Mechanical Vibrations in SI Units

Singiresu S. Rao

Chapter 12

Finite Element Method - all with Video Answers

Educators

DT

Chapter Questions

Problem 1

Derive the stiffness matrix of the tapered bar element (which deforms in the axial direction) shown in Fig. 12.13. The diameter of the bar decreases from $D$ to $d$ over its length.

James Kiss

James Kiss

Numerade Educator

Problem 2

Derive the stiffness matrix of the bar element in longitudinal vibration whose cross-sectional area varies as $A(x)=A_{0} e^{-(x / l)},$ where $A_{0}$ is the area at the root (see Fig. 12.14).

James Kiss

Numerade Educator

Problem 3

The tapered cantilever beam shown in Fig. 12.15 is used as a spring to carry a load $P$. Derive the stiffness matrix of the beam using a one-element idealization. Assume $B=25 \mathrm{~cm}$, $b=10 \mathrm{~cm}, t=2.5 \mathrm{~cm}, l=2 \mathrm{~m}, E=2.07 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2},$ and $P=1000 \mathrm{~N}$.

James Kiss

Numerade Educator

Problem 4

Derive the stiffness and mass matrices of the planar frame element (general beam element) shown in Fig. 12.16 in the global $X Y$ -coordinate system.

James Kiss

Numerade Educator

Problem 5

A multiple-leaf spring used in automobiles is shown in Fig. 12.17. It consists of five leaves, each of thickness $t=0.65 \mathrm{~cm}$ and width $w=3.8 \mathrm{~cm} .$ For the multiple-leaf spring described in Fig. $12.17,$ derive the assembled stiffness and mass matrices. Consider only one-half of the spring for modeling using five beam elements of equal length.

James Kiss

Numerade Educator

Problem 6

A seven-member planar truss (with pin joints) is shown in Fig. 12.18. Each of the seven members has an area of cross section of $4 \mathrm{~cm}^{2}$ and a Young's modulus of $207 \mathrm{GPa}$.
a. Label the complete set of local and global nodal displacement degrees of freedom of the truss. Assume the $X$ and $Y$ coordinates shown in Fig. 12.18 as global coordinates.
b. Find the coordinate transformation matrix of each member.
c. Find the local and global stiffness matrices of each member.

James Kiss

Numerade Educator

Problem 7

Find the stiffness and mass matrices of the beam supported on springs as shown in Fig. $12.19 .$ Model the beam using one finite element. Assume the material of the beam as steel with a Young's modulus of $207 \mathrm{GPa}$ and weight density of $7650 \mathrm{~N} / \mathrm{m}^{3} .$ Neglect the weights of the springs.

James Kiss

Numerade Educator

Problem 8

For the beam shown in Fig. $12.20,$ one end (point $A$ ) is fixed and a spring-mass system is attached to the other end (point $B$ ). Assume the cross section of the beam to be circular with radius $2 \mathrm{~cm}$ and the material of the beam to be steel with Young's modulus of $207 \mathrm{GPa}$ and weight density of $7650 \mathrm{~N} / \mathrm{m}^{3}$. Using two beam elements of equal length, derive the element stiffness and mass matrices of the two elements.

James Kiss

Numerade Educator

Problem 9

Find the global stiffness matrix of each of the four bar elements of the truss shown in Fig. 12.5 using the following data:
Nodal coordinates: $\left(X_{1}, Y_{1}\right)=(0,0) \mathrm{m},\left(X_{2}, Y_{2}\right)=(1.25,2.5) \mathrm{m},\left(X_{3}, Y_{3}\right)=(2.5,0) \mathrm{m},$
$\left(X_{4}, Y_{4}\right)=(5,3.75) \mathrm{m}$
Cross-sectional areas: $A_{1}=A_{2}=A_{3}=A_{4}=13 \mathrm{~cm}^{2}$
Young's modulus of all members: 200 GPa.

James Kiss

Numerade Educator

Problem 10

For the seven-member planar truss considered in Problem 12.6 (Fig. 12.18), determine the assembled stiffness matrix of the system before applying the boundary conditions.

James Kiss

Numerade Educator

Problem 11

Using the result of Problem $12.9,$ find the assembled stiffness matrix of the truss and formulate the equilibrium equations if the vertical downward load applied at node 4 is $5 \mathrm{kN}$.

James Kiss

Numerade Educator

Problem 12

For the beam considered in Problem 12.8 (Fig. 12.20), derive the assembled stiffness and mass matrices of the system.

James Kiss

Numerade Educator

Problem 13

For the tapered beam considered in Problem 12.3 (Fig. 12.15$),$ find the stress induced in the beam using a one-element idealization.

James Kiss

Numerade Educator

Problem 14

For the multiple-leaf spring described in Problem 12.5 (Fig. 12.17), consider only one-half of the spring for modeling using five beam elements of equal length and derive the stiffness and mass matrices of each of the five beam elements. The Young's modulus is $200 \mathrm{GPa}$ and the density is $7800 \mathrm{~kg} / \mathrm{m}^{3}$ for the material.

James Kiss

Numerade Educator

Problem 15

Find the nodal displacements of the crane shown in Fig. 12.21 when a vertically downward load of $4500 \mathrm{~N}$ is applied at node $4 .$ The Young's modulus is $200 \mathrm{GPa}$ and the cross- $\mathrm{sec}-$ tional area is $13 \times 10^{-4} \mathrm{~m}^{2}$ for elements 1 and 2 and $6.5 \times 10^{-4} \mathrm{~m}^{2}$ for elements 3 and 4 .

James Kiss

Numerade Educator

Problem 16

Find the tip deflection of the cantilever beam shown in Fig. 12.22 when a vertical load of $P=500 \mathrm{~N}$ is applied at point $Q$ using (a) a one-element approximation and (b) a two-element approximation. Assume $l=0.25 \mathrm{~m}, h=25 \mathrm{~mm}, b=50 \mathrm{~mm}, E=2.07 \times 10^{11} \mathrm{~Pa},$ and $k=10^{5} \mathrm{~N} / \mathrm{m}$.

James Kiss

Numerade Educator

Problem 17

Find the stresses in the stepped beam shown in Fig. 12.23 when a moment of $1000 \mathrm{~N}-\mathrm{m}$ is applied at node 2 using a two-element idealization. The beam has a square cross section $50 \mathrm{~mm} \times 50 \mathrm{~mm}$ between nodes 1 and 2 and $25 \mathrm{~mm} \times 25 \mathrm{~mm}$ between nodes 2 and $3 .$ Assume the Young's modulus as $2.1 \times 10^{11} \mathrm{~Pa}$.

James Kiss

Numerade Educator

Problem 18

Find the transverse deflection and slope of node 2 of the beam shown in Fig. 12.24 using a two-element idealization. Compare the solution with that of simple beam theory.

James Kiss

Numerade Educator

Problem 19

Find the displacement of node 3 and the stresses in the two members of the truss shown in Fig. 12.25. Assume that the Young's modulus and the cross-sectional areas of the two members are the same with $E=200 \mathrm{GPa}$ and $A=0.5 \times 10^{-3} \mathrm{~m}^{2}$.

James Kiss

Numerade Educator

Problem 20

A simplified model of a radial drilling machine is shown in Fig. $12.26 .$ If a vertical force of $5000 \mathrm{~N}$ along the $z$ -direction and a bending moment of $500 \mathrm{~N}-\mathrm{m}$ in the $x z$ -plane are developed at point $A$ during a metal cutting operation, find the stresses developed in the machine. Use two beam elements for the column and one beam element for the arm. Assume the material of the machine as steel.

James Kiss

Numerade Educator

Problem 21

The crank in the slider-crank mechanism shown in Fig. 12.27 rotates at a constant clockwise angular speed of 1000 rpm. Find the stresses in the connecting rod and the crank when the pressure acting on the piston is $1 \mathrm{MPa}$ and $\theta=30^{\circ} .$ The diameter of the piston is $0.3 \mathrm{~m}$ and the material of the mechanism is steel. Model the connecting rod and the crank by one beam element each. The lengths of the crank and connecting rod are $0.3 \mathrm{~m}$ and $1.2 \mathrm{~m}$, respectively.

James Kiss

Numerade Educator

Problem 22

A water tank of mass $W$ is supported by a hollow circular steel column of inner diameter $d,$ wall thickness $t,$ and height $l$. The wind pressure acting on the column can be assumed to vary linearly from 0 to $p_{\text {max }}$ as shown in Fig. $12.28 .$ Find the bending stress induced in the column under the loads using a one-beam element idealization. Data: $W=5000 \mathrm{~kg}, l=12 \mathrm{~m}, d=0.6 \mathrm{~m}, t=0.02 \mathrm{~m},$ and $p_{\max }=700 \mathrm{kPa}$.

James Kiss

Numerade Educator

Problem 23

For the seven-member planar truss considered in Problem 12.6 (Fig. 12.18), determine the following:
a. The system stiffness matrix after applying the boundary conditions.
b. The nodal displacements of the truss under the loads indicated in Fig. 12.18 .

James Kiss

Numerade Educator

Problem 24

Using one beam element, find the natural frequencies of the uniform pinned-free beam shown in Fig. 12.29 .

James Kiss

Numerade Educator

Problem 25

Using one beam element and one spring element, find the natural frequencies of the uniform, spring-supported cantilever beam shown in Fig. $12.22 .$

DT

Deepanshu Kumar Tibrewal

Numerade Educator

Problem 26

Using one beam element and one spring element, find the natural frequencies of the system shown in Fig. 12.30 .

James Kiss

Numerade Educator

Problem 27

Using two beam elements, find the natural frequencies and mode shapes of the uniform fixedfixed beam shown in Fig. 12.31 .

James Kiss

Numerade Educator

Problem 28

An electric motor, of mass $m=100 \mathrm{~kg}$ and operating speed $=1800 \mathrm{rpm}$, is fixed at the middle of a clamped-clamped steel beam of rectangular cross section, as shown in Fig. $12.32 .$ Design the beam such that the natural frequency of the system exceeds the operating speed of the motor.

James Kiss

Numerade Educator

Problem 29

Find the natural frequencies of the beam shown in Fig. $12.33,$ using three finite elements of length $l$ each.

James Kiss

Numerade Educator

Problem 30

Find the natural frequencies of the cantilever beam carrying an end mass $M$ shown in Fig. 12.34, using a one-beam element idealization.

James Kiss

Numerade Educator

Problem 31

Find the natural frequencies of vibration of the beam shown in Fig. $12.35,$ using two beam elements. Also find the load vector if a uniformly distributed transverse load $p$ is applied to element $1 .$

James Kiss

Numerade Educator

Problem 32

Find the natural frequencies of a beam of length $l$, which is pin connected at $x=0$ and fixed at $x=l,$ using one beam element.

James Kiss

Numerade Educator

Problem 33

Find the natural frequencies of torsional vibration of the stepped shaft shown in Fig. 12.36. Assume that $\rho_{1}=\rho_{2}=\rho, G_{1}=G_{2}=G, I_{p 1}=2 I_{p 2}=2 I_{p}, J_{1}=2 J_{2}=2 J$, and $l_{1}=l_{2}=l$.

James Kiss

Numerade Educator

Problem 34

Find the dynamic response of the stepped bar shown in Fig. $12.37(\mathrm{a})$ when its free end is subjected to the load given in Fig. $12.37(\mathrm{~b})$.

James Kiss

Numerade Educator

Problem 35

Find the natural frequencies of a cantilever beam of length $l,$ cross-sectional area $A,$ moment of inertia $I$, Young's modulus $E$, and density $\rho$, using one finite element.

James Kiss

Numerade Educator

Problem 36

Find the natural frequencies of vibration of the radial drilling machine considered in Problem $12.20($ Fig. 12.26$)$.

James Kiss

Numerade Educator

Problem 37

Find the natural frequencies of the water tank considered in Problem 12.22 (Fig. 12.28 ) using a one-beam element idealization.

James Kiss

Numerade Educator

Problem 38

Find the natural frequencies of vibration of the beam considered in Problem 12.7 using one finite element (Fig. 12.19).

James Kiss

Numerade Educator

Problem 39

Derive the consistent- and lumped-mass matrices of the tapered bar element (which deforms in the axial direction) shown in Fig. 12.13. The diameter of the bar decreases from $D$ to $d$ over its length.

James Kiss

Numerade Educator

Problem 40

Find the natural frequencies of the stepped bar shown in Fig. 12.38 with the following data using consistent- and lumped-mass matrices: $A_{1}=0.001 \mathrm{~m}^{2}, \quad A_{2}=0.0006 \mathrm{~m}^{2}$, $E=200 \mathrm{GPa}, \rho_{w}=7750 \mathrm{~kg} / \mathrm{m}^{3},$ and $l_{1}=l_{2}=1 \mathrm{~m}$.

James Kiss

Numerade Educator

Problem 41

Find the undamped natural frequencies of longitudinal vibration of the stepped bar shown in Fig. 12.39 with the following data using consistent- and lumped-mass matrices: $l_{1}=l_{2}=l_{3}=0.2 \mathrm{~m}, \quad A_{1}=2 A_{2}=4 A_{3}=0.4 \times 10^{-3} \mathrm{~m}^{2}, E=2.1 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2},$ and $\rho=7.8 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$

James Kiss

Numerade Educator

Problem 42

Consider the stepped bar shown in Fig. 12.11 with the following data: $A_{1}=25 \times 10^{-4} \mathrm{~m}^{2}$, $A_{2}=16 \times 10^{-4} \mathrm{~m}^{2}, A_{3}=9 \times 10^{-4} \mathrm{~m}^{2}, E_{i}=2 \times 10^{11} \mathrm{~Pa}, i=1,2,3, \rho_{i}=7.8 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ $i=1,2,3, l_{1}=3 \mathrm{~m}, l_{2}=2 \mathrm{~m}, l_{3}=1 \mathrm{~m} .$ Using MATLAB, find the axial displacements $u_{1}, u_{2},$ and $u_{3}$ under the load $p_{3}=500 \mathrm{~N}$.

James Kiss

Numerade Educator

Problem 43

Using MATLAB, find the natural frequencies and mode shapes of the stepped bar described in Problem 12.42 .

James Kiss

Numerade Educator

Problem 44

Use Program1 7 .m to find the natural frequencies of a fixed-fixed stepped beam, similar to the one shown in Fig. $12.12,$ with the following data:
Cross sections of elements: 1,2,$3: 0.1 \mathrm{~m} \times 0.1 \mathrm{~m}, 0.08 \mathrm{~m} \times 0.08 \mathrm{~m}, 0.05 \mathrm{~m} \times 0.05 \mathrm{~m}$
Lengths of elements: 1,2,$3: 0.8 \mathrm{~m}, 0.5 \mathrm{~m}, 0.2 \mathrm{~m}$
Young's modulus of all elements: $70 \mathrm{GPa}$ Density of all elements: $3000 \mathrm{~kg} / \mathrm{m}^{3}$

James Kiss

Numerade Educator

Problem 45

Write a computer program for finding the assembled stiffness matrix of a general planar truss.

James Kiss

Numerade Educator

Problem 46

Derive the stiffness and mass matrices of a uniform beam element in transverse vibration rotating at an angular velocity of $\Omega \mathrm{rad} / \mathrm{s}$ about a vertical axis as shown in Fig. $12.40(\mathrm{a})$. Using these matrices, find the natural frequencies of transverse vibration of the rotor blade of a helicopter (see Fig. $12.40(\mathrm{~b})$ ) rotating at a speed of $300 \mathrm{rpm}$. Assume a uniform rectangular cross section $0.02 \mathrm{~m} \times 0.3 \mathrm{~m}$ and a length $1.2 \mathrm{~m}$ for the blade. The material of the blade is aluminum.

James Kiss

Numerade Educator

Problem 47

An electric motor of mass $500 \mathrm{~kg}$ operates on the first floor of a building frame that can be modeled by a steel girder supported by two reinforced concrete columns, as shown in Fig. 12.41 . If the operating speed of the motor is 1500 rpm, design the girder and the columns such that the fundamental frequency of vibration of the building frame is greater than the operating speed of the motor. Use two beam and two bar elements for the idealization. Assume the following data:
$$
\begin{aligned}
\text { Girder: } E &=200 \mathrm{GPa}, & \rho &=250 \mathrm{~kg} / \mathrm{m}^{3}, & h / b=2 \\
\text { Columns: } E &=30 \mathrm{GPa}, & \rho &=75 \mathrm{~kg} / \mathrm{m}^{3}
\end{aligned}
$$

James Kiss

Numerade Educator