Chapter 13
Finite Difference Method
for the Vibration Analysis of Beams
and Plates
13.1 Introduction to the Finite Difference Method
The finite difference technique is another versatile numerical method for the solution
of vibration problems. The method has been explained in Chapter 4 with respect to
the time function. It is applied here with respect to spatial variables. The differential
equation is the starting point of the method. The continuum is divided into the form
of a mesh, and the unknowns in the problem are those at the nodes. The derivatives
of the equation are expressed in the finite difference form. The differential equation,
split in this discrete form is applied at each node. This results in a set of simultaneous
equations.
In the finite element method, the final differential equation is surpassed, whereas it
is this differential equation, which is the starting point in the finite difference method.
The approximation involved in the finite element method is physical in nature, as the
actual continuum is replaced by finite elements. The element formulation is mathematically exact. The finite difference technique involves the exact representation
of the continuum, in terms of the differential equation, and on this actual physical
problem, the mathematical model is approximated [1–6].
13.2 Central Difference Method
A variable w, which is a function of x, is shown in Fig. 13.1. The function is divided
into equally spaced intervals of h. Let i be any station. Stations to its right are indicated
as i + 1, i + 2, . . ., and the corresponding values of the function are w i+1 , w i+2 , . . ..
Stations to the left of i are designated as i − 1, i − 2…, and the corresponding values
of the function are w i−1 , w i−2 , . . ..
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. Mukhopadhyay, Structural Dynamics,
https://doi.org/10.1007/978-3-030-69674-0_13
491
Finite Difference Method
for the Vibration Analysis of Beams
and Plates
13.1 Introduction to the Finite Difference Method
The finite difference technique is another versatile numerical method for the solution
of vibration problems. The method has been explained in Chapter 4 with respect to
the time function. It is applied here with respect to spatial variables. The differential
equation is the starting point of the method. The continuum is divided into the form
of a mesh, and the unknowns in the problem are those at the nodes. The derivatives
of the equation are expressed in the finite difference form. The differential equation,
split in this discrete form is applied at each node. This results in a set of simultaneous
equations.
In the finite element method, the final differential equation is surpassed, whereas it
is this differential equation, which is the starting point in the finite difference method.
The approximation involved in the finite element method is physical in nature, as the
actual continuum is replaced by finite elements. The element formulation is mathematically exact. The finite difference technique involves the exact representation
of the continuum, in terms of the differential equation, and on this actual physical
problem, the mathematical model is approximated [1–6].
13.2 Central Difference Method
A variable w, which is a function of x, is shown in Fig. 13.1. The function is divided
into equally spaced intervals of h. Let i be any station. Stations to its right are indicated
as i + 1, i + 2, . . ., and the corresponding values of the function are w i+1 , w i+2 , . . ..
Stations to the left of i are designated as i − 1, i − 2…, and the corresponding values
of the function are w i−1 , w i−2 , . . ..
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. Mukhopadhyay, Structural Dynamics,
https://doi.org/10.1007/978-3-030-69674-0_13
491