Explanation Consider a vertical column with uniform cross section of length L subject to a constant axial load P . If constant vertical compressive force P is applied, then the deflection of vertical column function y ( x ) satisfies the following differential equation. E I d 2 y d x 2 + P y = 0 Consider the column is supported at both ends. The initial conditions are as follows: y ( 0 ) = 0 y ′ ( L ) = 0 The auxiliary equation for the given differential equation is m 2 + ( P E I ) = 0 . Solve the auxiliary equation as follows: m 2 = − P E I m = ± P E I i Therefore, the general solution of the differential equation is as follows: y ( x ) = c 1 cos ( P E I x ) + c 2 sin ( P E I x ) ...... ( a ) Apply the boundary conditions and obtain the values of the constants c 1 and c 2 . Substitute the first condition y ( 0 ) = 0 in equation (a) and calculate as follows: y ( 0 ) = c 1 cos ( P E I 0 ) + c 2 sin ( P E I 0 ) 0 = c 1 cos 0 c 1 = 0 Substitute the value for c 1 = 0 in the equation (a) and apply the initial condition y ( L ) = 0 as follows: y ( L ) = c 2 sin ( P E I L ) 0 = c 2 sin ( P E I L ) 0 = sin ( P E I L ) ( For non trivial solution c 2 ≠ 0 ) Sin n π = sin ( P E I L ) Further, solve the above equation as follows: n π = P n E I L n 2 π 2 = P n E I L 2 P n E I L 2 = n 2 π 2 P n = n 2 π 2 E I L 2 n = 1 , 2 , 3 ,

Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th

11th Edition

ISBN: 9781305965737

Author: Dennis G. Zill

Publisher: Brooks Cole

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Chapter 5.2, Problem 23E

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To sketch: The deflection curve for critical load P4 and state the position of all critical load.

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Chapter 5.2, Problem 22E

Chapter 5.2, Problem 24E

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Chapter 5 Solutions

Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th

Ch. 5.1 - 5.1.1 Spring/Mass systems: Free Undamped Motion A...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A mass...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A mass...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A force...Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A mass...Ch. 5.1 - 5.1.1Spring/Mass Systems: Free Undamped Motion A...

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The deflection curve for critical load P 4 and state the position of all critical load.

Solution Summary: The author illustrates the deflection curve for critical load P_4 and states the position of all critical loads.

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To sketch: The deflection curve for critical load P4 and state the position of all critical load.

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