In this problem we will analyze a model of a flexible beam that is being forced. A common experimental methodology in vibration analysis is the forcing of a structure at several different frequencies. The structure is mounted to a piston-style shaker, which forces the structure periodically. The input periodic forcing is typically computer-controlled. See Figure 12.6.2.
The The Euler-Bernoulli Beam Equation models the dynamics of this situation.
[imath]\rho\frac{\partial^2 u}{\partial t^2} + \frac{\partial^2}{\partial x^2}\left(EI\frac{\partial^2 u}{\partial x^2}\right) = f(x,t).[/imath]
The ends are free, leading to "no moment/no shear force" boundary conditions:
[imath]\frac{\partial^2 u}{\partial x^2}\big|_{x=0} = \frac{\partial^2 u}{\partial x^2}\big|_{x=L} = 0, \ \ \ \ \ \ \frac{\partial^3 u}{\partial x^3}\big|_{x=0} = \frac{\partial^3 u}{\partial x^3}\big|_{x=L} = 0,[/imath]
The parameter definitions are as follows. The linear mass density (which is the volumetric mass density times the cross-sectional area) of the material of the beam is [imath]\rho[/imath]. Young's modulus is [imath]E[/imath], and the moment of inertial is [imath]I[/imath]. Each of these parameters is known for the beam of interest. The moment of inertial for a rectangular cross section is [imath]I = wh^3/12[/imath], where [imath]h[/imath] is the thickness (measured in the direction of motion of the beam) and [imath]w[/imath] is the width (measured in the direction orthogonal to motion).
In undertaking this problem, there are several tasks whose solution will require computational assistance. A computer algebra system such as Mathematics or Maple will be very helpful. Here are your tasks:
(a) Apply separation of variables to solve the homogenous equation
[imath]\rho\frac{\partial^2 u}{\partial t^2} + \frac{\partial^2}{\partial x^2}\left(EI\frac{\partial^2 u}{\partial x^2}\right) = 0.[/imath]
The solution, as discussed in the separation of variables sections for the heat and wave equations takes the form [imath]u(x,t) = \sum_{n=1}^{\infty}u_n(x,t),[/imath] where [imath]u_n(x,t) = X_n(x)T_n(t).[/imath] This task has several subtasks:
(i) Find the general formula for the [imath]T(t)[/imath] function. Your answer should be of the form [imath]T(t) = P\cos(\omega t) + Q\sin(\omega t)[/imath] where [imath]P[/imath] and [imath]Q[/imath] are unknown constants and [imath]\omega[/imath] depends on [imath]\rho[/imath], [imath]E[/imath], [imath]I[/imath], [imath]L[/imath], and the spatial frequencies you will get from the [imath]X(x)[/imath] equation.
(ii) Find the general formula for the [imath]X(x)[/imath] function. Your answer should be of the form [imath]X(x) = Ae^{\beta x} + Be^{-\beta x} + C\cos \beta x + D\sin \beta x,[/imath] where [imath]A[/imath], [imath]B[/imath], [imath]C[/imath], and [imath]D[/imath] are unknown constants and [imath]\beta[/imath] depends on [imath]\rho[/imath], [imath]E[/imath], [imath]I[/imath], [imath]L[/imath] and the spatial frequencies.
(iii) Use the boundary conditions to find four equations that include the five unknowns of part (ii) [imath](A, B, C, D,[/imath] and [imath]\beta).[/imath] Write these equations as [imath]4 \times 4[/imath] matrix (that depends on [imath]\beta[/imath]) times the vector of coefficients [imath]A[/imath], [imath]B[/imath], [imath]C[/imath], and [imath]D.[/imath]
(iv) Since the right-hand side of your equation system is the zero vector, you have two possibilities: All the coefficients are zero, or the determinant of the matrix is zero. Plot the determinant as a function of [imath]\beta.[/imath] Plot it carefully so that you can see the oscillations. Find the smallest ten numbers [imath]\beta[/imath] that make the determinant equal to zero.
(v) What constraints must hold for [imath]A[/imath], [imath]B[/imath], [imath]C[/imath], [imath]D?[/imath] They are unknown parameters, but some relationships must be established.
(vi) Use those values of [imath]\beta[/imath] to determine the smallest five values of [imath]\omega[/imath] from part (i).
(b) Plot the 10 mode shapes you found.
(c) Use separation of variables to solve the forced equation,
[imath]\rho\frac{\partial^2 u}{\partial t^2} + \frac{\partial^2}{\partial x^2}\left(EI\frac{\partial^2 u}{\partial x^2}\right) = f(x,t).[/imath]
The forcing function is (approximately) [imath]f(x,t) = F_0 \sin(\alpha t)\delta(x - L/2)[/imath], a periodic function that is concentrated at the beam's midpoint. To use the separation of variables approach, we need to expand the forcing function in terms of the [imath]X_n(x)[/imath] functions. As described in the context of the wave equation on page 479 of the text and using the orthogonal function expansion techniques of Secion 11.1, the forcing function can be written as
[imath]f(x,t) = \sum_{n=1}^{\infty}\frac{\int_{0}^{L} f(x,t)X_n(x) \ dx}{\int_{0}^{L}X_n^2(x) \ dx}X_n(x).[/imath]
(d) The material parameters for the beam, a 6061-T6 aluminum beam with rectangular cross section, are as follows:
[imath]L = 1.22 \ m[/imath]
[imath]w = 0.019 \ m[/imath]
[imath]h = 0.0033 \ m[/imath]
[imath]E = 7.310 \times 10^{10} \ m = 73.10 \ GPa[/imath]
[imath]\rho = 0.1693 \ kg/m.[/imath]
Using these material parameters, plot the solution as a function of space and time.
(e) Plot the acceleration of the model and the data (obtained from the website) and compare the results.
(f) Generate a more exact forcing function representation based on the setup of the system and apply it to solve the force differential equation.
I have to hand over this simple assignment before July and I am sure you guys can do it?
Any help would be so appreciated.
The The Euler-Bernoulli Beam Equation models the dynamics of this situation.
[imath]\rho\frac{\partial^2 u}{\partial t^2} + \frac{\partial^2}{\partial x^2}\left(EI\frac{\partial^2 u}{\partial x^2}\right) = f(x,t).[/imath]
The ends are free, leading to "no moment/no shear force" boundary conditions:
[imath]\frac{\partial^2 u}{\partial x^2}\big|_{x=0} = \frac{\partial^2 u}{\partial x^2}\big|_{x=L} = 0, \ \ \ \ \ \ \frac{\partial^3 u}{\partial x^3}\big|_{x=0} = \frac{\partial^3 u}{\partial x^3}\big|_{x=L} = 0,[/imath]
The parameter definitions are as follows. The linear mass density (which is the volumetric mass density times the cross-sectional area) of the material of the beam is [imath]\rho[/imath]. Young's modulus is [imath]E[/imath], and the moment of inertial is [imath]I[/imath]. Each of these parameters is known for the beam of interest. The moment of inertial for a rectangular cross section is [imath]I = wh^3/12[/imath], where [imath]h[/imath] is the thickness (measured in the direction of motion of the beam) and [imath]w[/imath] is the width (measured in the direction orthogonal to motion).
In undertaking this problem, there are several tasks whose solution will require computational assistance. A computer algebra system such as Mathematics or Maple will be very helpful. Here are your tasks:
(a) Apply separation of variables to solve the homogenous equation
[imath]\rho\frac{\partial^2 u}{\partial t^2} + \frac{\partial^2}{\partial x^2}\left(EI\frac{\partial^2 u}{\partial x^2}\right) = 0.[/imath]
The solution, as discussed in the separation of variables sections for the heat and wave equations takes the form [imath]u(x,t) = \sum_{n=1}^{\infty}u_n(x,t),[/imath] where [imath]u_n(x,t) = X_n(x)T_n(t).[/imath] This task has several subtasks:
(i) Find the general formula for the [imath]T(t)[/imath] function. Your answer should be of the form [imath]T(t) = P\cos(\omega t) + Q\sin(\omega t)[/imath] where [imath]P[/imath] and [imath]Q[/imath] are unknown constants and [imath]\omega[/imath] depends on [imath]\rho[/imath], [imath]E[/imath], [imath]I[/imath], [imath]L[/imath], and the spatial frequencies you will get from the [imath]X(x)[/imath] equation.
(ii) Find the general formula for the [imath]X(x)[/imath] function. Your answer should be of the form [imath]X(x) = Ae^{\beta x} + Be^{-\beta x} + C\cos \beta x + D\sin \beta x,[/imath] where [imath]A[/imath], [imath]B[/imath], [imath]C[/imath], and [imath]D[/imath] are unknown constants and [imath]\beta[/imath] depends on [imath]\rho[/imath], [imath]E[/imath], [imath]I[/imath], [imath]L[/imath] and the spatial frequencies.
(iii) Use the boundary conditions to find four equations that include the five unknowns of part (ii) [imath](A, B, C, D,[/imath] and [imath]\beta).[/imath] Write these equations as [imath]4 \times 4[/imath] matrix (that depends on [imath]\beta[/imath]) times the vector of coefficients [imath]A[/imath], [imath]B[/imath], [imath]C[/imath], and [imath]D.[/imath]
(iv) Since the right-hand side of your equation system is the zero vector, you have two possibilities: All the coefficients are zero, or the determinant of the matrix is zero. Plot the determinant as a function of [imath]\beta.[/imath] Plot it carefully so that you can see the oscillations. Find the smallest ten numbers [imath]\beta[/imath] that make the determinant equal to zero.
(v) What constraints must hold for [imath]A[/imath], [imath]B[/imath], [imath]C[/imath], [imath]D?[/imath] They are unknown parameters, but some relationships must be established.
(vi) Use those values of [imath]\beta[/imath] to determine the smallest five values of [imath]\omega[/imath] from part (i).
(b) Plot the 10 mode shapes you found.
(c) Use separation of variables to solve the forced equation,
[imath]\rho\frac{\partial^2 u}{\partial t^2} + \frac{\partial^2}{\partial x^2}\left(EI\frac{\partial^2 u}{\partial x^2}\right) = f(x,t).[/imath]
The forcing function is (approximately) [imath]f(x,t) = F_0 \sin(\alpha t)\delta(x - L/2)[/imath], a periodic function that is concentrated at the beam's midpoint. To use the separation of variables approach, we need to expand the forcing function in terms of the [imath]X_n(x)[/imath] functions. As described in the context of the wave equation on page 479 of the text and using the orthogonal function expansion techniques of Secion 11.1, the forcing function can be written as
[imath]f(x,t) = \sum_{n=1}^{\infty}\frac{\int_{0}^{L} f(x,t)X_n(x) \ dx}{\int_{0}^{L}X_n^2(x) \ dx}X_n(x).[/imath]
(d) The material parameters for the beam, a 6061-T6 aluminum beam with rectangular cross section, are as follows:
[imath]L = 1.22 \ m[/imath]
[imath]w = 0.019 \ m[/imath]
[imath]h = 0.0033 \ m[/imath]
[imath]E = 7.310 \times 10^{10} \ m = 73.10 \ GPa[/imath]
[imath]\rho = 0.1693 \ kg/m.[/imath]
Using these material parameters, plot the solution as a function of space and time.
(e) Plot the acceleration of the model and the data (obtained from the website) and compare the results.
(f) Generate a more exact forcing function representation based on the setup of the system and apply it to solve the force differential equation.
I have to hand over this simple assignment before July and I am sure you guys can do it?
Any help would be so appreciated.
