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2/7/2014
1
03. Single DOF Systems:
Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Vibrations 3.01 Single DOF Systems: Governing Equations
Β§1.Chapter Objectives
β’ Obtain the governing equation of motion for single degree-of-
freedom (dof) translating and rotating systems by using force
balance and moment balance methods
β’ Obtain the governing equation of motion for single dof
translating and rotating systems by using Lagrangeβs
equations
β’ Determine the equivalent mass, equivalent stiffness, and
equivalent damping of a single dof system
β’ Determine the natural frequency and damping factor of a
system
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Vibrations 3.02 Single DOF Systems: Governing Equations
Β§2.Force-Balance and Moment-Balance Methods
1.Force Balance Method
Newtonian principle of linear momentum
πΉ β π = 0 (3.1a)
πΉ : the net external force vector acting on the system
π : the absolute linear momentum of the considered system
For a system of constant mass π whose center of mass is
moving with absolute acceleration π, the rate of change of
linear momentum π = π π
πΉ β π π = 0 (3.1b)
βπ π : inertial force
βΉThe sum of the external forces and inertial forces acting on
the system is zero; that is, the system is in equilibrium
under the action of external and inertial forces
Vibrations 3.03 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2.Force-Balance and Moment-Balance Methods
Vertical Vibrations of a Spring-Mass-Damper System
- Obtain an equation to describe the motions of the spring-mass-
damper system in the vertical
The position vector of
the mass from the fixed
point π π = π π= (πΏ + πΏπ π‘ + π₯) π
Force balance along
the π direction
π π‘ π + ππ π β ππ₯ + ππΏπ π‘ π β πππ
ππ‘ π β π
π2π
ππ‘2 π = 0
Vibrations 3.04 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2.Force-Balance and Moment-Balance Methods
- Noting that πΏ and πΏπ π‘ are constants, rearranging terms to get
the following scalar differential equation
ππ2π₯
ππ‘2 + πππ₯
ππ‘+ π π₯ + πΏπ π‘ = π π‘ + ππ
Vibrations 3.05 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2.Force-Balance and Moment-Balance Methods
Static Equilibrium Position
- The static-equilibrium position of a system is the position that
corresponds to the systemβs rest state; that is, a position with
zero velocity and zero acceleration
- The static-equilibrium position is the solution of
π π₯ + πΏπ π‘ = ππ
- The static displacement
πΏπ π‘ =ππ
πβΉ π₯ = 0 is the static-equilibrium position of the system
- The spring has an unstretched length πΏ, the static-equilibrium
position measured from the origin π is given by
π₯π π‘ = π₯π π‘ π = (πΏ + πΏπ π‘) π
Vibrations 3.06 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
2
Β§2.Force-Balance and Moment-Balance Methods
Equation of Motion for Oscillations about the Static-EquilibriumPosition
ππ2π₯
ππ‘2 + πππ₯
ππ‘+ π π₯ + πΏπ π‘ = π π‘ + ππ
πΏπ π‘ =ππ
π
βΉ ππ2π₯
ππ‘2 + πππ₯
ππ‘+ ππ₯ = π π‘
Equation (3.8) is the governing equation of motion of a single
dof system for oscillations about the static-equilibrium position
β’ The left-hand side: the forces from the components that
comprise a single dof system
β’ The right-hand side: the external force acting on the mass
Vibrations 3.07 Single DOF Systems: Governing Equations
(3.8)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2.Force-Balance and Moment-Balance Methods
Horizontal Vibrations of a Spring-Mass-Damper System
Consider a mass moving in a direction normal
to the direction of gravity
β’ It is assumed that the mass moves without
friction
β’ The unstretched length of the spring is πΏ, and
a fixed point π is located at the unstretched
position of the spring
β’ The spring does not undergo any static
deflection and carrying out a force balance
along the π direction
β’ The static-equilibrium position π₯ = 0 coincides with the
position corresponding to the unstretched spring
Vibrations 3.08 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2.Force-Balance and Moment-Balance Methods
Force Transmitted to Fixed Surface
The total reaction force due to the spring and
the damper on the fixed surface is the sum of
the static and dynamic forces
πΉπ = ππΏπ π‘ + ππ₯ + πππ₯
ππ‘
If considering only the dynamic part of the
reaction force-that is, only those forces created
by the motion π₯(π‘) from its static equilibrium
position, then
πΉπ π = ππ₯ + πππ₯
ππ‘
Vibrations 3.09 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2.Force-Balance and Moment-Balance Methods
- Ex.3.1 Wind-drivenOscillationsaboutaSystemβsStatic-EquilibriumPosition
The wind flow across civil structures typically generates a
excitation force π(π‘) on the structure that consists of a steady-
state part and a fluctuating part
π π‘ = ππ π + ππ(π‘)
ππ π : the time-independent steady-state force
ππ(π‘) : the fluctuating time-dependent portion of the force
A single dof model of the vibrating structure
ππ2π₯
ππ‘2 + πππ₯
ππ‘+ ππ₯ = ππ π + ππ π‘ βΉ π₯ π‘ = π₯0 + π₯π(π‘)
π₯0 : the static equilibrium position, π₯0 = ππ π /π
π₯π(π‘) : motions about the static position
βΉ ππ2π₯
ππ‘2 + πππ₯
ππ‘+ ππ₯ = ππ π‘
Vibrations 3.10 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2.Force-Balance and Moment-Balance Methods
- Ex.3.2 EardrumOscillations:NonlinearOscillatorandLinearizedSystems
Determine the static-equilibrium positions of this system and
illustrate how the governing nonlinear equation can be
linearized to study oscillations local to an equilibrium position
Solution
The governing nonlinear equation
ππ2π₯
ππ‘2 + ππ₯ + ππ₯2 = 0
The restoring force of the eardrum has a component with a
quadratic nonlinearity
Static-Equilibrium Positions
Equilibrium positions π₯ = π₯0 are solutions of the algebraic equation
π π₯0 + π₯02 = 0 βΉ π₯0 = 0, π₯0 = β1
Vibrations 3.11 Single DOF Systems: Governing Equations
(π)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2.Force-Balance and Moment-Balance Methods
Linearization
Equilibrium positions π₯ = π₯0 are solutions of the algebraic equation
π π₯0 + π₯02 = 0 βΉ π₯0 = 0, π₯0 = β1
Subtitute π₯ π‘ = π₯0 + π₯(π‘) into (a) with note that
π₯2 π‘ = π₯0 + π₯ π‘2
β π₯02 + 2π₯0 π₯ π‘ + β―
π2π₯
ππ‘2 =π2 π₯0 + π₯ π‘
ππ‘2 =π2 π₯
ππ‘2
βΉ ππ2 π₯
ππ‘2 + π π₯0 + π₯(π‘) + π π₯02 + 2π₯0 π₯ π‘ = 0
π₯0 = 0 βΉ ππ2 π₯
ππ‘2 + π π₯(π‘) = 0
π₯0 = β1 βΉ ππ2 π₯
ππ‘2 β π π₯(π‘) = 0
βΉ the equations have different stiffness terms
Vibrations 3.12 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
3
Β§2.Force-Balance and Moment-Balance Methods
2. Moment-Balance Methods
For single dof systems that undergo rotational motion, the
moment balance method is useful in deriving the governing
equation
The angular momentum about the center of mass of the disc
π» = π½πΊ ππ
βΉ π = π½πΊ ππ
Vibrations 3.13 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2.Force-Balance and Moment-Balance Methods
The governing equation of motion
π π‘ π β ππ‘ ππ β ππ‘
ππ
ππ‘π β π½πΊ
π2π
ππ‘2 = 0
βΉ π½πΊπ2π
ππ‘2 + ππ‘
ππ
ππ‘+ ππ‘π = π π‘
Vibrations 3.14 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2.Force-Balance and Moment-Balance Methods
All linear single dof vibratory systems are governed by a linear
second-order ordinary differential equation with an inertia term,
a stiffness term, a damping term, and a term related to the
external forcing imposed on the system
β’ Translational motion
ππ2π₯
ππ‘2 + πππ₯
ππ‘+ ππ₯ = π π‘
β’ Rotational motion
π½πΊπ2π
ππ‘2 + ππ‘
ππ
ππ‘+ ππ‘π = π π‘
Vibrations 3.15 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2.Force-Balance and Moment-Balance Methods
Ex.3.3 Hand Biomechanics
The moment balance about
point π
π β π½0 ππ = 0
π½0: the rotary inertia of the
forearm and the object
held in the hand
The net moment π acting
about the point π due to gravity loading and the forces due to
the biceps and triceps
π = βππππππ ππ β πππ
2πππ ππ + πΉπππ β πΉπ‘ππ
βΉ βππππππ ππ β πππ
2πππ ππ + πΉπππ β π½0 ππ = 0
Vibrations 3.16 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2.Force-Balance and Moment-Balance Methods
βππππππ ππ β πππ
2πππ ππ + πΉπππ β π½0 ππ = 0
Note that: πΉπ = βπππ, πΉπ‘ = πΎπ‘π£ = πΎπ‘π π, πΉ0 = ππ2/3 + ππ2
βΉ π +π
3π2 π + πΎπ‘π
2 π + ππππ + π +π
2πππππ π = 0
Static-Equilibrium Position
The equilibrium position π = π0 is a solution of the
transcendental equation
ππππ0 + π +π
2πππππ π0 = 0
Vibrations 3.17 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2.Force-Balance and Moment-Balance Methods
Linear System Governing βSmallβ Oscillations about the Static-
Equilibrium Position
Consider oscillations about the static-equilibrium position and
expand the angular variable π π‘ = π0 + π π‘ with note that
πππ π = cos π0 + π β πππ π0 β ππ πππ0 + β―
ππ(π‘)
ππ‘=
π(π0 + π)
ππ‘= π(π‘)
π2π(π‘)
ππ‘2 =π2(π0 + π)
ππ‘2 = π(π‘)
βΉ π +π
3π2 π + πΎπ‘π
2 π + ππ π = 0
where
ππ = πππ β π +π
2πππ πππ0
Vibrations 3.18 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
4
Β§3.Natural Frequency and Damping Factor
1.Natural Frequency
Translation Vibrations: Natural Frequency
ππ = 2πππ =π
π(πππ/π )
π : the stiffness of the system, π/π
π : the system mass, ππ
ππ : the natural frequency, π»π§
For the mass-damper-spring system
ππ = 2πππ =π
πΏπ π‘(πππ/π )
πΏπ π‘: the static deflection of the system, π
Vibrations 3.19 Single DOF Systems: Governing Equations
(3.15)
(3.14)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3.Natural Frequency and Damping Factor
Rotational Vibrations: Natural Frequency
ππ = 2πππ =ππ‘
π½(πππ/π )
ππ‘ : the torsion stiffness of the system, ππ/πππ
π½ : the system mass, πππ/π 2
ππ : the natural frequency, π»π§
Period of Undamped Free Oscillations
For an unforced and undamped system, the period of free
oscillation of the system is given by
π =1
ππ=
2π
ππ
Vibrations 3.20 Single DOF Systems: Governing Equations
(3.16)
(3.17)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ππ = 2πππ =π
πΏπ π‘(πππ/π ) (3.15)
Β§3.Natural Frequency and Damping Factor
Ex.3.4 Natural Frequency from Static Deflection of a Machine System
The static deflections of a machinery are found to be 0.1, 1,
10(ππ). Determine the natural frequency for vertical vibrations
Solution
ππ1 =1
2π
π
πΏπ π‘1=
1
2π
9.81
0.1 Γ 10β3 = 49.85π»π§
ππ2 =1
2π
π
πΏπ π‘2=
1
2π
9.81
1 Γ 10β3 = 15.76π»π§
ππ3 =1
2π
π
πΏπ π‘3=
1
2π
9.81
10 Γ 10β3 = 4.98π»π§
Vibrations 3.21 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3.Natural Frequency and Damping Factor
- Ex.3.5 Static Deflection and Natural Frequency of the Tibia
Bone in a Human Leg
Consider a person of 100ππ mass standing upright. The tibia
has a length of 40ππ, and it is modeled as a hollow tube with an
inner diameter of 2.4ππ and an outer diameter of 3.4ππ. The
Youngβs modulus of elasticity of the bone material is 2 Γ1010π/π2. Determine the static deflection in the tibia bone and
an estimate of the natural frequency of axial vibrations
Solution
Assume that both legs support the weight of the person
equally, so that the weight supported by the tibia
ππ = 100/2 Γ 9.81 = 490.5π
Vibrations 3.22 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ππ = 2πππ =π
πΏπ π‘(πππ/π ) (3.15)
Β§3.Natural Frequency and Damping Factor
The stiffness of the tibia
π =π΄πΈ
πΏ=
1 Γ 1010 Γπ4
3.4 Γ 10β2 2 β 2.4 Γ 10β2 2
40 Γ 10β2
= 22.78 Γ 106π/π2
The static deflection
πΏπ π‘ =ππ
π=
490.5
22.78 Γ 106 = 21.53 Γ 10β6π
The natural frequency
ππ =1
2π
π
πΏπ π‘=
1
2π
9.81
21.53 Γ 10β6 = 107.4π»π§
Vibrations 3.23 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3.Natural Frequency and Damping Factor
Ex.3.6 System with A Constant Natural Frequency
Examine how the spring-mounting system can be designed and
discuss a realization of this spring in practice
Solution
In order to realize the desired objective of constant natural
frequency regardless of the system weight, we need a
nonlinear spring whose equivalent spring constant is given by
π = π΄π
π΄: a constant, π = ππ: the weight, π: the gravitational constant
The natural frequency
ππ =1
2π
π
π=
1
2π
ππ
π=
1
2ππ΄ππ»π§
βΉ ππ is constant irrespective of the weight of the mass
Vibrations 3.24 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
5
Β§3.Natural Frequency and Damping Factor
Nonlinear Spring Mounting
When the side walls of a rubber cylindrical tube are
compressed into the nonlinear region, the equivalent spring
stiffness of this system approximates the characteristic given
by π = π΄π
For illustrative purposes, consider a spring that has the
general force-displacement relationship
πΉ π₯ = ππ₯
π
π
π, π: scale factors, π: shape factor
The static deflection
π₯0 = ππ
π
1/π
Vibrations 3.25 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3.Natural Frequency and Damping Factor
For βsmallβ amplitude vibrations about π₯0, the linear equivalent
stiffness of this spring is determined
πππ = ππΉ(π₯)
ππ₯π₯=π₯0
=ππ
π
π₯π
π
πβ1
=ππ
π
π
π
πβ1π
The natural frequency of this system
ππ =1
2π
πππ
π/π
=1
2π
ππ
π
π
π
β1/π
=1
2π
ππ
π
π
π
β1/2π
π»π§
Vibrations 3.26 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3.Natural Frequency and Damping Factor
Representative Spring Data
Consider the representative data of a
nonlinear spring shown in the figure
Using lsqcurvefit in Matlab to identify
π = 2500π, π = 0.011π, π = 2.77
βΉ ππ =1
2π
ππ
π
π
π
β1/2π
= 32.4747πβ1/3.54π»π§
Plot ππ(π)
Vibrations 3.27 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3.Natural Frequency and Damping Factor
Representative Spring Data
From the figure of ππ(π)
β’ over a sizable portion of the load
range, the natural frequency of the
system varies within the range of 8.8%
β’ The natural frequency of a system with
a linear spring whose static
displacement ranges from 12 Γ· 5ππvaries approximately from 4.5 Γ· 7.0π»π§or approximately 22% about a
frequency of 5.8π»π§
1
2π
9.8
0.012β 4.5π»π§,
1
2π
9.8
0.005β 7π»π§
of 5.8 Hz
Vibrations 3.28 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3.Natural Frequency and Damping Factor
2.Damping Factor
Translation Vibrations: Damping Factor
For translating single dof systems, the damping factor or
damping ratio π is defined as
π =π
2πππ=
π
2 ππ=
πππ
2π
π: the system damping coefficient, ππ /π
π: the system stiffness, π/π
π: the system mass, ππ
Critical Damping, Underdamping, and Overdamping
Defining the critical damping ππ
ππ = 2πππ = 2 ππ, π = π/ππ (3.19)
0 < π < 1: underdamped,π > 1: overdamped,π = 1: criticallydamped
Vibrations 3.29 Single DOF Systems: Governing Equations
(3.18)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3.Natural Frequency and Damping Factor
Rotational Vibrations: Damping Factor
For rotating single dof systems, the damping factor or damping
ratio π is defined as
π =ππ‘
2π½ππ=
ππ‘
2 ππ‘π½
ππ‘: the system damping coefficient, πππ /πππ
ππ‘: the system stiffness, ππ/πππ
π½: the system moment of inertia, πππ2
Vibrations 3.30 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
6
Β§3.Natural Frequency and Damping Factor
Governing Equation of Motion in Terms of Natural Frequency
and Damping Factor
Rewriting the equation of motion
π2π₯
ππ‘2 + 2πππ
ππ₯
ππ‘+ ππ
2π₯ =π(π‘)
πIf we introduce the dimensionless time π = πππ‘ , then the
equation can be written
π2π₯
ππ2 + 2πππ₯
ππ+ π₯ =
π(π)
π
Vibrations 3.31 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3.Natural Frequency and Damping Factor
- Ex.3.7 Effect of Mass on the Damping Factor
A system is initially designed to be critically damped - that is,
with a damping factor of π = 1. Due to a design change, the
mass of the system is increased 20% - that is, from π to 1.2π.
Will the system still be critically damped if the stiffness and the
damping coefficient of the system are kept the same?
Solution
The damping factor of the system after the design change
ππππ€ =π
2 π(1.2π)= 0.91
π
2 ππ= 0.91
π
ππ= 0.91
βΉ The system with the increased mass is no longer critically
damped; rather, it is now underdamped
Vibrations 3.32 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3.Natural Frequency and Damping Factor
- Ex.3.8 Effects of System Parameters on the Damping Ratio
An engineer finds that a single dof system with mass π ,
damping π, and spring constant π has too much static deflection
πΏπ π‘. The engineer would like to decrease πΏπ π‘ by a factor of 2,
while keeping the damping ratio constant. Determine the
different options
Solution
The problem involves vertical vibrations
πΏπ π‘ =ππ
π
2π =π
π
πΏπ π‘
π= π
πΏπ π‘
ππ2 =1
π
π2πΏπ π‘
π
βΉ there are three ways that one can achieve the goal
Vibrations 3.33 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3.Natural Frequency and Damping Factor
First choice
Let π remain constant, reduce πΏπ π‘ by one-half
πΏπ π‘ =ππ
π
πΏπ π‘β² =
πΏπ π‘
2=
ππ
2π=
πβ²π
πβ²Comparing (a) and (b)
πβ²π
πβ²=
ππ
2π=
π/ 2 π
π 2βΉ π β πβ² =
π
2, π β πβ² = π 2
Check the damping ratio
2πβ² = ππΏβ²
π π‘
ππβ²2 = ππΏπ π‘
2π π/ 22 = π
πΏπ π‘
ππ2 = 2π
Vibrations 3.34 Single DOF Systems: Governing Equations
Before (a)
After (b)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3.Natural Frequency and Damping Factor
Second choice
Let π remain constant, reduce πΏπ π‘ by one-half
2π = ππΏπ π‘
ππ2 =1
π
π2πΏπ π‘
π
2πβ² =1
π
πβ²2πΏπ π‘β²
π=
1
π
πβ²2πΏπ π‘
2π
Comparing (c) and (d)
πβ²2
2= π2 βΉ π β πβ² = π 2
The static deflection
πΏπ π‘β² =
ππ
πβ²=
πΏπ π‘
2=
ππ
2πβΉ π β πβ² = 2π
Vibrations 3.35 Single DOF Systems: Governing Equations
Before (c)
After (d)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3.Natural Frequency and Damping Factor
Third choice
Let π remain constant, reduce πΏπ π‘ by one-half
πΏπ π‘ =ππ
π
πΏπ π‘β² =
πΏπ π‘
2=
ππ
2π=
πβ²π
πComparing (e) and (f)
πβ² =π
2βΉ π β πβ² =
π
2The constant damping ratio
2πβ² = πβ²πΏβ²
π π‘
ππβ²2 = πβ²πΏπ π‘
2π π/2 2 = πβ²2πΏπ π‘
ππ2 = ππΏπ π‘
ππ2 = 2π
βΉ π β πβ² = π 2
Vibrations 3.36 Single DOF Systems: Governing Equations
Before (e)
After (f)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
7
πΉ π₯ = ππππ ππ( π₯) (2.52)
ππ2π₯
ππ‘2 + πππ₯
ππ‘+ ππ₯ = π π‘ (3.8)
Β§4.Governing Equations for Different Type of Damping
The governing equations of motion for systems with different
types of damping are obtained by replacing the term
corresponding to the force due to viscous damping with the force
due to either the fluid, structural, or dry friction type damping
Coulomb or Dry Friction Damping
Using Eq. (2.52) and Eq. (3.8), the governing equation of motion
takes the form
ππ2π₯
ππ‘2 + πππ₯
ππ‘+ ππππ ππ( π₯) = π(π‘)
which is a nonlinear equation because the damping
characteristic is piecewise linear
Vibrations 3.37 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
πππππππππ πππ¦ πππππ‘πππ πππππ
πΉ π₯ = ππ π₯2π ππ π₯ = ππ| π₯| π₯ (2.54)
πΉ = πππ½βπ ππ π₯ |π₯| (2.57)
ππ2π₯
ππ‘2 + πππ₯
ππ‘+ ππ₯ = π π‘ (3.8)
Β§4.Governing Equations for Different Type of Damping
Fluid Damping
Using Eq. (2.54) and Eq. (3.8), the governing equation of motion
ππ2π₯
ππ‘2 + ππ| π₯| π₯ + ππ₯ = π(π‘)
which is a nonlinear equation due to the nature of the damping
Structural Damping
Using Eq. (2.57) and Eq. (3.8), the governing equation of motion
ππ2π₯
ππ‘2 + πππ½βπ ππ π₯ |π₯| + ππ₯ = π(π‘)
Vibrations 3.38 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
πππππππππ πππ’ππ πππππππ πππππ
Β§5.Governing Equations for Different Type of Applied Forces
1.System with Base excitation
- The base-excitation model is a prototype that is useful for studying
β’ buildings subjected to earthquakes
β’ packaging during transportation
β’ vehicle response, and
β’ designing accelerometers
- The physical system of interest is represented by a single dof
system whose base is subjected to a displacement
disturbance π¦(π‘), and an equation governing the motion of
this system is sought to determine the response of the
system π₯(π‘)
Vibrations 3.39 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§5.Governing Equations for Different Type of Applied Forces
- A prototype of a single dof system subjected to a base excitation
β’ The vehicle provides the base excitation π¦(π‘) to the
instrumentation package modeled as a single dof
β’ The displacement response π₯(π‘) is measured from the
systemβs static-equilibrium position
Assume that no external force is applied directly to the mass;
that is, π π‘ = 0
Vibrations 3.40 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§5.Governing Equations for Different Type of Applied Forces
- The following governing equation of motion
ππ2π₯
ππ‘2 + πππ₯
ππ‘+ ππ₯ = π
ππ¦
ππ‘+ ππ¦
βΉ ππ2π₯
ππ‘2 + 2πππ
ππ₯
ππ‘+ ππ
2π₯ = 2πππ
ππ¦
ππ‘+ ππ
2π¦
π¦(π‘) and π₯(π‘) are measured from a fixed point π located in an
inertial reference frame and a fixed point located at the
systemβs static equilibrium position, respectively
Vibrations 3.41 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§5.Governing Equations for Different Type of Applied Forces
- If the relative displacement is desired, the governing equation
of motion
ππ2π§
ππ‘2 + πππ§
ππ‘+ ππ§ = βπ
π2π¦
ππ‘2
with π§ π‘ β‘ π₯ π‘ β π¦(π‘)
βΉπ2π§
ππ‘2 + 2πππ
ππ§
ππ‘+ ππ
2π§ = βπ2π¦
ππ‘2
Vibrations 3.42 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
8
Β§5.Governing Equations for Different Type of Applied Forces
2.System with Unbalanced Rotating Mass
- Assume that the unbalance generates a force that acts on the
systemβs mass. This force, in turn, is transmitted through the
spring and damper to the fixed base
- The unbalance is modeled as a mass π0 that rotates with an
angular speed π, and this mass is located a fixed distance πfrom the center of rotation
Vibrations 3.43 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§5.Governing Equations for Different Type of Applied Forces
- From the free-body diagram (FBD) of the unbalanced mass π0
ππ₯ = βπ0( π₯ β ππ2π ππππ‘)
ππ¦ = π0ππ2πππ ππ‘
- From the FBD of mas π
ππ2π₯
ππ‘2 + πππ₯
ππ‘+ ππ₯ = ππ₯
βΉ (π + π0)π2π₯
ππ‘2 + πππ₯
ππ‘+ ππ₯ = π0ππ
2π ππππ‘
βΉπ2π₯
ππ‘2 + 2πππ
ππ₯
ππ‘+ ππ
2π₯ =πΉ(π)
ππ ππππ‘
where π = π + π0, ππ = π/π, πΉ π = π0ππ2
- The static displacement of the spring
πΏπ π‘ =π + π0 π
π=
ππ
π
Vibrations 3.44 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§5.Governing Equations for Different Type of Applied Forces
3.System with Added Mass Due to a Fluid
- The equation of motion of the system
ππ2π₯
ππ‘2 + ππ₯ = π π‘ + π1(π‘)
π₯(π‘) : measured from the unstretched position of the spring
π(π‘) : the externally applied force
π1(π‘) : the force exerted by the fluid on the mass due to the
motion of the mass
Vibrations 3.45 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§5.Governing Equations for Different Type of Applied Forces
- The force generated by the fluid on the rigid body
π1 π‘ = βπΎ0ππ2π₯
ππ‘2 β πΆπ
ππ₯
ππ‘
π : the mass of the fluid displaced by the body
πΎ0 : an added mass coefficient
πΆπ : a positive fluid damping coefficient
- The governing equation of motion
π + πΎ0ππ2π₯
ππ‘2 + πΆπ
ππ₯
ππ‘+ ππ₯ = π π‘
πΎ0π : the added mass due to the fluid
Vibrations 3.46 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
Consider a system with π degrees of freedom that is described
by a set of π generalized coordinates ππ , π = 1,2,β¦π. In terms
of the chosen generalized coordinates, Lagrangeβs equations
have the form
π
ππ‘
ππ
π ππβ
ππ
πππ+
ππ·
π ππ+
ππ
πππ= ππ , π = 1,2,β¦ , π
ππ : generalized coordinate
ππ : generalized velocity
π : the kinetic energy of the system
π : the potential energy of the system
π· : the Rayleigh dissipation function
ππ : the generalized force that appears in the ππ‘β equation
Vibrations 3.47 Single DOF Systems: Governing Equations
(3.41)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
The generalized force ππ that appears in the ππ‘β equation
ππ =
π
πΉπ
π πππππ
+
π
ππ
πππ
π ππ
πΉπ, ππ : the vector representations of the externally
applied forces and moments on the ππ‘β body
ππ : the position vector to the location where the force
πΉπ is applied
ππ : the ππ‘β bodyβs angular velocity about the axis
along which the considered moment is applied
Vibrations 3.48 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
9
Β§6.Lagrangeβs Equations
Linear Vibratory Systems
For vibratory systems with linear characteristics
π =1
2
π=1
π
π=1
π
πππ ππ ππ
π =1
2
π=1
π
π=1
π
πππππππ
π· =1
2
π=1
π
π=1
π
πππ ππ ππ
πππ : the inertia coefficients
πππ : the stiffness coefficients
πππ : the damping coefficients
Vibrations 3.49 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
Single Degree-Of-Freedom
The case of a single degree-of-freedom system, π = 1, the
Lagrangeβs equation
π
ππ‘
ππ
π π1β
ππ
ππ1+
ππ·
π π1+
ππ
ππ1= π1
where the generalized force is obtained from
π1 =
π
πΉπ
π ππππ1
+
π
ππ
πππ
π π1
Vibrations 3.50 Single DOF Systems: Governing Equations
(3.44)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
Linear Single Degree-Of-Freedom Systems
The expressions for the system kinetic energy, the system
potential energy, and the system dissipation function reduce to
π =1
2
π=1
1
π=1
1
πππ ππ ππ =1
2π11 π1
2 β‘1
2ππ π1
2
π =1
2
π=1
1
π=1
1
πππππππ =1
2π11π1
2 β‘1
2πππ1
2
π· =1
2
π=1
1
π=1
1
πππ ππ ππ =1
2π11 π1
2 β‘1
2ππ π1
2
ππ, ππ, ππ : the equivalent mass, stiffness, and damping
From Lagrangeβs equation
ππ π1 + ππ π1 + πππ1 = π1
Vibrations 3.51 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
(3.46)
Β§6.Lagrangeβs Equations
To obtain the governing equation of motion of a linear vibrating
system with viscous damping
β’ Obtains expressions for the system kinetic energy π ,
system potential energy π, and system dissipation function π·
β’ Identify the equivalent mass ππ, equivalent stiffness ππ,
and equivalent damping ππ
β’ Determine the generalized force
β’ Apply the governing equation
ππ π1 + ππ π1 + πππ1 = π1
β’ Determine the system natural frequency
ππ =ππ
ππ, π =
ππ
2ππππ=
ππ
2 ππππ
Vibrations 3.52 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
- Ex.3.9 Motion of A Linear Single Degree-Of-Freedom System
Obtain the governing equation for the mass-damper-spring
system
Solution
Identify the following
π1 = π₯, πΉπ = π(π‘) π, ππ = π₯ π, ππ = 0
Determine the generalized force
π1 =
π
πΉπ
π ππππ1
+ 0 = π π‘ πππ₯ π
ππ₯= π(π‘)
The system kinetic energy π, system potential energy π, and
system dissipation function π·
π =1
2π π₯2, π =
1
2ππ₯2, π· =
1
2π π₯2
Vibrations 3.53 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
Identify the following
π1 = π₯, πΉπ = π(π‘) π, ππ = π₯ π, ππ = 0
Determine the generalized force
π1 =
π
πΉπ
π ππππ1
+ 0 = π π‘ πππ₯ π
ππ₯= π(π‘)
The system kinetic energy π, system potential energy
π, and system dissipation function π·
π =1
2π π₯2, π =
1
2ππ₯2, π· =
1
2π π₯2
βΉ ππ = π, ππ = π, ππ = π
The governing equation
ππ2π₯
ππ‘2 + πππ¦
ππ‘+ ππ₯ = π(π‘)
Vibrations 3.54 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
10
Β§6.Lagrangeβs Equations
- Ex.3.10 Motion of A System that Translates and Rotates
Obtain the governing equation of motion for βsmallβ oscillations
about the upright position
Solution
Choose the generalized coordinate
π1 = π, πΉπ = 0, ππ = π π‘ π, ππ = ππ
The generalized force
π1 =
π
ππ βπππ
π π1= π π‘ π β
π ππ
π π= π(π‘)
Vibrations 3.55 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
π½πΊ =1
2ππ2
Β§6.Lagrangeβs Equations
The potential energy
π =1
2ππ₯2 =
1
2π(ππ)2=
1
2ππ2π2
βΉ the equivalent stiffness
The kinetic energy of the system
π =1
2π π₯2 +
1
2π½πΊ π2
βΉ π =1
2ππ2 + π½πΊ π2 =
1
2
3
2ππ2 π2
βΉ the equivalent mass of the system
Vibrations 3.56 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
πππ‘ππ‘ππππππππππ‘ππ ππππππ¦
π‘ππππ πππ‘ππππππππ‘ππ ππππππ¦
ππ = ππ2
ππ =3
2ππ2
Β§6.Lagrangeβs Equations
The dissipation function
π· =1
2π π₯2 =
1
2π(π π)2=
1
2(ππ2) π2
βΉ the equivalent damping coefficient
ππ = ππ2
The governing equation of motion3
2ππ2 π + ππ2 π + ππ2π = π(π‘)
Natural frequency and damping factor
ππ =ππ
ππ=
ππ2
3ππ2/2=
2π
3π
π =ππ
2ππππ=
ππ2
2(3ππ2/2) 2π/3π=
6
6ππ
Vibrations 3.57 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
- Ex.3.11 Inverted Pendulum
Obtain the governing equation of motion for βsmallβ oscillations
about the upright position
Solution
The total rotary inertia of the system
π½π = π½π1+ π½π2
π½π1: mass momentof inertia of π1 about pointπ
π½π2: massmomentof inertiaof thebaraboutpointπ
π½π1=
2
5π1π
2 + π1πΏ12
π½π2=
1
12π2πΏ2
2 + π2
πΏ2
2
2
=1
3π2πΏ2
2
Vibrations 3.58 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
Choosing π1 = π as the generalized coordinate, the system
kinetic energy takes the form
π =1
2π½π π2 =
1
2π½π1
+ π½π2 π2
=1
2
2
5π1π
2 + π1πΏ12 +
1
3π2πΏ2
2 π2
For small π βΉ π₯1 β πΏ1π
The system potential energy
π =1
2ππ₯1
2 β1
2π1ππΏ1π
2 β1
2π2π
πΏ2
2π2
=1
2ππΏ1
2 β π1ππΏ1 β π2ππΏ2
2π2
π· =1
2π π₯1
2 =1
2ππΏ1
2 π2
Vibrations 3.59 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
The dissipation function
Β§6.Lagrangeβs Equations
The equivalent inertia, the equivalent stiffness, and the
equivalent damping properties of the system
π =1
2
2
5π1π
2 +π1πΏ12 +
1
3π2πΏ2
2 π2
π =1
2ππΏ1
2 β π1ππΏ1 β π2ππΏ2
2π2
π· =1
2π π₯1
2 =1
2ππΏ1
2 π2
The governing equation of motion ππ π + ππ
π + πππ = 0
Natural frequency
ππ =ππ
ππ=
ππΏ12 β π1ππΏ1 β π2ππΏ2/2
π½π1+ π½π2
ππ can be negative, which affects the stability of the system
Vibrations 3.60 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
βΉππ =2
5π1π
2 +π1πΏ12 +
1
3π2πΏ2
2
βΉππ =ππΏ12 βπ1ππΏ1 βπ2π
πΏ2
2
βΉ ππ = ππΏ12
2/7/2014
11
Β§6.Lagrangeβs Equations
β’ Natural Frequency of Pendulum System
Now locate the pivot point π on the top, the
equivalent stiffness of this system
ππ = ππΏ12 + π1ππΏ1 + π2π
πΏ2
2and the natural frequency of this system
ππ =ππ
ππ=
ππΏ12 + π1ππΏ1 + π2ππΏ2/2
π½π1+ π½π2
If π2 βͺ π1, π βͺ πΏ1, and π = 0, then
ππ =π1ππΏ1 1 + π2πΏ2/π1πΏ1
π1πΏ12 1 + 2π2/5πΏ1
2 βπ
πΏ
β the natural frequency of a pendulum composed of a rigid,
weightless rod carrying a mass a distance πΏ1 from its pivot
Vibrations 3.61 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
- Ex.3.12 Motion of A Disk Segment
Derive the governing equation of motion of a disk segment
Solution
The position vector from the fixed point π to the
center of mass πΊ
π = βπ π + ππ πππ π + (π β ππππ π) π
The absolute velocity of the center of mass π = β π β ππππ π π π + ππ πππ π π
Selecting the generalized coordinate π1 = π ,
the system kinetic energy
π =1
2π½πΊ π2 +
1
2π π β π
βΉ π =1
2π½πΊ π2 +
1
2π π 2 + π2 β 2ππ πππ π π2
Vibrations 3.62 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Taylor series expansion
πππ π = πππ π0 + π β πππ π0 β ππ πππ0 β1
2 π2πππ π0 + β―
π πππ = π ππ π0 + π β π πππ0 β ππππ π0 β1
2 π2π πππ0 + β―
Β§6.Lagrangeβs Equations
Choosing the fixed ground as the datum, the system potential
energy
π = ππ π β ππππ π
Small Oscillations about the Upright Position
Express the angular displacement as
π(π‘) = π0 + π(π‘)
Since π0 = 0 , and small π , using π πππ β π ,
πππ π β 1 β1
2 π2, rewrite the energy functions
π β1
2π½πΊ + π π β π 2 π2, π β ππ π β π +
1
2πππ π2
Vibrations 3.63 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
The equivalent inertia of the system
π β1
2π½πΊ + π π β π 2 π2
The potential energy is not in standard form because of the
constant term ππ π β π
π β ππ π β π +1
2πππ π2
However, since the datum for the potential energy is not
unique, we can shift the datum for the potential energy from
the fixed ground to a distance (π β π) above the ground
π =1
2πππ π2
Then, the equivalent stiffness can be defined
ππ = πππ
Vibrations 3.64 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
βΉ ππ = π½πΊ + π π β π 2
Β§6.Lagrangeβs Equations
The governing equation
π½πΊ + π π β π 2 π + πππ π = 0
Natural Frequency
ππ =ππ
ππ
=πππ
π½πΊ + π π β π 2
=π
π½πΊ + π π β π 2
ππ
Vibrations 3.65 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
- Ex.3.13 Translating System with a Pre-tensioned/compressedSpring
Derive the governing equation of motion for vertical
translations π₯ of the mass about the static
equilibrium position of the system
Solution
The equation of motion will be derived for
βsmallβ amplitude vertical oscillations; that is,
π₯/πΏ βͺ 1
The horizontal spring is pretensioned with a tension, which is
produced by an initial extension of the spring by an amount πΏ0
π1 = π1πΏ0
The kinetic energy of the system
π =1
2π π₯2
Vibrations 3.66 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
12
Binomial expansion 1 + π₯ π = 1 + ππ₯ +1
2π(π β 1)π₯2 + β―
Β§6.Lagrangeβs Equations
The potential energy of the system
π =1
2π1 πΏ0 + βπΏ 2
πππ π πππππ π1
+1
2π2π₯
2
πππ π πππππ π2
βπΏ : the change in the length of the spring with
stiffness π1 due to the motion π₯ of the mass
βπΏ = πΏ2 + π₯2 β πΏ = πΏ 1 + (π₯/πΏ)2β πΏ
Assume that |π₯/πΏ| βͺ 1, using binomial expansion
1+(π₯/πΏ)2= 1+(π₯/πΏ)2 1/2 = 1+1
2(π₯/πΏ)2+
1
8(π₯/πΏ)4+β―
βΉ βπΏ β πΏ 1+(π₯/πΏ)2/2 β πΏ = πΏ(π₯/πΏ)2/2
βΉ π =1
2π1 πΏ0 +
πΏ
2
π₯
πΏ
2 2
+1
2π2π₯
2
Vibrations 3.67 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
π
ππ‘
ππ
π π1β
ππ
ππ1+
ππ·
π π1+
ππ
ππ1= π1 (3.44)
Β§6.Lagrangeβs Equations
Chose the generalize coordinate π1 = π₯
π =1
2π π₯2
βΉπ
ππ‘
ππ
π π₯=
π
ππ‘π π₯ = π π₯
ππ
ππ₯= 0,
ππ·
π π₯= 0, π = 0
π =1
2π1 πΏ0 +
πΏ
2
π₯
πΏ
2 2
+1
2π2π₯
2
βΉππ
ππ₯=π1 πΏ0 +
πΏ
2
π₯
πΏ
2 2π₯
πΏ+π2π₯= π1 +
π1πΏ0
πΏπ₯+
π1
2
π₯3
πΏ2β π2 +
π1πΏ
π₯
Vibrations 3.68 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
The governing equation of motion
π π₯ + π2 +π1
πΏπ₯ = 0
The natural frequency
ππ = ππ/ππ = π2 + π1/πΏ /π
If the spring of constant π1 is compressed instead of being in
tension, then we can replace π1 by βπ1 , and the natural
frequency
ππ = ππ/ππ = π2 β π1/πΏ /π
The natural frequency ππ can be made very low by adjusting
the compression of the spring with stiffness π1. At the same
time, the spring with stiffness π2 can be made stiff enough so
that the static displacement of the system is not excessive
Vibrations 3.69 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
- Ex.3.14 Equation of Motion for a Disk with An Extended Mass
Determine the governing equation of motion
and the natural frequency for the system
Solution
The velocity of π
π£π =π ππππ‘
=π
ππ‘π₯ + πΏπ πππ π + πΏ β πΏπππ π π
= βπ π + πΏ ππππ π π + πΏ ππ πππ π
Vibrations 3.70 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
The kinetic energy of the system π = ππ + ππ
ππ =1
2ππ π₯2 +
1
2π½πΊ π2
=1
2πππ 2 π2 +
1
2π½πΊ π2
ππ =1
2ππ£π
2
=1
2π βπ π + πΏ ππππ π π + πΏ ππ πππ π
2
=1
2π(π 2 + πΏ2 β 2πΏπ πππ π) π2
β1
2π πΏ β π 2 π2
βΉ π = ππ + ππ =1
2π πΏ β π 2 + πππ 2 + π½πΊ π2 β‘
1
2ππ
π2
Vibrations 3.71 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
The potential energy of the system
π =1
2ππ₯2 + ππ(πΏ β πΏπππ π)
=1
2ππ 2π2 + πππΏ 1 β πππ π
βΉ π =1
2ππ 2π2 +
1
2πππΏπ2 πππ π β 1 β
π2
2
=1
2ππ 2 + πππΏ π2 β‘
1
2πππ
2
The dissipation function
π· =1
2π π₯2 =
1
2ππ 2 π2 β‘
1
2ππ
π2
ππ π + ππ
π + πππ = 0,ππ =ππ
ππ=
ππ 2 + πππΏ
π(πΏ β π )2+πππ 2 + π½πΊ
Vibrations 3.72 Single DOF Systems: Governing Equations
The governing equation
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
13
Β§6.Lagrangeβs Equations
- Ex.3.15 Micro-Electromechanical System
Determine the governing equation of motion and the natural
frequency for the micro-electromechanical system
Solution
The potential energy
π =1
2ππ‘π
2 +1
2π π₯0 π‘ βπ₯1
2 +1
4π2π(πΏ2 βπΏ1)π
2
=1
2ππ‘π
2 +1
2π π₯0 π‘ βπΏ2π
2 +1
4π2π(πΏ2 βπΏ1)π
2
The kinetic energy
π =1
2π½0 π2 +
1
2π1 π₯1
2 =1
2π½0 +π1πΏ2
2 π2 β‘1
2ππ π2
Dissipation function
Vibrations 3.73 Single DOF Systems: Governing Equations
π· =1
2π π₯2
2 =1
2ππΏ1
2 π2 β‘1
2ππ π2
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
π =1
2ππ‘π
2 +1
2π π₯0 π‘ β πΏ2π
2 +1
4π2π(πΏ2 β πΏ1)π
2
The potential energy is not in the standard form βΉ the
governing equation must be derived from Lagrangeβs equationππ
ππ= ππ‘ + ππΏ2
2 +1
2π2π(πΏ2 β πΏ1) π β ππΏ2π₯0 π‘
= πππ β ππΏ2π₯0 π‘
The governing equation of motion
ππ π + ππ π + πππ = ππΏ2π₯0(π‘)
The natural frequency
ππ =ππ
ππ=
ππ‘ + ππΏ22 + π2π(πΏ2 β πΏ1)/2
π½0 + π1πΏ22
Vibrations 3.74 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
Ex.3.16 Slider Mechanism
Obtain the equation of motion of the slider mechanism
Solution
The geometric constraints on the motion
π2 π = π2 + π2 β 2πππππ π (a)
βΉ π π =ππ
π(π) ππ πππ
π π π πππ½ = ππ πππ (b)
π = π π πππ π½ + ππππ π (c)
βΉ π π πππ π½ β π π π½π πππ½ β π ππ πππ = 0
βΉ π½ = π π πππ π½ β π ππ πππ
π(π)π πππ½=
π
π2(π)πππππ π β π2
Vibrations 3.75 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
System Kinetic Energy
π =1
2π½ππ +π½ππ π2 +
1
2π½ππ
π½2 +1
2ππ π2 +
1
2ππ π
2 π½2
π½ππ =1
3πππ
2
π½ππ =1
3πππ
2
π½ππ =1
3πππ
2
π π β‘ π½ππ + π½ππ + π½ππ + ππ π2
πππππ π β π2
π2(π)
2
+ ππ
πππ πππ
π(π)
2
βΉ π =1
2π(π) π2
Vibrations 3.76 Single DOF Systems: Governing Equations
where
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
π
ππ‘
ππ
π π1β
ππ
ππ1+
ππ·
π π1+
ππ
ππ1= π1 (3.44)
Β§6.Lagrangeβs Equations
System Kinetic Energy
π =1
2π(π) π2
System Potential Energy
π =1
2ππ2(π) +
1
2ππ π π‘ β ππ 2
Equation of motion
π
ππ‘
ππ
π π=
π
ππ‘π(π) π = π(π) π,
ππ
ππ= πβ² π π2
ππ
ππ= ππ π πβ² π + πππ2π β πππ2π(π‘)
βΉ π π π +1
2πβ² π π2 + ππ π πβ² π + πππ2π = πππ2π(π‘)
Vibrations 3.77 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
- Ex.3.17 Oscillations of A Crankshaft
Obtain the equation of motion of the crankshaft
Solution
β’ Kinematics
The position vector of the
slider mass ππ
ππ = ππππ π + ππππ πΎ π + π π
The position vector of the center of mass πΊ of the crank
ππΊ = ππππ π + ππππ πΎ π + ππ πππ + ππ πππΎ π
From geometry
ππ πππ = π + ππ πππΎ
The slider velocity
π£π = ππ = βπ ππ πππ β π πΎπ πππΎ π = βπ π π πππ +π‘πππΎπππ π π
Vibrations 3.78 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
βΉ π ππππ π = π πΎπππ πΎ βΉ πΎ =π
π
πππ π
πππ πΎ π
2/7/2014
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Β§6.Lagrangeβs Equations
The velocity of the center of mass πΊ of the crank
π£πΊ = βπ ππ πππ β π πΎπ πππΎ π + π ππππ π β π πΎπππ πΎ π
βΉ π£πΊ = β π πππ +π
ππ‘πππΎπππ π π π π +
π
ππππ π π π π
β’ System Kinetic Energy
The total kinetic energy of the system
π =1
2π½π π2 +
1
2ππΊπ£πΊ
2 +1
2π½πΊ πΎ2 +
1
2πππ£π
2 β‘1
2π½(π) π2
π½ π = π½π + π2ππΊ π πππ +π
ππ‘πππΎπππ π
2
+π
ππππ π
2
+π½πΊπ
π
πππ π
πππ πΎ
2
+ π2ππ π πππ + π‘πππΎπππ π 2
Vibrations 3.79 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
where,
πΎ = π ππβ1π
ππ πππ β
π
π
Β§6.Lagrangeβs Equations
β’ Equation of Motion
The governing equation of motion has the form
π
ππ‘
ππ
π πβ
ππ
ππ= βπ(π‘)
After performing the differentiation operations
π½ π π +1
2
ππ½(π)
ππ π2 = βπ(π‘)
The angle π can be expressed
π π‘ = ππ‘ + π(π‘)
Then
π½ π π +1
2
ππ½(π)
πππ + π
2= βπ(π‘)
Vibrations 3.80 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
- Ex.3.18 Vibration of A Centrifugal Governor
Derive the equation of motion of
governor by usingLagrangeβsequation
Solution
The velocity vector relative to point
π of the left hand mass
ππ = βπΏ ππππ π π + πΏ ππ πππ π
+(π + πΏπ πππ)ππ
The kinetic energy
π π, π = 21
2π ππππ
= π βπΏ ππππ π 2 + πΏ ππ πππ 2 + π + πΏπ πππ π 2
= ππ2 π + πΏπ πππ 2 + π π2πΏ2
Vibrations 3.81 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
The potential energy with respect to the static equilibrium position
π π =1
2π 2πΏ 1βπππ π 2 β2πππΏπππ π
Using equation (3.44) with
π1 = π
π· = 0
π1 = 0
and performing the required
operations, to obtain the following
governing equation
ππΏ2 π β πππΏπ2πππ π β ππ2 + 2π πΏ2π ππππππ π
+πΏ ππ + 2ππΏ π πππ = 0
Vibrations 3.82 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
π
ππ‘
ππ
π π1β
ππ
ππ1+
ππ·
π π1+
ππ
ππ1= π1 (3.44)
Β§6.Lagrangeβs Equations
ππΏ2 π β πππΏπ2πππ π β ππ2 + 2π πΏ2π ππππππ π
+πΏ ππ + 2ππΏ π πππ = 0
Introducing the quantities
πΎ β‘π
πΏ, ππ
2 β‘π
πΏ, ππ
2 β‘2π
πRewrite the equation
π β πΎπ2πππ π
β π2 + ππ2 π ππππππ π
+ ππ2 + ππ
2 π πππ = 0
Assume that the oscillation π about π = 0 are small (πππ π β 1,π πππ β π) to get the final equation
π + ππ2 β π2 π = πΎπ2
Vibrations 3.83 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§6.Lagrangeβs Equations
- Ex.3.19 Oscillations of A Rotating System
Determine the change in the equilibrium position of
the wheel and the natural frequency of the system
about this equilibrium position
Solution
The spring force = the centrifugal force
ππΏ = π(π + πΏ)Ξ©2 βΉ πΏ =π
π1π2
Ξ©2 β 1
, π1π2 =
π
π
For small angles of rotation, the kinetic energy
π =1
2
1
2ππ2
π₯
π
2
+1
2π π₯2 =
1
2
3
2π π₯2
The potential energy for oscillations about the equilibrium
position
Vibrations 3.84 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
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Β§6.Lagrangeβs Equations
The potential energy for oscillations about the equilibrium position
π =1
2ππ₯2
The Lagrange equation for this undamped system
π
ππ‘
ππ
π π₯β
ππ
ππ₯+
ππ·
π π₯+
ππ
ππ₯= ππ₯ = ππ₯Ξ©2
where the centrifugal force ππ₯Ξ©2 is treated as an external force
The governing equation
3
2π π₯ + π β πΞ©2 π₯ = 0
The natural frequency
ππ =π
π=
2
3π1π
2 β Ξ©2
Vibrations 3.85 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Excercises
Vibrations 3.86 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien