All Chapters Covered
SOLUTIONS
,Table of Contents
1. Single-Degree-of-Freedom Systems
2. Random Vibrations
3. Dynamic Response of SDOF Systems Using Numerical Methods
4. Systems with Several Degrees of Freedom
5. Equations of Motion of Continuous Systems
6. Vibration of Strings and Bars
7. Beam Vibrations
8. Continuous Beams and Frames
9. Vibrations of Plates
10. Vibration of Shells
11. Finite Elements and Time Integration Numerical Techniques
12. Shock Spectra
, Chapter 1
1.1 Write the equations of motion for the one-degree-of-freedom systems shown in Figures1.72 (a) … (i). Assume
that the loading is in the form of a force P(t), a given displacement a(t), or a given rotation (t ) as indicated in
the figure.
Figure 1.72 One-degree-of-freedom systems
, Solutions
(a) (b)
(
spring force = 3EI / L3 u )
(
spring force = 48EI / L3 u ) 3EI
mu + u = P(t)
48EI L3
mu + 3 u = P(t)
L
(c) (d)
( ) ( )
spring force = 3EI / L3 u − 3EI / L2 (t)
(
spring force = 3EI / L3 )(u − a) mu +
3EI
u=
3EI
(t)
L3 L2
3EI
mu +
L3
(u − a ) = 0
3EI 3EI
mu + u= a(t)
3
L L3
(e) (f)
spring force = (EA / L)u
EA ( ) (
spring force = 2 3EI / L3 u = 6EI / L3 u )
mu + u = P(t) 6EI
L mu + u = P(t)
L3
SOLUTIONS
,Table of Contents
1. Single-Degree-of-Freedom Systems
2. Random Vibrations
3. Dynamic Response of SDOF Systems Using Numerical Methods
4. Systems with Several Degrees of Freedom
5. Equations of Motion of Continuous Systems
6. Vibration of Strings and Bars
7. Beam Vibrations
8. Continuous Beams and Frames
9. Vibrations of Plates
10. Vibration of Shells
11. Finite Elements and Time Integration Numerical Techniques
12. Shock Spectra
, Chapter 1
1.1 Write the equations of motion for the one-degree-of-freedom systems shown in Figures1.72 (a) … (i). Assume
that the loading is in the form of a force P(t), a given displacement a(t), or a given rotation (t ) as indicated in
the figure.
Figure 1.72 One-degree-of-freedom systems
, Solutions
(a) (b)
(
spring force = 3EI / L3 u )
(
spring force = 48EI / L3 u ) 3EI
mu + u = P(t)
48EI L3
mu + 3 u = P(t)
L
(c) (d)
( ) ( )
spring force = 3EI / L3 u − 3EI / L2 (t)
(
spring force = 3EI / L3 )(u − a) mu +
3EI
u=
3EI
(t)
L3 L2
3EI
mu +
L3
(u − a ) = 0
3EI 3EI
mu + u= a(t)
3
L L3
(e) (f)
spring force = (EA / L)u
EA ( ) (
spring force = 2 3EI / L3 u = 6EI / L3 u )
mu + u = P(t) 6EI
L mu + u = P(t)
L3
