PA
S
SV
IB
ES
PASSVIBES
, CHAPTER 1
Problem 1.1
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.
PA
Figure P1.1
Solution:
If ke is the effective stiffness,
fS keu
u
S
k1
k1 u
fS fS
k2 u
k2
SV
Equilibrium of forces: fS ( k1 k2 ) u
Effective stiffness: ke fS u k1 k2
Equation of motion: mu keu p ( t )
IB
ES
1
Copyright © 2023 Pearson Education, Inc. PASSVIBES
,Problem 1.2
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:
PA
If ke is the effective stiffness,
fS keu (a)
u
k1 k2
fS
S
If the elongations of the two springs are u1 and u2 ,
u u1 u2 (b)
SV
Because the force in each spring is fS ,
fS k1u1 fS k2u2 (c)
Solving for u1 and u2 and substituting in Eq. (b) gives
fS f f 1 1 1
S S
ke k1 k2 ke k1 k2
k1 k2
ke
IB
k1 k2
Equation of motion: mu keu p ( t ) .
ES
2
Copyright © 2023 Pearson Education, Inc. PASSVIBES
, Problem 1.3
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:
PA
k1 k3
m
Figure P1.3a
k2
k 1+ k 2 k3
m
S
Figure P1.3b
u
ke
SV
m
Figure P1.3c
This problem can be solved either by starting from the
definition of stiffness or by using the results of Problems
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.
IB
First, using Problem 1.1, the parallel arrangement of
k1 and k2 is replaced by a single spring, as shown in Fig.
1.3b. Second, using the result of Problem 1.2, the series
arrangement of springs in Fig. 1.3b is replaced by a single
spring, as shown in Fig. 1.3c:
ES
1 1 1
ke k1 k2 k3
Therefore the effective stiffness is
( k1 k2 ) k3
ke
k1 k2 k3
The equation of motion is mu keu p ( t ) .
3
Copyright © 2023 Pearson Education, Inc. PASSVIBES
S
SV
IB
ES
PASSVIBES
, CHAPTER 1
Problem 1.1
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.
PA
Figure P1.1
Solution:
If ke is the effective stiffness,
fS keu
u
S
k1
k1 u
fS fS
k2 u
k2
SV
Equilibrium of forces: fS ( k1 k2 ) u
Effective stiffness: ke fS u k1 k2
Equation of motion: mu keu p ( t )
IB
ES
1
Copyright © 2023 Pearson Education, Inc. PASSVIBES
,Problem 1.2
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:
PA
If ke is the effective stiffness,
fS keu (a)
u
k1 k2
fS
S
If the elongations of the two springs are u1 and u2 ,
u u1 u2 (b)
SV
Because the force in each spring is fS ,
fS k1u1 fS k2u2 (c)
Solving for u1 and u2 and substituting in Eq. (b) gives
fS f f 1 1 1
S S
ke k1 k2 ke k1 k2
k1 k2
ke
IB
k1 k2
Equation of motion: mu keu p ( t ) .
ES
2
Copyright © 2023 Pearson Education, Inc. PASSVIBES
, Problem 1.3
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:
PA
k1 k3
m
Figure P1.3a
k2
k 1+ k 2 k3
m
S
Figure P1.3b
u
ke
SV
m
Figure P1.3c
This problem can be solved either by starting from the
definition of stiffness or by using the results of Problems
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.
IB
First, using Problem 1.1, the parallel arrangement of
k1 and k2 is replaced by a single spring, as shown in Fig.
1.3b. Second, using the result of Problem 1.2, the series
arrangement of springs in Fig. 1.3b is replaced by a single
spring, as shown in Fig. 1.3c:
ES
1 1 1
ke k1 k2 k3
Therefore the effective stiffness is
( k1 k2 ) k3
ke
k1 k2 k3
The equation of motion is mu keu p ( t ) .
3
Copyright © 2023 Pearson Education, Inc. PASSVIBES
