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The behaviour of a swinging door can be modelled by the second order, constant coefficient ODE d²0 b de k + + 0=0 dt² I dt Ī (1) where (t) gives the angular position of the door, and I, b, k are positive mechanical parameters that affect the motion of the door. wall door Ө wall 00 e(t) (i) (a) Substitute ( et into equation (1) to find the characteristic equation. (b) Determine the relationship between I, b, and k that will ensure the roots of the characteristic equation are not complex. That is, we seek solutions that have no oscillatory behaviour. (c) State the general solution corresponding to the case of repeated real roots in (b). AUGU t

The behaviour of a swinging door can be modelled by the second order, constant coefficient ODE d²0 b de k + + 0=0 dt² I dt Ī (1) where (t) gives the angular position of the door, and I, b, k are positive mechanical parameters that affect the motion of the door. wall door Ө wall 00 e(t) (i) (a) Substitute ( et into equation (1) to find the characteristic equation. (b) Determine the relationship between I, b, and k that will ensure the roots of the characteristic equation are not complex. That is, we seek solutions that have no oscillatory behaviour. (c) State the general solution corresponding to the case of repeated real roots in (b). AUGU t

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The behaviour of a swinging door can be modelled by the second order, constant coefficient ODE d²0 b de k + I dt I dt² (1) where (t) gives the angular position of the door, and I, b, k are positive mechanical parameters that affect the motion of the door. wall door (i) (a) Substitute ( Ө + = 0 wall e(t) = = et into equation (1) to find the characteristic equation. (b) Determine the relationship between I, b, and k that will ensure the roots of the characteristic equation are not complex. That is, we seek solutions that have no oscillatory behaviour. (c) State the general solution corresponding to the case of repeated real roots in (b). (ii) (a) Letting y₁(t) = : 0(t) and y₂(t) = 0'(t) = y₁ (t), show that equation (1) can be written as a system of first order ODEs involving y₁ and Y2. Summarise the system in the matrix equation y' = M y. (b) Show that the eigenvalues of M are identical to the roots computed in (i)(b). (c) Find the eigenvector for the case of repeated eigenvalues. For this matrix you should only be able to find a single eigenvector. Repeated eigenvalues having a single eigenvector are referred to as defective eigenvalues. The matrix is also said to be defective. They will be explored in other units. It is common to revert to the procedure of (i)(a)-(c) when a defective eigenvalue is found.
Transcribed Image Text:The behaviour of a swinging door can be modelled by the second order, constant coefficient ODE d²0 b de k + I dt I dt² (1) where (t) gives the angular position of the door, and I, b, k are positive mechanical parameters that affect the motion of the door. wall door (i) (a) Substitute ( Ө + = 0 wall e(t) = = et into equation (1) to find the characteristic equation. (b) Determine the relationship between I, b, and k that will ensure the roots of the characteristic equation are not complex. That is, we seek solutions that have no oscillatory behaviour. (c) State the general solution corresponding to the case of repeated real roots in (b). (ii) (a) Letting y₁(t) = : 0(t) and y₂(t) = 0'(t) = y₁ (t), show that equation (1) can be written as a system of first order ODEs involving y₁ and Y2. Summarise the system in the matrix equation y' = M y. (b) Show that the eigenvalues of M are identical to the roots computed in (i)(b). (c) Find the eigenvector for the case of repeated eigenvalues. For this matrix you should only be able to find a single eigenvector. Repeated eigenvalues having a single eigenvector are referred to as defective eigenvalues. The matrix is also said to be defective. They will be explored in other units. It is common to revert to the procedure of (i)(a)-(c) when a defective eigenvalue is found.
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