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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
ยง1.Chapter Objectives
โข Obtain the governing equation of motion for single degree-of-
freedom (dof) translating and rotating systems by using force
balance and moment balance methods
โข Obtain the governing equation of motion for single dof
translating and rotating systems by using Lagrangeโs
equations
โข Determine the equivalent mass, equivalent stiffness, and
equivalent damping of a single dof system
โข 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
ยง2.Force-Balance and Moment-Balance Methods
1.Force Balance Method
Newtonian principle of linear momentum
๐น โ ๐ = 0 (3.1a)
๐น : the net external force vector acting on the system
๐ : the absolute linear momentum of the considered system
For a system of constant mass ๐ whose center of mass is
moving with absolute acceleration ๐, the rate of change of
linear momentum ๐ = ๐ ๐
๐น โ ๐ ๐ = 0 (3.1b)
โ๐ ๐ : inertial force
โนThe 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
ยง2.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 ๐ ๐ = ๐ ๐= (๐ฟ + ๐ฟ๐ ๐ก + ๐ฅ) ๐
Force balance along
the ๐ direction
๐ ๐ก ๐ + ๐๐ ๐ โ ๐๐ฅ + ๐๐ฟ๐ ๐ก ๐ โ ๐๐๐
๐๐ก ๐ โ ๐
๐2๐
๐๐ก2 ๐ = 0
Vibrations 3.04 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง2.Force-Balance and Moment-Balance Methods
- Noting that ๐ฟ and ๐ฟ๐ ๐ก are constants, rearranging terms to get
the following scalar differential equation
๐๐2๐ฅ
๐๐ก2 + ๐๐๐ฅ
๐๐ก+ ๐ ๐ฅ + ๐ฟ๐ ๐ก = ๐ ๐ก + ๐๐
Vibrations 3.05 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง2.Force-Balance and Moment-Balance Methods
Static Equilibrium Position
- The static-equilibrium position of a system is the position that
corresponds to the systemโs rest state; that is, a position with
zero velocity and zero acceleration
- The static-equilibrium position is the solution of
๐ ๐ฅ + ๐ฟ๐ ๐ก = ๐๐
- The static displacement
๐ฟ๐ ๐ก =๐๐
๐โน ๐ฅ = 0 is the static-equilibrium position of the system
- The spring has an unstretched length ๐ฟ, the static-equilibrium
position measured from the origin ๐ is given by
๐ฅ๐ ๐ก = ๐ฅ๐ ๐ก ๐ = (๐ฟ + ๐ฟ๐ ๐ก) ๐
Vibrations 3.06 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
2
ยง2.Force-Balance and Moment-Balance Methods
Equation of Motion for Oscillations about the Static-EquilibriumPosition
๐๐2๐ฅ
๐๐ก2 + ๐๐๐ฅ
๐๐ก+ ๐ ๐ฅ + ๐ฟ๐ ๐ก = ๐ ๐ก + ๐๐
๐ฟ๐ ๐ก =๐๐
๐
โน ๐๐2๐ฅ
๐๐ก2 + ๐๐๐ฅ
๐๐ก+ ๐๐ฅ = ๐ ๐ก
Equation (3.8) is the governing equation of motion of a single
dof system for oscillations about the static-equilibrium position
โข The left-hand side: the forces from the components that
comprise a single dof system
โข 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
ยง2.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
โข It is assumed that the mass moves without
friction
โข The unstretched length of the spring is ๐ฟ, and
a fixed point ๐ is located at the unstretched
position of the spring
โข The spring does not undergo any static
deflection and carrying out a force balance
along the ๐ direction
โข The static-equilibrium position ๐ฅ = 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
ยง2.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
๐น๐ = ๐๐ฟ๐ ๐ก + ๐๐ฅ + ๐๐๐ฅ
๐๐ก
If considering only the dynamic part of the
reaction force-that is, only those forces created
by the motion ๐ฅ(๐ก) from its static equilibrium
position, then
๐น๐ ๐ = ๐๐ฅ + ๐๐๐ฅ
๐๐ก
Vibrations 3.09 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง2.Force-Balance and Moment-Balance Methods
- Ex.3.1 Wind-drivenOscillationsaboutaSystemโsStatic-EquilibriumPosition
The wind flow across civil structures typically generates a
excitation force ๐(๐ก) on the structure that consists of a steady-
state part and a fluctuating part
๐ ๐ก = ๐๐ ๐ + ๐๐(๐ก)
๐๐ ๐ : the time-independent steady-state force
๐๐(๐ก) : the fluctuating time-dependent portion of the force
A single dof model of the vibrating structure
๐๐2๐ฅ
๐๐ก2 + ๐๐๐ฅ
๐๐ก+ ๐๐ฅ = ๐๐ ๐ + ๐๐ ๐ก โน ๐ฅ ๐ก = ๐ฅ0 + ๐ฅ๐(๐ก)
๐ฅ0 : the static equilibrium position, ๐ฅ0 = ๐๐ ๐ /๐
๐ฅ๐(๐ก) : motions about the static position
โน ๐๐2๐ฅ
๐๐ก2 + ๐๐๐ฅ
๐๐ก+ ๐๐ฅ = ๐๐ ๐ก
Vibrations 3.10 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง2.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
๐๐2๐ฅ
๐๐ก2 + ๐๐ฅ + ๐๐ฅ2 = 0
The restoring force of the eardrum has a component with a
quadratic nonlinearity
Static-Equilibrium Positions
Equilibrium positions ๐ฅ = ๐ฅ0 are solutions of the algebraic equation
๐ ๐ฅ0 + ๐ฅ02 = 0 โน ๐ฅ0 = 0, ๐ฅ0 = โ1
Vibrations 3.11 Single DOF Systems: Governing Equations
(๐)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง2.Force-Balance and Moment-Balance Methods
Linearization
Equilibrium positions ๐ฅ = ๐ฅ0 are solutions of the algebraic equation
๐ ๐ฅ0 + ๐ฅ02 = 0 โน ๐ฅ0 = 0, ๐ฅ0 = โ1
Subtitute ๐ฅ ๐ก = ๐ฅ0 + ๐ฅ(๐ก) into (a) with note that
๐ฅ2 ๐ก = ๐ฅ0 + ๐ฅ ๐ก2
โ ๐ฅ02 + 2๐ฅ0 ๐ฅ ๐ก + โฏ
๐2๐ฅ
๐๐ก2 =๐2 ๐ฅ0 + ๐ฅ ๐ก
๐๐ก2 =๐2 ๐ฅ
๐๐ก2
โน ๐๐2 ๐ฅ
๐๐ก2 + ๐ ๐ฅ0 + ๐ฅ(๐ก) + ๐ ๐ฅ02 + 2๐ฅ0 ๐ฅ ๐ก = 0
๐ฅ0 = 0 โน ๐๐2 ๐ฅ
๐๐ก2 + ๐ ๐ฅ(๐ก) = 0
๐ฅ0 = โ1 โน ๐๐2 ๐ฅ
๐๐ก2 โ ๐ ๐ฅ(๐ก) = 0
โน 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
ยง2.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
๐ป = ๐ฝ๐บ ๐๐
โน ๐ = ๐ฝ๐บ ๐๐
Vibrations 3.13 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง2.Force-Balance and Moment-Balance Methods
The governing equation of motion
๐ ๐ก ๐ โ ๐๐ก ๐๐ โ ๐๐ก
๐๐
๐๐ก๐ โ ๐ฝ๐บ
๐2๐
๐๐ก2 = 0
โน ๐ฝ๐บ๐2๐
๐๐ก2 + ๐๐ก
๐๐
๐๐ก+ ๐๐ก๐ = ๐ ๐ก
Vibrations 3.14 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง2.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
โข Translational motion
๐๐2๐ฅ
๐๐ก2 + ๐๐๐ฅ
๐๐ก+ ๐๐ฅ = ๐ ๐ก
โข Rotational motion
๐ฝ๐บ๐2๐
๐๐ก2 + ๐๐ก
๐๐
๐๐ก+ ๐๐ก๐ = ๐ ๐ก
Vibrations 3.15 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง2.Force-Balance and Moment-Balance Methods
Ex.3.3 Hand Biomechanics
The moment balance about
point ๐
๐ โ ๐ฝ0 ๐๐ = 0
๐ฝ0: the rotary inertia of the
forearm and the object
held in the hand
The net moment ๐ acting
about the point ๐ due to gravity loading and the forces due to
the biceps and triceps
๐ = โ๐๐๐๐๐๐ ๐๐ โ ๐๐๐
2๐๐๐ ๐๐ + ๐น๐๐๐ โ ๐น๐ก๐๐
โน โ๐๐๐๐๐๐ ๐๐ โ ๐๐๐
2๐๐๐ ๐๐ + ๐น๐๐๐ โ ๐ฝ0 ๐๐ = 0
Vibrations 3.16 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง2.Force-Balance and Moment-Balance Methods
โ๐๐๐๐๐๐ ๐๐ โ ๐๐๐
2๐๐๐ ๐๐ + ๐น๐๐๐ โ ๐ฝ0 ๐๐ = 0
Note that: ๐น๐ = โ๐๐๐, ๐น๐ก = ๐พ๐ก๐ฃ = ๐พ๐ก๐ ๐, ๐น0 = ๐๐2/3 + ๐๐2
โน ๐ +๐
3๐2 ๐ + ๐พ๐ก๐
2 ๐ + ๐๐๐๐ + ๐ +๐
2๐๐๐๐๐ ๐ = 0
Static-Equilibrium Position
The equilibrium position ๐ = ๐0 is a solution of the
transcendental equation
๐๐๐๐0 + ๐ +๐
2๐๐๐๐๐ ๐0 = 0
Vibrations 3.17 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง2.Force-Balance and Moment-Balance Methods
Linear System Governing โSmallโ Oscillations about the Static-
Equilibrium Position
Consider oscillations about the static-equilibrium position and
expand the angular variable ๐ ๐ก = ๐0 + ๐ ๐ก with note that
๐๐๐ ๐ = cos ๐0 + ๐ โ ๐๐๐ ๐0 โ ๐๐ ๐๐๐0 + โฏ
๐๐(๐ก)
๐๐ก=
๐(๐0 + ๐)
๐๐ก= ๐(๐ก)
๐2๐(๐ก)
๐๐ก2 =๐2(๐0 + ๐)
๐๐ก2 = ๐(๐ก)
โน ๐ +๐
3๐2 ๐ + ๐พ๐ก๐
2 ๐ + ๐๐ ๐ = 0
where
๐๐ = ๐๐๐ โ ๐ +๐
2๐๐๐ ๐๐๐0
Vibrations 3.18 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
4
ยง3.Natural Frequency and Damping Factor
1.Natural Frequency
Translation Vibrations: Natural Frequency
๐๐ = 2๐๐๐ =๐
๐(๐๐๐/๐ )
๐ : the stiffness of the system, ๐/๐
๐ : the system mass, ๐๐
๐๐ : the natural frequency, ๐ป๐ง
For the mass-damper-spring system
๐๐ = 2๐๐๐ =๐
๐ฟ๐ ๐ก(๐๐๐/๐ )
๐ฟ๐ ๐ก: the static deflection of the system, ๐
Vibrations 3.19 Single DOF Systems: Governing Equations
(3.15)
(3.14)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง3.Natural Frequency and Damping Factor
Rotational Vibrations: Natural Frequency
๐๐ = 2๐๐๐ =๐๐ก
๐ฝ(๐๐๐/๐ )
๐๐ก : the torsion stiffness of the system, ๐๐/๐๐๐
๐ฝ : the system mass, ๐๐๐/๐ 2
๐๐ : the natural frequency, ๐ป๐ง
Period of Undamped Free Oscillations
For an unforced and undamped system, the period of free
oscillation of the system is given by
๐ =1
๐๐=
2๐
๐๐
Vibrations 3.20 Single DOF Systems: Governing Equations
(3.16)
(3.17)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
๐๐ = 2๐๐๐ =๐
๐ฟ๐ ๐ก(๐๐๐/๐ ) (3.15)
ยง3.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(๐๐). Determine the natural frequency for vertical vibrations
Solution
๐๐1 =1
2๐
๐
๐ฟ๐ ๐ก1=
1
2๐
9.81
0.1 ร 10โ3 = 49.85๐ป๐ง
๐๐2 =1
2๐
๐
๐ฟ๐ ๐ก2=
1
2๐
9.81
1 ร 10โ3 = 15.76๐ป๐ง
๐๐3 =1
2๐
๐
๐ฟ๐ ๐ก3=
1
2๐
9.81
10 ร 10โ3 = 4.98๐ป๐ง
Vibrations 3.21 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง3.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 100๐๐ mass standing upright. The tibia
has a length of 40๐๐, and it is modeled as a hollow tube with an
inner diameter of 2.4๐๐ and an outer diameter of 3.4๐๐. The
Youngโs modulus of elasticity of the bone material is 2 ร1010๐/๐2. 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
๐๐ = 100/2 ร 9.81 = 490.5๐
Vibrations 3.22 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
๐๐ = 2๐๐๐ =๐
๐ฟ๐ ๐ก(๐๐๐/๐ ) (3.15)
ยง3.Natural Frequency and Damping Factor
The stiffness of the tibia
๐ =๐ด๐ธ
๐ฟ=
1 ร 1010 ร๐4
3.4 ร 10โ2 2 โ 2.4 ร 10โ2 2
40 ร 10โ2
= 22.78 ร 106๐/๐2
The static deflection
๐ฟ๐ ๐ก =๐๐
๐=
490.5
22.78 ร 106 = 21.53 ร 10โ6๐
The natural frequency
๐๐ =1
2๐
๐
๐ฟ๐ ๐ก=
1
2๐
9.81
21.53 ร 10โ6 = 107.4๐ป๐ง
Vibrations 3.23 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง3.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
๐ = ๐ด๐
๐ด: a constant, ๐ = ๐๐: the weight, ๐: the gravitational constant
The natural frequency
๐๐ =1
2๐
๐
๐=
1
2๐
๐๐
๐=
1
2๐๐ด๐๐ป๐ง
โน ๐๐ 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
ยง3.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 ๐ = ๐ด๐
For illustrative purposes, consider a spring that has the
general force-displacement relationship
๐น ๐ฅ = ๐๐ฅ
๐
๐
๐, ๐: scale factors, ๐: shape factor
The static deflection
๐ฅ0 = ๐๐
๐
1/๐
Vibrations 3.25 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง3.Natural Frequency and Damping Factor
For โsmallโ amplitude vibrations about ๐ฅ0, the linear equivalent
stiffness of this spring is determined
๐๐๐ = ๐๐น(๐ฅ)
๐๐ฅ๐ฅ=๐ฅ0
=๐๐
๐
๐ฅ๐
๐
๐โ1
=๐๐
๐
๐
๐
๐โ1๐
The natural frequency of this system
๐๐ =1
2๐
๐๐๐
๐/๐
=1
2๐
๐๐
๐
๐
๐
โ1/๐
=1
2๐
๐๐
๐
๐
๐
โ1/2๐
๐ป๐ง
Vibrations 3.26 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง3.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
๐ = 2500๐, ๐ = 0.011๐, ๐ = 2.77
โน ๐๐ =1
2๐
๐๐
๐
๐
๐
โ1/2๐
= 32.4747๐โ1/3.54๐ป๐ง
Plot ๐๐(๐)
Vibrations 3.27 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง3.Natural Frequency and Damping Factor
Representative Spring Data
From the figure of ๐๐(๐)
โข over a sizable portion of the load
range, the natural frequency of the
system varies within the range of 8.8%
โข The natural frequency of a system with
a linear spring whose static
displacement ranges from 12 รท 5๐๐varies approximately from 4.5 รท 7.0๐ป๐งor approximately 22% about a
frequency of 5.8๐ป๐ง
1
2๐
9.8
0.012โ 4.5๐ป๐ง,
1
2๐
9.8
0.005โ 7๐ป๐ง
of 5.8 Hz
Vibrations 3.28 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง3.Natural Frequency and Damping Factor
2.Damping Factor
Translation Vibrations: Damping Factor
For translating single dof systems, the damping factor or
damping ratio ๐ is defined as
๐ =๐
2๐๐๐=
๐
2 ๐๐=
๐๐๐
2๐
๐: the system damping coefficient, ๐๐ /๐
๐: the system stiffness, ๐/๐
๐: the system mass, ๐๐
Critical Damping, Underdamping, and Overdamping
Defining the critical damping ๐๐
๐๐ = 2๐๐๐ = 2 ๐๐, ๐ = ๐/๐๐ (3.19)
0 < ๐ < 1: underdamped,๐ > 1: overdamped,๐ = 1: criticallydamped
Vibrations 3.29 Single DOF Systems: Governing Equations
(3.18)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง3.Natural Frequency and Damping Factor
Rotational Vibrations: Damping Factor
For rotating single dof systems, the damping factor or damping
ratio ๐ is defined as
๐ =๐๐ก
2๐ฝ๐๐=
๐๐ก
2 ๐๐ก๐ฝ
๐๐ก: the system damping coefficient, ๐๐๐ /๐๐๐
๐๐ก: the system stiffness, ๐๐/๐๐๐
๐ฝ: the system moment of inertia, ๐๐๐2
Vibrations 3.30 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
6
ยง3.Natural Frequency and Damping Factor
Governing Equation of Motion in Terms of Natural Frequency
and Damping Factor
Rewriting the equation of motion
๐2๐ฅ
๐๐ก2 + 2๐๐๐
๐๐ฅ
๐๐ก+ ๐๐
2๐ฅ =๐(๐ก)
๐If we introduce the dimensionless time ๐ = ๐๐๐ก , then the
equation can be written
๐2๐ฅ
๐๐2 + 2๐๐๐ฅ
๐๐+ ๐ฅ =
๐(๐)
๐
Vibrations 3.31 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง3.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 ๐ = 1. Due to a design change, the
mass of the system is increased 20% - that is, from ๐ to 1.2๐.
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
๐๐๐๐ค =๐
2 ๐(1.2๐)= 0.91
๐
2 ๐๐= 0.91
๐
๐๐= 0.91
โน 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
ยง3.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 ๐ ,
damping ๐, and spring constant ๐ has too much static deflection
๐ฟ๐ ๐ก. The engineer would like to decrease ๐ฟ๐ ๐ก by a factor of 2,
while keeping the damping ratio constant. Determine the
different options
Solution
The problem involves vertical vibrations
๐ฟ๐ ๐ก =๐๐
๐
2๐ =๐
๐
๐ฟ๐ ๐ก
๐= ๐
๐ฟ๐ ๐ก
๐๐2 =1
๐
๐2๐ฟ๐ ๐ก
๐
โน 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
ยง3.Natural Frequency and Damping Factor
First choice
Let ๐ remain constant, reduce ๐ฟ๐ ๐ก by one-half
๐ฟ๐ ๐ก =๐๐
๐
๐ฟ๐ ๐กโฒ =
๐ฟ๐ ๐ก
2=
๐๐
2๐=
๐โฒ๐
๐โฒComparing (a) and (b)
๐โฒ๐
๐โฒ=
๐๐
2๐=
๐/ 2 ๐
๐ 2โน ๐ โ ๐โฒ =
๐
2, ๐ โ ๐โฒ = ๐ 2
Check the damping ratio
2๐โฒ = ๐๐ฟโฒ
๐ ๐ก
๐๐โฒ2 = ๐๐ฟ๐ ๐ก
2๐ ๐/ 22 = ๐
๐ฟ๐ ๐ก
๐๐2 = 2๐
Vibrations 3.34 Single DOF Systems: Governing Equations
Before (a)
After (b)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง3.Natural Frequency and Damping Factor
Second choice
Let ๐ remain constant, reduce ๐ฟ๐ ๐ก by one-half
2๐ = ๐๐ฟ๐ ๐ก
๐๐2 =1
๐
๐2๐ฟ๐ ๐ก
๐
2๐โฒ =1
๐
๐โฒ2๐ฟ๐ ๐กโฒ
๐=
1
๐
๐โฒ2๐ฟ๐ ๐ก
2๐
Comparing (c) and (d)
๐โฒ2
2= ๐2 โน ๐ โ ๐โฒ = ๐ 2
The static deflection
๐ฟ๐ ๐กโฒ =
๐๐
๐โฒ=
๐ฟ๐ ๐ก
2=
๐๐
2๐โน ๐ โ ๐โฒ = 2๐
Vibrations 3.35 Single DOF Systems: Governing Equations
Before (c)
After (d)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง3.Natural Frequency and Damping Factor
Third choice
Let ๐ remain constant, reduce ๐ฟ๐ ๐ก by one-half
๐ฟ๐ ๐ก =๐๐
๐
๐ฟ๐ ๐กโฒ =
๐ฟ๐ ๐ก
2=
๐๐
2๐=
๐โฒ๐
๐Comparing (e) and (f)
๐โฒ =๐
2โน ๐ โ ๐โฒ =
๐
2The constant damping ratio
2๐โฒ = ๐โฒ๐ฟโฒ
๐ ๐ก
๐๐โฒ2 = ๐โฒ๐ฟ๐ ๐ก
2๐ ๐/2 2 = ๐โฒ2๐ฟ๐ ๐ก
๐๐2 = ๐๐ฟ๐ ๐ก
๐๐2 = 2๐
โน ๐ โ ๐โฒ = ๐ 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
๐น ๐ฅ = ๐๐๐๐ ๐๐( ๐ฅ) (2.52)
๐๐2๐ฅ
๐๐ก2 + ๐๐๐ฅ
๐๐ก+ ๐๐ฅ = ๐ ๐ก (3.8)
ยง4.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
๐๐2๐ฅ
๐๐ก2 + ๐๐๐ฅ
๐๐ก+ ๐๐๐๐ ๐๐( ๐ฅ) = ๐(๐ก)
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
๐๐๐๐๐๐๐๐๐ ๐๐๐ฆ ๐๐๐๐๐ก๐๐๐ ๐๐๐๐๐
๐น ๐ฅ = ๐๐ ๐ฅ2๐ ๐๐ ๐ฅ = ๐๐| ๐ฅ| ๐ฅ (2.54)
๐น = ๐๐๐ฝโ๐ ๐๐ ๐ฅ |๐ฅ| (2.57)
๐๐2๐ฅ
๐๐ก2 + ๐๐๐ฅ
๐๐ก+ ๐๐ฅ = ๐ ๐ก (3.8)
ยง4.Governing Equations for Different Type of Damping
Fluid Damping
Using Eq. (2.54) and Eq. (3.8), the governing equation of motion
๐๐2๐ฅ
๐๐ก2 + ๐๐| ๐ฅ| ๐ฅ + ๐๐ฅ = ๐(๐ก)
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
๐๐2๐ฅ
๐๐ก2 + ๐๐๐ฝโ๐ ๐๐ ๐ฅ |๐ฅ| + ๐๐ฅ = ๐(๐ก)
Vibrations 3.38 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
๐๐๐๐๐๐๐๐๐ ๐๐๐ข๐๐ ๐๐๐๐๐๐๐ ๐๐๐๐๐
ยง5.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
โข buildings subjected to earthquakes
โข packaging during transportation
โข vehicle response, and
โข designing accelerometers
- The physical system of interest is represented by a single dof
system whose base is subjected to a displacement
disturbance ๐ฆ(๐ก), and an equation governing the motion of
this system is sought to determine the response of the
system ๐ฅ(๐ก)
Vibrations 3.39 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง5.Governing Equations for Different Type of Applied Forces
- A prototype of a single dof system subjected to a base excitation
โข The vehicle provides the base excitation ๐ฆ(๐ก) to the
instrumentation package modeled as a single dof
โข The displacement response ๐ฅ(๐ก) is measured from the
systemโs static-equilibrium position
Assume that no external force is applied directly to the mass;
that is, ๐ ๐ก = 0
Vibrations 3.40 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง5.Governing Equations for Different Type of Applied Forces
- The following governing equation of motion
๐๐2๐ฅ
๐๐ก2 + ๐๐๐ฅ
๐๐ก+ ๐๐ฅ = ๐
๐๐ฆ
๐๐ก+ ๐๐ฆ
โน ๐๐2๐ฅ
๐๐ก2 + 2๐๐๐
๐๐ฅ
๐๐ก+ ๐๐
2๐ฅ = 2๐๐๐
๐๐ฆ
๐๐ก+ ๐๐
2๐ฆ
๐ฆ(๐ก) and ๐ฅ(๐ก) are measured from a fixed point ๐ located in an
inertial reference frame and a fixed point located at the
systemโs static equilibrium position, respectively
Vibrations 3.41 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง5.Governing Equations for Different Type of Applied Forces
- If the relative displacement is desired, the governing equation
of motion
๐๐2๐ง
๐๐ก2 + ๐๐๐ง
๐๐ก+ ๐๐ง = โ๐
๐2๐ฆ
๐๐ก2
with ๐ง ๐ก โก ๐ฅ ๐ก โ ๐ฆ(๐ก)
โน๐2๐ง
๐๐ก2 + 2๐๐๐
๐๐ง
๐๐ก+ ๐๐
2๐ง = โ๐2๐ฆ
๐๐ก2
Vibrations 3.42 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
8
ยง5.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
systemโs mass. This force, in turn, is transmitted through the
spring and damper to the fixed base
- The unbalance is modeled as a mass ๐0 that rotates with an
angular speed ๐, and this mass is located a fixed distance ๐from the center of rotation
Vibrations 3.43 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง5.Governing Equations for Different Type of Applied Forces
- From the free-body diagram (FBD) of the unbalanced mass ๐0
๐๐ฅ = โ๐0( ๐ฅ โ ๐๐2๐ ๐๐๐๐ก)
๐๐ฆ = ๐0๐๐2๐๐๐ ๐๐ก
- From the FBD of mas ๐
๐๐2๐ฅ
๐๐ก2 + ๐๐๐ฅ
๐๐ก+ ๐๐ฅ = ๐๐ฅ
โน (๐ + ๐0)๐2๐ฅ
๐๐ก2 + ๐๐๐ฅ
๐๐ก+ ๐๐ฅ = ๐0๐๐
2๐ ๐๐๐๐ก
โน๐2๐ฅ
๐๐ก2 + 2๐๐๐
๐๐ฅ
๐๐ก+ ๐๐
2๐ฅ =๐น(๐)
๐๐ ๐๐๐๐ก
where ๐ = ๐ + ๐0, ๐๐ = ๐/๐, ๐น ๐ = ๐0๐๐2
- The static displacement of the spring
๐ฟ๐ ๐ก =๐ + ๐0 ๐
๐=
๐๐
๐
Vibrations 3.44 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง5.Governing Equations for Different Type of Applied Forces
3.System with Added Mass Due to a Fluid
- The equation of motion of the system
๐๐2๐ฅ
๐๐ก2 + ๐๐ฅ = ๐ ๐ก + ๐1(๐ก)
๐ฅ(๐ก) : measured from the unstretched position of the spring
๐(๐ก) : the externally applied force
๐1(๐ก) : 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
ยง5.Governing Equations for Different Type of Applied Forces
- The force generated by the fluid on the rigid body
๐1 ๐ก = โ๐พ0๐๐2๐ฅ
๐๐ก2 โ ๐ถ๐
๐๐ฅ
๐๐ก
๐ : the mass of the fluid displaced by the body
๐พ0 : an added mass coefficient
๐ถ๐ : a positive fluid damping coefficient
- The governing equation of motion
๐ + ๐พ0๐๐2๐ฅ
๐๐ก2 + ๐ถ๐
๐๐ฅ
๐๐ก+ ๐๐ฅ = ๐ ๐ก
๐พ0๐ : 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
ยง6.Lagrangeโs Equations
Consider a system with ๐ degrees of freedom that is described
by a set of ๐ generalized coordinates ๐๐ , ๐ = 1,2,โฆ๐. In terms
of the chosen generalized coordinates, Lagrangeโs equations
have the form
๐
๐๐ก
๐๐
๐ ๐๐โ
๐๐
๐๐๐+
๐๐ท
๐ ๐๐+
๐๐
๐๐๐= ๐๐ , ๐ = 1,2,โฆ , ๐
๐๐ : generalized coordinate
๐๐ : generalized velocity
๐ : the kinetic energy of the system
๐ : the potential energy of the system
๐ท : the Rayleigh dissipation function
๐๐ : the generalized force that appears in the ๐๐กโ equation
Vibrations 3.47 Single DOF Systems: Governing Equations
(3.41)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs Equations
The generalized force ๐๐ that appears in the ๐๐กโ equation
๐๐ =
๐
๐น๐
๐ ๐๐๐๐๐
+
๐
๐๐
๐๐๐
๐ ๐๐
๐น๐, ๐๐ : the vector representations of the externally
applied forces and moments on the ๐๐กโ body
๐๐ : the position vector to the location where the force
๐น๐ is applied
๐๐ : the ๐๐กโ bodyโs 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
ยง6.Lagrangeโs Equations
Linear Vibratory Systems
For vibratory systems with linear characteristics
๐ =1
2
๐=1
๐
๐=1
๐
๐๐๐ ๐๐ ๐๐
๐ =1
2
๐=1
๐
๐=1
๐
๐๐๐๐๐๐๐
๐ท =1
2
๐=1
๐
๐=1
๐
๐๐๐ ๐๐ ๐๐
๐๐๐ : the inertia coefficients
๐๐๐ : the stiffness coefficients
๐๐๐ : the damping coefficients
Vibrations 3.49 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs Equations
Single Degree-Of-Freedom
The case of a single degree-of-freedom system, ๐ = 1, the
Lagrangeโs equation
๐
๐๐ก
๐๐
๐ ๐1โ
๐๐
๐๐1+
๐๐ท
๐ ๐1+
๐๐
๐๐1= ๐1
where the generalized force is obtained from
๐1 =
๐
๐น๐
๐ ๐๐๐๐1
+
๐
๐๐
๐๐๐
๐ ๐1
Vibrations 3.50 Single DOF Systems: Governing Equations
(3.44)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs 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
๐ =1
2
๐=1
1
๐=1
1
๐๐๐ ๐๐ ๐๐ =1
2๐11 ๐1
2 โก1
2๐๐ ๐1
2
๐ =1
2
๐=1
1
๐=1
1
๐๐๐๐๐๐๐ =1
2๐11๐1
2 โก1
2๐๐๐1
2
๐ท =1
2
๐=1
1
๐=1
1
๐๐๐ ๐๐ ๐๐ =1
2๐11 ๐1
2 โก1
2๐๐ ๐1
2
๐๐, ๐๐, ๐๐ : the equivalent mass, stiffness, and damping
From Lagrangeโs equation
๐๐ ๐1 + ๐๐ ๐1 + ๐๐๐1 = ๐1
Vibrations 3.51 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
(3.46)
ยง6.Lagrangeโs Equations
To obtain the governing equation of motion of a linear vibrating
system with viscous damping
โข Obtains expressions for the system kinetic energy ๐ ,
system potential energy ๐, and system dissipation function ๐ท
โข Identify the equivalent mass ๐๐, equivalent stiffness ๐๐,
and equivalent damping ๐๐
โข Determine the generalized force
โข Apply the governing equation
๐๐ ๐1 + ๐๐ ๐1 + ๐๐๐1 = ๐1
โข Determine the system natural frequency
๐๐ =๐๐
๐๐, ๐ =
๐๐
2๐๐๐๐=
๐๐
2 ๐๐๐๐
Vibrations 3.52 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs 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
๐1 = ๐ฅ, ๐น๐ = ๐(๐ก) ๐, ๐๐ = ๐ฅ ๐, ๐๐ = 0
Determine the generalized force
๐1 =
๐
๐น๐
๐ ๐๐๐๐1
+ 0 = ๐ ๐ก ๐๐๐ฅ ๐
๐๐ฅ= ๐(๐ก)
The system kinetic energy ๐, system potential energy ๐, and
system dissipation function ๐ท
๐ =1
2๐ ๐ฅ2, ๐ =
1
2๐๐ฅ2, ๐ท =
1
2๐ ๐ฅ2
Vibrations 3.53 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs Equations
Identify the following
๐1 = ๐ฅ, ๐น๐ = ๐(๐ก) ๐, ๐๐ = ๐ฅ ๐, ๐๐ = 0
Determine the generalized force
๐1 =
๐
๐น๐
๐ ๐๐๐๐1
+ 0 = ๐ ๐ก ๐๐๐ฅ ๐
๐๐ฅ= ๐(๐ก)
The system kinetic energy ๐, system potential energy
๐, and system dissipation function ๐ท
๐ =1
2๐ ๐ฅ2, ๐ =
1
2๐๐ฅ2, ๐ท =
1
2๐ ๐ฅ2
โน ๐๐ = ๐, ๐๐ = ๐, ๐๐ = ๐
The governing equation
๐๐2๐ฅ
๐๐ก2 + ๐๐๐ฆ
๐๐ก+ ๐๐ฅ = ๐(๐ก)
Vibrations 3.54 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
10
ยง6.Lagrangeโs Equations
- Ex.3.10 Motion of A System that Translates and Rotates
Obtain the governing equation of motion for โsmallโ oscillations
about the upright position
Solution
Choose the generalized coordinate
๐1 = ๐, ๐น๐ = 0, ๐๐ = ๐ ๐ก ๐, ๐๐ = ๐๐
The generalized force
๐1 =
๐
๐๐ โ๐๐๐
๐ ๐1= ๐ ๐ก ๐ โ
๐ ๐๐
๐ ๐= ๐(๐ก)
Vibrations 3.55 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
๐ฝ๐บ =1
2๐๐2
ยง6.Lagrangeโs Equations
The potential energy
๐ =1
2๐๐ฅ2 =
1
2๐(๐๐)2=
1
2๐๐2๐2
โน the equivalent stiffness
The kinetic energy of the system
๐ =1
2๐ ๐ฅ2 +
1
2๐ฝ๐บ ๐2
โน ๐ =1
2๐๐2 + ๐ฝ๐บ ๐2 =
1
2
3
2๐๐2 ๐2
โน 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
๐๐๐ก๐๐ก๐๐๐๐๐๐๐๐๐๐ก๐๐ ๐๐๐๐๐๐ฆ
๐ก๐๐๐๐ ๐๐๐ก๐๐๐๐๐๐๐๐ก๐๐ ๐๐๐๐๐๐ฆ
๐๐ = ๐๐2
๐๐ =3
2๐๐2
ยง6.Lagrangeโs Equations
The dissipation function
๐ท =1
2๐ ๐ฅ2 =
1
2๐(๐ ๐)2=
1
2(๐๐2) ๐2
โน the equivalent damping coefficient
๐๐ = ๐๐2
The governing equation of motion3
2๐๐2 ๐ + ๐๐2 ๐ + ๐๐2๐ = ๐(๐ก)
Natural frequency and damping factor
๐๐ =๐๐
๐๐=
๐๐2
3๐๐2/2=
2๐
3๐
๐ =๐๐
2๐๐๐๐=
๐๐2
2(3๐๐2/2) 2๐/3๐=
6
6๐๐
Vibrations 3.57 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs Equations
- Ex.3.11 Inverted Pendulum
Obtain the governing equation of motion for โsmallโ oscillations
about the upright position
Solution
The total rotary inertia of the system
๐ฝ๐ = ๐ฝ๐1+ ๐ฝ๐2
๐ฝ๐1: mass momentof inertia of ๐1 about point๐
๐ฝ๐2: massmomentof inertiaof thebaraboutpoint๐
๐ฝ๐1=
2
5๐1๐
2 + ๐1๐ฟ12
๐ฝ๐2=
1
12๐2๐ฟ2
2 + ๐2
๐ฟ2
2
2
=1
3๐2๐ฟ2
2
Vibrations 3.58 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs Equations
Choosing ๐1 = ๐ as the generalized coordinate, the system
kinetic energy takes the form
๐ =1
2๐ฝ๐ ๐2 =
1
2๐ฝ๐1
+ ๐ฝ๐2 ๐2
=1
2
2
5๐1๐
2 + ๐1๐ฟ12 +
1
3๐2๐ฟ2
2 ๐2
For small ๐ โน ๐ฅ1 โ ๐ฟ1๐
The system potential energy
๐ =1
2๐๐ฅ1
2 โ1
2๐1๐๐ฟ1๐
2 โ1
2๐2๐
๐ฟ2
2๐2
=1
2๐๐ฟ1
2 โ ๐1๐๐ฟ1 โ ๐2๐๐ฟ2
2๐2
๐ท =1
2๐ ๐ฅ1
2 =1
2๐๐ฟ1
2 ๐2
Vibrations 3.59 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
The dissipation function
ยง6.Lagrangeโs Equations
The equivalent inertia, the equivalent stiffness, and the
equivalent damping properties of the system
๐ =1
2
2
5๐1๐
2 +๐1๐ฟ12 +
1
3๐2๐ฟ2
2 ๐2
๐ =1
2๐๐ฟ1
2 โ ๐1๐๐ฟ1 โ ๐2๐๐ฟ2
2๐2
๐ท =1
2๐ ๐ฅ1
2 =1
2๐๐ฟ1
2 ๐2
The governing equation of motion ๐๐ ๐ + ๐๐
๐ + ๐๐๐ = 0
Natural frequency
๐๐ =๐๐
๐๐=
๐๐ฟ12 โ ๐1๐๐ฟ1 โ ๐2๐๐ฟ2/2
๐ฝ๐1+ ๐ฝ๐2
๐๐ 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
โน๐๐ =2
5๐1๐
2 +๐1๐ฟ12 +
1
3๐2๐ฟ2
2
โน๐๐ =๐๐ฟ12 โ๐1๐๐ฟ1 โ๐2๐
๐ฟ2
2
โน ๐๐ = ๐๐ฟ12
2/7/2014
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ยง6.Lagrangeโs Equations
โข Natural Frequency of Pendulum System
Now locate the pivot point ๐ on the top, the
equivalent stiffness of this system
๐๐ = ๐๐ฟ12 + ๐1๐๐ฟ1 + ๐2๐
๐ฟ2
2and the natural frequency of this system
๐๐ =๐๐
๐๐=
๐๐ฟ12 + ๐1๐๐ฟ1 + ๐2๐๐ฟ2/2
๐ฝ๐1+ ๐ฝ๐2
If ๐2 โช ๐1, ๐ โช ๐ฟ1, and ๐ = 0, then
๐๐ =๐1๐๐ฟ1 1 + ๐2๐ฟ2/๐1๐ฟ1
๐1๐ฟ12 1 + 2๐2/5๐ฟ1
2 โ๐
๐ฟ
โ the natural frequency of a pendulum composed of a rigid,
weightless rod carrying a mass a distance ๐ฟ1 from its pivot
Vibrations 3.61 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs 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 ๐ to the
center of mass ๐บ
๐ = โ๐ ๐ + ๐๐ ๐๐๐ ๐ + (๐ โ ๐๐๐๐ ๐) ๐
The absolute velocity of the center of mass ๐ = โ ๐ โ ๐๐๐๐ ๐ ๐ ๐ + ๐๐ ๐๐๐ ๐ ๐
Selecting the generalized coordinate ๐1 = ๐ ,
the system kinetic energy
๐ =1
2๐ฝ๐บ ๐2 +
1
2๐ ๐ โ ๐
โน ๐ =1
2๐ฝ๐บ ๐2 +
1
2๐ ๐ 2 + ๐2 โ 2๐๐ ๐๐๐ ๐ ๐2
Vibrations 3.62 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Taylor series expansion
๐๐๐ ๐ = ๐๐๐ ๐0 + ๐ โ ๐๐๐ ๐0 โ ๐๐ ๐๐๐0 โ1
2 ๐2๐๐๐ ๐0 + โฏ
๐ ๐๐๐ = ๐ ๐๐ ๐0 + ๐ โ ๐ ๐๐๐0 โ ๐๐๐๐ ๐0 โ1
2 ๐2๐ ๐๐๐0 + โฏ
ยง6.Lagrangeโs Equations
Choosing the fixed ground as the datum, the system potential
energy
๐ = ๐๐ ๐ โ ๐๐๐๐ ๐
Small Oscillations about the Upright Position
Express the angular displacement as
๐(๐ก) = ๐0 + ๐(๐ก)
Since ๐0 = 0 , and small ๐ , using ๐ ๐๐๐ โ ๐ ,
๐๐๐ ๐ โ 1 โ1
2 ๐2, rewrite the energy functions
๐ โ1
2๐ฝ๐บ + ๐ ๐ โ ๐ 2 ๐2, ๐ โ ๐๐ ๐ โ ๐ +
1
2๐๐๐ ๐2
Vibrations 3.63 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs Equations
The equivalent inertia of the system
๐ โ1
2๐ฝ๐บ + ๐ ๐ โ ๐ 2 ๐2
The potential energy is not in standard form because of the
constant term ๐๐ ๐ โ ๐
๐ โ ๐๐ ๐ โ ๐ +1
2๐๐๐ ๐2
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 (๐ โ ๐) above the ground
๐ =1
2๐๐๐ ๐2
Then, the equivalent stiffness can be defined
๐๐ = ๐๐๐
Vibrations 3.64 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
โน ๐๐ = ๐ฝ๐บ + ๐ ๐ โ ๐ 2
ยง6.Lagrangeโs Equations
The governing equation
๐ฝ๐บ + ๐ ๐ โ ๐ 2 ๐ + ๐๐๐ ๐ = 0
Natural Frequency
๐๐ =๐๐
๐๐
=๐๐๐
๐ฝ๐บ + ๐ ๐ โ ๐ 2
=๐
๐ฝ๐บ + ๐ ๐ โ ๐ 2
๐๐
Vibrations 3.65 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs Equations
- Ex.3.13 Translating System with a Pre-tensioned/compressedSpring
Derive the governing equation of motion for vertical
translations ๐ฅ of the mass about the static
equilibrium position of the system
Solution
The equation of motion will be derived for
โsmallโ amplitude vertical oscillations; that is,
๐ฅ/๐ฟ โช 1
The horizontal spring is pretensioned with a tension, which is
produced by an initial extension of the spring by an amount ๐ฟ0
๐1 = ๐1๐ฟ0
The kinetic energy of the system
๐ =1
2๐ ๐ฅ2
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 + ๐ฅ ๐ = 1 + ๐๐ฅ +1
2๐(๐ โ 1)๐ฅ2 + โฏ
ยง6.Lagrangeโs Equations
The potential energy of the system
๐ =1
2๐1 ๐ฟ0 + โ๐ฟ 2
๐๐๐ ๐ ๐๐๐๐๐ ๐1
+1
2๐2๐ฅ
2
๐๐๐ ๐ ๐๐๐๐๐ ๐2
โ๐ฟ : the change in the length of the spring with
stiffness ๐1 due to the motion ๐ฅ of the mass
โ๐ฟ = ๐ฟ2 + ๐ฅ2 โ ๐ฟ = ๐ฟ 1 + (๐ฅ/๐ฟ)2โ ๐ฟ
Assume that |๐ฅ/๐ฟ| โช 1, using binomial expansion
1+(๐ฅ/๐ฟ)2= 1+(๐ฅ/๐ฟ)2 1/2 = 1+1
2(๐ฅ/๐ฟ)2+
1
8(๐ฅ/๐ฟ)4+โฏ
โน โ๐ฟ โ ๐ฟ 1+(๐ฅ/๐ฟ)2/2 โ ๐ฟ = ๐ฟ(๐ฅ/๐ฟ)2/2
โน ๐ =1
2๐1 ๐ฟ0 +
๐ฟ
2
๐ฅ
๐ฟ
2 2
+1
2๐2๐ฅ
2
Vibrations 3.67 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
๐
๐๐ก
๐๐
๐ ๐1โ
๐๐
๐๐1+
๐๐ท
๐ ๐1+
๐๐
๐๐1= ๐1 (3.44)
ยง6.Lagrangeโs Equations
Chose the generalize coordinate ๐1 = ๐ฅ
๐ =1
2๐ ๐ฅ2
โน๐
๐๐ก
๐๐
๐ ๐ฅ=
๐
๐๐ก๐ ๐ฅ = ๐ ๐ฅ
๐๐
๐๐ฅ= 0,
๐๐ท
๐ ๐ฅ= 0, ๐ = 0
๐ =1
2๐1 ๐ฟ0 +
๐ฟ
2
๐ฅ
๐ฟ
2 2
+1
2๐2๐ฅ
2
โน๐๐
๐๐ฅ=๐1 ๐ฟ0 +
๐ฟ
2
๐ฅ
๐ฟ
2 2๐ฅ
๐ฟ+๐2๐ฅ= ๐1 +
๐1๐ฟ0
๐ฟ๐ฅ+
๐1
2
๐ฅ3
๐ฟ2โ ๐2 +
๐1๐ฟ
๐ฅ
Vibrations 3.68 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs Equations
The governing equation of motion
๐ ๐ฅ + ๐2 +๐1
๐ฟ๐ฅ = 0
The natural frequency
๐๐ = ๐๐/๐๐ = ๐2 + ๐1/๐ฟ /๐
If the spring of constant ๐1 is compressed instead of being in
tension, then we can replace ๐1 by โ๐1 , and the natural
frequency
๐๐ = ๐๐/๐๐ = ๐2 โ ๐1/๐ฟ /๐
The natural frequency ๐๐ can be made very low by adjusting
the compression of the spring with stiffness ๐1. At the same
time, the spring with stiffness ๐2 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
ยง6.Lagrangeโs 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 ๐
๐ฃ๐ =๐ ๐๐๐๐ก
=๐
๐๐ก๐ฅ + ๐ฟ๐ ๐๐๐ ๐ + ๐ฟ โ ๐ฟ๐๐๐ ๐ ๐
= โ๐ ๐ + ๐ฟ ๐๐๐๐ ๐ ๐ + ๐ฟ ๐๐ ๐๐๐ ๐
Vibrations 3.70 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs Equations
The kinetic energy of the system ๐ = ๐๐ + ๐๐
๐๐ =1
2๐๐ ๐ฅ2 +
1
2๐ฝ๐บ ๐2
=1
2๐๐๐ 2 ๐2 +
1
2๐ฝ๐บ ๐2
๐๐ =1
2๐๐ฃ๐
2
=1
2๐ โ๐ ๐ + ๐ฟ ๐๐๐๐ ๐ ๐ + ๐ฟ ๐๐ ๐๐๐ ๐
2
=1
2๐(๐ 2 + ๐ฟ2 โ 2๐ฟ๐ ๐๐๐ ๐) ๐2
โ1
2๐ ๐ฟ โ ๐ 2 ๐2
โน ๐ = ๐๐ + ๐๐ =1
2๐ ๐ฟ โ ๐ 2 + ๐๐๐ 2 + ๐ฝ๐บ ๐2 โก
1
2๐๐
๐2
Vibrations 3.71 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs Equations
The potential energy of the system
๐ =1
2๐๐ฅ2 + ๐๐(๐ฟ โ ๐ฟ๐๐๐ ๐)
=1
2๐๐ 2๐2 + ๐๐๐ฟ 1 โ ๐๐๐ ๐
โน ๐ =1
2๐๐ 2๐2 +
1
2๐๐๐ฟ๐2 ๐๐๐ ๐ โ 1 โ
๐2
2
=1
2๐๐ 2 + ๐๐๐ฟ ๐2 โก
1
2๐๐๐
2
The dissipation function
๐ท =1
2๐ ๐ฅ2 =
1
2๐๐ 2 ๐2 โก
1
2๐๐
๐2
๐๐ ๐ + ๐๐
๐ + ๐๐๐ = 0,๐๐ =๐๐
๐๐=
๐๐ 2 + ๐๐๐ฟ
๐(๐ฟ โ ๐ )2+๐๐๐ 2 + ๐ฝ๐บ
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
ยง6.Lagrangeโs 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
๐ =1
2๐๐ก๐
2 +1
2๐ ๐ฅ0 ๐ก โ๐ฅ1
2 +1
4๐2๐(๐ฟ2 โ๐ฟ1)๐
2
=1
2๐๐ก๐
2 +1
2๐ ๐ฅ0 ๐ก โ๐ฟ2๐
2 +1
4๐2๐(๐ฟ2 โ๐ฟ1)๐
2
The kinetic energy
๐ =1
2๐ฝ0 ๐2 +
1
2๐1 ๐ฅ1
2 =1
2๐ฝ0 +๐1๐ฟ2
2 ๐2 โก1
2๐๐ ๐2
Dissipation function
Vibrations 3.73 Single DOF Systems: Governing Equations
๐ท =1
2๐ ๐ฅ2
2 =1
2๐๐ฟ1
2 ๐2 โก1
2๐๐ ๐2
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs Equations
๐ =1
2๐๐ก๐
2 +1
2๐ ๐ฅ0 ๐ก โ ๐ฟ2๐
2 +1
4๐2๐(๐ฟ2 โ ๐ฟ1)๐
2
The potential energy is not in the standard form โน the
governing equation must be derived from Lagrangeโs equation๐๐
๐๐= ๐๐ก + ๐๐ฟ2
2 +1
2๐2๐(๐ฟ2 โ ๐ฟ1) ๐ โ ๐๐ฟ2๐ฅ0 ๐ก
= ๐๐๐ โ ๐๐ฟ2๐ฅ0 ๐ก
The governing equation of motion
๐๐ ๐ + ๐๐ ๐ + ๐๐๐ = ๐๐ฟ2๐ฅ0(๐ก)
The natural frequency
๐๐ =๐๐
๐๐=
๐๐ก + ๐๐ฟ22 + ๐2๐(๐ฟ2 โ ๐ฟ1)/2
๐ฝ0 + ๐1๐ฟ22
Vibrations 3.74 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs Equations
Ex.3.16 Slider Mechanism
Obtain the equation of motion of the slider mechanism
Solution
The geometric constraints on the motion
๐2 ๐ = ๐2 + ๐2 โ 2๐๐๐๐๐ ๐ (a)
โน ๐ ๐ =๐๐
๐(๐) ๐๐ ๐๐๐
๐ ๐ ๐ ๐๐๐ฝ = ๐๐ ๐๐๐ (b)
๐ = ๐ ๐ ๐๐๐ ๐ฝ + ๐๐๐๐ ๐ (c)
โน ๐ ๐ ๐๐๐ ๐ฝ โ ๐ ๐ ๐ฝ๐ ๐๐๐ฝ โ ๐ ๐๐ ๐๐๐ = 0
โน ๐ฝ = ๐ ๐ ๐๐๐ ๐ฝ โ ๐ ๐๐ ๐๐๐
๐(๐)๐ ๐๐๐ฝ=
๐
๐2(๐)๐๐๐๐๐ ๐ โ ๐2
Vibrations 3.75 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs Equations
System Kinetic Energy
๐ =1
2๐ฝ๐๐ +๐ฝ๐๐ ๐2 +
1
2๐ฝ๐๐
๐ฝ2 +1
2๐๐ ๐2 +
1
2๐๐ ๐
2 ๐ฝ2
๐ฝ๐๐ =1
3๐๐๐
2
๐ฝ๐๐ =1
3๐๐๐
2
๐ฝ๐๐ =1
3๐๐๐
2
๐ ๐ โก ๐ฝ๐๐ + ๐ฝ๐๐ + ๐ฝ๐๐ + ๐๐ ๐2
๐๐๐๐๐ ๐ โ ๐2
๐2(๐)
2
+ ๐๐
๐๐๐ ๐๐๐
๐(๐)
2
โน ๐ =1
2๐(๐) ๐2
Vibrations 3.76 Single DOF Systems: Governing Equations
where
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
๐
๐๐ก
๐๐
๐ ๐1โ
๐๐
๐๐1+
๐๐ท
๐ ๐1+
๐๐
๐๐1= ๐1 (3.44)
ยง6.Lagrangeโs Equations
System Kinetic Energy
๐ =1
2๐(๐) ๐2
System Potential Energy
๐ =1
2๐๐2(๐) +
1
2๐๐ ๐ ๐ก โ ๐๐ 2
Equation of motion
๐
๐๐ก
๐๐
๐ ๐=
๐
๐๐ก๐(๐) ๐ = ๐(๐) ๐,
๐๐
๐๐= ๐โฒ ๐ ๐2
๐๐
๐๐= ๐๐ ๐ ๐โฒ ๐ + ๐๐๐2๐ โ ๐๐๐2๐(๐ก)
โน ๐ ๐ ๐ +1
2๐โฒ ๐ ๐2 + ๐๐ ๐ ๐โฒ ๐ + ๐๐๐2๐ = ๐๐๐2๐(๐ก)
Vibrations 3.77 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs Equations
- Ex.3.17 Oscillations of A Crankshaft
Obtain the equation of motion of the crankshaft
Solution
โข Kinematics
The position vector of the
slider mass ๐๐
๐๐ = ๐๐๐๐ ๐ + ๐๐๐๐ ๐พ ๐ + ๐ ๐
The position vector of the center of mass ๐บ of the crank
๐๐บ = ๐๐๐๐ ๐ + ๐๐๐๐ ๐พ ๐ + ๐๐ ๐๐๐ + ๐๐ ๐๐๐พ ๐
From geometry
๐๐ ๐๐๐ = ๐ + ๐๐ ๐๐๐พ
The slider velocity
๐ฃ๐ = ๐๐ = โ๐ ๐๐ ๐๐๐ โ ๐ ๐พ๐ ๐๐๐พ ๐ = โ๐ ๐ ๐ ๐๐๐ +๐ก๐๐๐พ๐๐๐ ๐ ๐
Vibrations 3.78 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
โน ๐ ๐๐๐๐ ๐ = ๐ ๐พ๐๐๐ ๐พ โน ๐พ =๐
๐
๐๐๐ ๐
๐๐๐ ๐พ ๐
2/7/2014
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ยง6.Lagrangeโs Equations
The velocity of the center of mass ๐บ of the crank
๐ฃ๐บ = โ๐ ๐๐ ๐๐๐ โ ๐ ๐พ๐ ๐๐๐พ ๐ + ๐ ๐๐๐๐ ๐ โ ๐ ๐พ๐๐๐ ๐พ ๐
โน ๐ฃ๐บ = โ ๐ ๐๐๐ +๐
๐๐ก๐๐๐พ๐๐๐ ๐ ๐ ๐ ๐ +
๐
๐๐๐๐ ๐ ๐ ๐ ๐
โข System Kinetic Energy
The total kinetic energy of the system
๐ =1
2๐ฝ๐ ๐2 +
1
2๐๐บ๐ฃ๐บ
2 +1
2๐ฝ๐บ ๐พ2 +
1
2๐๐๐ฃ๐
2 โก1
2๐ฝ(๐) ๐2
๐ฝ ๐ = ๐ฝ๐ + ๐2๐๐บ ๐ ๐๐๐ +๐
๐๐ก๐๐๐พ๐๐๐ ๐
2
+๐
๐๐๐๐ ๐
2
+๐ฝ๐บ๐
๐
๐๐๐ ๐
๐๐๐ ๐พ
2
+ ๐2๐๐ ๐ ๐๐๐ + ๐ก๐๐๐พ๐๐๐ ๐ 2
Vibrations 3.79 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
where,
๐พ = ๐ ๐๐โ1๐
๐๐ ๐๐๐ โ
๐
๐
ยง6.Lagrangeโs Equations
โข Equation of Motion
The governing equation of motion has the form
๐
๐๐ก
๐๐
๐ ๐โ
๐๐
๐๐= โ๐(๐ก)
After performing the differentiation operations
๐ฝ ๐ ๐ +1
2
๐๐ฝ(๐)
๐๐ ๐2 = โ๐(๐ก)
The angle ๐ can be expressed
๐ ๐ก = ๐๐ก + ๐(๐ก)
Then
๐ฝ ๐ ๐ +1
2
๐๐ฝ(๐)
๐๐๐ + ๐
2= โ๐(๐ก)
Vibrations 3.80 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs Equations
- Ex.3.18 Vibration of A Centrifugal Governor
Derive the equation of motion of
governor by usingLagrangeโsequation
Solution
The velocity vector relative to point
๐ of the left hand mass
๐๐ = โ๐ฟ ๐๐๐๐ ๐ ๐ + ๐ฟ ๐๐ ๐๐๐ ๐
+(๐ + ๐ฟ๐ ๐๐๐)๐๐
The kinetic energy
๐ ๐, ๐ = 21
2๐ ๐๐๐๐
= ๐ โ๐ฟ ๐๐๐๐ ๐ 2 + ๐ฟ ๐๐ ๐๐๐ 2 + ๐ + ๐ฟ๐ ๐๐๐ ๐ 2
= ๐๐2 ๐ + ๐ฟ๐ ๐๐๐ 2 + ๐ ๐2๐ฟ2
Vibrations 3.81 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs Equations
The potential energy with respect to the static equilibrium position
๐ ๐ =1
2๐ 2๐ฟ 1โ๐๐๐ ๐ 2 โ2๐๐๐ฟ๐๐๐ ๐
Using equation (3.44) with
๐1 = ๐
๐ท = 0
๐1 = 0
and performing the required
operations, to obtain the following
governing equation
๐๐ฟ2 ๐ โ ๐๐๐ฟ๐2๐๐๐ ๐ โ ๐๐2 + 2๐ ๐ฟ2๐ ๐๐๐๐๐๐ ๐
+๐ฟ ๐๐ + 2๐๐ฟ ๐ ๐๐๐ = 0
Vibrations 3.82 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
๐
๐๐ก
๐๐
๐ ๐1โ
๐๐
๐๐1+
๐๐ท
๐ ๐1+
๐๐
๐๐1= ๐1 (3.44)
ยง6.Lagrangeโs Equations
๐๐ฟ2 ๐ โ ๐๐๐ฟ๐2๐๐๐ ๐ โ ๐๐2 + 2๐ ๐ฟ2๐ ๐๐๐๐๐๐ ๐
+๐ฟ ๐๐ + 2๐๐ฟ ๐ ๐๐๐ = 0
Introducing the quantities
๐พ โก๐
๐ฟ, ๐๐
2 โก๐
๐ฟ, ๐๐
2 โก2๐
๐Rewrite the equation
๐ โ ๐พ๐2๐๐๐ ๐
โ ๐2 + ๐๐2 ๐ ๐๐๐๐๐๐ ๐
+ ๐๐2 + ๐๐
2 ๐ ๐๐๐ = 0
Assume that the oscillation ๐ about ๐ = 0 are small (๐๐๐ ๐ โ 1,๐ ๐๐๐ โ ๐) to get the final equation
๐ + ๐๐2 โ ๐2 ๐ = ๐พ๐2
Vibrations 3.83 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ยง6.Lagrangeโs 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
๐๐ฟ = ๐(๐ + ๐ฟ)ฮฉ2 โน ๐ฟ =๐
๐1๐2
ฮฉ2 โ 1
, ๐1๐2 =
๐
๐
For small angles of rotation, the kinetic energy
๐ =1
2
1
2๐๐2
๐ฅ
๐
2
+1
2๐ ๐ฅ2 =
1
2
3
2๐ ๐ฅ2
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
2/7/2014
15
ยง6.Lagrangeโs Equations
The potential energy for oscillations about the equilibrium position
๐ =1
2๐๐ฅ2
The Lagrange equation for this undamped system
๐
๐๐ก
๐๐
๐ ๐ฅโ
๐๐
๐๐ฅ+
๐๐ท
๐ ๐ฅ+
๐๐
๐๐ฅ= ๐๐ฅ = ๐๐ฅฮฉ2
where the centrifugal force ๐๐ฅฮฉ2 is treated as an external force
The governing equation
3
2๐ ๐ฅ + ๐ โ ๐ฮฉ2 ๐ฅ = 0
The natural frequency
๐๐ =๐
๐=
2
3๐1๐
2 โ ฮฉ2
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