⵜ W 共 x,y,t 兲 ⫹ h 2 W 共 x,y,t 兲 ⫽F 共 x,y,t 兲 ⫺q 共 x,y,t 兲 t 2
˜f 共 兲 ⫽F关 f 共 x兲兴 ⫽
⬁
⬁
⫺⬁
⫺⬁
⫺⬁
f 共 x兲 exp共 ⫺i x兲 dx
⬁
⬁
⬁
⫺⬁
⫺⬁
⫺⬁
where the Laplace operator ⵜ 2 ⫽ 2 / x 2 ⫹ 2 / y 2 , D ⫽Eh 3 / 关 12(1⫺ 2 ) 兴 is stiffness of the plate, h is thickness of the plate, is density of the plate, and E and are Young’s elastic modulus and Poisson ratio of the plate, respectively. The most widely used foundation model in rigid pavement design is Winkler foundation, 关6,20,21兴, which assumes the reactive pressure to be proportional to the deflection of the plate, i.e., q ⫽KW where K is the modulus of subgrade reaction. A constant K implies a linear elasticity of the subgrade. When the damping effect of the subgrade is considered, the restoring force becomes q⫽KW⫹C W/ t. This is a viscoelastic foundation consisting of a spring of strength K and a dashpot of strength, C, placed parallel, as shown in Fig. 1. Substitution of the restoring force into Eq. 共1兲 gives
3
W 共 x,y,t 兲 t
(5b)
˜ 共 ;x0 兲 ⫹KG ˜ 共 ;x0 兲 ⫹iC G ˜ 共 ;x0 兲 ⫺ h 2 G ˜ 共 ;x0 兲 D共 2⫹ 2 兲2G (6)
in which ˜F( ) is the Fourier transform of F(x), and the displacement response W(x) has been replaced by the symbol G(x;x0 ) to indicate the Green’s function. In the derivation of Eq. 共6兲 the following property of Fourier transform is used: F关 f 共 n 兲 共 t 兲兴 ⫽ 共 i 兲 n F关 f 共 t 兲兴 .
˜F 共 兲 ⫽
冕冕冕 ⬁
⬁
⬁
⫺⬁
⫺⬁
⫺⬁
␦ 共 x⫺x0 兲 exp共 ⫺i x兲 dx⫽exp共 ⫺i x兲 (8)
in which the property of the Dirac-delta function, i.e., Eq. 共4兲, is utilized while evaluating the above integral. Substituting this result 共8兲 into Eq. 共6兲 gives (9)
(2)
The Green’s Function
(3)
in which x⫽(x,y,t), x0 ⫽(x 0 ,y 0 ,t 0 ), ␦ (x⫺x0 )⫽ ␦ (x⫺x 0 ) ␦ (y ⫺y 0 ) ␦ (t⫺t 0 ), and ␦共•兲 is the Dirac-delta function, defined by
冕
⬁
⫺⬁
␦ 共 x⫺x 0 兲 f 共 x 兲 dx⫽ f 共 x 0 兲 .
(4)
Define the three-dimensional Fourier transform and its inversion, 关24兴, 596 Õ Vol. 70, JULY 2003
(7)
Since ˜F ( ) is the representation of F(x) in the frequency domain, ˜F ( ) needs to be evaluated as well. This can be achieved by applying three-dimensional Fourier transform on both sides of 共3兲
˜ 共 ;x0 兲 ⫽exp共 ⫺i x0 兲关 D 共 2 ⫹ 2 兲 2 ⫹K⫹iC ⫺ h 2 兴 ⫺1 . G
According to the mathematical physics theory, the Green’s function is a fundamental solution of a partial differential equation, 关22,23兴 For the present problem, the Green’s function is defined as the solution of Eq. 共1兲 given that the external excitation F(x,y,t) is characterized by F 共 x兲 ⫽ ␦共 x⫺x0 兲
˜f 共 兲 exp共 i x兲 d
where ⫽( , , ), F关 • 兴 and F⫺1 关 • 兴 are the Fourier transform and its inversion, respectively. To solve the Green’s function, we apply three-dimensional Fourier transform to both sides of Eq. 共2兲
2
⫹h
(5a)
˜ 共 兲 ⫽F (1)
W 共 x,y,t 兲 ⫽F 共 x,y,t 兲 . t2
⬁
f 共 x兲 ⫽F⫺1 关˜f 共 兲兴 ⫽ 共 2 兲 ⫺3
2
Dⵜ 2 ⵜ 2 W 共 x,y,t 兲 ⫹KW 共 x,y,t 兲 ⫹C
冕冕冕 冕冕冕
The Green’s function given by 共9兲 is in the frequency domain and needs to be converted back to the time domain. To this end, take the inverse Fourier transform of Eq. 共9兲 G 共 x;x0 兲 ⫽ 共 2 兲 ⫺3
冕冕冕 ⬁
⬁
⬁
⫺⬁
⫺⬁
⫺⬁
exp关 i 共 x⫺x0 兲兴关 D 共 2 ⫹ 2 兲 2
⫹K⫹iC ⫺ h 2 兴 ⫺1 d .
(10)
Equation 共10兲 is the Green’s function of a plate on the viscoelastic foundation. The Green’s function serves as a fundamental solution of a partial differential equation. It can be very useful when dealing with circular loads.
4
The Frequency Response Function
4.1 Integral Representation. We use the Green’s function obtained in the previous section to construct the frequency response function 共FRF兲. Denote W(x) as the solution of Eq. 共1兲 in Transactions of the ASME
which the external load is a harmonic circular load with its center located at the origin of the coordinate system, i.e., F FRF共 x兲 ⫽ 共 r 20 兲 ⫺1 H 共 r 20 ⫺x 2 ⫺y 2 兲 exp共 i⍀t 兲
冕冕冕 t
⬁
⬁
⫺⬁
⫺⬁
⫺⬁
F 共 x0 兲 G 共 x;x0 兲 dx0 .
(13)
Take 共10兲 and 共11兲 into 共13兲 and apply the property of the Diracdelta function twice. W 共 r,⍀ 兲 1 ⫽ 2
冕冕冕 ⬁
⬁
⫺⬁
⫺⬁
⬁
0
J 1 共 r 0 兲 J 0 共 r 兲 exp共 i t 兲 exp关 i 共 ⫺⍀ 兲 t 0 兴 r 0 共 D 4 ⫹K⫹iC ⫺ h 2 兲
⫻d d dt 0
(14)
Here, J 0 (•) and J 1 (•) are the Bessel functions of the first kind, 关25兴. Since
冕
⬁
⫺⬁
exp关 i 共 ⍀⫺ 兲 t 0 兴 dt 0 ⫽2 ␦ 共 ⍀⫺ 兲 ,
(15)
冕
⬁
0
J 1共 r 0 兲 J 0共 r 兲 dd. r 0 共 D 4 ⫹K⫹iC⍀⫺ h⍀ 2 兲 (16)
Comparing 共16兲 to 共12兲 it is straightforward that H Circle共 r,⍀ 兲 ⫽
冕
⬁
0
J 1共 r 0 兲 J 0共 r 兲 d. r 0 共 D 4 ⫹K⫹iC⍀⫺ h⍀ 2 兲
(17)
The frequency response function of the plate to a concentrated harmonic load F Point(x)⫽ ␦ (x) ␦ (y)exp(i⍀t) can be obtained by simply taking the limit r 0 →0 on both sides of 共17兲, i.e., H Point共 r,⍀ 兲 ⫽
1 2
冕
⬁
0
J 0共 r 兲 d. 4 D ⫹K⫹iC⍀⫺ h⍀ 2
(18)
Now we have obtained the frequency response function H Circle(x,⍀) and H Point(r,⍀) in the rectangular and cylindrical coordinate systems, respectively. In general, the frequency response function given by 共17兲 and 共18兲 are complex functions. The Bessel function of the first kind J 0 (z) can be given in integration representation, 关25兴, i J 0共 z 兲 ⫽
冕
⬁
4 ⫹ 共 K⫺ h⍀ 2 ⫹iC⍀ 兲 /D⫽0.
Realizing the property of even function, we can also rewrite identity 共19兲 as
冕
⬁
⫺⬁
exp共 ⫺iz cosh ␥ 兲 d ␥ .
Substituting this expression into Eq. 共18兲 gives Journal of Applied Mechanics
(22)
Characteristic Eq. 共22兲 is a fourth-order algebraic equation, roots of which are dependent upon parameters related to plate structure, foundation, and loading condition. We separate our discussion into two categories: no damping effect 共Winkler foundation兲 and with damping effect 共viscoelastic foundation兲. Within each individual category three cases are separately addressed because the relationship between load frequency and eigenfrequencies may result in different scenarios. 4.2.1 No damping (C⫽0). This case corresponds to a plate ¯ ⫽ 兩 (K on a Winkler foundation. Define equivalent stiffness as K ⫺ h⍀ 2 )/D 兩 and resonance frequency as ⍀ 0 ⫽ 冑K/ h. ¯ ⫽0. All the four a. ⍀⬍⍀ 0 . Equation 共22兲 becomes 4 ⫹K roots of this equation possess complex values and can be ¯ exp关i(1⫹2j)/4兴 with j⫽0,1,2,3. given by j ⫽ 4冑K b. ⍀⫽⍀ 0 . Equation 共22兲 becomes 4 ⫽0 and all four roots degrade and becomes ⫽0. ¯ ⫽0. Two of the four c. ⍀⬎⍀ 0 . Equation 共22兲 becomes 4 ⫺K roots are imaginary and the other two are real valued, which ¯ exp关i(j/2) 兴 with j⫽0,1,2,3, respecare given by j ⫽ 4冑K tively.
¯ ⫹iC ¯ ⫽0. All four a. ⍀⬍⍀ 0 . Equation 共22兲 becomes 4 ⫹K roots possess complex values and can be given by j ¯ 2 ⫹C ¯ 2 exp关i(⫹⫹2j)/4兴 with j⫽0,1,2,3 in which ⫽ 8冑K ¯ ¯ ⬎0. tan ⫽C/K ¯ ⫽0. In this case all b. ⍀⫽⍀ 0 . Equation 共22兲 becomes 4 ⫹iC four roots possess complex values and can be given by j ¯ exp关i(3⫹4j)/8兴 with j⫽0,1,2,3, respectively. ⫽ 4冑C ¯ ⫹iC ¯ ⫽0. All four c. ⍀⬎⍀ 0 . Equation 共22兲 becomes 4 ⫺K roots possess complex values and can be given by j ¯ 2 ⫹C ¯ 2 exp关i(⫹2j)/4兴 with j⫽0,1,2,3 in which ⫽ 8冑K ¯ /K ¯ ⬍0. tan ⫽⫺C 4.3 Closed-Form Representation. According to the residue theorem 共Saff and Snider, 关26兴兲 the residues of the integrand of 共32兲 in the upper half-plane contribute to the following integration:
冕
⬁
J 1 共 r 0 兲 exp共 ⫺i r cosh ␥ 兲
⫺⬁
¯ ⫹iC ¯ 4 ⫾K J
⫽2 i
0
i
0
J 1 共 r 0 兲 exp共 ⫺i r cosh ␥ 兲 dd␥. 4 2 ⫺⬁ ⫹ 共 K⫺ h⍀ ⫹iC⍀ 兲 /D (21) ⬁
4.2 Roots of the Characteristic Equation. Before the integration 共21兲 can be further evaluated, it is necessary to investigate the roots of the characteristic equation of type
关 exp共 ⫺iz cosh ␥ 兲 ⫺exp共 iz cosh ␥ 兲兴 d ␥ . (19)
J 0共 z 兲 ⫽
⬁
4.2.2 With Damping (C⫽0). Define the equivalent damping ¯ ⫽C⍀/D. Three cases are discussed as follows: coefficient C
substituting 共15兲 into formula 共14兲 gives W 共 r,⍀ 兲 ⫽exp共 i⍀t 兲
冕冕
(12)
Denote H Circle(x,⍀) the frequency response function 共FRF兲 of the plate. Expression 共12兲 simply says that both the response and external excitation possess identical frequency ⍀, though response of the plate may have a phase difference with the external excitation reflected in the H Circle(x,⍀). The solution of Eqs. 共1兲 and 共11兲 can be constructed by integrating the Green’s function over all dimensions, i.e., W 共 x兲 ⫽
i r 0D 2
(11)
in which ⍀ is frequency of the harmonic load. The steady-state response can be expressed as W 共 x兲 ⫽H Circle共 x,⍀ 兲 exp共 i⍀t 兲 .
H Circle共 r,⍀ 兲 ⫽
(20)
⫽
i 2
兺
j⫽1 J
兺
j⫽1
冋
Res
d
J 1 共 r 0 兲 exp共 ⫺i r cosh ␥ 兲 ¯ ⫹iC ¯ 4 ⫾K
冏
册冏
⫽ j
J 1 共 r 0 兲 exp共 ⫺i r cosh ␥ 兲
3
(23) ⫽ j ,Im共 兲 ⬎0
in which J is the number of poles whose imaginary parts are positive, and j represents the poles of the integrand of Eq. 共23兲. Based on the previous analysis, it is clear that two complex JULY 2003, Vol. 70 Õ 597
roots exist in the upper half-plane of the complex -plane for all the cases of C⫽0 and the case ⍀⬍⍀ 0 of C⫽0. For ⍀⬎⍀ 0 and C⫽0 two poles are located on the real axis, while for ⍀⫽⍀ 0 and ⍀⫽⍀ 0 only one pole is located on the real axis. Since the residue theorem cannot be directly applied in the sense of Riemann integral, the concept of Cauchy principal value 共p.v.兲 of the integration 共23兲 has to be introduced for these two cases, 关26兴. In Fig. 2共a兲 and 共b兲 two integral contours are respectively provided for ⍀⫽⍀ 0 of C⫽0 and ⍀⬎⍀ 0 of C⫽0. As shown in Fig. 2共a兲, the contour consists of three portions for the case ⍀⫽⍀ 0 and C⫽0. The integral of the left-hand side ⬁ of Eq. 共23兲 now becomes 兰 ⫺⬁ J 1 ( r 0 )exp(⫺ir cosh ␥)/4 d.
lim →0
冕
0
Since no pole is embraced by the closed contour, the theorem of residue says that the integral along this closed contour becomes zero, i.e.,
冖
冕 冕 冕 R
⫽p.v.
⫺R
⫹
⫹
C1
⫽0
(24)
C2
in which the abbreviation p.v. means the Cauchy principal value of the integration. For those values on C 1 , they can be expressed by ⫽ exp(i) and d ⫽i exp(i)d. The integration 兰 C 1 as →0 is then given by
J 1 关 exp共 i  兲 r 0 兴 exp关 ⫺ir exp共 i  兲 cosh ␥ 兴 i exp共 i  兲 d  ⫽ lim i 4 exp共 i4  兲 →0
冕
0
J 1 关 exp共 i  兲 r 0 兴 exp关 ⫺ir exp共 i  兲 cosh ␥ 兴 d. 3 exp共 i3  兲 (25)
By applying the Maclaurin expansion and the L’Hospital rule to the limit 共25兲, it is found that a singularity with an order O( ⫺1 ) exists. For those values on C 2 , they can be expressed by ⫽R exp(i) and d ⫽iR exp(i)d. The integration 兰 C 2 as R→⬁ is then given by
lim R→⬁
冕
0
J 1 关 R exp共 i  兲 r 0 兴 exp关 ⫺iRr exp共 i  兲 cosh ␥ 兴 iR exp共 i  兲 d  ⫽ lim i R 4 exp共 i4  兲 R→⬁
p.v.
冕
⫺⬁
⫽ lim R→⬁
冕
R
⫺R
⫽⫺ lim
→0
冕
⬃O 共 ⫺1 兲 .
0
冖
Comparison among 共24兲, 共25兲, and 共26兲 shows
⬁
冕
(27)
C1
For the case ⍀⬎⍀ 0 and C⫽0, as shown in Fig. 2共b兲, the contour consists of four portions. One pole is within the range of the closed contour. Suppose that a tiny amount of damping is present, which is the case in reality, the pole that is currently exactly located on the positive part of the real axis will be actually pushed down into the fourth quadrant of the complex plane. Hence, its contribution to the integral vanishes. So we now have
J 1 关 R exp共 i  兲 r 0 兴 exp关 ⫺iRr exp共 i  兲 cosh ␥ 兴 d  ⫽0. R 3 exp共 i3  兲 (26)
冕 冕 冕 冕 R
⫽p.v.
⫺R
⫹
冋
⫽2 iRes
⫽
⫹
C1
⫹
C2
C3
J 1 共 r 0 兲 exp共 ⫺i r cosh ␥ 兲 ¯ 4 ⫺K
冏
册冏
iJ 1 共 r 0 兲 exp共 ⫺i r cosh ␥ 兲 23
¯ ⫽i 冑K
.
¯ ⫽i 冑K 4
(28)
4
For those values on C 2 and C 3 , one can prove limR→0 兰 C 2 ⫽0 and lim 3 →0 兰 C 3 ⫽0 in the same manner as in the previous discussion. For those values on C 1 , they can be expressed by ¯ ⫹ 1 exp(i) and d ⫽i 1 exp(i)d. The integration 兰 C ⫽⫺ 4冑K 1 as 1 →0 is then given by
Fig. 2 The integral contours for evaluating the Cauchy principal value in Eq. „23…
598 Õ Vol. 70, JULY 2003
Transactions of the ASME
lim 1 →0
0
¯ ⫹ 1 exp共 i  兲兴 其 exp兵 ⫺ir 关 ⫺ 冑K ¯ ⫹ 1 exp共 i  兲兴 cosh ␥ 其 J 1 兵 r 0 关 ⫺ 冑K
4 ¯ ⫹ 1 exp共 i  兲兴 4 ⫺K ¯ 关⫺ K
冕
4
冋
⫽⫺ iRes
4
冑
J 1 共 r 0 兲 exp共 ⫺i r cosh ␥ 兲 ¯ ⫺K 4
Comparison among 共28兲 and 共29兲 gives p.v.
冕
⬁
⫺⬁
冕
⫽ lim
⫺R
R→⬁
⫹
R
⫽
冏
iJ 1 共 r 0 兲 exp共 ⫺i r cosh ␥ 兲 23
iJ 1 共 r 0 兲 exp共 ⫺i r cosh ␥ 兲 43
冏
¯ ⫽⫺ 冑K
4
冕
⬁
43
H Circle共 r,⍀ 兲 ⫽
4
冕
⬁
0
exp共 ⫺izrch ␥ 兲 d ␥ ⫽H 共02 兲 共 zr 兲
(31)
0
where H 0(2 ) (•) is the Hankel function of the second kind 共i.e., the Bessel function of the third kind兲. Applying this expression and the closed-form representation of the inner integral of Eq. 共21兲, we are eventually able to write the integration 共21兲 in a closed-form expression, as illustrated by 共32兲: J 1 共 n r 0 兲 H 共02 兲 共 n r 兲 i H Circle共 r,⍀ 兲 ⫽ 4r 0 D n⫽a,b 3n
¯ ⫽⫺ 冑K
.
(29)
4
H Point共 r,⍀ 兲 ⫽
J 1共 r 0 兲 J 0共 r 兲 d r 0 共 D 4 ⫹K⫺ h⍀ 2 兲
(35)
冕
(36)
1 2
⬁
0
J 0共 r 兲 d. D 4 ⫹K⫺ h⍀ 2
Clearly, H Circle(r,⍀) and H Point(r,⍀) given by 共35兲 and 共36兲 are real functions. It implies that no phase difference exists between the response and the external excitation. If the radius of the circular load approaches zero, the frequency response functions given by 共32兲 becomes H Point共 r,⍀ 兲 ⫽
H 共02 兲 共 n r 兲 i 8D n⫽a,b 2n
兺
(37)
where the following limit is used: J 1共 nr 0 兲 ⫽1/2. →0 n r 0
lim
兺
(32)
r0
(38)
In practice, the vibratory devices used for pavement nondestructive test generate harmonic loads with frequency 5⬃60 Hz, 关3兴. These frequencies usually fall into the low frequency range of ⍀⬍⍀ 0 . Under such condition, it is appropriate to write 共37兲 as
where poles a and b are provided in Table 1.
5
冏
⫺ iJ 1 共 r 0 兲 exp共 ⫺i r cosh ␥ 兲
4
(30)
So far, all cases have been analyzed and the left-hand side of Eq. 共23兲 has been represented in closed-form expressions for different cases. With the help of these closed-form expressions, the frequency response function given by 共21兲 is now ready to be further evaluated. Realize that an integral representation of Hankel function is, 关25兴, 2i
¯ ⫽⫺ 冑K
⫽
This expression 共34兲 is exactly identical to a known result, 关27– 29兴. If damping is ignored in Eqs. 共17兲 and 共18兲, the frequency response functions become
¯ ⫽i 冑K
.
册冏
i 1 exp共 i  兲 d 
Verification Through Special Cases
¯ D 兲 ⫺1/2兵 H 共02 兲 关 冑K ¯ exp共 i /4兲 r 兴 H Point共 r,⍀ 兲 ⫽ 共 64K
It is of interest to examine the static solution through applying the results 共17兲 and 共18兲. The derivation of static solution can also be of great value in terms of verifying if the general result given by Eq. 共32兲 is correct. For a static load, F Circle sta(x)
4
¯ exp共 i3 /4兲 r 兴 其 . ⫹H 共02 兲 关 冑K 4
(39)
–
⫽( r 20 ) ⫺1 H(r 20 ⫺x 2 ⫺y 2 ) and F Point sta(x)⫽ ␦ (x) ␦ (y). The static – solution of Eq. 共1兲 corresponding to the static load can be achieved by letting ⍀⫽0 in 共17兲 and 共18兲 H Circle
共 r 兲⫽ – sta
冕
共 r 兲⫽ – sta
1 2
H Point
⬁
0
J 1共 r 0 兲 J 0共 r 兲 d r 0 共 D 4 ⫹K 兲
(33)
冕
(34)
⬁
0
J 0共 r 兲 d D 4 ⫹K
6
Conclusion
In this paper we derived a closed-form solution of dynamic response of a Kirchhoff plate on a viscoelastic foundation subjected to impulse and harmonic circular loads. The solution utilizes the Bessel and Hankel functions. The result has been partially verified by comparing the static solution of a point source obtained in this paper to a well-known result. This analytical expression permits one to construct fast algorithms for parameter identification in pavement nondestructive test.
Table 1 Poles that contribute to the harmonic response of the plate Damping
Frequency
C⫽0 C⫽0 C⫽0 C⫽0
⍀⬍⍀ 0 ⍀⫽⍀ 0 ⍀⬎⍀ 0 ⍀⬎⍀ 0
C⫽0 C⫽0
⍀⫽⍀ 0 ⍀⬎⍀ 0
Journal of Applied Mechanics
Poles a and b 1/4 i /4
¯ 1/4e i3 /4 and b ⫽K O( ⫺1 ) ¯ 1/4 and b ⫽⫺K ¯ 1/4 a ⫽iK ¯ 2 ⫹C ¯ 2 ) 1/8e i( ⫹ )/4, b ⫽(K ¯ 2 ⫹C ¯ 2 ) 1/8e i( ⫹3 )/4 a ⫽(K ¯ ¯ and tan ⫽C/K ¯ 1/4e i3 /8 and b ⫽C ¯ 1/4e i7 /8 a ⫽C ¯ 2 ⫹C ¯ 2 ) 1/8e i( ⫹2 )/4, b ⫽(K ¯ 2 ⫹C ¯ 2 ) 1/8e i( ⫹4 )/4 a ⫽(K ¯ /K ¯ and tan ⫽⫺C ¯ e a ⫽K
JULY 2003, Vol. 70 Õ 599
Acknowledgments The author is grateful to Prof. Vikram Kinra, the Associate Editor, for his constructive comments and suggestions, which enhance the content and the presentation of the original manuscript.
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