IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD I.U. Udo-Akpan1 and J. U. Chukwuchekwa2 1Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Rivers State, Nigeria. 2Department of Mathematics, Federal University of Technology, Owerri, Imo State, Nigeria. Abstract In this investigation, we extend our search on the dynamic buckling loads of some elastic structures to that of a clamped column lying on a nonlinear (cubic) elastic foundation but impacted upon axially by a step load. In order to ensure a uniformly valid solution, we employ multi–scaling two–timing regular perturbation procedures in asymptotic expansions of the variables. It is shown that (a) clamped columns buckle at higher buckling loads than columns with simply–supported ends irrespective of whether the columns are loaded statically or dynamically and whether damped or undamped, (b) Specifically, the inequalities satisfied by the static buckling load �� and dynamic buckling load�� in the clamped case are respectively given as 1 < �� < 2.125 and 1 < �� < 2.125 as against 0 < �� < 1 and 0 < �� < 1 for simply– supported end conditions, (c) At low values of the static buckling load ��, there is no appreciable change in the values of the dynamic buckling load �� but at higher values of ��, �� increases sharply with increased static buckling load ��. However, the increase seems to decrease with increased damping. We are able to mathematically relate the dynamic buckling load �� to the static buckling load �� and thereby by–passing the labour of repeating the entire process for different imperfection parameters. Thus, given either �� or ��, we can predict either value without the actual knowledge of the size of the small imperfection parameter. Keywords: Dynamic buckling, viscously damped clamped column, elastic structures, step load, two-timing perturbation 2010 Mathematical Subject Classification: 74B20, 74H10, 34E10 1. INTRODUCTION Investigations into the static or dynamic stability (or otherwise) of columns (finite or infinite) are age old problems that have been embarked upon by researchers for some years now. As observed by [1] and [2], buckling of structures is one of the structural instabilities that have been known for centuries ever since the equation to derive the critical buckling of a column was derived by Leonhard Euler [3]. As a result of this, previous studies on the subject matter are indeed enormous and varied, and include investigations by [4]-[7], among others. We must stress that columns, are in themselves, indispensable structural materials and their utility cuts across all cultures in human history. This investigation is concerned with analytical determination of the dynamic buckling load of a viscously damped but clamped imperfect column that rests on a nonlinear (cubic) elastic foundation, where the column is struck by a step load. The viscous https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 1 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD damping, though small in magnitude, is not in any way related either physically or mathematically, to an equally small imperfection that is stress-free and twice- differentiable. The formulation therefore contains two small but dimensionally independent parameters upon which asymptotic expansions are initiated using a two- timing multi-scaling perturbation procedure. The analysis contained here is an extension of a similar study espoused by [8], where, in that study, damping was taken to be related in some way, to the imperfection �� (�), so that once the imperfection was fixed, the damping was equally fixed. However, we note, from physical reasoning, that damping need not in any way, be related to the imperfection in all probabilities. Similar studies were done by [9]-[12], among others. Apart from addressing the phenomenon of viscous damping in a special fashion, this investigation is related, in spirit, to similar studies by [13]-[25]. 2. FORMULATION OF THE PROBLEM As in [6], the dimensional differential equation satisfied by the deflection �(�, �) of a finite viscously damped column lying on a nonlinear (cubic) elastic foundation but struck by a load �(�) is ���,�� + ��,� + ���,���� + 2�(�)�,�� + ��� − ����� = −2�(�) ���� ��� , � > 0, (2.1a) �(�, 0) = �,�(�, 0) = 0, (2.1b) � = �,� = 0, �� � = 0, π, � > 0 (2.1c) Here, �� is the mass per unit length, � is the damping coefficient, EI is the bending stiffness, where E and I are the Young’s modulus and the moment of inertia respectively. The nonlinear elastic foundation exerts a force per unit length given by ��� − ����� on the column, where �� and �� are constants such that ��>0, ��>0, and � is the imperfection-sensitivity parameter which is such that for � = 1, the nonlinear elastic foundation is said to “softening”, whereas for � = -1, the foundation is said to be “hardening”. We have here excluded all nonlinearities of �(�, �) higher than the cubic and have also excluded all nonlinear derivatives of�(�, �). 3. NON-DIMENSIONALIZATION OF THE GOVERNING EQUATION We shall non-dimensionalize the equations (2.1a - c) by using the following non- dimensional quantities � = � �� �� � � � �, � = � �� �� � � � �, ��(�) = �(�) 2(����) � � , ��� = � �� �� � � � �� , 2� = � (����) � � , � = � �� �� � � � �, 0 < � << 1, 0 < δ ≪ 1; 0 < � < 2.125; https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 2 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD On introducing these non-dimensional quantities into (2.1a-c) and simplifying, we get �,�� + 2��,� + �,���� + 2��(�)�,�� + � − ��� = −2���(�) ���� ��� , � > 0, 0 < � < π (3.1a) �(�, 0) = �,�(�, 0) = 0, 0 < � < π (3.1b) � = �,� = 0, �� � = 0, π, � > 0 (3.1c) Here, a subscript following a comma indicates partial differentiation and �(�) indicates the actual time dependence of the load having its magnitude as λ. In our case, �(�) is a step load characterized by �(�) = � 1, � > 0 0, � < 0 � (3.2) 4. CLASSICAL BUCKLING LOAD, �� The classical buckling load �� is the load that the associated linear perfect column buckles statically. The equations required are obtained from (3.1a) as �,���� + 2��,�� + � = 0, 0 < � < � (4.1a) � = �,� = 0, �� � = 0, π. (4.1b) We note that the deflection � at this stage depends only on�. To solve (4.1a,b), we let �(�) = ∑ (1 − ���2��)�� � ��� (4.2) On substituting (4.2) into (4.1a),multiplying by ���2�� for fixed� and integrating form 0 to �, we see that for � = �, we get (16�� − 8��� + 1)�� = 0 (4.3) where, m is a fixed value of n. According to [18], the condition for static buckling is �� �� = 0 (4.4) where, � is the displacement (or deflection). This gives the classical buckling load �� as λ� = ����� � ��� (4.5a) The least value of �� is when � = 1 and in this case, we get λ� = �� � = 2.125 (4.5b) In retrospect, a similar column with simply-supported end conditions that satisfy the same equation as (4.1a) but, instead of (4.1b), it would satisfy the conditions � = �,�� = 0, �� � = 0, π (4.5c) has the classical buckling load as �� = 1 which is different from (4.5a,b). Thus, a clamped column has a higher classical buckling load than the same column with simply – supported end conditions. https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 3 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 5. STATIC BUCKLING LOAD, �� This is the load that the column would require to buckle statically. The required differential equation is �,���� + 2��,�� + � − ��� = −2�� ���� ��� , 0 < � < � (5.1a) � = �,� = 0, �� � = 0, π, (5.1b) To determine the displacement (or deflection) �(�) in (5.1a,b), we set �(�) ≡ 1and let �� = ���(1 − ���2��), │���│ ≪ 1 Next, we let ∑ �(�)� ��� �� ; �(�) = �(�)(�) (5.2) By substituting (5.2) into (5.1a) and equating the coefficients of powers of ��, � = 1, 2, 3, …, we get O(ϵ):��(�) ≡ �,���� (�) + 2��,�� (�) + �(�) = −8���������2�� (5.3) O(��):��(�) = 0 (5.4) O(��): ��(�) = ���(�)� � (5.5) etc. �(�)(�) = �,� (�) �� � = 0, π (5.6) We seek for the solutions of (5.3) – (5.6) by letting �(�)(�) = 2 ∑ �� (�)� ��� sin� �� = ∑ (1 − ���2��)�� (�)� ��� (5.7) On substituting (5.7) into (5.3), we get ∑ ��� (�)(8��� − 16��)���2�� + �� (�)(1 − ���2��)�� ��� = −8���������2�� (5.8) On multiplying (5.8) through by Cos2mx and integrating from 0 to π, we see that for � = �, (16�� − 8��� + 1)�� (�) = 8������ This gives �� (�) = ������� ����� ������ = � (5.9a) ∴ �(�) = �� (�)(1 − ���2��) (5.9b) On substituting (5.7) into (5.4), we easily get �� (�) = 0 (5.10) Next, we substitute (5.7) in (5.5) and use (5.9a,b) to get ���� (�)(8��� − 16��)���2�� + �� (�)(1 − ���2��)� = ���(�)� � (1 − ���2��)� � ��� = ���(�)� � � � � − �� � ���2�� + � � ���4�� − � � ���6���(5.11) We multiply (5.11) through by Cos2mx and integrate from 0 to π, to get �� (�) = ����� �(����� ������) (5.12) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 4 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD Next, we multiply (5.11) by Cos4mx and integrate from 0 to π to get ��� (�) = ����� �(������ �������) (5.13) Lastly, we multiply (5.11) by Cos6mx and integrate from 0 to π and get, ��� (�) = � ��� �������������� � (5.14) Thus, we get �(�) = �� (�)(1 − ���2��) + ��� (�)(1 − ���4��) + ��� (�)(1 − ���6��)(5.15) So that �(�) = ��� (�)(1 − ���2��) + ����� (�)(1 − ���2��)� + ��� (�)(1 − ���4��) �+ ��� (�)(1 − ���6��)� + … (5.16) Before determining the static buckling load, we need to evaluate (5.16) at � = � �� . This is informed by the fact that eventually, we shall need to determine the associated dynamic problem at � = � �� (which also means finding the maximum of (5.16)). Such evaluation yields � = 2��� (�) + 2����� (�) − ��� (�) � + … (5.17) We can write (5.17) as � = ��� + ���� + … (5.18) where, �� = 2�� (�) , �� = 2��� (�) − ��� (�) � (5.19) As in [18], the static buckling load is obtained by first reversing the series (5.18) in the form, � = ��� + ���� + … (5.20) On substituting for w from (5.18) in (5.20) and equating the coefficients of powers of ϵ, we get �� = � �� , �� = ��� �� � (5.21) The maximization (4.4) is now easily executed from (5.20) to yield �� + 3���� � = 0 (5.22) where, �� is the value of � evaluated at static buckling. From (5.18), we get �� = � ��� ��� = � √� � �� � �� � � � (5.23) Next, we determine (5.20) at static buckling and get � = ��� + ���� + … = ��(�� + ���� �) = � �√� � �� �� � � � (5.24) On substituting in (5.24) for �� and ��, we get (16�� − 8���� + 1) � � = 18√5��� � ������� �1 + � �� � ����� ������� ��������������� �� � � (5.25) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 5 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD where �� is the static buckling load. The least value of �� is obtained when � = 1, and for this, we get (17 − 8��) � � = 18√5��� � ����� �1 + � �� � ��� ��� ����� ��� �� � � (5.26) If it is required that the buckling mode be strictly in the shape of imperfection, then the results corresponding to (5.25) and (5.26) respectively become (16�� − 8���� + 1) � � = 18√5����� � ����� (5.27) and (17 − 8��) � � = 18√5��� � ����� (5.28) By way of comparison, we can perform a similar analysis on the same column but with simply-supported end conditions and the results corresponding to those of (5.25), (5.26), (5.27) and (5.28) are respectively given by (�� − 2���� + 1) � � = � � ��������� � � �1 + � ��� ������� ������������� �� � � (5.29) (1 − ��) � � = � �√� ������� � � �1 + � ���� ������ �� � � (5.30) (�� − 2���� + 1) � � = � � ��������� � � (5.31) and (1 − ��) � � = � �√� ������� � � (5.32) The result (5.32) was first obtained by [6]. So far, we conclude that the static buckling load of structures largely depends, among other things, on the type of end constraints of the structures. 6. THE DYNAMIC PROBLEM The associated dynamic problem follows from (3.1a) which we now recast as �,�� + 2��,� + �,���� + 2��(�)�,�� + � − ��� = −2���(�) ���� ��� , � > 0, 0 < � < π (6.1a) �(�, 0) = �,�(�, 0) = 0, 0 < � < π (6.1b) � = �,� = 0, �� � = 0, π, � > 0 (6.1c) Henceforth, we shall substitute for the step load �(�) as in (3.2). We let � = ��, (6.2a) �̂ = � + � ��(�)��� ��(�)���⋯ � � (6.2b) where, ��(0) = 0, i = 1, 2, 3, …, �� = ��(�) We stress that ϵ and δ are small and unrelated parameters. Thus, from (6.2a,b), we get �� �� = �� ��̂ ��̂ �� + �� ��̂ ��̂ �� �� �� + �� �� �� �� = (1 + �� � �� + �� � �� + ⋯ )�,�� + ��,� (6.3a) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 6 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD where,(… )� = �(… ) �� and a subscript after a comma indicates partial differentiation. ∴ ��� ��� = (1 + �� � �� + �� � �� + ⋯ )��,���� + ���,�� + 2�(1 + �� � �� + �� � �� + ⋯ )�,��� + �(�� ���� + �� ���� + ⋯ )�,�� (6.3b) On substituting (6.3a,b) into (6.1a) for �(�) = 1, we get �(1 + �� � �� + �� � �� + ⋯ )��,���� � + ���,�� + 2�(1 + �� � �� + �� � �� + ⋯ )�,��� +��(�� ���� + �� ���� + ⋯ )�,��� + 2��(1 + �� � �� + �� � �� + ⋯ )�,�� + ��,�� + �,���� + 2��(�)�,�� + � − ��� = −2���(�) ���� ��� (6.4) Next, we adopt the asymptotic series � = ∑ ∑ �(��)(�, �̂� ��� � ��� , �)���� (6.5) and substitute same into (6.4), and afterwards, equate powers of ���� to get �(�): ��(��) = �,���� (��) + �,���� (��) + 2��,�� (��) + �(��) = −2��(�) ���� ��� (6.6) �(��): ��(��) = −2 ��,��� (��) + �,�� (��) � (6.7) �(���): ��(��) = −2 ��,��� (��) + �,�� (��) � − �,�� (��) (6.8) �(��): ��(��) = 0 (6.9) �(���): ��(��) = −2 ��,��� (��) + �,�� (��) � (6.10) �(����): ��(��) = −2 ��,��� (��) + �,�� (��) � − �,�� (��) (6.11) �(��): ��(��) = ���(��)� � − 2�� � �,���� (��) (6.12) �(���): ��(��) = 3���(��)� � �(��) − 2 ��,��� �� + �,�� (��) � − �� ���,�� (��) − 2�� � �,�� (��) (6.13) �(����): ��(��) = 3� ��(��)��(��)� � + ��(��)� � �(��)� − �,�� (��) −2 ��,��� (��) + �� � �,��� (��) � − �� ���,�� (��) − 2 ��,�� (��) + �� � �,�� (��) + �,� (��) � (6.14) The initial conditions, which are evaluated at �̂ = 0 = � are �(��)(�, 0, 0) = 0, � = 1, 2, 3, … ; � = 0, 1, 2, 3, … (6.15) Ο(ϵ): � ,�� (��)(x, 0, 0) = 0 (6.16) Ο(ϵδ): � ,�� (��)(x, 0, 0) + �,� (��)(x, 0, 0) = 0 (6.17) Ο(ϵδ�): � ,�� (��)(x, 0, 0) + �,� (��)(x, 0, 0) = 0 (6.18) In general, we have � ,�� (��)(x, 0, 0) + �,� �(���)(x, 0, 0) = 0, k = 1, 2, 3, … (6.19) Ο(ϵ�): � ,�� (��)(x, 0, 0) = 0 (6.20) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 7 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD In general, we have � ,�� (��)(x, 0, 0) + �,� �(���)(x, 0, 0) = 0, k = 1, 2, 3, … (6.21) Ο(ϵ�): � ,�� (��)(x, 0, 0) + �� � (0)� ,�� (��)(x, 0, 0) = 0 (6.22) O(ϵ��): � ,�� (��)(x, 0, 0) + �� � (0)� ,�� (��)(x, 0, 0) + �,� (��)(x, 0, 0) = 0 (6.23) Ο(ϵ���): �,�� ��(x, 0, 0) + �� � (0)� ,�� (��)(x, 0, 0) + �,� (��)(0, 0) = 0 (6.24) Generally, we have �,�� (��) (x, 0, 0) + �� � (0)�,�� (��) (x, 0, 0) + �,� ��(���)� (x, 0, 0) = 0 , k = 1, 2, 3, … (6.25) The boundary conditions are �(��) = U,� (��) = 0 �� � = 0, �. For solution to all the systems of equation involved here, we set �(��)(x, t̂, τ) = 2 ∑ U� (��)(t̂, τ)sin�nx = ∑ U� (��) (t̂, τ)(1 − cos 2��)� ��� � ��� (6.26a) We shall assume ��(�) = ���(1 − cos 2��) (6.26b) for �, fixed. On substituting (6.26a,b) into (6.6) and simplifying, we get � �U�,���� (��) + U� (��) � (1 − cos 2��) � ��� + �(8��� − 16��)U� (��) cos 2�� � ��� = −8���������2�� (6.27a) Next, we multiply (6.27a) by ���2�� and integrate from 0 to π , and note that for � = �, we get U�,���� (��) + ��U� (��) = 8������ (6.27b) U� (��)(0,0) = 0, U�,�� (��) (0,0) = 0 (6.27c) where, �� = (16�� − 8��� + 1) > 0, ∀ � (6.27d) The solution of (6.27b-d) is U� (��) (t̂, τ) = ��(�) cos ��̂ + ��(�) sin ��̂ + � (6.28a) � = ������� �� = ������� ����� ������ (6.28b) where, ��(0) = −�, ��(0) = 0 (6.28c) This means that �(��) = U� (��) (1 − cos 2��) (6.28d) We next substitute (6.28d) into (6.7), using (6.25), and thereafter, multiply by cos 2�� and note that for � = 2�, we get U �,���� (��) + ��U� (��) = −2 � U�,��� (��) + U�,�� (��) � (6.29a) U� (��)(0,0) = 0, U�,�� (��)(0,0) + U�,� (��)(0,0) (6.29b) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 8 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD On substituting for U� (��) in (6.29) from (6.28a), we ensure a uniformly valid solution in �̂ by equating to zero the coefficients of cos ��̂ and sin ��̂ and getting �� � + �� = 0, �� � + �� = 0 (6.30a) The solution of (6.30a) using (6.28c) yields ��(�) = 0, ��(�) = −���� (6.30b) The remaining equation in (6.29a,b) is solved to get U� (��) = ��(�) cos ��̂ + ��(�) sin ��̂ (6.31a) ��(0) = 0, ��(0) = � � (6.31b) We however note that �� �(0) = B, �� ��(0) = −B (6.31c) We equally note at this stage that (6.32) �(��) = U� (��) (1 − cos 2��) On substituting from (6.32) and (6.28d) in (6.8), multiplying thereafter by cos 2�� and integrating from 0 to π, we get, (for � = �) U �,���� (��) + ��U� (��) = −2 � U�,��� (��) + U�,�� (��) � − U�,�� (��) (6.33a) U� (��)(0,0) = 0, U�,�� (��) (0,0) + U�,� (��)(0,0) (6.33b) Next, we substitute in (6.33a) for U� (��) and U� (��) from (6.31a) and (6.28a) and ensure a uniformly valid solution in �̂ by equating to zero the coefficients of cos ��̂ and sin ��̂ and so, get, respectively �� � + �� = ��� �� �� and �� � + �� = 0 (6.33c) On solving (6.33c), we get ��(τ) = ��� �∫ ��� ��(�) �� � � �� + ��(0)�, ��(τ) = 0 (6.33d) The remaining equation in the substitution into (6.33a) is solved to yield U� (��)(t̂, τ) = ��(�) cos ��̂ + ��(�) sin ��̂ (6.34a) where, ��(0) = 0, ��(0) = 0 (6.34b) In passing, we note from (6.33c) that �� � (0) = −��(0) ��� ��(�) �� = ��� �� , �� ��(0) = � � (6.34c) We conclude that �(��) = U� (��) (1 − cos 2��) (6.35) On solving equations (6.9) to (6.11), using the appropriate initial and boundary conditions, we get �(��)(�, �̂, �) = �(��)(�, �̂, �) = �(��)(�, �̂, �) = 0 (6.36) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 9 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD We next substitute on the right hand side of (6.12), using (6.25) and simplify to get ��(��) = � ���� + 3�� �� 2 � + 3 ����� + �� � 4 � cos � �̂ + 3��� � cos 2��̂ 2 � + � �� � cos 3��̂ 4 � � 5 2 − 15 cos 2�� 4 + 3 2 cos 4�� − 1 4 cos 6��� −2�� � U�,���� (��) (1 − cos 2��) (6.37) Next, we assume �(��) = ∑ U� (��) (1 − cos 2��) � ��� (6.38) On substituting (6.38) into (6.37) and first multiplying through by cos 2��, and thereafter integrating from 0 to π, we see that, for � = �, we get U�,���� (��) + ��U� (��) = 15� 4 ���� + 3�� �� 2 � + 3 ����� + �� � 4 � cos � �̂ + 3��� � cos 2��̂ 2 � + � �� � ��� ���� � � − 2�� � U�,���� (��) (6.39a) U� (��)(0,0) = 0, � �,�� (��)(0, 0) + �� � (0)� �,�� (��)(0, 0) = 0 (6.39b) Next, if in the substitution in (6.12), we multiply through by ���4�� and thereafter integrate from 0 to π, we get (for� = 2�) U��,���� (��) + ��� � U�� (��) = −3� 2 ���� + 3�� �� 2 � + 3 ����� + �� � 4 � cos � �̂ + 3��� � cos 2��̂ 2 � + � �� � ��� ���� � � (6.40a) U�� (��)(0,0) = 0, � ��,�� (��)(0, 0) = 0 (6.40b) where, ��� � = (256�� − 32��� + 1) > 0, ∀ � (6.40�) Lastly, if in the substitution into (6.12), we multiply through by cos 6��, and integrate from 0 to π, we see that, for � = 3�, we get U��,���� (��) + ��� � U�� (��) = � 4 ���� + 3�� �� 2 � + 3 ����� + �� � 4 � cos � �̂ + 3��� � cos 2��̂ 2 � + ��� � ��� ���� � � (6.41a) U�� (��)(0,0) = 0, � ��,�� (��) (0, 0) = 0 (6.41b) where, ��� � = (1296�� − 72��� + 1) > 0, ∀ � (6.41�) To solve (6.39a,b), we substitute for U� (��) from (6.28a), (noting (6.28c)), and ensure a uniformly valid solution in �̂ by equating to zero, the coefficient of cos � �̂ and get �� � (�) = ���� �� ��� + �� � � �, �� � (0) = ������� ���� (6.42a) �� ��(0) = ����� ���� (6.42b) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 10 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD The remaining equation in (6.39a) is now solved to get U� (��)(�̂, �) = �� cos � �̂ + �� sin � �̂ + 15� 4 � �� �� − 3��� � cos 2��̂ 2�� � − ��� � cos 3��̂ 32�� � (6.43�) where ��(�) = ��� + ��� �� � � (6.43b) ��(0) = ���� ����� , (0) = 0 (6.43c) ��(0) = ��� � , �� �(0) = −3��, �� ��(0) = 6�� (6.43d) Turning to (6.40a,b), we solve to get U�� (��) (�̂, �) = �� cos ��� �̂ + �� sin ��� �̂ − 3� 2 � �� ��� � + �� cos ��̂ ��� � − �� + 3��� � cos 2��̂ 2(��� � − 4��) � + � �� � ��� ���� ����� � ����� � (6.44a) where, ��(�) = 3 ����� + �� � � �, ��(0) = ����� � , �� �(0) = ���� � , �� ��(0) = ����� � (6.44b) ��(0) = ������ � , ��(0) = 0 (6.44c) �� = � ���� � + �� ����� � ����� − � ��� � ���� + � ����� � ����� (6.44d) We next solve (6.41a) and get U�� (��)(�̂, �) = �� cos ��� �̂ + �� sin ��� �̂ + � 4 � �� ��� � + �� cos ��̂ ��� � − �� + 3��� � cos 2��̂ 2(��� � − 4��)� + � �� � ��� ���� ����� � ����� � (6.45a) where, ��(0) = ����� 4 , �� = � 1 ��� � + 15 ��� � − �� − 3 ��� � − 4�� � + � 1 2(��� � − 9��) �, ��(0) = 0 (6.45�) Thus, we have �(��) = U� (��)(1 − cos 2��) + U�� (��)(1 − cos 4��) + U�� (��)(1 − cos 6��) (6.45�) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 11 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD We next substitute on the right hand side of (6.13) and get ��(��) = 3� 4 ��� ���� + �� � 4 � sin � �̂ + ��� sin2 � �̂ − �� � 4 sin3 � �̂�� � 5 2 − 15 cos 2�� 4 � ��+ 3 cos 4�� 2 − cos 6�� 4 �� − 2 �U�,��� (��) (1 − cos 2��) + U��,��� (��) (1 − cos 4��)� + U��,��� (��) (1 − cos 6��) + U�,�� (��) (1 − cos 2��) + U��,�� (��) (1 − cos 4��) �+ U��,�� (��) (1 − cos 6��)� − �� �� U�,�� (��) (1 − cos 2��) −2�� � U�,�� (��) (1 − cos 2��) (6.46) Let �(��) = � U� (��) (1 − cos 2��) (6.47) � ��� On substituting (6.47) into (6.46), first multiplying through by cos 2�� and integrating from 0 to π, we get, for � = � U�,���� (��) + ��U� (��) = 45� 4 ��� ���� + �� � 4 � sin � �̂ + ��� sin2 � �̂ − �� � 4 sin3 � �̂�� − �� �� U�,�� (��) − 2�� � U�,�� (��) − 2 �U��,��� (��) + U��,�� (��) � (6.48a) U� (��)(0,0) = 0, � �,�� (��)(0, 0) + �� � (0)� �,�� (��)(0, 0) + ��,� (��)(0, 0) = 0 (6.48b) Next, we multiply (6.46) by cos 4��, integrate from 0 to π, we get, for � = 2� U��,���� (��) + ��� � U�� (��) = −9��� 8 ����� + �� � 4 � sin � �̂ + ��� sin2 � �̂ − �� � 4 sin3 � �̂�� − 2 �U��,��� (��) + U��,�� (��) � (6.49a) U�� (��)(0,0) = 0, � ��,�� (��)(0, 0) + ���,� (��) (0, 0) = 0 (6.49b) Similarly, we multiply (6.46) by cos 6��, integrate from 0 to π, we get, for � = 3� U��,���� (��) + ��� � U�� (��) = 3��� 16 ����� + �� � 4 � sin � �̂ + ��� sin2 � �̂ − �� � 4 sin3 � �̂�� − 2 �U��,��� (��) + U��,�� (��) � (6.50a) U�� (��)(0,0) = 0, � ��,�� (��) (0, 0) + ���,� (��) (0, 0) = 0 (6.50b) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 12 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD To solve (6.48a), we substitute for terms there and after, ensure a uniformly valid solution by equating to zero the coefficients of cos � �̂ and sin � �̂. This gives, for cos � �̂ �� � + �� = 0 (6.51�) Forsin � �̂: �� � + �� = �� 2 (�� �� + 2�� � ) − 45� 32� (�� + �� �)�� (6.51�) On solving (6.51a,b), we get �� = 0; �� = ��� ���(0) − � �� � �� 2 (�� �� + 2�� � ) + 45� 32� (�� + �� �)��� � � ��� (6.51�) �� �(0) = �������� ����� (6.52d) The remaining equations, having ensured a uniformly valid solution in (6.48a) are U�,���� (��) + ��U� (��) = �� sin2 � �̂ + ��sin3 � �̂ (6.52�) U� (��)(0,0) = 0, � �,�� (��)(0, 0) + �� � (0)� �,�� (��)(0, 0) + ��,� (��)(0, 0) = 0 (6.52b) where, �� = �������� �� − ���� �� {(�� �)� + (�� �)} (6.52c) �� = −45������ � 64 − 15� 32� {(�� �)� + (�� �)} (6.52�) (0) = 75��� 16� , ��(0) = −225��� 64� (6.52�) We note the following r� � (0) = −345��� 128� , r� � (0) = 465��� 32� , r� � (0) = −45��� 128� (6.52�) On solving (6.52a,b), we get U� (��) = �� cos � �̂ + �� sin � �̂ − �� sin2 � �̂ 3�� − ��sin3 � �̂ 8�� (6.53�) where, ��(0) = 0 (6.53�) ���(0) − 2��(0) 3� − 3��(0) 8� + ��(0)���(0) + 15� 4 � �� � �� − ��� ��� �� − 3�� ��� � �� � │��� = 0 This yields ��(0) = −5695�� 512�� (6.53�) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 13 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD To ensure a uniformly valid solution in terms of �̂ in (6.49a), we equate to zero the coefficients of cos2 ��� �̂and sin2 ��� �̂ and respectively get �� � + �� = 0 (6.53�) and �� � + �� = � ���� � � �� � ��� + �� � � � + ����� �� ��� ��� � � �� � (6.53e) On solving (6.53d, e), we get �� = 0, �� = ������(0) + ∫ ���� � � ��� (6.53f) where, �� = � ���� � � �� � ��� + �� � � � + ����� �� ��� ��� � � �� � (6.53g) From (6.53e), we get �� � (0) = �����, �� = � ���� � �� ��� + �� ����� � � ��� � − ��� � (6.53h) The remaining equation in (6.49a) is now written as U ��,���� (��) + ��� � U�� (��) = �� sin � �̂ + �� sin2 � �̂ + ��sin3 � �̂ (6.54a) U�� (��)(0,0) = 0, � ��,�� (��) (0, 0) + ���,� (��) (0, 0) = 0 (6.54b) where, �� = ����� � ��� + �� � � � + ������ �� ��� ��� � � �� (6.54c) �� = �������� � − �������� �� � � ��� ��� ��� � � ��� (6.54d) �� = ���� ��� �� − ������� �� � � ��� ��� ����� � � ���� (6.54e) where, ��(0) = �����, �� = ��� ��� + �� ����� � � ��� (6.54f) ��(0) = �����, �� = 9 � � �� + � ��� � � ��� � (6.54g) ��(0) = ������, ��� = � ��� − �� ����� � � ���� (6.54h) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 14 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD On solving (6.54a,b), we get U�� (��) = �� cos ����̂ + �� sin ����̂ + �� ��� ��� ��� � � �� + �� ��� ���� ��� � � ��� + �� ��� ���� ��� � � ��� (6.55a) ��(0) = 0, ��(0) = − 1 2��� � ��� ��� � − �� + 2��� ��� � − 4�� + 3��� ��� � − 9�� + �� � (0)� = ��− �� � � �� � ��� � + �� � ��� � � �� + ���� � �� ��� � � ��� + ��� ��� ����� � � ���� ��� ��� (6.55b) That is ��(0) = ������, (6.55c) ��� = − 1 2��� �� � �� ��� � − �� + 2�� ��� � − 4�� + 3��� ��� � − 9�� �� + �� � � � ��� � − �� ����� � � ��� + � ��� � � ��� − � ��� � � ����� (6.55d) Next we ensure a uniformly valid solution in �̂ in (6.50a) by first substituting for the relevant terms there, and equating to zero the coefficients of cos ����̂ and sin ����̂ to get �� � + �� = 0, �� � + �� = 0 (6.56a) On solving (6.56a), we get ��(�) = 0, ��(�) = ��(0)��� (6.56b) where, �� � (0) = −��(0), �� ��(0) = ��(0) (6.56c) The remaining equation in (6.50a) is now written as U��,���� (��) + ��� � U�� (��) = �� sin � �̂ + �� sin2 � �̂ + ���sin3 � �̂ (6.57�) U�� (��)(0,0) = 0, � ��,�� (��)( 0, 0) + ���,� (��) (0, 0) = 0 (6.57b) where, �� = ���� �� ��� + �� � � � + �������� � � ��� ��� � � � ����� � � ��� (6.57c) �� = ������� �� + ������� � ��� �� �� ����� � � ���� (6.57d) ��� = ����� ��� �� + ��� � (�� ��� � + �� �) (6.57e) ��(0) = �����, �� = �� ��� + �� ����� � � ��� (6.57f) ��(0) = ����� ��� , ���(0) = ����� �� (6.57g) �� �(0) = ������ ���� − ����� ����� � � ��� (6.57h) �� �(0) = ��� � �� ��� − �� ��� � � ��� (6.57i) ��� � (0) = ���� ��� (6.57j) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 15 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD On solving (6.57a,b), we get U�� (��) = �� cos ����̂ + �� sin ����̂ + �� ��� ��� ��� � � �� + �� ��� ���� ��� � � ��� + ��� ��� ���� ��� � � ��� (6.58a) with ��(0) = 0, (6.58b) ��(0) = − 1 ��� � � � �� ��� � − �� + 2�� ��� � − 4�� + 3��� ��� � − 9�� � + �� � � + ��� � � �� � ��� � + �� � ��� � � �� + ����� �� � �� ����� � � ���� + ��� ��� � ����� � � ���� ��� ��� (6.58c) We note from (6.56a) that �� � (0) = −��(0) = ������ � (6.58d) On simplifying (6.58c), we get ��(0) = �����, (6.58e) �� = − 1 ��� �� � �� ��� � − �� − 3 8(��� � − 4��) − 9 56(��� � − 9��) �� − �� 8 + �� � � � ���� � + �� ����� � � ��� − � ��� � � ��� + � ����� � � ���� �� (6.58f) We next substitute on the right hand side of (6.14) and get ��(��) = 3� ��� (��) ��� (��) � � + ��� (��) � � �� (��) � (1 − cos 2��) − ���,�� (��)(1 − cos 2��) + ���,�� (��) (1 − cos 4��) + ���,�� (��) (1 − cos 6��)� −2 ���,��� (��) (1 − cos 2��) + ���,��� (��) (1 − cos 4��) + ���,��� (��) (1 − cos 6��)� + ���,�� (��) (1 − cos 2��) + ���,�� (��) (1 − cos 4��) + ���,�� (��) (1 − cos 6��)� + �� � ��,��� (��) (1 − cos 2��) − �� ����,�� (��) (1 − cos 2��) −2 ��� � ��,�� (��) (1 − cos 2��) + ��,� (��)(1 − cos 2��) + ���,� (��) (1 − cos 4��)� +����,� (��) (1 − cos 6��)� (6.59a) By letting �(��) = ∑ U� (��) (1 − cos 2��) � ��� (6.59b) and substituting same into (6.59a), multiplying the resultant equation by cos 2�� and integrating from 0 to π, we see that, for � = �, we get U�,���� (��) + ��U� (��) = 45� 4 ��� (��) ��� (��) � � + ��� (��) � � �� (��) � − ��,�� (��) − 2 ���,��� (��) + ��,�� (��) + �� � ��,��� (��) � − �� ����,�� (��) − 2�� � ��,�� (��) − 2��,� (��) (6.60a) U� (��)(0,0) = 0, � �,�� (��)(0, 0) + ��,� (��)(0, 0) + �� � ��,�� (��) = 0 (6.60b) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 16 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD For � = 2� in the substitution into (6.59a), we get U��,���� (��) + ��� � U�� (��) = −9� 4 ��� (��) ��� (��) � � + ��� (��) � � �� (��) � − ���,�� (��) − 2 ����,��� (��) + ���,�� (��) � − 2���,� (��) (6.61a) U�� (��)(0,0) = 0, � ��,�� (��)(0, 0) + ���,� (��) (0, 0) (6.61b) For � = 3� in the substitution into (6.59a), we get U��,���� (��) + ��� � U�� (��) = 3� 4 ��� (��) ��� (��) � � + ��� (��) � � �� (��) � − ���,�� (��) −2 ����,��� (��) + ���,�� (��) � − 2���,� (��) (6.62a) U�� (��)(0,0) = 0, � ��,�� (��) (0, 0) + ���,� (��) (0, 0) (6.62�) If we substitute for terms on the right hand side of (6.60a), we get U �,���� (��) + ��U� (��) = 45� 4 ��� � � �� cos ��̂ 4 + � 2 (1 − cos 2��̂) − �� 4 cos 3��̂�� + ������ + �� ��� + ��� � � � cos ��̂ + �� ��� + �� � � � ��� ��̂� + ������� cos 2��̂ + �������� 2��̂ + ���� � � cos 3��̂ + ���� � � sin 3��̂�� − ��� �� cos ��̂ + ��� � � �� �� �� − ���� �� �� ��� cos 2��̂ − ��� �� �� ���� cos 3��̂�� −2 ����� � cos ��̂ − ��� � sin ��̂ − 2�� � 3� cos 2��̂ − 3�� � 8� cos 3��̂�� −��� � �� � � cos ��̂] − �� ��� ��cos ��̂ − 2�� � ��� cos ��̂ −2 ��� � cos ��̂ + ��� � � �� � �� − ���� �� � ��� cos 2��̂ − ��� �� � ���� cos 3��̂�� (6.63a) To ensure a uniformly valid solution in �̂ as far as (6.63a) is concerned, we equate to zero the coefficients of cos ��̂ and sin ��̂ and respectively get �� � + �� = −ℎ�(�) = − � �� �� ��� + �� � � � (6.63b) and �� � + �� = ℎ�(�) = − � �� � ��� � � ���� � � + �� ��� + ��� � � �� − 2�� � − �� �� − �� ����� � −��� � �(2�� + �� � )] (6.63c) On solving (6.63b,c), we get �� = ��� ���(0) − � ℎ�(�)���� � � � �� = ��� ���(0) + � ℎ�(�)���� � � � https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 17 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD The remaining equation in (6.60a) is U �,���� (��) + ��U� (��) = ��� + ��� cos 2��̂ + ��� sin 2��̂ + ��� cos 3��̂ + ��� sin 3��̂ (6.64�) U� (��)(0,0) = 0, � �,�� (��)(0, 0) + ��,� (��)(0, 0) + �� � ��,�� (��) = 0 (6.64b) where, ��� = ��� � ������ − ��� � � � − ���� �� ��� − ���� ��� ��� (6.64c) ��� = ��� � ������ − ��� � � � + ������ �� �� ��� + ����� �� �� �� + ������� �� � ��� (6.64d) ��� = �������� � (6.64e) ��� = ��� �� (�� ��� − �� ���) + ������ �� �� ����� + ���� ����� �� (6.64f) ��� = ������� � �� + ������ �� � ���� (6.64g) ���(0) = ������ ��� , ���(0) = ������ ���� , ���(0) = ������ �� , (6.64h) ���(0) = ������� ����� , ���(0) = ������ ���� (6.64i) On solving (6.64a), using (6.64b) we get U� (��)(�̂, �) = ��� cos ��̂ + ��� sin ��̂ + ��� �� − 1 3�� (��� cos 2��̂ + ��� sin 2��̂) − � ��� (��� cos 3��̂ + ��� sin 3��̂) (6.65a) where, ���(0) = −���(0) �� + ���(0) 3�� + ���(0) 8�� ���(0) = ��������� ����� , ���(0) = 0 (6.65b) As was in the case of U� (��) in (6.60a) which eventually led to (6.65a), we now substitute for terms into (6.61a) and to ensure a uniformly valid solution in �̂, we equate to zero the coefficients of cos ����̂ and sin ����̂ and respectively get �� � + �� = �� �� (�� �� + 2�� � ) (6.66a) and �� � + �� = �� �� (�� �� + 2�� � ) (6.66b) On solving, we get �� = ��� ���(0) − � �� ∫ ��(�� �� + 2�� � )�� � � � (6.66b) �� = ��� ���(0) − � �� ∫ ��(�� �� + 2�� � )�� � � � (6.66c) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 18 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD The remaining equation in the substitution into (6.61a) is U��,���� (��) + ��� � U�� (��) = ��� + ��� cos � �̂ + ��� sin � �̂ + ���cos 2� �̂ + ��� sin 2� �̂ + ���cos 3� �̂ + ���sin 3� �̂ (6.67a) U�� (��)(0,0) = 0, � ��,�� (��)(0, 0) + ���,� (��) (0, 0) (6.67b) where, ��� = ��� � (��� � + �����) + ��� �� ��� � + ��� ��� ���� � (6.67c) ��� = ��� � � ���� � � + �� � ��� � � + ���� + ���� �� ����� � ���� + ���� � ����� � ���� − ����� ����� ��� � ��� (6.67d) ��� = ����� � � ��� � � + ��� (6.67e) ��� = −9� � ��� � � + ������ + ������ �� �� ����� � ����� + ������ �� � ����� � ����� − ����� ����� ��� � ���� (6.67f) ��� = ������� � (6.67g) ��� = −9� � �� ��� � − ���� � � + ����� �� �� ����� � ����� + ����� �� � ����� � ����� − ����� ����� ��� � ���� (6.67h) ��� = ������� � � (6.67i) On solving (6.67a,b), we get U�� (��)(�̂, �) = ��� cos ����̂ + ��� sin ����̂ + ��� ��� � + � 1 ��� � − �� � (��� cos ��̂ + ��� sin ��̂) + � � ��� � � ���� (��� cos 2��̂ + ��� sin 2��̂) + � � ��� � � ���� (��� cos 3��̂ + ��� sin 3��̂) (6.68a) We may not need ���(0) and ���(0). We next substitute into (6.62a) and to ensure a uniformly valid solution in �̂, equate to zero, the coefficients of cos ����̂ and sin ����̂ and respectively get �� � + �� = �� ���� (�� �� + 2�� � ) (6.68b) and �� � + �� = � ���� (�� �� + 2�� � ) (6.68c) On solving, we get �� = ��� ���(0) − � ���� ∫ ��(�� �� + 2�� � )�� � � � (6.68d) �� = ��� ���(0) + �� ���� ∫ ��(�� �� + 2�� � )�� � � � (6.68e) The remaining equation in the substitution into (6.62a) is U��,���� (��) + ��� � U�� (��) = ��� + ��� cos � �̂ + ��� sin � �̂ + ���cos 2� �̂ + ��� sin 2� �̂ + ���cos 3� �̂ + ���sin 3� �̂ (6.69a) U�� (��)(0,0) = 0, � ��,�� (��) (0, 0) + ���,� (��) (0, 0) (6.69b) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 19 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD where, ��� = ����� � � + ������� � − � ��� ��� �� �� ���� � (6.69c) ��� = ���� ��� �� + ���� � � ��� � � + ��� − ��� �� ����� � ���� − ����� �� ��� ��� � ��� − ���� � ����� � ���� (6.69d) ��� = 3��� ��� − �� � � � (6.69e) ��� = ������ � � + ������� � − ������ �� �� ����� � ����� − ������ �� � ��� � ���� − ����� ����� ��� � ���� (6.69f) ��� = ������� � (6.69g) ��� = − ������� � � + ��� ���� �� − ���� �� �� ����� � ����� − ���� �� � ����� � ����� − ������ � ����� ��� � ���� (6.69h) ��� = ������ � �� (6.69i) ���(0) = ���� ��� , ���(0) = 3������ (6.69j) ��� = �� ���� + �� ������ � ���� + ��� ��� � ��� � �� ���� + �� ����� � ����� � − �� ����� � ���� (6.69k) ���(0) = 0, ���(0) = 3������ (6.69l) ��� = �� ��� − � ����� � ���� − �� ��� � ��� � �� ��� − �� ��� � ���� + � ��� � ��� (6.69m) ���(0) = 0, ���(0) = 3������, ���(0) = 0 (6.69n) ��� = � ���� + �� ������ � ����� (6.69o) On solving (6.69a,b), we get ��� �� (�̂, �) = ��� cos ����̂ + ��� sin ����̂ + ��� ��� � + � 1 ��� � − �� � (��� cos ��̂ + ��� sin ��̂) + � � ��� � � ���� (��� cos 2��̂ + ��� sin 2��̂) + � � ��� � � ���� (��� cos 3��̂ + ��� sin 3��̂) (6.70a) where, ���(0) = 3������, (6.70b) ��� = − � � ������ � + ��� ��� � � �� + ��� ��� � � ��� + ��� ��� � � ���� (6.70c) ���(0) = 0 (6.70d) So far, we write the deflection as �(�, �̂, �) = ���� (��) + ��� (��) + ���� (��) + ⋯ �(1 − cos 2��) +����� (��)(1 − cos 2��) + ��� (��)(1 − cos 4��) + ��� (��)(1 − cos 6��)� +���� (��)(1 − cos 2��) + ��� (��)(1 − cos 4��) + ��� (��)(1 − cos 6��)� +����� (��)(1 − cos 2��) + ��� (��)(1 − cos 4��) + ��� (��)(1 − cos 6��)� �+ ⋯ ] + ⋯ (6.71) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 20 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 7. CRITICAL VALUES OF THE DEPENDENT VARIABLES AT MAXIMUM DEFLECTION As in [18, 19], the dynamic buckling load ��, which is defined as the largest load parameter for the solution of the problem to be bounded, is obtained from the maximization �� ��� = 0 (7.1) where, �� is the maximum deflection whose maximum values of the dependent variables we shall now determine. Let ��, �̂�, �� be the values, at maximum displacement of �, �̂and � respectively and let us now assume the following asymptotic series �̂� = �̂� + ��̂�� + ���̂�� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + ⋯ ) + ⋯ (7.2a) �� = �� + ���� + ����� + ⋯ + ��(��� + ���� + ����� + ⋯ ) + ⋯ �� = ��� = �[�� + ���� + ����� + ⋯ + ��(��� + ���� + ����� + ⋯ ) + ⋯ ] (7.2b) As a function of �, �̂, �, the conditions for �(�, �̂, �) to have a maximum are �,� = 0, (1 + �� � �� + �� � �� + ⋯ )�,�� + ��,� = 0 (7.3) On substituting (6.71) into the first of (7.3), we get the value of � at maximum deflection, namely ��, as �� = � �� , � = 1, 2, 3, … (7.4) On evaluating (6.71) at � = �� = � �� , we get � = 2���� (��) + ��� (��) + ���� (��) + ⋯ � + 2�� ��� (��) + ��� (��) + ���� (��) + ��� (��) �� +�� ���� (��) + ��� (��) � + ⋯ � (7.5) We shall now expand each of the terms of (7.3), (which is evaluated at (��, �̂�, ��)) by using (7.2a-c), as well as (7.4) and (7.5). Thus, we get ���,�� (��) = � ���,�� (��) + {��̂�� + ���̂�� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + ⋯ ) + ⋯ }��,���� (��)� +�{�� + ���� + ����� + ⋯ + ��(��� + ���� + ����� + ⋯ ) + ⋯ }� �,��� (��) + 1 2 �{��̂�� + ���̂�� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + ⋯ ) + ⋯ }�� �,������ (��) � +2�{��̂�� + ⋯ + ��(�̂�� + ��̂�� + ⋯ ) + ⋯ } × {�� + ���� + ⋯ + ��(��� + ���� + ⋯ )}� �,����� (��) +��{�� + ���� + ⋯ + ��(��� + ���� + ⋯ ) + ⋯ } �����,���� (��) ��� (���,�) (7.6a) ��� �,�� (��) = ���� �,�� (��) + {��̂�� + ⋯ + ��(�̂�� + ��̂�� + … )}� �,���� (��)� +�{�� + ⋯ + ��(��� + ���� + ⋯ ) + ⋯ }� �,��� (��) + 1 2 �{��̂�� + … + ��(�̂�� + ��̂�� + ⋯ ) + ⋯ }���,������ (��) � +2�{��̂�� + ⋯ + ��(�̂�� + ��̂�� + ⋯ ) + ⋯ } × {�� + ���� + ⋯ + ��(��� + ���� + ⋯ )}� �,����� (��) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 21 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD +��{�� + ���� + ⋯ + ��(��� + ���� + ⋯ ) + ⋯ } �����,���� (��) ��� (���,�) (7.6b) �����,�� (��) = ��� ���,�� (��) + {��̂�� + ⋯ + ��(�̂�� + ��̂�� + … )}��,���� (��)� + �{�� + ⋯ + ��(��� + ���� + ⋯ ) + ⋯ } ��� �,��� (��) + ⋯ �� (���,�) (7.6c) �� ���,�� (��) + ���,�� (��) � = �� ����,�� (��) + ���,�� (��) � + { ��̂�� + ���̂�� + ⋯ } ���,���� (��) + ���,���� (��) �� + �{�� + ���� + ⋯ } ���� �,��� (��) + � ��,��� (��) � + ⋯ �� (���,�) (7.6d) ����� �,�� (��) + � ��,�� (��) � = ��� ����,�� (��) + ���,�� (��) � + { ��̂�� + ⋯ } ���,���� (��) + ���,���� (��) �� + �{�� + ���� + ⋯ } ���� �,��� (��) + � ��,��� (��) � + ⋯ �� (���,�) (7.6e) ������ �,�� (��) + � ��,�� (��) � = ���� ���� �,�� (��) + � ��,�� (��) � + ⋯ �� (���,�) (7.6f) ���� � � �,�� (��) = �� ��� � � �,�� (��) + �� � {��̂�� + ���̂�� + ⋯ }� �,���� (��) + �{�� + ���� + ⋯ }��� � � �,�� (��) � ,� � + � � {�̂�� + ⋯ }��� � � �,���� (��) + �{�� + ���� + ⋯ }{��̂�� + ���̂�� + ⋯ }��� � � �,���� (��) � ,� + � � ��(�� + ⋯ )� ����� � � �,�� (��) � ,�� + ⋯ �� (���,�) (7.6g) ����� � ��,�� (��) = ��� ��� � ��,�� (��) + (��̂�� + ⋯ )�� � ��,���� (��)� + ���(�� + ⋯ )��� � � �,���� (��) � ,� + ⋯ �� (���,�) (7.6h) ������ � � �,�� (��) = ���� ���� � � �,�� (��) + ⋯ �� (���,�) (7.6i) ����,� (��) = �� ���,� (��) + {��̂�� + ⋯ + ��(�̂�� + ��̂�� + … )}��,��� (��) � + � � �{��̂�� + … + ��(�̂�� + ��̂�� + ⋯ ) + ⋯ }�� �,����� (��) � +2�{��̂�� + ⋯ + ��(�̂�� + … ) + ⋯ } × {�� + ���� + ⋯ + ��(��� + ���� + ⋯ )} ���� �,���� (��) + ⋯ ��� (���,�) (7.6j) �����,� (��) = ������,� (��) + ⋯ + {��̂�� + ⋯ + ��(�̂�� + ��̂�� + ⋯ )}� ����,��� (��) + ⋯ �� (���,�) (7.6k) ������,� (��) + ���,� (��) � = ��� ����,� (��) + ���,� (��) � + �(��̂�� + … ) ���,��� (��) + ���,��� (��) � + ⋯ �� ��+�(�� + … )���,�� (��) + ���,�� (��) � + ⋯ �� (���,�) (7.6l) �������,� (��) + ���,� (��) � = ������,� (��) + ���,� (��) �│(���,�) + ⋯ (7.6m) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 22 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD By substituting (7.6a – m) into the second equation of (7.3) and equating the coefficients of ����, we get �(�): ��,�� (��) (�̂�, 0) = 0 (7.7a) �(��): �̂��� �,���� (��) + ��� �,��� (��) + ��,� (��) = 0 (7.7b) �(���): �̂����,���� (��) + �����,��� (��) + (�̂��)� 2 ��,���� (��) + �̂������,����� (��) + (��)� 2 ��,���� (��) +�̂����,���� (��) + ����,��� (��) + ��,�� (��) + �̂����,��� (��) + ����,�� (��) + ��,� (��) (7.7c) �(��): �̂��� �,���� (��) + �� �,���� (��) + � ��,���� (��) � + �� � (0)� �,�� (��) = 0 (7.7d) �(���): �̂����,���� (��) + �����,��� (��) + 1 2 �2�̂���̂����,������ (��) + 2�̂������,����� (��) � + �̂����,���� (��) + �̂�� ���,���� (��) + ���,���� (��) � + �� ���,��� (��) + ���,��� (��) � + ���,�� (��) + ���,�� (��) � + �̂��� �,��� (��) + ���,� (��) + ���,� (��) � = 0 (7.7e) etc., where equations (7.7a) to (7.7e) are evaluated at (�̂�, 0). From (7.7a), we get ��̂� = ��, � = 1, 2, 3, … ∴ �̂� = � � , (� = 1) (7.8a) where we have taken � = 1. On substituting (7.8) in (7.7b) and simplifying, we get �̂�� = � ��� �,�� (��) � ��,� (��) � � �,��� (��) � � �,���� (��) � (���,�) = 0 (7.8b) On substituting for terms in (7.7c) and simplifying, we get �̂�� = �������,�� (��) � �,���� (��) � (���,�) = ��� �� = � �� (7.8c) We next substitute into (7.7d) and get �̂�� = � ��� �,���� (��) � � ��,���� (��) � � �,���� (��) � (���,�) (7.8d) Now, ��,���� (��) (�̂�, 0) = ����� �� , ���,���� (��) (�̂�, 0) = ������, (7.8e) ��� = � � �� � ���� � − �� ����� � � ��� + � ����� � � ���� − � ����� � � ���� �� − �� ���� ����� � � ��� + ���� ���� � � ���� − ��� ����� � � ���� �� (7.8f) � �,���� (��)(�̂�, 0) = −��� (7.8g) On substituting for terms in (7.8d), we get �̂�� = ������, ��� = ���� + �� �� � (7.8h) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 23 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD After substituting for terms in (7.7e) and simplifying, many terms vanish and the remaining terms give �̂�� = �� �� � �,���� (��) ��̂���� + � �,����� (��) + � �,�� (��) + � ��,�� (��) + ��,� (��) + ���,� (��) ��� (���,�) (7.8i) We now evaluate each of the terms in (7.8i) as follows: ��,�� (��) (�̂�, 0) = ������� ��� (7.9a) ���,�� (��) (�̂�, 0) = ������ (7.9b) ��� = �� cos ����̂� − ��� ����� � � ��� − � ������ � � ���� − � ������ � � ���� (7.9c) ��,� (��)(�̂�, 0) = ������ ���� (7.9d) ���,� (��) (�̂�, 0) = ������ (7.9e) ��� = − �� ��� ������ � − � � � � ��� � + �� ����� � � ��� + � ������ � � ���� + � ������ � � ���� � (7.9f) � �,����� (��) (�̂�, 0) = ���� �(0) = ��� (7.9g) On substituting for terms in (7.8i), we get �̂�� = ������, ��� = � �� �������� + ���� ��� + ��� − ����� ����� + ���� (7.10) Later, we shall also need terms like ��, ���, ���, ��� ��� ��� which we now evaluate directly from (6.2b) (evaluated at maximum values of the variables). Thus, at maximum deflection, (6.2b) becomes �̂� = �� + � ��(��)��� ��(��)���⋯ � � (7.11) By using (7.2a – c), we can write (7.11) as �̂� + ��̂�� + ���̂�� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + ⋯ ) + ⋯ = �� + ���� + ����� + ⋯ + ��(��� + ���� + ����� + ⋯ ) + �� ���(0) + ���� � (0) + ��� � 2 �� ��(0)� +�� ���(0) + ���� � (0) + ��� � � �� ��(0) + ⋯ � + ⋯ (7.12) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 24 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD We note the fact that ��(0) = 0 and equally note that �� can be expanded using (7.2b). By equating coefficients of ���� in (7.12), we get �(�): �̂� = � � = �� ∴ �� = � � (7.13a) �(��): �̂�� = 0 = ��� ∴ ��� = 0 (7.13b) �(���): �̂�� = �� � = � �� = ��� ∴ ��� = � �� (7.13c) �(��): �̂�� = ������ = ��� + �� � (0)�� ∴ ��� = ������ − �� � (0)�� = ������ (�) (7.13d) ��� (�) = ��� + ���� ���� (7.14) �(���): �̂�� = ������ = ��� + ����� � (0) + �� � 2 �� ��(0) ∴ ��� = ������ − ����� � (0) − �� � � �� ��(0) = ������ (�) (7.15a) ��� (�) = ��� − ���� ���� (7.15b) 8. MAXIMUM DEFLECTION, �� To determine the maximum deflection w�, we evaluate (7.5) at �̂�, �� ��� ��. Thus, we get w� = 2����� (��) + ���� (��) + ����� (��) � + 2�� ����� (��) + ���� (��) � + ����� (��) + ���� (��) �� + ������� (��) + ���� (��) � + ⋯ � + ⋯ (8.1) where,��� (��) = �� (��) (�̂�, ��). We now expand each term of (8.1) using (7.2a – c). Thus, we have ���� (��) = ��� (��)(�̂�, ��, ��) = ���� (��) + {�̂��� + ���̂�� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + ⋯ )}� + � � {{�̂��� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + … )}�� + �{�� + ���� + ⋯ + ��(��� + ���� + ⋯ )} × {�̂��� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + … )}��,�� (��) ��+��{�� + ���� + ⋯ + ��(��� + ���� + ⋯ )}� + ⋯ ]|(���,�) (8.2) ����� (��) = �� ��� (��) + {�̂��� + ���̂�� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + … )}��,�� (��)� + �{�� + ���� + ⋯ + ��(��� + ���� + ⋯ )} + � � {{�̂��� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + … )}�� + 2�{�� + ���� + ⋯ + ��(��� + ���� + ⋯ )} × {�̂��� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + … )}��,��� (��) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 25 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD ��+��{�� + ���� + ⋯ + ��(��� + ���� + ⋯ )}� + ⋯ ]|(���,�) (8.3) ������ (��) = ��� ��� (��) + {�̂��� + ⋯ + ��(�̂�� + ⋯ )}��,�� (��)� ��+�{�� + ⋯ + ��(��� + ���� + ⋯ )} + ⋯ ]|(���,�) (8.4) ����� (��) + ��� (��) � = �� ���� (��) + ��� (��) � + {�̂��� + ⋯ }��� (��) + ��� (��) � ,�� � ��+�{�� + ⋯ + ��(��� + ���� + ⋯ )}���,� (��) + ���,� (��) ��� (���,�) (8.5) ������ (��) + ��� (��) � = ��� ���� (��) + ��� (��) � + {�̂��� + ⋯ }��� (��) + ��� (��) � ,�� � ��+ �(�� + ⋯ )���,� (��) + ���,� (��) ��� (���,�) (8.6) ������� (��) + ��� (��) � = ���������,� (��) + ���,� (��) ��� (���,�) (8.7) On substituting (8.2) – (8.7) into (8.1), we observe that most terms vanish and the remaining ones are given as w� = 2� ����� (��) + �����,� (��) + ���� � 2 ��,�� (��) ��� (���,�) +2�� ���� (��) + ��� (��) � + � ������,� (��) + �̂����,�� (��) + �����,� (��) + ���,� (��) ��� +��������,� (��) + �̂���̂��� �,���� (��) + �������,�� (��) + �̂��� �,�� (��) + ���̂��� �,��� (��) � ���+ �� � � ���,� (��) + ���,� (��) � + ��� (��) + ��� (��) ���� (���,�) + ⋯ (8.8) We however note that �� (��)(�̂�, 0) = ����� �� , (8.9a) ��� (��)(�̂�, 0) = ����� (8.9b) �� = � � � �(����� ������) ��� + �(����� ������) ����� � ���� + �(����� ������) ����� � ����� + (����� ������) ����� � ����� � (8.9c) �� (��)(�̂�, 0) = ������ ��� , (8.10a) ��� (��)(�̂�, 0) = ������ (8.10b) ��� = � �(����� ������) ���� � �� − ���(����� ������) ��� � ��� + ���(����� ������) ��� � ���� − ���(����� ������) ����� � ����� � (8.10c) On substituting (8.9a) – (8.10c) into (8.8) and simplifying, we get w� = 4�� �1 − �� �� + � � � �� � � � � + ������� �� ��1 + ���� �� � + ���� + ������ + ⋯ (8.11a) where, ��� = �� �� � � � � ��� ���� + ���� − ��� (�) − ���� (8.11b) ��� = �� �� �� � � � ��� (�) − ��� (�) − � � � � ��� − ��� + � � � � ��� � ��� ���� + ���� − �� ��� + ���� (8.11c) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 26 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 9. DYNAMIC BUCKLING LOAD, �� Having determined the maximum deflection, w� as in (8.11a – c), we shall now determine the dynamic buckling load �� from the maximization (7.1). We shall first reverse the series (8.10a) which we now write as w� = ��� + ���� + ⋯ (9.1a) where, �� = 4 �1 − �� �� + � � � �� � � � � (9.1b) �� = ����� �� ��1 + ���� �� � + ���� + ������ (9.1c) Thus, for the reversal, we write � = ��w� + ���� � + ⋯ (9.2) Upon substituting in (8.13) for w� from (8.12a), and equating the coefficients of �, we get �� = � �� , �� = − � �� �� �� (9.3a) The maximization (7.1) is now easily executed from (9.2) to yield �� + 3����� � = 0, (9.3b) where w�� is the maximum value of the deflection at dynamic buckling. Thus, we have w�� = � ��� ��� (9.4) On substituting from (9.3a) in (9.4) we get w�� = � √� � �� � �� � (9.5) If we next evaluate (9.2) at dynamic buckling, we get � = ��w�� + ����� � + ⋯ = w��( �� + 3����� � ) (9.6) On substituting in (9.6) for ��, �� ��� w��, we get � = � �√� � �� �� � � � (9.7) On substituting in (9.7), we get the equation for determining the dynamic buckling load λ� as (16�� − 8��λ� + 1) � � = 18√10(����)��λ�� � � � ���� ���� �� �� ����� ������ �� �� �� � � � � �� � � � � � � (9.9) Equation (9.9) gives an implicit formula for determining the dynamic buckling load λ�. By using equation (5.25), we can eliminate the imperfection parameter ϵ and hence relate λ� to λ�. This gives � ������������ ������������ � � � = √2 � �� �� � ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ ���� ���� �� �� ����� ������ �� �� �� � � � � �� � � � �� � �� � ����� ������� ��������������� � ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ � � (9.10) https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 27 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD The least value of the dynamic buckling load λ� is obtained when � = 1 and for this value, equations (9.9) and (9.10) respectively become (17 − 8λ�) � � = 18√10(����)λ�� � � � ���� ���� �� �� ����� ������ �� �� �� � � � � �� � � � � � � (9.11) and � ������ ������ � � � = √2 � �� �� � ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ ���� ���� �� �� ����� ������ �� �� �� � � � � �� � � � �� � �� � ������ ��������� � ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ � � (9.12) where the right hand sides of (9.11) and (9.12) are to be evaluated at � = 1. 10. ANALYSIS OF RESULTS The graphical plots of the results were done using Q-Basic codes and the results are hereby presented in Fig. 1, Fig. 2 and Fig. 3. Fig. 1: Relationship between the Static buckling load, �� and Imperfection factor, ϵ using Eqn. (5.26), Eqn.(5.28), Eqn.(5.30) and Eqn.(5.32). 0 0.5 1 1.5 2 2.5 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 ST A TI C B U C K LI N G L O A D IMPERFECTION FACTOR STATIC BUCKLING LOAD (EQN 5.28) STATIC BUCKLING LOAD (EQN 5.30) STATIC BUCKLING LOAD (EQN 5.32) STATIC BUCKLING LOAD (EQN. 5.26) 〖λ�〗^ ϵ https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 28 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD Fig. 2: Relationship between the Dynamic buckling loads and Imperfection parameters at some fixed values of the damping factor, δ, using Eqn. (9.11). Fig. 3: Relationship between the Dynamic buckling loads and the Static buckling loads at some fixed values of the damping factor, δ, using Eqn. (9.12). From Fig. 1, we observe that the static buckling load of a clamped column is always higher than that of the same column with simply–supported boundary conditions. In general, the static buckling load of a column with either clamped or simply-supported boundary conditions and whose deflections are strictly in the shape of imperfection always has the least static buckling load. However, while the static buckling load of a clamped column satisfies the inequality, 1 < �� < 2.125, a similar column with simply– supported boundary supports satisfies the inequality 0 < �� < 1. Fig. 2 shows that the dynamic buckling load �� decreases with increased imperfection for any value of the damping parameter. From Fig. 3, we observe that at low values of the static buckling load �� (precisely for 1 < �� < 1.45), there is no significant difference in the value of the dynamic buckling load �� of the column, whether damped or undamped. However, at higher values of �� (i.e, 1.45 < �� < 2.125), the dynamic buckling load �� rises with �� and the highest of 0 0.2 0.4 0.6 0.8 1 1.2 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 D Y N A M IC B U C K LI N G L O A D IMPERFECTION FACTOR DYNAMIC BUCKLING LOAD, δ = 0 DYNAMIC BUCKLING LOAD, δ = 0.01 DYNAMIC BUCKLING LOAD, δ = 0.02 DYNAMIC BUCKLING LOAD, δ = 0.03 λD ϵ 0 0.5 1 1.5 2 2.5 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1D Y N A M IC B U C K LI N G L O A D STATIC BUCKLING LOAD DYNAMIC BUCKLING LOAD , δ = 0 DYNAMIC BUCKLING LOAD , δ = 0.01 DYNAMIC BUCKLING LOAD , δ = 0.02 DYNAMIC BUCKLING LOAD , δ = 0.03 λD λS https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 29 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD such rise is the undamped case. It is not clear whether such a result is specific to clamped boundary conditions or whether it is general. As in the static loading case, the inequality satisfied by the clamped column is 1 < �� < 2.125. We thus observe that generally, whether in the static or dynamic loading cases, clamped columns buckle at much more higher loads than similar columns with simply–supported boundary supports. For reasons attributed to nonlinearity and imperfection, clamped columns on nonlinear elastic foundations buckle at lower values of buckling loads than the corresponding classical buckling load of the column. We observe that while the buckling modes split into three distinct modes proportional to (1 − cos 2��), (1 − cos 4��) and (1 − cos 6��), it is only the buckling modes in the shapes of (1 − cos 2��) and (1 − cos 6��) that eventually contribute to dynamic buckling. The mode in the shape of (1 − cos 4��) does not contribute. Lastly, as seen in equations (8.20) and (8.22), we are able to directly relate �� to �� even without the knowledge of the size of the associated imperfection. In this way, we have circumvented the process of repeating the arduous manipulations for different imperfection parameters. 11. CONCLUSION We have carried out an analytical investigation of the buckling of an imperfect clamped column lying on a nonlinear elastic foundation but struck axially by a step load. It is our contention that similar enquiries can be extended to other loading conditions apart from step load and to other structures apart from columns. ACKNOWLEDGEMENTS: Authors acknowledge the valuable contribution their mentor, Prof. A.M. Ette (FNMS), for his guardian and support on this work. REFERENCES [1] Timoshenko, S.P. and Gere, J.M., Theory of elastic stability, Dover Publications, London. 2012. [2] Thompson, J. M. T. and Hunt, G. W.,A general theory of elastic stability, John Wiley and Sons Ltd., London. 1973. [3] Hu, N. and Burgue��o, R.,Elastic postbuckling response of axially – loaded cylindrical shells with seeded geometric imperfection design; Thin – walled structures, 96, 256 – 268, 2015. [4] Amazigo, J. C., Budiansky, B. and Carrier, G. F.,Asymptotic Analyses of the buckling of imperfect columns on nonlinear elastic foundations, Int. J. Solids Structures, 10, 1342 – 1356, 1970. https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 30 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) I.U. Udo-Akpan1 * https://ijojournals.com/ Volume 07 Issue 02 || February, 2024 || ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD [5] Qiang, H., Zhang, S. and Yang, G.,The Asymptotic solution of a dynamic buckling problem in elastic column, Appl. Maths. and Mech. (English edition)20(8), 867 – 872, 1999. [6] Amazigo, J. C. and Frank, D.,Dynamic buckling of an imperfect column on a nonlinear foundation, Quart. Appl. Math., 31(6), 1 – 9, 1973. [7] Artem, H. S. and Aydin, L.,Exact solution and dynamic buckling analysis of a beam – column loading. Appl. Math. and Mech. 31(10), 1317 – 1324, 2010. [8] Ette, A. M., Chukwuchekwa, J. U. and Udo – Akpan, I. U.,On the buckling of a clamped viscously damped column trapped by a step load. Int. J. of Appl. Sciences and Mathematics, 3 (2), 117 – 123, 2016. 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UdoAkpan, (2024) Efficient Solution of Nonhomogeneous Linear Differential Equations with Constant coefficients: the method of undetermined coefficient approach, IEEE-SEM, 12(1), 21-36 https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 32