Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver, Chehrzad Shakiban

Chapter 6 Equilibrium - all with Video Answers

Educators

Chapter Questions

Problem 1

If a bar in a structure compresses $2 \mathrm{~cm}$ under a force of 5 newtons applied to a node, how far will it compress under a force of 20 newtons applied at the same node?

Khushbu Rani

Numerade Educator

00:44

A mass-spring chain consists of two masses connected to two fixed supports. The spring constants are $c_1=c_3=1$ and $c_2=2$. (a) Find the stiffness matrix $K$. (b) Solve the equilibrium equations $K \mathbf{u}=\mathbf{f}$ when $\mathbf{f}=(4,3)^T$. (c) Which mass moved the farthest? (d) Which spring has been stretched the most? Compressed the most?

Ze-Han Lee

Numerade Educator

04:27

Problem 1

Draw the electrical networks corresponding to the following incidence matrices.
(a) $\left(\begin{array}{rrrr}1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1\end{array}\right)$,
(b) $\left(\begin{array}{rrrr}0 & 0 & 1 & -1 \\ 1 & 0 & 0 & -1 \\ 0 & -1 & 1 & 0 \\ 1 & 0 & -1 & 0\end{array}\right)$,
(c) $\left(\begin{array}{rrrrr}0 & 1 & 0 & 0 & -1 \\ -1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 1 \\ 0 & -1 & 1 & 0 & 0\end{array}\right)$,
(d)
$\left(\begin{array}{rrrrr}-1 & 0 & 1 & 0 & 0 \\ 0 & -1 & 0 & 1 & 0 \\ 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & -1 & 0 & 1\end{array}\right)$
$\left(\begin{array}{rrrrrrr}0 & -1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 1 & 0 & 0 \\ 1 & 0 & -1 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 1 \\ 0 & 0 & -1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & -1\end{array}\right)$.

Cory Glover

Numerade Educator

00:30

Problem 2

Solve Exercise 6.1.1 when the first and second springs are interchanged, $c_1=2$, $c_2=c_3=1$. Which of your conclusions changed?

Cory Kuzinski

Numerade Educator

03:03

Problem 2

An individual bar in a structure experiences a stress of 3 under a unit horizontal force applied to all the nodes and a stress of -2 under a unit vertical force applied to all nodes. What combinations of horizontal and vertical forces will make the bar stress-free?

Khoobchandra Agrawal

Numerade Educator

03:35

Problem 2

Suppose that all wires in the illustrated network have unit resistivity. (a) Write down the incidence matrix $A$. (b) Write down the equilibrium system for the network when node 4 is grounded and there is a current source of magnitude 3 at node 1 . (c) Solve the system for the voltage potentials at the ungrounded nodes. (d) If you connect a light bulb to the network, which wire should you connect it to so that it shines the brightest?

Jonathan Everett

Numerade Educator

05:40

Problem 3

What happens in the network in Figure 6.5 if we ground both nodes 3 and 4 ? Set up and solve the system and compare the currents for the two cases.

Jacob Denson

Numerade Educator

01:04

Problem 3

Redo Exercises 6.1.1-2 when the bottom support and spring are removed.

Sherrie Fenner

Numerade Educator

02:49

Problem 3

(a) For the reinforced structure illustrated in Figure 6.16, determine the displacements of the nodes and the stresses in the bars under a uniform horizontal force, and interpret physically. (b) Answer the same question for the doubly reinforced structure in Figure 6.17.

Khoobchandra Agrawal

Numerade Educator

02:09

Problem 4

Discuss the effect of a uniform horizontal force in the direction of the horizontal bar on the swing set and its reinforced version in Example 6.9.

Lucas Finney

Numerade Educator

03:00

Problem 4

A mass-spring chain consists of four masses suspended between two fixed supports. The spring stiffnesses are $c_1=1, c_2=\frac{1}{2}, c_3=\frac{2}{3}, c_4=\frac{1}{2}, c_5=1$. (a) Determine the equilibrium positions of the masses and the elongations of the springs when the external force is $\mathbf{f}=(0,1,1,0)^T$. Is your solution unique? $(b)$ Suppose we fix only the top support. Solve the problem with the same data and compare your results.

Ben Nicholson

Numerade Educator

02:19

Problem 4

(a) Write down the incidence matrix $A$ for the illustrated electrical network. (b) Suppose all the wires contain unit resistors, except for $R_4=2$. Let there be a unit current source at node 1 , and assume node 5 is grounded. Find the voltage potentials at the nodes and the currents through the wires.
(c) Which wire would shock you the most?

Khoobchandra Agrawal

Numerade Educator

01:43

Problem 5

All the bars in the illustrated square planar structure have unit stiffness. (a) Write down the reduced incidence matrix $A$. (b) Write down the equilibrium equations for the structure when subjected to external forces at the free nodes. (c) Is the structure stable? statically determinate? Explain in detail. (d) Find a set of external forces with the property that the upper left node moves horizontally, while the upper right node stays in place. Which bar is under the most stress?

Manish Jain

Numerade Educator

02:48

Problem 5

(a) Show that, in a mass-spring chain with two fixed ends, under any external force, the average elongation of the springs is zero: $\frac{1}{n+1}\left(e_1+\cdots+e_{n+1}\right)=0 .(b)$ What can you say about the average elongation of the springs in a chain with one fixed end?

Khoobchandra Agrawal

Numerade Educator

00:53

Problem 5

Answer Exercise 6.2.4 if, instead of the current source, you put a 1.5 volt battery on wire 1 .

Prabhu Ramji

Numerade Educator

03:51

Problem 6

Suppose we subject the $i^{\text {th }}$ mass (and no others) in a chain to a unit force, and then measure the resulting displacement of the $j^{\text {th }}$ mass. Prove that this is the same as the displacement of the $i^{\text {th }}$ mass when the chain is subject to a unit force on the $j^{\text {th }}$ mass. Hint: See Exercise 1.6.20.

Khoobchandra Agrawal

Numerade Educator

02:07

Problem 6

Consider an electrical network running along the sides of a tetrahedron. Suppose that each wire contains a $3 \mathrm{ohm}$ resistor and there is a 10 volt battery source on one wire. Determine how much current flows through the wire directly opposite the battery.

Vishal Gupta

Numerade Educator

00:52

Problem 6

In the square structure of Exercise 6.3.5, the diagonal struts simply cross each other. We could also try joining them at an additional central node. Compare the stresses in the two structures under a uniform horizontal and a uniform vertical force at the two upper nodes, and discuss what you observe.

Mayukh Banik

Numerade Educator

01:41

Problem 7

Now suppose that each wire in the tetrahedral network in Exercise 6.2 .6 contains a $1 \mathrm{ohm}$ resistor and there are two 5 volt battery sources located on two non-adjacent wires. Determine how much current flows through the wires in the network.

Mark Scythian

Numerade Educator

01:43

Problem 7

(a) Write down the reduced incidence matrix $A^{\star}$ for the pictured structure with 4 bars and 2 fixed supports. The width and the height of the vertical sides are each 1 unit, while the top node is 1.5 units above the base. (b) Predict the number of independent solutions to $A^{\star} \mathbf{u}=\mathbf{0}$, and then solve to describe them both numerically and geometrically. (c) What condition(s) must be imposed on the external forces to maintain equilibrium in the structure? (d) Add in just enough additional bars so that the resulting reinforced structure has only the trivial solution to $A^{\star} \mathbf{u}=\mathbf{0}$. Is your reinforced structure stable?

Manish Jain

Numerade Educator

04:06

Problem 7

Find the displacements $u_1, u_2, \ldots, u_{100}$ of 100 masses connected in a row by identical springs, with spring constant $c=1$. Consider the following three types of force functions: (a) Constant force: $f_1=\cdots=f_{100}=.01$; (b) Linear force: $f_i=.0002 i$; (c) Quadratic force: $f_i=6 \cdot 10^{-6} i(100-i)$. Also consider two different boundary conditions at the bottom: (i) spring 101 connects the last mass to a support; (ii) mass 100 hangs free at the end of the line of springs. Graph the displacements and elongations in all six cases. Discuss your results; in particular, comment on whether they agree with your physical intuition.

Stephen Zaffke

Numerade Educator

10:30

Problem 8

(a) Suppose you are given three springs with respective stiffnesses $c=1, c^{\prime}=2, c^{\prime \prime}=3$. In what order should you connect them to three masses and a top support so that the bottom mass goes down the farthest under a uniform gravitational force?
(b) Answer Exercise 6.1.8 when the springs connect two masses to top and bottom supports.

Vishal Gupta

Numerade Educator

01:26

Problem 8

(a) How do the currents change if the resistances in the wires in the cubical network in Example 6.4 are all equal to $1 \mathrm{ohm}$ ?
(b) What if wire $k$ has resistance $R_k=k$ ohms?

Hubert Agamasu

Numerade Educator

01:43

Problem 8

Consider the two-dimensional "house" constructed out of bars, as in the accompanying picture. The bottom nodes are fixed. The width of the house is 3 units, the height of the vertical sides 1 unit, and the peak is 1.5 units above the base.
(a) Determine the reduced incidence matrix $A$ for this structure.
(b) How many distinct modes of instability are there? Describe them geometrically, and indicate whether they are mechanisms or rigid motions.
(c) Suppose we apply a combination of forces to each non-fixed node in the structure. Determine conditions such that the structure can support the forces. Write down an explicit nonzero set of external forces that satisfy these conditions, and compute the corresponding elongations of the individual bars. Which bar is under the most stress? (d) Add in a minimal number of bars so that the resulting structure can support any force. Before starting, decide, from general principles, how many bars you need to add. (e) With your new stable configuration, use the same force as before, and recompute the forces on the individual bars. Which bar now has the most stress? How much have you reduced the maximal stress in your reinforced building?

Manish Jain

Numerade Educator

01:50

Problem 9

Answer Exercise 6.3.8 for the illustrated two- and three-dimensional houses. In the twodimensional case, the width and total height of the vertical bars is 2 units, and the peak is an additional .5 unit higher. In the threedimensional house, the width and vertical heights are equal to 1 unit, the length is 3 units, while the peaks are 1.5 units above the base.

Trinity Steen

Numerade Educator

02:39

Problem 9

Suppose you are given six resistors with respective resistances $1,2,3,4,5$, and 6 . How should you connect them in a tetrahedral network (one resistor per wire) so that a light bulb on the wire opposite the battery burns the brightest?

Nishant Kumar

Numerade Educator

10:30

Problem 9

Generalizing Exercise 6.1.8, suppose you are given $n$ different springs. (a) In which order should you connect them to $n$ masses and a top support so that the bottom mass goes down the farthest under a uniform gravitational force? Does your answer depend upon the relative sizes of the spring constants? (b) Answer the same question when the springs connect $n-1$ masses to both top and bottom supports.

Vishal Gupta

Numerade Educator

02:22

Problem 10

Consider a structure consisting of three bars joined in a vertical line hanging from a top support. (a) Write down the equilibrium equations for this system when only forces and displacements in the vertical direction are allowed, i.e., a one-dimensional structure. Is the problem statically determinate, statically indeterminate, or unstable? If the latter, describe all possible mechanisms and the constraints on the forces required to maintain equilibrium. (b) Answer part (a) when the structure is two-dimensional, i.e., is allowed to move in a plane. (c) Answer the same question for the fully three-dimensional version.

Keshav Singh

Numerade Educator

02:14

Problem 10

The nodes in an electrical network lie on the vertices $\left(\frac{i}{n}, \frac{j}{n}\right)$ for $-n \leq i, j \leq n$ in a square grid centered at the origin; the wires run along the grid lines. The boundary nodes, when $x$ or $y= \pm 1$, are all grounded. A unit current source is introduced at the origin.
(a) Compute the potentials at the nodes and currents along the wires for $n=2,3,4$.
(b) Investigate and compare the solutions for large $n$, i.e., as the grid size becomes small. Do you detect any form of limiting behavior?

Victor Salazar

Numerade Educator

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Problem 10

Find the $L D L^T$ factorization of an $n \times n$ tridiagonal matrix whose diagonal entries are all equal to 2 and whose sub- and super-diagonal entries are all equal to -1 . Hint: Start with the $3 \times 3$ case (6.13), and then analyze a slightly larger one to spot the pattern.

Nick Johnson

Numerade Educator

09:14

Problem 11

A space station is built in the shape of a three-dimensional simplex whose nodes are at the positions $\mathbf{0}, \mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 \in \mathbb{R}^3$, and each pair of nodes is connected by a bar. (a) Sketch the space station and find its incidence matrix $A$. (b) Show that ker $A$ is six-dimensional, and find a basis. (c) Explain which three basis vectors correspond to rigid translations. (d) Find three basis vectors that correspond to linear approximations to rotations around the three coordinate axes. (e) Suppose the bars all have unit stiffness. Compute the full stiffness matrix for the space station. (f) What constraints on external forces at the four nodes are required to maintain equilibrium? Can you interpret them physically? $(g$ ) How many nodes do you need to fix to stabilize the structure? (h) Suppose you fix the three nodes in the $x y$-plane. How much internal force does each bar experience under a unit vertical force on the upper vertex?

Maria Gabriela Cota Moreira

Numerade Educator

06:11

Problem 11

Show that, in a network with all unit resistors, the currents $\mathbf{y}$ can be characterized as the unique solution to the Kirchhoff equations $A^T \mathbf{y}=\mathbf{f}$ of minimum Euclidean norm.

Sophie S

Numerade Educator

03:32

Problem 11

In a statically indeterminate situation, the equations $A^T \mathbf{y}=\mathbf{f}$ do not have a unique solution for the internal forces $\mathbf{y}$ in terms of the external forces $\mathbf{f}$. (a) Prove that, nevertheless, if $C=\mathrm{I}$, the internal forces are the unique solution of minimal Euclidean norm, as given by Theorem 4.50. (b) Use this method to directly find the internal force for the system in Example 6.1. Make sure that your values agree with those in the example.

Chai Santi

Numerade Educator

02:14

Problem 12

Suppose a space station is built in the shape of a regular tetrahedron with all sides of unit length. Answer all questions in Exercise 6.3.11.

Allison Knapp

Numerade Educator

03:19

Problem 12

True or false: (a) The nodal voltage potentials in a network with batteries $\mathbf{b}$ are the same as in the same network with the current sources $\mathbf{f}=-A^T C \mathbf{b}$. (b) Are the currents the same?

Vishal Gupta

Numerade Educator

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Problem 12

Prove directly that the stiffness matrices in Examples 6.1 and 6.2 are positive definite.

Victor Salazar

Numerade Educator

02:25

Problem 13

A mass-spring ring consists of $n$ masses connected in a circle by $n$ identical springs, and the masses are allowed only to move in the angular direction. (a) Derive the equations of equilibrium. (b) Discuss stability, and characterize the external forces that will maintain equilibrium. (c) Find such a set of nonzero external forces in the case of a four-mass ring and solve the equilibrium equations. What does the nonuniqueness of the solution represent?

Ajay Singhal

Numerade Educator

01:46

Problem 13

(a) Assuming all wires have unit resistance, find the voltage potentials at all the nodes and the currents along the wires of the following trees when the bottom node is grounded and a unit current source is introduced at the top node.
(i)
(ii)
(iii)
(iv)
(v)
(b) Can you make any general predictions about electrical currents in trees?

Prem Bijarniya

Numerade Educator

01:49

Problem 13

Write down the potential energy for the following mass-spring chains with identical unit springs when subject to a uniform gravitational force: (a) three identical masses connected to only a top support. (b) four identical masses connected to top and bottom supports. (c) four identical masses connected only to a top support.

Averell Hause

Carnegie Mellon University

01:13

Problem 14

A node in a tree is called terminating if it has only one edge. Repeat the preceding exercise when all terminating nodes except for the top one are grounded.

Vysakh M

Numerade Educator

06:28

Problem 14

(a) Find the total potential energy of the equilibrium configuration of the mass-spring chain in Exercise 6.1.1. (b) Test the minimum principle by substituting three other possible displacements of the masses and checking that they all have larger potential energy.

Supratim Pal

Numerade Educator

Problem 14

A structure in $\mathbb{R}^3$ has $n$ movable nodes, admits no rigid motions, and is statically determinate. (a) How many bars must it have? (b) Find an example with $n=3$.

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Problem 15

Suppose the graph of an electrical network is a tree, as in Exercise 2.6.9. Show that if one of the nodes in the tree is grounded, the system is statically determinate.

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06:30

Problem 15

Prove that if we apply a unit force to node $i$ in a structure and measure the displacement of node $j$ in the direction of the force, then we obtain the same value if we apply the force to node $j$ and measure the displacement at node $i$ in the same direction. Hint: First, solve Exercise 6.1.6.

Jonah Han

Numerade Educator

01:19

Problem 15

Answer Exercise 6.1.14 for the mass-spring chain in Exercise 6.1.4.

Sherrie Fenner

Numerade Educator

05:05

Problem 16

Suppose two wires in a network join the same pair of nodes. Explain why their effect on the rest of the network is the same as a single wire whose conductance $c=c_1+c_2$ is the sum of the individual conductances. How are the resistances related?

Anisha Sinha

Numerade Educator

07:15

Problem 16

True or false: A structure in $\mathbb{R}^3$ will admit no rigid motions if and only if at least 3 nodes are fixed.

Aidan Mcnabb

Numerade Educator

13:12

Problem 16

Describe the mass-spring chains that gives rise to the following potential energy functions, and find their equilibrium configuration: (a) $3 u_1^2-4 u_1 u_2+3 u_2^2+u_1-3 u_2$, (b) $5 u_1^2-6 u_1 u_2+3 u_2^2+2 u_2$, (c) $2 u_1^2-3 u_1 u_2+4 u_2^2-5 u_2 u_3+\frac{5}{2} u_3^2-u_1-u_2+u_3$, (d) $2 u_1^2-u_1 u_2+u_2^2-u_2 u_3+u_3^2-u_3 u_4+2 u_4^2+u_1-2 u_3$.

Samuel Hannah

Numerade Educator

03:20

Problem 17

(a) Write down the equilibrium equations for a network that contains both batteries and current sources. (b) Formulate a general superposition principle for such situations.
(c) Write down a formula for the power in the network.

Morgan Sizemore

Numerade Educator

Problem 17

Suppose all bars have unit stiffness. Explain why the internal forces in a structure form the solution of minimal Euclidean norm among all solutions to $A^T \mathbf{y}=\mathbf{f}$.

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00:49

Problem 17

Explain why the columns of the reduced incidence matrices (6.4) and (6.14) are linearly independent.

James Chok

Numerade Educator

01:14

Problem 18

Prove that the voltage potential at node $i$ due to a unit current source at node $j$ is the same as the voltage potential at node $j$ due to a unit current source at node $i$. Can you give a physical explanation of this reciprocity relation?

Ajay Singhal

Numerade Educator

Problem 18

Let $A$ be the reduced incidence matrix for a structure and $C$ the diagonal bar stiffness matrix. Suppose $\mathbf{f}$ is a set of external forces that maintain equilibrium of the structure. (a) Prove that $\mathbf{f}=A^T \mathrm{C} \mathbf{g}$ for some $\mathbf{g}$. (b) Prove that an allowable displacement $\mathbf{u}$ is a least squares solution to the system $A \mathbf{u}=\mathbf{g}$ with respect to the weighted norm $\|\mathbf{v}\|^2=\mathbf{v}^T C \mathbf{v}$.

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01:44

Problem 18

Suppose that when subject to a nonzero external force $\mathbf{f} \neq \mathbf{0}$, a mass-spring chain has equilibrium position $\mathbf{u}^{\star}$. Prove that the potential energy is strictly negative at equilibrium: $p\left(\mathbf{u}^{\star}\right)<0$.

Manish Kumar

Numerade Educator

01:19

Problem 19

Return to the situation investigated in Exercise 6.1.8. How should you arrange the springs in order to minimize the potential energy in the resulting mass-spring chain?

Sherrie Fenner

Numerade Educator

00:35

Problem 19

What is the analogue of condition (6.33) for a disconnected graph?

Julie Silva

Numerade Educator

00:52

Problem 19

Suppose an unstable structure admits no rigid motions - only mechanisms. Let $\mathbf{f}$ be an external force on the structure that maintains equilibrium. Suppose that you stabilize the structure by adding in the minimal number of reinforcing bars. Prove that the given force $\mathbf{f}$ induces the same stresses in the original bars, while the reinforcing bars experience no stress. Are the displacements necessarily the same? Does the result continue to hold when more reinforcing bars are added to the structure? Hint: Use Exercise 6.3.18.

Mayukh Banik

Numerade Educator

00:26

Problem 20

True or false: The potential energy function uniquely determines the mass-spring chain.

Monica Miller

Numerade Educator

Problem 20

When a node is fixed to a roller, it is permitted to move only along a straight line the direction of the roller. Consider the three-bar structure in Example 6.5. Suppose node 1 is fixed, but node 4 is attached to a roller that permits it to move only in the horizontal direction. (a) Construct the reduced incidence matrix and the equilibrium equations in this situation. You should have a system of 5 equations in 5 unknowns - the horizontal and vertical displacements of nodes 2 and 3 and the horizontal displacement of node 4 . (b) Is your structure stable? If not, how many rigid motions and how many mechanisms does it permit?

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05:00

Problem 21

Answer Exercise 6.3 .20 when the roller at node 4 allows it to move in only the vertical direction.

Naman Kumar

Numerade Educator

02:42

Problem 22

Redo Exercises 6.3.20-21 for the reinforced structure in Figure 6.16.

Tom Rutherford

Numerade Educator

01:18

Problem 23

(a) Suppose that we fix one node in a planar structure and put a second node on a roller. Does the structure admit any rigid motions? (b) How many rollers are needed to prevent all rigid motions in a three-dimensional structure? Are there any restrictions on the directions of the rollers?

Satpal Satpal

Numerade Educator

00:31

Problem 24

True or false: If a structure is statically indeterminate, then every non-zero applied force will result in (a) one or more nodes having a non-zero displacement; (b) one or more bars having a non-zero elongation.

Jonathon Brumley

Numerade Educator

Problem 25

True or false: If a structure constructed out of bars with identical stiffnesses is stable, then the same structure constructed out of bars with differing stiffnesses is also stable.

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