1 Introduction
Type of system | Left end condition | BCs at x = 0 |
---|---|---|
Classical | ||
Pinned |
×
|
\(u = 0, EIu_{xx} = 0.\)
|
Sliding |
×
|
\(u_{x} = 0, ~ EIu_{xxx} = 0.\)
|
Clamped |
×
|
\(u = 0, \, u_{x} = 0.\)
|
Non-classical | ||
Damper |
×
|
\(EIu_{xx} = 0, \, EIu_{xxx}=-\alpha {u}_{t}.\)
|
2 Governing equations of motion
3 The Laplace transform method
-
[G1] The Green’s function \(g_{\xi }\) satisfies the fourth order ordinary differential equation in each of the two subintervals \(0< x< \xi\) and \(\xi<x< \infty\), that is, \(Lg_{\xi }=0\) except when \(x=\xi\).
-
[G2] The Green’s function \(g_{\xi }\) satisfies at x = 0 one of the homogeneous boundary conditions, as given in Table 1.
-
[G3] The Green’s function \(g_{\xi }\) and its first and second order derivatives exist and are continuous at \(x=\xi\).
-
[G4] The third order derivative of the Green’s function \(g_{\xi }\) with respect to x has a jump discontinuity which is defined as$$\lim _{\epsilon \rightarrow \ 0} \left[ g^{\prime \prime \prime }_{\xi }(\xi +\epsilon )-g^{\prime \prime \prime }_{\xi }(\xi -\epsilon )\right] =1.$$(10)
4 Classical boundary conditions
4.1 Pinned end, \(u=u_{xx}=0\)
4.2 Sliding end, \(u_{x}=u_{xxx}=0\)
4.3 Clamped end, \(u=u_{x}=0\)
5 Non-classical boundary condition
5.1 Damper end, \(u_{xx} = 0\), \(u_{xxx}=-\tilde{\lambda } {u}_{t}\)
5.2 Damper-clamped ends, \(u_{xxx}(0,t)=-\tilde{\lambda } {u}_{t}(0,t)\), \(u_{xx}(0,t) =u(L,t) = u_{x}(L,t) = 0\)
n
|
\(\beta _{num,n}\)
|
\(p_{num,n}\)
|
\((n-\frac{1}{2}) \frac{\pi }{L}\)
|
---|---|---|---|
−1 | 0.03887 + 0.03887i | −0.00302 + 0i | – |
0 | 1.00000 + 1.00000i | −2.00000 + 0i | – |
1 | – | – | 0.15708 |
2 | 0.39535 + 0.01861i | −0.01471 + 0.15596i | 0.47124 |
3 | 0.71834 + 0.03367i | −0.04837 + 0.51488i | 0.78540 |
4 | 1.04526 + 0.04286i | −0.08960 + 1.09073i | 1.09956 |
5 | 1.37292 + 0.04574i | −0.12560 + 1.88282i | 1.41372 |
6 | 1.69789 + 0.04416i | −0.14996 + 2.88088i | 1.72788 |
7 | 2.01967 + 0.04084i | −0.16497 + 4.07740i | 2.04204 |
8 | 2.33906 + 0.03727i | −0.17435 + 5.46981i | 2.35619 |
9 | 2.65687 + 0.03395i | −0.18040 + 7.05781i | 2.67035 |
10 | 2.97365 + 0.03104i | −0.18460 + 8.84163i | 2.98451 |
11 | 3.28974 + 0.02850i | −0.18752 + 10.82158i | 3.39867 |
12 | 3.60537 + 0.02631i | −0.18971 + 12.99800i | 3.61283 |
13 | 3.92066 + 0.02440i | −0.19133 + 15.37098i | 3.92699 |
14 | 4.23571 + 0.02274i | −0.19264 + 17.94072i | 4.24115 |
15 | 4.55059 + 0.02127i | −0.19358 + 20.70742i | 4.55531 |
16 | 4.86533 + 0.01998i | −0.19442 + 23.67104i | 4.86947 |
17 | 5.17997 + 0.01883i | −0.19508 + 26.83173i | 5.18363 |
18 | 5.49453 + 0.01780i | −0.19561 + 30.18954i | 5.49779 |
19 | 5.80903 + 0.01688i | −0.19611 + 33.74454i | 5.81195 |
20 | 6.12348 + 0.01605i | −0.19656 + 37.49675i | 6.12611 |