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The Finite Element Method in Engineering

Singiresu S. Rao

Chapter 12

Dynamic Analysis - all with Video Answers

Educators

Chapter Questions

01:07

Problem 1

Find the solution of Example 12.7 using the lumped mass matrix.

James Kiss
James Kiss
Numerade Educator

Problem 2

Find the solution of Example 12.8 using the lumped mass matrix.

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01:30

Problem 3

Find the natural frequencies and modes of vibration for a two-element fixed-fixed beam.

James Kiss
James Kiss
Numerade Educator
02:41

Problem 4

Find the matural frequencies and modes of vibration for a one-element simply supported beam.

James Kiss
James Kiss
Numerade Educator
01:30

Problem 5

Find the natural frequencies and modes of vibration for a two-element simply supported beam by taking advantage of the symmetry about the midpoint.

James Kiss
James Kiss
Numerade Educator
01:53

Problem 6

Find the natural frequencies and mode shapes of the rod shown in Figure 12.11 in axial vibration.

James Kiss
James Kiss
Numerade Educator

Problem 7

Sometimes it is desimble to suppress less important or unwanted degrees of freedom from the original system of equations
$$
\begin{aligned}
& {[K \mid=\vec{X}=\vec{p}} \\
& n \times n \times 1=n \times 1
\end{aligned}
$$
to reduce the size of the problem to be solved. This procedure, known as static condensation or condenantion of anwunted degres of frendom, consists of partitioning Eq. $(\mathrm{P} .1$ ) as follows:
$$
\left[\begin{array}{c|c}
K_{11} & K_{12} \\
p \times p & p \times q \\
\hdashline K_{21} & K_{22} \\
q \times p & q \times q
\end{array}\right]\left\{\begin{array}{c}
\vec{X}_1 \\
p \times 1 \\
\hdashline \vec{X}_2 \\
q \times 1
\end{array}\right\}=\left\{\begin{array}{c}
\vec{p}_1 \\
p \times 1 \\
\hdashline \vec{p}_2 \\
q \times 1
\end{array}\right\}, p+q=n
$$
where $\vec{X}_2$ is the vertor of umwanted degrees of freedom. Equation (P.2) gives
$$
\begin{aligned}
& {\left[K_{11}\right] \vec{X}_1+\left[K_{12} \mid \vec{X}_2=\vec{P}_1\right.} \\
& {\left[K_{21}\right] \vec{X}_1+\left|K_{22}\right| \vec{X}_2=\vec{P}_2}
\end{aligned}
$$
Solving Eq. (P.4) for $\vec{X}_2$ and substituting the result in Eq. (P.3) lead to the desired condensed set of equations
$$
\underset{p \times p)_{p \times 1}}{\vec{X}_1}=\vec{p}
$$
Derive the expressions of $[\underline{K}]$ and $\vec{P}$

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04:00

Problem 8

Modify the program CST3D.m (see section 23.5) to find the displacements and the first two natural frequencies of a box beam (similar to the one shown in Figure 10.9) with the following data:
Length $=100 \mathrm{in}$, width $=20 \mathrm{in}$, depth $=10 \mathrm{in}, \mathrm{t}_{\mathrm{c}}=0.5 \mathrm{in}, \mathrm{t}_{\mathrm{w}}=1.0 \mathrm{in}, E=30 \times 10^5 \mathrm{poi}, v=0.3, P_1=P_2=1000$ lb

James Kiss
James Kiss
Numerade Educator
02:10

Problem 9

Find the natural frequencies of longitudinal vibration of the stepped bar shown in Figure 12.12 using consistent mass matrices.

James Kiss
James Kiss
Numerade Educator
01:07

Problem 10

Solve Problem 12.9 using lumped mass matrices.

James Kiss
James Kiss
Numerade Educator
02:10

Problem 11

Find the natural frequencies of longitudinal vibration of the stepped bar shown in Figure 12.13 using consistent mass matrices.

James Kiss
James Kiss
Numerade Educator

Problem 12

Solve Problem 12.11 using lumped mass matrices.

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02:34

Problem 13

Find the mode shapes of the stepped bar shown in Fagure 12.12 corresponding to the natural frequencies found in Problem 12.9.

James Kiss
James Kiss
Numerade Educator
02:34

Problem 14

Find the mode shapes of the stepped bar shown in Figure 12.12 corresponding to the natural frequencies found in Problem 12.10.

James Kiss
James Kiss
Numerade Educator

Problem 15

Orthogonalixe the mode shapes found in Problem 12.13 with respect to the corresponding mass matrix.

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Problem 16

Orthogonalize the mode shapes found in Problem 12.14 with respect to the corresponding mass matrix.

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Problem 17

Find the consistent and lumped mass matrices of the bar element shown in Frgure 12.14 in the $X Y 2$ coordinate system.

Nick Johnson
Nick Johnson
Numerade Educator
02:19

Problem 18

a. Derive the stiffness and consistent mass matrices of the two-bar truss shown in Fagure 12.15.
b. Determine the natural frequencies of the truss (using the consistent mass matrix).

James Kiss
James Kiss
Numerade Educator

Problem 19

a. Derive the lumped mass matrix of the two-bar truss shown in Figure 12.15.
b. Determine the natural frequencies of the truss (using the lumped mass matrix).

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03:38

Problem 20

The properties of the two elements in the stepped beam shown in Figure 12.16 are given below:
Element $1: E=30 \times 10^6$ psi, $\rho=0.283 \mathrm{lbf} / \mathrm{in}^3$, cross-section $=$ circular, 2 -in. diameter
Element 2: $E=11 \times 10^6 p s i, \rho=0.1 \mathrm{lbf} / \mathrm{in}^3$, cross-section = circular, 1 -in diameter
Find the natural frequencies of the stepped beam.

James Kiss
James Kiss
Numerade Educator
02:50

Problem 21

Find the mode shapes of the stepped beam considered in Problem 12.20.

Surendra Kumar
Surendra Kumar
Numerade Educator

Problem 22

Find the natural frequencies of the triangular plate shown in Figure 12.17 using the consistent mass matrix. Use one triangular membrane element for modeling

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01:07

Problem 23

Solve Problem 12.22 using the lumped mass matrix.

James Kiss
James Kiss
Numerade Educator
02:41

Problem 24

Consider the tetrahedron element shown in Figure 12.18. Find the natural frequencies of the element by fixing the fare 123 .

James Kiss
James Kiss
Numerade Educator
01:12

Problem 25

Consider the stepped bar shown in Figure 12.13. If the force shown in Figure 12.19 is applied along $Q 1$. determine the dynamic response, $Q_1(\mathrm{r})$.

Subhakanta Sahoo
Subhakanta Sahoo
Numerade Educator

Problem 26

The cantilever beam shown in Fagure $12.20(a)$ is subjected to the force indicated in Figure 12.20(b) along the direction of $Q_1$. Determine the responses $Q_1(t)$ and $Q_2(t)$.

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Problem 27

a. Derive the consistent mass matrix of a bar element shown in Eq. (12.26) starting from the matrix of shape functions, $[\mathrm{N}]$, given by Eq. $[12.24)$.
b. Derive the consistent mass matrix of a space truss element shown in Eq. (12.30) starting from the matrix of shape functions, $[\mathrm{N} \mid$, defined by Eq. $(12.28)$

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Problem 28

Derive the consistent mass matrix of a uniform beam element shown in Figure 12.4 by evaluating the integrals shown in Eq. $(12.34)$.

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00:59

Problem 29

Derive the consistent mass matrix of a planar frame element shown in Figure 9.14.

James Kiss
James Kiss
Numerade Educator

Problem 30

Derive the consistent mass matrix of a uniform rod (or bar) element under torsion shown in Figure 9.9(c).

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Problem 31

Derive the consistent mass matrix given by Eq. (12.39) of a space frame element shown in Figure 9.9[a].

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Problem 32

Derive the consistent mass matrix given by Eq. (12.45), for a triangular membrane element with nine degrees of freedom shown in Figure 10.2.

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Problem 33

Derive the consistent mass matrix given by Eq. (12.55) for a tetrahedron element.

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03:26

Problem 34

Find the lumped mass matrix of a uniform beam element, shown in Figure 12.4, by including its mass moment of inertia.

Stanley Enemuo
Stanley Enemuo
Numerade Educator

Problem 35

Find the lumped mass matrix of a uniform membrane element shown in Figure 10.2 .

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Problem 36

Derive the consistent mass matrix of a two-node tapered bar element (with an axial degree of freedom at each node) with the area of cross-section varying linearly along $x$ as $A(x)=A_i\left(1-\frac{x}{l}\right)+A_1\left(\frac{x}{1}\right)$.

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Problem 37

Find the lumped mass matrix of a two-node tapered bar element (with an axial degree of freedom at each node] with the area of cross-section varying linearly along $x$ as $A(x)=A_1\left(1-\frac{x}{1}\right)+A_1\left(\frac{x}{I}\right)$.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 38

Derive the consistent mass matrix of a two-node beam element with two degrees of freedom at each node (as shown in Figure 12.4) when the area of cooss-section of the beam varies linearly as $A(x)=A_i\left(1-\frac{x}{I}\right)+A_i\left(\frac{x}{I}\right)$.

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Problem 39

Find the lumped mass matrix of a two-node beam element with two degrees of freedom at each node (as shown in Figure 12.4) when the area of cross-section of the beam vanies linearly as $A(x)=A_1\left(1-\frac{x}{1}\right)+A_1\left(\frac{x}{1}\right)$.

Victor Salazar
Victor Salazar
Numerade Educator
02:19

Problem 40

Find the natural frequencies of the free uniform bar shown in Fagure 12.1(a) using one finite element. Lse the consistent mass matrix of the element.

James Kiss
James Kiss
Numerade Educator
02:19

Problem 41

Find the natural frequencies of the free uniform bar shown in Fagure 12.1 (a) using one finite element. Lse the lumped mass matrix of the element.

James Kiss
James Kiss
Numerade Educator
01:27

Problem 42

Find the natural frequencies of vibration of the constrained tapered bar element shown in Figure 12.21 with the area of cross-section varying as $A(x)=2 A\left(1-\frac{x}{21}\right)$ using a one-bar element with the consistent mass matrix.

James Kiss
James Kiss
Numerade Educator
01:27

Problem 43

Find the natural frequencies of vibration of the constrained tapered bar dement shown in Figure 12.21 with the cross-section area varying as $A(x)=2 A\left(1-\frac{x}{2 i}\right)$ using a one-bar element with the lumped mass matrix.

James Kiss
James Kiss
Numerade Educator
01:27

Problem 44

Find the natural frequencies of vibration of the free tapered bar element shown in Figure 12.22 with the cross-saction area varying as $A(x)=2 A\left(1-\frac{x}{21}\right)$ using a one-bar element with the consistent mass matrix.

James Kiss
James Kiss
Numerade Educator
01:27

Problem 45

Find the natural frequencies of vibration of the free tapered bar element shown in Figure 12.22 with the cross-section area varying as $A(x)=2 A\left(1-\frac{x}{21}\right)$ using a one-bar element with the lumped mass matrix.

James Kiss
James Kiss
Numerade Educator
01:27

Problem 46

Figure 12.23 shows a unifom beam fixed at $x=0$ and simply supported at $x=1$. Find the natural frequency of vibration of the beam using a one-beam element with the consistent mass matrix.

James Kiss
James Kiss
Numerade Educator
01:27

Problem 47

Figure 12.23 shows a unifonn beam fixed at $x=0$ and simply supported at $x=1$, Find the natural frequency of vibration of the beam using a one-beam element with the lumped mass matrix.

James Kiss
James Kiss
Numerade Educator
04:00

Problem 48

Consider a stepped beam fixed at both the ends as shown in Figure 12.24 with the following data: $\rho=7850 \mathrm{~kg} / \mathrm{m}^3, E=207 \mathrm{GPa}, A_1=$ cross-sectional area of step $1=10 \mathrm{~cm} \times 10 \mathrm{~cm}$, and $A_2=$ cross-sectional area of step $2=5 \mathrm{~cm} \times 5 \mathrm{~cm}$. Using a one-beam element for each step of the beam, determine the following:
a. The stiffness matrix and the mass matrix, using consisient mass matrices of elements, of the stepped beam. b. The natural frequencies of vibration of the stepped beam.

James Kiss
James Kiss
Numerade Educator
04:00

Problem 49

Consider a stepped beam fixed at both the ends as shown in Figure 12.24 with the following data: $\rho=7850 \mathrm{~kg}^{\prime} \mathrm{m}^3, E=207 \mathrm{GPa}, A_1=$ cross-sectional area of $\operatorname{step} 1=10 \mathrm{~cm} \times 10 \mathrm{~cm}$, and $A_2=$ cross-sectional area of $\operatorname{step} 2=5 \mathrm{~cm} \times 5 \mathrm{~cm}$. Using one beam element for each step of the beam, determine the following:
a. The stiffness matrix and the mass matrix, using lumped mass matrices of the stepped beam elements.
b. The natural frequencies of vibration for the stepped beam.

James Kiss
James Kiss
Numerade Educator
04:00

Problem 50

Using a ane-beam element idealization of the beam column shown in Figure 12.25, find the natural frequencies of vibration. Use a lumped mass matrix of the beam column.
Data: $E=207 \mathrm{GPa}, \rho=7850 \mathrm{~kg} / \mathrm{m}^3, l=1 \mathrm{~m}$, area of cross-section = round with $10 \cdot \mathrm{cm}$ diameter

James Kiss
James Kiss
Numerade Educator
04:00

Problem 51

Using a one-beam element idealization of the beam column shown in Figure 12.25, find the natural frequencies of vibration. Use a consistent mass matrix of the beam column.
Data: $\mathrm{E}=207 \mathrm{GPa}, \rho=7850 \mathrm{~kg} / \mathrm{m}^3, l=1 \mathrm{~m}$, area of cross-section = round with $10-\mathrm{cm}$ diameter

James Kiss
James Kiss
Numerade Educator
01:04

Problem 52

Consider a rectangular plate in plane stress shown in Figure 12.26. Using two triangular-plane membrane elements for idealization, derive the eigenvalue problem to find the natural frequencies of vibration of the plate. Use lumped mass matrices.

Raj Bala
Raj Bala
Numerade Educator
01:04

Problem 53

Consider a rectangular plate in plane stress shown in Figure 12.26. Using two triangular plane membrane elements for idealization, derive the eigenvalue problem to find the natural frequencies of vibration of the plate. Use consistent mass matrices.

Raj Bala
Raj Bala
Numerade Educator