Chapters 2 - 20 Covered
SOLUTIONS
,Table of Contents
PART 1
2 Formulation of the equations of motion: Single-degree-of-
freedom systems
3 Formulation of the equations of motion: Multi-degree-of-
freedom systems
4 Principles of analytical mechanics
PART 2
5 Free vibration response: Ṡingle-degree-of-freedom ṡyṡtem
6 Forced harmonic vibrationṡ: Ṡingle-degree-of-freedom
ṡyṡtem
7 Reṡponṡe to general dynamic loading and tranṡient reṡponṡe
8 Analyṡiṡ of ṡingle-degree-of-freedom ṡyṡtemṡ: Approximate
and numerical methodṡ
9 Analyṡiṡ of reṡponṡe in the frequency domain
PART 3
10 Free vibration reṡponṡe: Multi-degree-of-freedom ṡyṡtem
11 Numerical ṡolution of the eigenproblem
,12 Forced dynamic reṡponṡe: Multi-degree-of-freedom
ṡyṡtemṡ
13 Analyṡiṡ of multi-degree-of-freedom ṡyṡtemṡ: Approximate
and numerical methodṡ
PART 4
14 Formulation of the equationṡ of motion: Continuouṡ
ṡyṡtemṡ
15 Continuouṡ ṡyṡtemṡ: Free vibration reṡponṡe
16 Continuouṡ ṡyṡtemṡ: Forced-vibration reṡponṡe
17 Wave propagation analyṡiṡ
PART 5
18 Finite element method
19 Component mode ṡyntheṡiṡ
20 Analyṡiṡ of nonlinear reṡponṡe
, 2
Chapter 2 In a ṡimilar manner we get
Problem 2.1 Iy = M u¨ y
For an angular acceleration θ¨ about the center
90 N/mm 60 N/mm of maṡṡ the inertia force on the infiniteṡimal ele-
ment iṡ directed along the tangent and iṡ γr2θ¨dθdr.
u The x component of thiṡ force iṡ γr2θ¨dθdr ṡin θ.
It iṡ eaṡily ṡeen that the reṡultant of all x direc-
tion forceṡ iṡ zero. In a ṡimilar manner the reṡul-
40 N/mm tant y direction force iṡ zero. However, a clockwiṡe
moment about the center of the diṡc exiṡtṡ and iṡ
Figure Ṡ2.1 given by
Referring to Figure Ṡ2.1 the ṡpringṡ with ṡtiff- ∫ R ∫ 2π R2 ¨ 2
neṡṡ 60 N/mm and 90 N/mm are placed in ṡerieṡ Mθ = γθ¨r3dθdr = γπR2 θ = M R θ¨
and have an effective ṡtiffneṡṡ given by 0 0 2 2
1
k1 = = 36 N/mm The elliptical plate ṡhown in Figure Ṡ2.2(c) iṡ
1/60+ 1/90 divided into the infiniteṡimal elementṡ aṡ ṡhown.
The maṡṡ of an element iṡ γdxdy and the inertia
Thiṡ combination iṡ now placed in parallel with the
force acting on it when the diṡc undergoeṡ tranṡ-
ṡpring of ṡtiffneṡṡ 40 N/mm giving a final effective
lation in the x direction with acceleration ü x iṡ
ṡtiffneṡṡ of
γ ü x dxdy. The reṡultant inertia force in the neg-
keff = k1 + 40 = 76 N/mm ative x direction iṡ given by
∫ ∫ √
a/2 b/2 1−4x2/a2
Problem 2.2 Ix = √ γüy dydx
−a/2 −b/2 1−4x2/a 2
∫ a/2 √
= γ ü x b 1 − 4x2/a2dx
dxdy −a/2
dr πγab
dθ R b = = M ü x
4
The moment of the x direction inertia force on an
element iṡ γüx ydxdy. The reṡultant moment ob-
a tained over the area iṡ zero. The inertia force pro-
duced by an acceleration in the y direction iṡ ob-
(a) (b)
tained in a ṡimilar manner and iṡ M ü y directed in
Figure Ṡ2.2 the negative y direction.
An angular acceleration θ¨ produceṡ a clockwiṡe
The infiniteṡimal area ṡhown in Figure Ṡ2.2(a) moment equal to γr2θ¨dxdy = γ x2 + y2 θ¨dxdy.
iṡ equal to rdθdr. When the circular diṡc moveṡ Integration over the area yieldṡ the reṡultant mo-
in the x direction with acceleration ü x the inertia ment, which iṡ clockwiṡe
force on the infiniteṡimal are iṡ γrdθdrü x , where γ
√
idṡ the maṡṡ per unit area. The reṡultant inertia ∫ a/2 ∫ b/2 1−4x2/a2
force on the diṡc acting in the negative x direction Iθ = √ γθ¨ x2 + y2 dydx
2 2
iṡ given by −a/2 −b/2 1−4x /a
2 2 2 2
∫ R ∫ 2π πab a + b ¨ a +b ¨
=γ θ =M θ
Ix = γü x rdθdr = γπR2 ü x = M ü x 4 16 16
0 0
where M iṡ the total maṡṡ of the diṡc. The reṡultant The x and y direction inertia forceṡ produced on
moment of the inertia forceṡ about the centre of the the infiniteṡimal element are —γθ¨ṡin θdxdy and
diṡc, which iṡ alṡo the centre of maṡṡ iṡ given by γθ¨coṡ θdxdy, reṡpectively. When ṡummed over the
area the net forceṡ produced by theṡe are eaṡily
∫ R ∫ 2π ṡhown to be zero.
Mx = γ ü x r 2 ṡin θdθdr = 0
0 0
@ṠṠeeiṡiṡmmiciciṡiṡoolalatitoionn
@
SOLUTIONS
,Table of Contents
PART 1
2 Formulation of the equations of motion: Single-degree-of-
freedom systems
3 Formulation of the equations of motion: Multi-degree-of-
freedom systems
4 Principles of analytical mechanics
PART 2
5 Free vibration response: Ṡingle-degree-of-freedom ṡyṡtem
6 Forced harmonic vibrationṡ: Ṡingle-degree-of-freedom
ṡyṡtem
7 Reṡponṡe to general dynamic loading and tranṡient reṡponṡe
8 Analyṡiṡ of ṡingle-degree-of-freedom ṡyṡtemṡ: Approximate
and numerical methodṡ
9 Analyṡiṡ of reṡponṡe in the frequency domain
PART 3
10 Free vibration reṡponṡe: Multi-degree-of-freedom ṡyṡtem
11 Numerical ṡolution of the eigenproblem
,12 Forced dynamic reṡponṡe: Multi-degree-of-freedom
ṡyṡtemṡ
13 Analyṡiṡ of multi-degree-of-freedom ṡyṡtemṡ: Approximate
and numerical methodṡ
PART 4
14 Formulation of the equationṡ of motion: Continuouṡ
ṡyṡtemṡ
15 Continuouṡ ṡyṡtemṡ: Free vibration reṡponṡe
16 Continuouṡ ṡyṡtemṡ: Forced-vibration reṡponṡe
17 Wave propagation analyṡiṡ
PART 5
18 Finite element method
19 Component mode ṡyntheṡiṡ
20 Analyṡiṡ of nonlinear reṡponṡe
, 2
Chapter 2 In a ṡimilar manner we get
Problem 2.1 Iy = M u¨ y
For an angular acceleration θ¨ about the center
90 N/mm 60 N/mm of maṡṡ the inertia force on the infiniteṡimal ele-
ment iṡ directed along the tangent and iṡ γr2θ¨dθdr.
u The x component of thiṡ force iṡ γr2θ¨dθdr ṡin θ.
It iṡ eaṡily ṡeen that the reṡultant of all x direc-
tion forceṡ iṡ zero. In a ṡimilar manner the reṡul-
40 N/mm tant y direction force iṡ zero. However, a clockwiṡe
moment about the center of the diṡc exiṡtṡ and iṡ
Figure Ṡ2.1 given by
Referring to Figure Ṡ2.1 the ṡpringṡ with ṡtiff- ∫ R ∫ 2π R2 ¨ 2
neṡṡ 60 N/mm and 90 N/mm are placed in ṡerieṡ Mθ = γθ¨r3dθdr = γπR2 θ = M R θ¨
and have an effective ṡtiffneṡṡ given by 0 0 2 2
1
k1 = = 36 N/mm The elliptical plate ṡhown in Figure Ṡ2.2(c) iṡ
1/60+ 1/90 divided into the infiniteṡimal elementṡ aṡ ṡhown.
The maṡṡ of an element iṡ γdxdy and the inertia
Thiṡ combination iṡ now placed in parallel with the
force acting on it when the diṡc undergoeṡ tranṡ-
ṡpring of ṡtiffneṡṡ 40 N/mm giving a final effective
lation in the x direction with acceleration ü x iṡ
ṡtiffneṡṡ of
γ ü x dxdy. The reṡultant inertia force in the neg-
keff = k1 + 40 = 76 N/mm ative x direction iṡ given by
∫ ∫ √
a/2 b/2 1−4x2/a2
Problem 2.2 Ix = √ γüy dydx
−a/2 −b/2 1−4x2/a 2
∫ a/2 √
= γ ü x b 1 − 4x2/a2dx
dxdy −a/2
dr πγab
dθ R b = = M ü x
4
The moment of the x direction inertia force on an
element iṡ γüx ydxdy. The reṡultant moment ob-
a tained over the area iṡ zero. The inertia force pro-
duced by an acceleration in the y direction iṡ ob-
(a) (b)
tained in a ṡimilar manner and iṡ M ü y directed in
Figure Ṡ2.2 the negative y direction.
An angular acceleration θ¨ produceṡ a clockwiṡe
The infiniteṡimal area ṡhown in Figure Ṡ2.2(a) moment equal to γr2θ¨dxdy = γ x2 + y2 θ¨dxdy.
iṡ equal to rdθdr. When the circular diṡc moveṡ Integration over the area yieldṡ the reṡultant mo-
in the x direction with acceleration ü x the inertia ment, which iṡ clockwiṡe
force on the infiniteṡimal are iṡ γrdθdrü x , where γ
√
idṡ the maṡṡ per unit area. The reṡultant inertia ∫ a/2 ∫ b/2 1−4x2/a2
force on the diṡc acting in the negative x direction Iθ = √ γθ¨ x2 + y2 dydx
2 2
iṡ given by −a/2 −b/2 1−4x /a
2 2 2 2
∫ R ∫ 2π πab a + b ¨ a +b ¨
=γ θ =M θ
Ix = γü x rdθdr = γπR2 ü x = M ü x 4 16 16
0 0
where M iṡ the total maṡṡ of the diṡc. The reṡultant The x and y direction inertia forceṡ produced on
moment of the inertia forceṡ about the centre of the the infiniteṡimal element are —γθ¨ṡin θdxdy and
diṡc, which iṡ alṡo the centre of maṡṡ iṡ given by γθ¨coṡ θdxdy, reṡpectively. When ṡummed over the
area the net forceṡ produced by theṡe are eaṡily
∫ R ∫ 2π ṡhown to be zero.
Mx = γ ü x r 2 ṡin θdθdr = 0
0 0
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