Solutions Manual for Dynamics of Structures Theory and Applications to Earthquake Engineering 6th Edition by Anil K. Chopra
12 views 1 purchase
Course
Earthquake Engineering 6th Edition
Institution
Earthquake Engineering 6th Edition
Solutions Manual for Dynamics of Structures Theory and Applications to Earthquake Engineering 6th Edition by Anil K. Chopra. PART I: SINGLE-DEGREE-OF-FREEDOM SYSTEMS Equations of Motion, Problem Statement, and Solution Methods Free Vibration Response to Harmonic and Periodic Excitations Response to...
Starting from the basic definition of stiffness, determine
the effective stiffness of the combined spring and write the
equation of motion for the spring–mass systems shown in
Fig. P1.1.
R
Figure P1.1
U
Solution:
If ke is the effective stiffness,
SE
fS keu
u
k1
k2
fS
k1 u
k2 u
fS
IS
O
N
Equilibrium of forces: fS ( k1 k2 ) u
N
Effective stiffness: ke fS u k1 k2
Equation of motion: mu keu p ( t )
O
C
ED
M
Starting from the basic definition of stiffness, determine
the effective stiffness of the combined spring and write the
equation of motion for the spring–mass systems shown in
Fig. P1.2.
Figure P1.2
Solution:
R
If ke is the effective stiffness,
U
fS keu (a)
SE
u
k1 k2
fS
If the elongations of the two springs are u1 and u2 ,
u u1 u2 (b) IS
O
Because the force in each spring is fS ,
fS k1u1 fS k2u2 (c)
N
Solving for u1 and u2 and substituting in Eq. (b) gives
fS f f 1 1 1
N
Starting from the basic definition of stiffness, determine
the effective stiffness of the combined spring and write the
equation of motion for the spring–mass systems shown in
Fig. P1.3.
Figure P1.3
Solution:
R
k1 k3
m
U
Figure P1.3a
k2
SE
k 1+ k 2 k3
m
Figure P1.3b
ke
u IS
O
m
Figure P1.3c
N
This problem can be solved either by starting from the
definition of stiffness or by using the results of Problems
N
P1.1 and P1.2. We adopt the latter approach to illustrate
the procedure of reducing a system with several springs to
a single equivalent spring.
O
First, using Problem 1.1, the parallel arrangement of
k1 and k2 is replaced by a single spring, as shown in Fig.
C
1.3b. Second, using the result of Problem 1.2, the series
arrangement of springs in Fig. 1.3b is replaced by a single
ED
spring, as shown in Fig. 1.3c:
1 1 1
ke k1 k2 k3
Therefore the effective stiffness is
M
( k1 k2 ) k3
ke
k1 k2 k3
The equation of motion is mu keu p ( t ) .
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller MedConnoisseur. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $19.99. You're not tied to anything after your purchase.
