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2/7/2014
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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
§1.Chapter Objectives
• Obtain the governing equation of motion for single degree-of-
freedom (dof) translating and rotating systems by using force
balance and moment balance methods
• Obtain the governing equation of motion for single dof
translating and rotating systems by using Lagrange’s
equations
• Determine the equivalent mass, equivalent stiffness, and
equivalent damping of a single dof system
• 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
§2.Force-Balance and Moment-Balance Methods
1.Force Balance Method
Newtonian principle of linear momentum
𝐹 − 𝑝 = 0 (3.1a)
𝐹 : the net external force vector acting on the system
𝑝 : the absolute linear momentum of the considered system
For a system of constant mass 𝑚 whose center of mass is
moving with absolute acceleration 𝑎, the rate of change of
linear momentum 𝑝 = 𝑚 𝑎
𝐹 − 𝑚 𝑎 = 0 (3.1b)
−𝑚 𝑎 : inertial force
⟹The 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
§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 𝑂 𝑟 = 𝑟 𝑗= (𝐿 + 𝛿𝑠𝑡 + 𝑥) 𝑗
Force balance along
the 𝑗 direction
𝑓 𝑡 𝑗 + 𝑚𝑔 𝑗 − 𝑘𝑥 + 𝑘𝛿𝑠𝑡 𝑗 − 𝑐𝑑𝑟
𝑑𝑡 𝑗 − 𝑚
𝑑2𝑟
𝑑𝑡2 𝑗 = 0
Vibrations 3.04 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§2.Force-Balance and Moment-Balance Methods
- Noting that 𝐿 and 𝛿𝑠𝑡 are constants, rearranging terms to get
the following scalar differential equation
𝑚𝑑2𝑥
𝑑𝑡2 + 𝑐𝑑𝑥
𝑑𝑡+ 𝑘 𝑥 + 𝛿𝑠𝑡 = 𝑓 𝑡 + 𝑚𝑔
Vibrations 3.05 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§2.Force-Balance and Moment-Balance Methods
Static Equilibrium Position
- The static-equilibrium position of a system is the position that
corresponds to the system’s rest state; that is, a position with
zero velocity and zero acceleration
- The static-equilibrium position is the solution of
𝑘 𝑥 + 𝛿𝑠𝑡 = 𝑚𝑔
- The static displacement
𝛿𝑠𝑡 =𝑚𝑔
𝑘⟹ 𝑥 = 0 is the static-equilibrium position of the system
- The spring has an unstretched length 𝐿, the static-equilibrium
position measured from the origin 𝑂 is given by
𝑥𝑠𝑡 = 𝑥𝑠𝑡 𝑗 = (𝐿 + 𝛿𝑠𝑡) 𝑗
Vibrations 3.06 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
2
§2.Force-Balance and Moment-Balance Methods
Equation of Motion for Oscillations about the Static-EquilibriumPosition
𝑚𝑑2𝑥
𝑑𝑡2 + 𝑐𝑑𝑥
𝑑𝑡+ 𝑘 𝑥 + 𝛿𝑠𝑡 = 𝑓 𝑡 + 𝑚𝑔
𝛿𝑠𝑡 =𝑚𝑔
𝑘
⟹ 𝑚𝑑2𝑥
𝑑𝑡2 + 𝑐𝑑𝑥
𝑑𝑡+ 𝑘𝑥 = 𝑓 𝑡
Equation (3.8) is the governing equation of motion of a single
dof system for oscillations about the static-equilibrium position
• The left-hand side: the forces from the components that
comprise a single dof system
• 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
§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
• It is assumed that the mass moves without
friction
• The unstretched length of the spring is 𝐿, and
a fixed point 𝑂 is located at the unstretched
position of the spring
• The spring does not undergo any static
deflection and carrying out a force balance
along the 𝑖 direction
• The static-equilibrium position 𝑥 = 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
§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
𝐹𝑅 = 𝑘𝛿𝑠𝑡 + 𝑘𝑥 + 𝑐𝑑𝑥
𝑑𝑡
If considering only the dynamic part of the
reaction force-that is, only those forces created
by the motion 𝑥(𝑡) from its static equilibrium
position, then
𝐹𝑅𝑑 = 𝑘𝑥 + 𝑐𝑑𝑥
𝑑𝑡
Vibrations 3.09 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§2.Force-Balance and Moment-Balance Methods
- Ex.3.1 Wind-drivenOscillationsaboutaSystem’sStatic-EquilibriumPosition
The wind flow across civil structures typically generates a
excitation force 𝑓(𝑡) on the structure that consists of a steady-
state part and a fluctuating part
𝑓 𝑡 = 𝑓𝑠𝑠 + 𝑓𝑑(𝑡)
𝑓𝑠𝑠 : the time-independent steady-state force
𝑓𝑑(𝑡) : the fluctuating time-dependent portion of the force
A single dof model of the vibrating structure
𝑚𝑑2𝑥
𝑑𝑡2 + 𝑐𝑑𝑥
𝑑𝑡+ 𝑘𝑥 = 𝑓𝑠𝑠 + 𝑓𝑑 𝑡 ⟹ 𝑥 𝑡 = 𝑥0 + 𝑥𝑑(𝑡)
𝑥0 : the static equilibrium position, 𝑥0 = 𝑓𝑠𝑠/𝑘
𝑥𝑑(𝑡) : motions about the static position
⟹ 𝑚𝑑2𝑥
𝑑𝑡2 + 𝑐𝑑𝑥
𝑑𝑡+ 𝑘𝑥 = 𝑓𝑑 𝑡
Vibrations 3.10 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§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
𝑚𝑑2𝑥
𝑑𝑡2 + 𝑘𝑥 + 𝑘𝑥2 = 0
The restoring force of the eardrum has a component with a
quadratic nonlinearity
Static-Equilibrium Positions
Equilibrium positions 𝑥 = 𝑥0 are solutions of the algebraic equation
𝑘 𝑥0 + 𝑥02 = 0 ⟹ 𝑥0 = 0, 𝑥0 = −1
Vibrations 3.11 Single DOF Systems: Governing Equations
(𝑎)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§2.Force-Balance and Moment-Balance Methods
Linearization
Equilibrium positions 𝑥 = 𝑥0 are solutions of the algebraic equation
𝑘 𝑥0 + 𝑥02 = 0 ⟹ 𝑥0 = 0, 𝑥0 = −1
Subtitute 𝑥 𝑡 = 𝑥0 + 𝑥(𝑡) into (a) with note that
𝑥2 𝑡 = 𝑥0 + 𝑥 𝑡2
≈ 𝑥02 + 2𝑥0 𝑥 𝑡 + ⋯
𝑑2𝑥
𝑑𝑡2 =𝑑2 𝑥0 + 𝑥 𝑡
𝑑𝑡2 =𝑑2 𝑥
𝑑𝑡2
⟹ 𝑚𝑑2 𝑥
𝑑𝑡2 + 𝑘 𝑥0 + 𝑥(𝑡) + 𝑘 𝑥02 + 2𝑥0 𝑥 𝑡 = 0
𝑥0 = 0 ⟹ 𝑚𝑑2 𝑥
𝑑𝑡2 + 𝑘 𝑥(𝑡) = 0
𝑥0 = −1 ⟹ 𝑚𝑑2 𝑥
𝑑𝑡2 − 𝑘 𝑥(𝑡) = 0
⟹ 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
§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
𝐻 = 𝐽𝐺 𝜃𝑘
⟹ 𝑀 = 𝐽𝐺 𝜃𝑘
Vibrations 3.13 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§2.Force-Balance and Moment-Balance Methods
The governing equation of motion
𝑀 𝑡 𝑘 − 𝑘𝑡 𝜃𝑘 − 𝑐𝑡
𝑑𝜃
𝑑𝑡𝑘 − 𝐽𝐺
𝑑2𝜃
𝑑𝑡2 = 0
⟹ 𝐽𝐺𝑑2𝜃
𝑑𝑡2 + 𝑐𝑡
𝑑𝜃
𝑑𝑡+ 𝑘𝑡𝜃 = 𝑀 𝑡
Vibrations 3.14 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§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
• Translational motion
𝑚𝑑2𝑥
𝑑𝑡2 + 𝑐𝑑𝑥
𝑑𝑡+ 𝑘𝑥 = 𝑓 𝑡
• Rotational motion
𝐽𝐺𝑑2𝜃
𝑑𝑡2 + 𝑐𝑡
𝑑𝜃
𝑑𝑡+ 𝑘𝑡𝜃 = 𝑀 𝑡
Vibrations 3.15 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§2.Force-Balance and Moment-Balance Methods
Ex.3.3 Hand Biomechanics
The moment balance about
point 𝑂
𝑀 − 𝐽0 𝜃𝑘 = 0
𝐽0: the rotary inertia of the
forearm and the object
held in the hand
The net moment 𝑀 acting
about the point 𝑂 due to gravity loading and the forces due to
the biceps and triceps
𝑀 = −𝑀𝑔𝑙𝑐𝑜𝑠𝜃𝑘 − 𝑚𝑔𝑙
2𝑐𝑜𝑠𝜃𝑘 + 𝐹𝑏𝑎𝑘 − 𝐹𝑡𝑎𝑘
⟹ −𝑀𝑔𝑙𝑐𝑜𝑠𝜃𝑘 − 𝑚𝑔𝑙
2𝑐𝑜𝑠𝜃𝑘 + 𝐹𝑏𝑎𝑘 − 𝐽0 𝜃𝑘 = 0
Vibrations 3.16 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§2.Force-Balance and Moment-Balance Methods
−𝑀𝑔𝑙𝑐𝑜𝑠𝜃𝑘 − 𝑚𝑔𝑙
2𝑐𝑜𝑠𝜃𝑘 + 𝐹𝑏𝑎𝑘 − 𝐽0 𝜃𝑘 = 0
Note that: 𝐹𝑏 = −𝑘𝑏𝜃, 𝐹𝑡 = 𝐾𝑡𝑣 = 𝐾𝑡𝑎 𝜃, 𝐹0 = 𝑚𝑙2/3 + 𝑀𝑙2
⟹ 𝑀 +𝑚
3𝑙2 𝜃 + 𝐾𝑡𝑎
2 𝜃 + 𝑘𝑏𝑎𝜃 + 𝑀 +𝑚
2𝑔𝑙𝑐𝑜𝑠𝜃 = 0
Static-Equilibrium Position
The equilibrium position 𝜃 = 𝜃0 is a solution of the
transcendental equation
𝑘𝑏𝑎𝜃0 + 𝑀 +𝑚
2𝑔𝑙𝑐𝑜𝑠𝜃0 = 0
Vibrations 3.17 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§2.Force-Balance and Moment-Balance Methods
Linear System Governing “Small” Oscillations about the Static-
Equilibrium Position
Consider oscillations about the static-equilibrium position and
expand the angular variable 𝜃 𝑡 = 𝜃0 + 𝜃 𝑡 with note that
𝑐𝑜𝑠𝜃 = cos 𝜃0 + 𝜃 ≈ 𝑐𝑜𝑠𝜃0 − 𝜃𝑠𝑖𝑛𝜃0 + ⋯
𝑑𝜃(𝑡)
𝑑𝑡=
𝑑(𝜃0 + 𝜃)
𝑑𝑡= 𝜃(𝑡)
𝑑2𝜃(𝑡)
𝑑𝑡2 =𝑑2(𝜃0 + 𝜃)
𝑑𝑡2 = 𝜃(𝑡)
⟹ 𝑀 +𝑚
3𝑙2 𝜃 + 𝐾𝑡𝑎
2 𝜃 + 𝑘𝑒 𝜃 = 0
where
𝑘𝑒 = 𝑘𝑏𝑎 − 𝑀 +𝑚
2𝑔𝑙𝑠𝑖𝑛𝜃0
Vibrations 3.18 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
4
§3.Natural Frequency and Damping Factor
1.Natural Frequency
Translation Vibrations: Natural Frequency
𝜔𝑛 = 2𝜋𝑓𝑛 =𝑘
𝑚(𝑟𝑎𝑑/𝑠)
𝑘 : the stiffness of the system, 𝑁/𝑚
𝑚 : the system mass, 𝑘𝑔
𝑓𝑛 : the natural frequency, 𝐻𝑧
For the mass-damper-spring system
𝜔𝑛 = 2𝜋𝑓𝑛 =𝑔
𝛿𝑠𝑡(𝑟𝑎𝑑/𝑠)
𝛿𝑠𝑡: the static deflection of the system, 𝑚
Vibrations 3.19 Single DOF Systems: Governing Equations
(3.15)
(3.14)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§3.Natural Frequency and Damping Factor
Rotational Vibrations: Natural Frequency
𝜔𝑛 = 2𝜋𝑓𝑛 =𝑘𝑡
𝐽(𝑟𝑎𝑑/𝑠)
𝑘𝑡 : the torsion stiffness of the system, 𝑁𝑚/𝑟𝑎𝑑
𝐽 : the system mass, 𝑘𝑔𝑚/𝑠2
𝑓𝑛 : the natural frequency, 𝐻𝑧
Period of Undamped Free Oscillations
For an unforced and undamped system, the period of free
oscillation of the system is given by
𝑇 =1
𝑓𝑛=
2𝜋
𝜔𝑛
Vibrations 3.20 Single DOF Systems: Governing Equations
(3.16)
(3.17)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
𝜔𝑛 = 2𝜋𝑓𝑛 =𝑔
𝛿𝑠𝑡(𝑟𝑎𝑑/𝑠) (3.15)
§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(𝑚𝑚). Determine the natural frequency for vertical vibrations
Solution
𝑓𝑛1 =1
2𝜋
𝑔
𝛿𝑠𝑡1=
1
2𝜋
9.81
0.1 × 10−3 = 49.85𝐻𝑧
𝑓𝑛2 =1
2𝜋
𝑔
𝛿𝑠𝑡2=
1
2𝜋
9.81
1 × 10−3 = 15.76𝐻𝑧
𝑓𝑛3 =1
2𝜋
𝑔
𝛿𝑠𝑡3=
1
2𝜋
9.81
10 × 10−3 = 4.98𝐻𝑧
Vibrations 3.21 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§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 100𝑘𝑔 mass standing upright. The tibia
has a length of 40𝑐𝑚, and it is modeled as a hollow tube with an
inner diameter of 2.4𝑐𝑚 and an outer diameter of 3.4𝑐𝑚. The
Young’s modulus of elasticity of the bone material is 2 ×1010𝑁/𝑚2. 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
𝑚𝑔 = 100/2 × 9.81 = 490.5𝑁
Vibrations 3.22 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
𝜔𝑛 = 2𝜋𝑓𝑛 =𝑔
𝛿𝑠𝑡(𝑟𝑎𝑑/𝑠) (3.15)
§3.Natural Frequency and Damping Factor
The stiffness of the tibia
𝑘 =𝐴𝐸
𝐿=
1 × 1010 ×𝜋4
3.4 × 10−2 2 − 2.4 × 10−2 2
40 × 10−2
= 22.78 × 106𝑁/𝑚2
The static deflection
𝛿𝑠𝑡 =𝑚𝑔
𝑘=
490.5
22.78 × 106 = 21.53 × 10−6𝑚
The natural frequency
𝑓𝑛 =1
2𝜋
𝑔
𝛿𝑠𝑡=
1
2𝜋
9.81
21.53 × 10−6 = 107.4𝐻𝑧
Vibrations 3.23 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§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
𝑘 = 𝐴𝑊
𝐴: a constant, 𝑊 = 𝑚𝑔: the weight, 𝑔: the gravitational constant
The natural frequency
𝑓𝑛 =1
2𝜋
𝑘
𝑚=
1
2𝜋
𝑘𝑔
𝑊=
1
2𝜋𝐴𝑔𝐻𝑧
⟹ 𝑓𝑛 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
§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 𝑘 = 𝐴𝑊
For illustrative purposes, consider a spring that has the
general force-displacement relationship
𝐹 𝑥 = 𝑎𝑥
𝑏
𝑐
𝑎, 𝑏: scale factors, 𝑐: shape factor
The static deflection
𝑥0 = 𝑏𝑊
𝑎
1/𝑐
Vibrations 3.25 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§3.Natural Frequency and Damping Factor
For “small” amplitude vibrations about 𝑥0, the linear equivalent
stiffness of this spring is determined
𝑘𝑒𝑞 = 𝑑𝐹(𝑥)
𝑑𝑥𝑥=𝑥0
=𝑎𝑐
𝑏
𝑥𝑜
𝑏
𝑐−1
=𝑎𝑐
𝑏
𝑊
𝑏
𝑐−1𝑐
The natural frequency of this system
𝑓𝑛 =1
2𝜋
𝑘𝑒𝑞
𝑊/𝑔
=1
2𝜋
𝑔𝑐
𝑏
𝑊
𝑎
−1/𝑐
=1
2𝜋
𝑔𝑐
𝑏
𝑊
𝑎
−1/2𝑐
𝐻𝑧
Vibrations 3.26 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§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
𝑎 = 2500𝑁, 𝑏 = 0.011𝑚, 𝑐 = 2.77
⟹ 𝑓𝑛 =1
2𝜋
𝑔𝑐
𝑏
𝑊
𝑎
−1/2𝑐
= 32.4747𝑊−1/3.54𝐻𝑧
Plot 𝑓𝑛(𝑊)
Vibrations 3.27 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§3.Natural Frequency and Damping Factor
Representative Spring Data
From the figure of 𝑓𝑛(𝑊)
• over a sizable portion of the load
range, the natural frequency of the
system varies within the range of 8.8%
• The natural frequency of a system with
a linear spring whose static
displacement ranges from 12 ÷ 5𝑚𝑚varies approximately from 4.5 ÷ 7.0𝐻𝑧or approximately 22% about a
frequency of 5.8𝐻𝑧
1
2𝜋
9.8
0.012≈ 4.5𝐻𝑧,
1
2𝜋
9.8
0.005≈ 7𝐻𝑧
of 5.8 Hz
Vibrations 3.28 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§3.Natural Frequency and Damping Factor
2.Damping Factor
Translation Vibrations: Damping Factor
For translating single dof systems, the damping factor or
damping ratio 𝜉 is defined as
𝜉 =𝑐
2𝑚𝜔𝑛=
𝑐
2 𝑘𝑚=
𝑐𝜔𝑛
2𝑘
𝑐: the system damping coefficient, 𝑁𝑠/𝑚
𝑘: the system stiffness, 𝑁/𝑚
𝑚: the system mass, 𝑘𝑔
Critical Damping, Underdamping, and Overdamping
Defining the critical damping 𝑐𝑐
𝑐𝑐 = 2𝑚𝜔𝑛 = 2 𝑘𝑚, 𝜉 = 𝑐/𝑐𝑐 (3.19)
0 < 𝜉 < 1: underdamped,𝜉 > 1: overdamped,𝜉 = 1: criticallydamped
Vibrations 3.29 Single DOF Systems: Governing Equations
(3.18)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§3.Natural Frequency and Damping Factor
Rotational Vibrations: Damping Factor
For rotating single dof systems, the damping factor or damping
ratio 𝜉 is defined as
𝜉 =𝑐𝑡
2𝐽𝜔𝑛=
𝑐𝑡
2 𝑘𝑡𝐽
𝑐𝑡: the system damping coefficient, 𝑁𝑚𝑠/𝑟𝑎𝑑
𝑘𝑡: the system stiffness, 𝑁𝑚/𝑟𝑎𝑑
𝐽: the system moment of inertia, 𝑘𝑔𝑚2
Vibrations 3.30 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
6
§3.Natural Frequency and Damping Factor
Governing Equation of Motion in Terms of Natural Frequency
and Damping Factor
Rewriting the equation of motion
𝑑2𝑥
𝑑𝑡2 + 2𝜉𝜔𝑛
𝑑𝑥
𝑑𝑡+ 𝜔𝑛
2𝑥 =𝑓(𝑡)
𝑚If we introduce the dimensionless time 𝜏 = 𝜔𝑛𝑡 , then the
equation can be written
𝑑2𝑥
𝑑𝜏2 + 2𝜉𝑑𝑥
𝑑𝜏+ 𝑥 =
𝑓(𝜏)
𝑘
Vibrations 3.31 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§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 𝜉 = 1. Due to a design change, the
mass of the system is increased 20% - that is, from 𝑚 to 1.2𝑚.
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
𝜉𝑛𝑒𝑤 =𝑐
2 𝑘(1.2𝑚)= 0.91
𝑐
2 𝑘𝑚= 0.91
𝑐
𝑐𝑐= 0.91
⟹ 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
§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 𝑚 ,
damping 𝑐, and spring constant 𝑘 has too much static deflection
𝛿𝑠𝑡. The engineer would like to decrease 𝛿𝑠𝑡 by a factor of 2,
while keeping the damping ratio constant. Determine the
different options
Solution
The problem involves vertical vibrations
𝛿𝑠𝑡 =𝑚𝑔
𝑘
2𝜉 =𝑐
𝑚
𝛿𝑠𝑡
𝑔= 𝑐
𝛿𝑠𝑡
𝑔𝑚2 =1
𝑚
𝑐2𝛿𝑠𝑡
𝑔
⟹ 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
§3.Natural Frequency and Damping Factor
First choice
Let 𝑐 remain constant, reduce 𝛿𝑠𝑡 by one-half
𝛿𝑠𝑡 =𝑚𝑔
𝑘
𝛿𝑠𝑡′ =
𝛿𝑠𝑡
2=
𝑚𝑔
2𝑘=
𝑚′𝑔
𝑘′Comparing (a) and (b)
𝑚′𝑔
𝑘′=
𝑚𝑔
2𝑘=
𝑚/ 2 𝑔
𝑘 2⟹ 𝑚 → 𝑚′ =
𝑚
2, 𝑘 → 𝑘′ = 𝑘 2
Check the damping ratio
2𝜉′ = 𝑐𝛿′
𝑠𝑡
𝑔𝑚′2 = 𝑐𝛿𝑠𝑡
2𝑔 𝑚/ 22 = 𝑐
𝛿𝑠𝑡
𝑔𝑚2 = 2𝜉
Vibrations 3.34 Single DOF Systems: Governing Equations
Before (a)
After (b)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§3.Natural Frequency and Damping Factor
Second choice
Let 𝑚 remain constant, reduce 𝛿𝑠𝑡 by one-half
2𝜉 = 𝑐𝛿𝑠𝑡
𝑔𝑚2 =1
𝑚
𝑐2𝛿𝑠𝑡
𝑔
2𝜉′ =1
𝑚
𝑐′2𝛿𝑠𝑡′
𝑔=
1
𝑚
𝑐′2𝛿𝑠𝑡
2𝑔
Comparing (c) and (d)
𝑐′2
2= 𝑐2 ⟹ 𝑐 → 𝑐′ = 𝑐 2
The static deflection
𝛿𝑠𝑡′ =
𝑚𝑔
𝑘′=
𝛿𝑠𝑡
2=
𝑚𝑔
2𝑘⟹ 𝑘 → 𝑘′ = 2𝑘
Vibrations 3.35 Single DOF Systems: Governing Equations
Before (c)
After (d)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§3.Natural Frequency and Damping Factor
Third choice
Let 𝑘 remain constant, reduce 𝛿𝑠𝑡 by one-half
𝛿𝑠𝑡 =𝑚𝑔
𝑘
𝛿𝑠𝑡′ =
𝛿𝑠𝑡
2=
𝑚𝑔
2𝑘=
𝑚′𝑔
𝑘Comparing (e) and (f)
𝑚′ =𝑚
2⟹ 𝑚 → 𝑚′ =
𝑚
2The constant damping ratio
2𝜉′ = 𝑐′𝛿′
𝑠𝑡
𝑔𝑚′2 = 𝑐′𝛿𝑠𝑡
2𝑔 𝑚/2 2 = 𝑐′2𝛿𝑠𝑡
𝑔𝑚2 = 𝑐𝛿𝑠𝑡
𝑔𝑚2 = 2𝜉
⟹ 𝑐 → 𝑐′ = 𝑐 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
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𝐹 𝑥 = 𝜇𝑚𝑔𝑠𝑔𝑛( 𝑥) (2.52)
𝑚𝑑2𝑥
𝑑𝑡2 + 𝑐𝑑𝑥
𝑑𝑡+ 𝑘𝑥 = 𝑓 𝑡 (3.8)
§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
𝑚𝑑2𝑥
𝑑𝑡2 + 𝑐𝑑𝑥
𝑑𝑡+ 𝜇𝑚𝑔𝑠𝑔𝑛( 𝑥) = 𝑓(𝑡)
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
𝑛𝑜𝑛𝑙𝑖𝑛𝑒𝑎𝑟 𝑑𝑟𝑦 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑓𝑜𝑟𝑐𝑒
𝐹 𝑥 = 𝑐𝑑 𝑥2𝑠𝑔𝑛 𝑥 = 𝑐𝑑| 𝑥| 𝑥 (2.54)
𝐹 = 𝑘𝜋𝛽ℎ𝑠𝑔𝑛 𝑥 |𝑥| (2.57)
𝑚𝑑2𝑥
𝑑𝑡2 + 𝑐𝑑𝑥
𝑑𝑡+ 𝑘𝑥 = 𝑓 𝑡 (3.8)
§4.Governing Equations for Different Type of Damping
Fluid Damping
Using Eq. (2.54) and Eq. (3.8), the governing equation of motion
𝑚𝑑2𝑥
𝑑𝑡2 + 𝑐𝑑| 𝑥| 𝑥 + 𝑘𝑥 = 𝑓(𝑡)
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
𝑚𝑑2𝑥
𝑑𝑡2 + 𝑘𝜋𝛽ℎ𝑠𝑔𝑛 𝑥 |𝑥| + 𝑘𝑥 = 𝑓(𝑡)
Vibrations 3.38 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
𝑛𝑜𝑛𝑙𝑖𝑛𝑒𝑎𝑟 𝑓𝑙𝑢𝑖𝑑 𝑑𝑎𝑚𝑝𝑖𝑛𝑔 𝑓𝑜𝑟𝑐𝑒
§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
• buildings subjected to earthquakes
• packaging during transportation
• vehicle response, and
• designing accelerometers
- The physical system of interest is represented by a single dof
system whose base is subjected to a displacement
disturbance 𝑦(𝑡), and an equation governing the motion of
this system is sought to determine the response of the
system 𝑥(𝑡)
Vibrations 3.39 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§5.Governing Equations for Different Type of Applied Forces
- A prototype of a single dof system subjected to a base excitation
• The vehicle provides the base excitation 𝑦(𝑡) to the
instrumentation package modeled as a single dof
• The displacement response 𝑥(𝑡) is measured from the
system’s static-equilibrium position
Assume that no external force is applied directly to the mass;
that is, 𝑓 𝑡 = 0
Vibrations 3.40 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§5.Governing Equations for Different Type of Applied Forces
- The following governing equation of motion
𝑚𝑑2𝑥
𝑑𝑡2 + 𝑐𝑑𝑥
𝑑𝑡+ 𝑘𝑥 = 𝑐
𝑑𝑦
𝑑𝑡+ 𝑘𝑦
⟹ 𝑚𝑑2𝑥
𝑑𝑡2 + 2𝜉𝜔𝑛
𝑑𝑥
𝑑𝑡+ 𝜔𝑛
2𝑥 = 2𝜉𝜔𝑛
𝑑𝑦
𝑑𝑡+ 𝜔𝑛
2𝑦
𝑦(𝑡) and 𝑥(𝑡) are measured from a fixed point 𝑂 located in an
inertial reference frame and a fixed point located at the
system’s static equilibrium position, respectively
Vibrations 3.41 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§5.Governing Equations for Different Type of Applied Forces
- If the relative displacement is desired, the governing equation
of motion
𝑚𝑑2𝑧
𝑑𝑡2 + 𝑐𝑑𝑧
𝑑𝑡+ 𝑘𝑧 = −𝑚
𝑑2𝑦
𝑑𝑡2
with 𝑧 𝑡 ≡ 𝑥 𝑡 − 𝑦(𝑡)
⟹𝑑2𝑧
𝑑𝑡2 + 2𝜉𝜔𝑛
𝑑𝑧
𝑑𝑡+ 𝜔𝑛
2𝑧 = −𝑑2𝑦
𝑑𝑡2
Vibrations 3.42 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
8
§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
system’s mass. This force, in turn, is transmitted through the
spring and damper to the fixed base
- The unbalance is modeled as a mass 𝑚0 that rotates with an
angular speed 𝜔, and this mass is located a fixed distance 𝑒from the center of rotation
Vibrations 3.43 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§5.Governing Equations for Different Type of Applied Forces
- From the free-body diagram (FBD) of the unbalanced mass 𝑚0
𝑁𝑥 = −𝑚0( 𝑥 − 𝜖𝜔2𝑠𝑖𝑛𝜔𝑡)
𝑁𝑦 = 𝑚0𝜖𝜔2𝑐𝑜𝑠𝜔𝑡
- From the FBD of mas 𝑀
𝑀𝑑2𝑥
𝑑𝑡2 + 𝑐𝑑𝑥
𝑑𝑡+ 𝑘𝑥 = 𝑁𝑥
⟹ (𝑀 + 𝑚0)𝑑2𝑥
𝑑𝑡2 + 𝑐𝑑𝑥
𝑑𝑡+ 𝑘𝑥 = 𝑚0𝜖𝜔
2𝑠𝑖𝑛𝜔𝑡
⟹𝑑2𝑥
𝑑𝑡2 + 2𝜉𝜔𝑛
𝑑𝑥
𝑑𝑡+ 𝜔𝑛
2𝑥 =𝐹(𝜔)
𝑚𝑠𝑖𝑛𝜔𝑡
where 𝑚 = 𝑀 + 𝑚0, 𝜔𝑛 = 𝑘/𝑚, 𝐹 𝜔 = 𝑚0𝜖𝜔2
- The static displacement of the spring
𝛿𝑠𝑡 =𝑀 + 𝑚0 𝑔
𝑘=
𝑚𝑔
𝑘
Vibrations 3.44 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§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
𝑚𝑑2𝑥
𝑑𝑡2 + 𝑘𝑥 = 𝑓 𝑡 + 𝑓1(𝑡)
𝑥(𝑡) : measured from the unstretched position of the spring
𝑓(𝑡) : the externally applied force
𝑓1(𝑡) : 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
§5.Governing Equations for Different Type of Applied Forces
- The force generated by the fluid on the rigid body
𝑓1 𝑡 = −𝐾0𝑀𝑑2𝑥
𝑑𝑡2 − 𝐶𝑓
𝑑𝑥
𝑑𝑡
𝑀 : the mass of the fluid displaced by the body
𝐾0 : an added mass coefficient
𝐶𝑓 : a positive fluid damping coefficient
- The governing equation of motion
𝑚 + 𝐾0𝑀𝑑2𝑥
𝑑𝑡2 + 𝐶𝑓
𝑑𝑥
𝑑𝑡+ 𝑘𝑥 = 𝑓 𝑡
𝐾0𝑀 : 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
§6.Lagrange’s Equations
Consider a system with 𝑁 degrees of freedom that is described
by a set of 𝑁 generalized coordinates 𝑞𝑖 , 𝑖 = 1,2,…𝑁. In terms
of the chosen generalized coordinates, Lagrange’s equations
have the form
𝑑
𝑑𝑡
𝜕𝑇
𝜕 𝑞𝑗−
𝜕𝑇
𝜕𝑞𝑗+
𝜕𝐷
𝜕 𝑞𝑗+
𝜕𝑉
𝜕𝑞𝑗= 𝑄𝑗 , 𝑗 = 1,2,… , 𝑁
𝑞𝑗 : generalized coordinate
𝑞𝑗 : generalized velocity
𝑇 : the kinetic energy of the system
𝑉 : the potential energy of the system
𝐷 : the Rayleigh dissipation function
𝑄𝑗 : the generalized force that appears in the 𝑗𝑡ℎ equation
Vibrations 3.47 Single DOF Systems: Governing Equations
(3.41)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s Equations
The generalized force 𝑄𝑗 that appears in the 𝑗𝑡ℎ equation
𝑄𝑗 =
𝑙
𝐹𝑙
𝜕 𝑟𝑙𝜕𝑞𝑗
+
𝑙
𝑀𝑙
𝜕𝜔𝑙
𝜕 𝑞𝑗
𝐹𝑙, 𝑀𝑙 : the vector representations of the externally
applied forces and moments on the 𝑙𝑡ℎ body
𝑟𝑙 : the position vector to the location where the force
𝐹𝑙 is applied
𝜔𝑙 : the 𝑙𝑡ℎ body’s 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
§6.Lagrange’s Equations
Linear Vibratory Systems
For vibratory systems with linear characteristics
𝑇 =1
2
𝑗=1
𝑁
𝑛=1
𝑁
𝑚𝑗𝑛 𝑞𝑗 𝑞𝑛
𝑉 =1
2
𝑗=1
𝑁
𝑛=1
𝑁
𝑘𝑗𝑛𝑞𝑗𝑞𝑛
𝐷 =1
2
𝑗=1
𝑁
𝑛=1
𝑁
𝑐𝑗𝑛 𝑞𝑗 𝑞𝑛
𝑚𝑗𝑛 : the inertia coefficients
𝑘𝑗𝑛 : the stiffness coefficients
𝑐𝑗𝑛 : the damping coefficients
Vibrations 3.49 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s Equations
Single Degree-Of-Freedom
The case of a single degree-of-freedom system, 𝑁 = 1, the
Lagrange’s equation
𝑑
𝑑𝑡
𝜕𝑇
𝜕 𝑞1−
𝜕𝑇
𝜕𝑞1+
𝜕𝐷
𝜕 𝑞1+
𝜕𝑉
𝜕𝑞1= 𝑄1
where the generalized force is obtained from
𝑄1 =
𝑙
𝐹𝑙
𝜕 𝑟𝑙𝜕𝑞1
+
𝑙
𝑀𝑙
𝜕𝜔𝑙
𝜕 𝑞1
Vibrations 3.50 Single DOF Systems: Governing Equations
(3.44)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s 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
𝑇 =1
2
𝑗=1
1
𝑛=1
1
𝑚𝑗𝑛 𝑞𝑗 𝑞𝑛 =1
2𝑚11 𝑞1
2 ≡1
2𝑚𝑒 𝑞1
2
𝑉 =1
2
𝑗=1
1
𝑛=1
1
𝑘𝑗𝑛𝑞𝑗𝑞𝑛 =1
2𝑘11𝑞1
2 ≡1
2𝑘𝑒𝑞1
2
𝐷 =1
2
𝑗=1
1
𝑛=1
1
𝑐𝑗𝑛 𝑞𝑗 𝑞𝑛 =1
2𝑐11 𝑞1
2 ≡1
2𝑐𝑒 𝑞1
2
𝑚𝑒, 𝑘𝑒, 𝑐𝑒 : the equivalent mass, stiffness, and damping
From Lagrange’s equation
𝑚𝑒 𝑞1 + 𝑐𝑒 𝑞1 + 𝑘𝑒𝑞1 = 𝑄1
Vibrations 3.51 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
(3.46)
§6.Lagrange’s Equations
To obtain the governing equation of motion of a linear vibrating
system with viscous damping
• Obtains expressions for the system kinetic energy 𝑇 ,
system potential energy 𝑉, and system dissipation function 𝐷
• Identify the equivalent mass 𝑚𝑒, equivalent stiffness 𝑘𝑒,
and equivalent damping 𝑐𝑒
• Determine the generalized force
• Apply the governing equation
𝑚𝑒 𝑞1 + 𝑐𝑒 𝑞1 + 𝑘𝑒𝑞1 = 𝑄1
• Determine the system natural frequency
𝜔𝑛 =𝑘𝑒
𝑚𝑒, 𝜉 =
𝑐𝑒
2𝑚𝑒𝜔𝑛=
𝑐𝑒
2 𝑘𝑒𝑚𝑒
Vibrations 3.52 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s 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
𝑞1 = 𝑥, 𝐹𝑙 = 𝑓(𝑡) 𝑗, 𝑟𝑙 = 𝑥 𝑗, 𝑀𝑙 = 0
Determine the generalized force
𝑄1 =
𝑙
𝐹𝑙
𝜕 𝑟𝑙𝜕𝑞1
+ 0 = 𝑓 𝑡 𝑗𝜕𝑥 𝑗
𝜕𝑥= 𝑓(𝑡)
The system kinetic energy 𝑇, system potential energy 𝑉, and
system dissipation function 𝐷
𝑇 =1
2𝑚 𝑥2, 𝑉 =
1
2𝑘𝑥2, 𝐷 =
1
2𝑐 𝑥2
Vibrations 3.53 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s Equations
Identify the following
𝑞1 = 𝑥, 𝐹𝑙 = 𝑓(𝑡) 𝑗, 𝑟𝑙 = 𝑥 𝑗, 𝑀𝑙 = 0
Determine the generalized force
𝑄1 =
𝑙
𝐹𝑙
𝜕 𝑟𝑙𝜕𝑞1
+ 0 = 𝑓 𝑡 𝑗𝜕𝑥 𝑗
𝜕𝑥= 𝑓(𝑡)
The system kinetic energy 𝑇, system potential energy
𝑉, and system dissipation function 𝐷
𝑇 =1
2𝑚 𝑥2, 𝑉 =
1
2𝑘𝑥2, 𝐷 =
1
2𝑐 𝑥2
⟹ 𝑚𝑒 = 𝑚, 𝑘𝑒 = 𝑘, 𝑐𝑒 = 𝑐
The governing equation
𝑚𝑑2𝑥
𝑑𝑡2 + 𝑐𝑑𝑦
𝑑𝑡+ 𝑘𝑥 = 𝑓(𝑡)
Vibrations 3.54 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/7/2014
10
§6.Lagrange’s Equations
- Ex.3.10 Motion of A System that Translates and Rotates
Obtain the governing equation of motion for “small” oscillations
about the upright position
Solution
Choose the generalized coordinate
𝑞1 = 𝜃, 𝐹𝑙 = 0, 𝑀𝑙 = 𝑀 𝑡 𝑘, 𝜔𝑙 = 𝜃𝑘
The generalized force
𝑄1 =
𝑙
𝑀𝑙 ∙𝜕𝜔𝑙
𝜕 𝑞1= 𝑀 𝑡 𝑘 ∙
𝜕 𝜃𝑘
𝜕 𝜃= 𝑀(𝑡)
Vibrations 3.55 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
𝐽𝐺 =1
2𝑚𝑟2
§6.Lagrange’s Equations
The potential energy
𝑉 =1
2𝑘𝑥2 =
1
2𝑘(𝑟𝜃)2=
1
2𝑘𝑟2𝜃2
⟹ the equivalent stiffness
The kinetic energy of the system
𝑇 =1
2𝑚 𝑥2 +
1
2𝐽𝐺 𝜃2
⟹ 𝑇 =1
2𝑚𝑟2 + 𝐽𝐺 𝜃2 =
1
2
3
2𝑚𝑟2 𝜃2
⟹ 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
𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦
𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦
𝑘𝑒 = 𝑘𝑟2
𝑚𝑒 =3
2𝑚𝑟2
§6.Lagrange’s Equations
The dissipation function
𝐷 =1
2𝑐 𝑥2 =
1
2𝑐(𝑟 𝜃)2=
1
2(𝑐𝑟2) 𝜃2
⟹ the equivalent damping coefficient
𝑐𝑒 = 𝑐𝑟2
The governing equation of motion3
2𝑚𝑟2 𝜃 + 𝑐𝑟2 𝜃 + 𝑘𝑟2𝜃 = 𝑀(𝑡)
Natural frequency and damping factor
𝜔𝑛 =𝑘𝑒
𝑚𝑒=
𝑘𝑟2
3𝑚𝑟2/2=
2𝑘
3𝑚
𝜉 =𝑐𝑒
2𝑚𝑒𝜔𝑛=
𝑐𝑟2
2(3𝑚𝑟2/2) 2𝑘/3𝑚=
6
6𝑘𝑚
Vibrations 3.57 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s Equations
- Ex.3.11 Inverted Pendulum
Obtain the governing equation of motion for “small” oscillations
about the upright position
Solution
The total rotary inertia of the system
𝐽𝑂 = 𝐽𝑂1+ 𝐽𝑂2
𝐽𝑂1: mass momentof inertia of 𝑚1 about point𝑂
𝐽𝑂2: massmomentof inertiaof thebaraboutpoint𝑂
𝐽𝑂1=
2
5𝑚1𝑟
2 + 𝑚1𝐿12
𝐽𝑂2=
1
12𝑚2𝐿2
2 + 𝑚2
𝐿2
2
2
=1
3𝑚2𝐿2
2
Vibrations 3.58 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s Equations
Choosing 𝑞1 = 𝜃 as the generalized coordinate, the system
kinetic energy takes the form
𝑇 =1
2𝐽𝑂 𝜃2 =
1
2𝐽𝑂1
+ 𝐽𝑂2 𝜃2
=1
2
2
5𝑚1𝑟
2 + 𝑚1𝐿12 +
1
3𝑚2𝐿2
2 𝜃2
For small 𝜃 ⟹ 𝑥1 ≈ 𝐿1𝜃
The system potential energy
𝑉 =1
2𝑘𝑥1
2 −1
2𝑚1𝑔𝐿1𝜃
2 −1
2𝑚2𝑔
𝐿2
2𝜃2
=1
2𝑘𝐿1
2 − 𝑚1𝑔𝐿1 − 𝑚2𝑔𝐿2
2𝜃2
𝐷 =1
2𝑐 𝑥1
2 =1
2𝑐𝐿1
2 𝜃2
Vibrations 3.59 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
The dissipation function
§6.Lagrange’s Equations
The equivalent inertia, the equivalent stiffness, and the
equivalent damping properties of the system
𝑇 =1
2
2
5𝑚1𝑟
2 +𝑚1𝐿12 +
1
3𝑚2𝐿2
2 𝜃2
𝑉 =1
2𝑘𝐿1
2 − 𝑚1𝑔𝐿1 − 𝑚2𝑔𝐿2
2𝜃2
𝐷 =1
2𝑐 𝑥1
2 =1
2𝑐𝐿1
2 𝜃2
The governing equation of motion 𝑚𝑒 𝜃 + 𝑐𝑒
𝜃 + 𝑘𝑒𝜃 = 0
Natural frequency
𝜔𝑛 =𝑘𝑒
𝑚𝑒=
𝑘𝐿12 − 𝑚1𝑔𝐿1 − 𝑚2𝑔𝐿2/2
𝐽𝑂1+ 𝐽𝑂2
𝑘𝑒 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
⟹𝑚𝑒 =2
5𝑚1𝑟
2 +𝑚1𝐿12 +
1
3𝑚2𝐿2
2
⟹𝑘𝑒 =𝑘𝐿12 −𝑚1𝑔𝐿1 −𝑚2𝑔
𝐿2
2
⟹ 𝑐𝑒 = 𝑐𝐿12
2/7/2014
11
§6.Lagrange’s Equations
• Natural Frequency of Pendulum System
Now locate the pivot point 𝑂 on the top, the
equivalent stiffness of this system
𝑘𝑒 = 𝑘𝐿12 + 𝑚1𝑔𝐿1 + 𝑚2𝑔
𝐿2
2and the natural frequency of this system
𝜔𝑛 =𝑘𝑒
𝑚𝑒=
𝑘𝐿12 + 𝑚1𝑔𝐿1 + 𝑚2𝑔𝐿2/2
𝐽𝑂1+ 𝐽𝑂2
If 𝑚2 ≪ 𝑚1, 𝑟 ≪ 𝐿1, and 𝑘 = 0, then
𝜔𝑛 =𝑚1𝑔𝐿1 1 + 𝑚2𝐿2/𝑚1𝐿1
𝑚1𝐿12 1 + 2𝑟2/5𝐿1
2 →𝑔
𝐿
→ the natural frequency of a pendulum composed of a rigid,
weightless rod carrying a mass a distance 𝐿1 from its pivot
Vibrations 3.61 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s 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 𝑂 to the
center of mass 𝐺
𝑟 = −𝑅𝜃 + 𝑏𝑠𝑖𝑛𝜃 𝑖 + (𝑅 − 𝑏𝑐𝑜𝑠𝜃) 𝑗
The absolute velocity of the center of mass 𝑟 = − 𝑅 − 𝑏𝑐𝑜𝑠𝜃 𝜃 𝑖 + 𝑏𝑠𝑖𝑛𝜃 𝜃 𝑗
Selecting the generalized coordinate 𝑞1 = 𝜃 ,
the system kinetic energy
𝑇 =1
2𝐽𝐺 𝜃2 +
1
2𝑚 𝑟 ∙ 𝑟
⟹ 𝑇 =1
2𝐽𝐺 𝜃2 +
1
2𝑚 𝑅2 + 𝑏2 − 2𝑏𝑅𝑐𝑜𝑠𝜃 𝜃2
Vibrations 3.62 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Taylor series expansion
𝑐𝑜𝑠𝜃 = 𝑐𝑜𝑠 𝜃0 + 𝜃 ≈ 𝑐𝑜𝑠𝜃0 − 𝜃𝑠𝑖𝑛𝜃0 −1
2 𝜃2𝑐𝑜𝑠𝜃0 + ⋯
𝑠𝑖𝑛𝜃 = 𝑠𝑖𝑛 𝜃0 + 𝜃 ≈ 𝑠𝑖𝑛𝜃0 − 𝜃𝑐𝑜𝑠𝜃0 −1
2 𝜃2𝑠𝑖𝑛𝜃0 + ⋯
§6.Lagrange’s Equations
Choosing the fixed ground as the datum, the system potential
energy
𝑉 = 𝑚𝑔 𝑅 − 𝑏𝑐𝑜𝑠𝜃
Small Oscillations about the Upright Position
Express the angular displacement as
𝜃(𝑡) = 𝜃0 + 𝜃(𝑡)
Since 𝜃0 = 0 , and small 𝜃 , using 𝑠𝑖𝑛𝜃 ≈ 𝜃 ,
𝑐𝑜𝑠𝜃 ≈ 1 −1
2 𝜃2, rewrite the energy functions
𝑇 ≈1
2𝐽𝐺 + 𝑚 𝑅 − 𝑏 2 𝜃2, 𝑉 ≈ 𝑚𝑔 𝑅 − 𝑏 +
1
2𝑚𝑔𝑏 𝜃2
Vibrations 3.63 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s Equations
The equivalent inertia of the system
𝑇 ≈1
2𝐽𝐺 + 𝑚 𝑅 − 𝑏 2 𝜃2
The potential energy is not in standard form because of the
constant term 𝑚𝑔 𝑅 − 𝑏
𝑉 ≈ 𝑚𝑔 𝑅 − 𝑏 +1
2𝑚𝑔𝑏 𝜃2
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 (𝑅 − 𝑏) above the ground
𝑉 =1
2𝑚𝑔𝑏 𝜃2
Then, the equivalent stiffness can be defined
𝑘𝑒 = 𝑚𝑔𝑏
Vibrations 3.64 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
⟹ 𝑚𝑒 = 𝐽𝐺 + 𝑚 𝑅 − 𝑏 2
§6.Lagrange’s Equations
The governing equation
𝐽𝐺 + 𝑚 𝑅 − 𝑏 2 𝜃 + 𝑚𝑔𝑏 𝜃 = 0
Natural Frequency
𝜔𝑛 =𝑘𝑒
𝑚𝑒
=𝑚𝑔𝑏
𝐽𝐺 + 𝑚 𝑅 − 𝑏 2
=𝑔
𝐽𝐺 + 𝑚 𝑅 − 𝑏 2
𝑚𝑏
Vibrations 3.65 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s Equations
- Ex.3.13 Translating System with a Pre-tensioned/compressedSpring
Derive the governing equation of motion for vertical
translations 𝑥 of the mass about the static
equilibrium position of the system
Solution
The equation of motion will be derived for
“small” amplitude vertical oscillations; that is,
𝑥/𝐿 ≪ 1
The horizontal spring is pretensioned with a tension, which is
produced by an initial extension of the spring by an amount 𝛿0
𝑇1 = 𝑘1𝛿0
The kinetic energy of the system
𝑇 =1
2𝑚 𝑥2
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 + 𝑥 𝑛 = 1 + 𝑛𝑥 +1
2𝑛(𝑛 − 1)𝑥2 + ⋯
§6.Lagrange’s Equations
The potential energy of the system
𝑉 =1
2𝑘1 𝛿0 + ∆𝐿 2
𝑓𝑜𝑟 𝑠𝑝𝑟𝑖𝑛𝑔 𝑘1
+1
2𝑘2𝑥
2
𝑓𝑜𝑟 𝑠𝑝𝑟𝑖𝑛𝑔 𝑘2
∆𝐿 : the change in the length of the spring with
stiffness 𝑘1 due to the motion 𝑥 of the mass
∆𝐿 = 𝐿2 + 𝑥2 − 𝐿 = 𝐿 1 + (𝑥/𝐿)2− 𝐿
Assume that |𝑥/𝐿| ≪ 1, using binomial expansion
1+(𝑥/𝐿)2= 1+(𝑥/𝐿)2 1/2 = 1+1
2(𝑥/𝐿)2+
1
8(𝑥/𝐿)4+⋯
⟹ ∆𝐿 ≈ 𝐿 1+(𝑥/𝐿)2/2 − 𝐿 = 𝐿(𝑥/𝐿)2/2
⟹ 𝑉 =1
2𝑘1 𝛿0 +
𝐿
2
𝑥
𝐿
2 2
+1
2𝑘2𝑥
2
Vibrations 3.67 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
𝑑
𝑑𝑡
𝜕𝑇
𝜕 𝑞1−
𝜕𝑇
𝜕𝑞1+
𝜕𝐷
𝜕 𝑞1+
𝜕𝑉
𝜕𝑞1= 𝑄1 (3.44)
§6.Lagrange’s Equations
Chose the generalize coordinate 𝑞1 = 𝑥
𝑇 =1
2𝑚 𝑥2
⟹𝑑
𝑑𝑡
𝜕𝑇
𝜕 𝑥=
𝑑
𝑑𝑡𝑚 𝑥 = 𝑚 𝑥
𝜕𝑇
𝜕𝑥= 0,
𝜕𝐷
𝜕 𝑥= 0, 𝑄 = 0
𝑉 =1
2𝑘1 𝛿0 +
𝐿
2
𝑥
𝐿
2 2
+1
2𝑘2𝑥
2
⟹𝜕𝑉
𝜕𝑥=𝑘1 𝛿0 +
𝐿
2
𝑥
𝐿
2 2𝑥
𝐿+𝑘2𝑥= 𝑘1 +
𝑘1𝛿0
𝐿𝑥+
𝑘1
2
𝑥3
𝐿2≈ 𝑘2 +
𝑇1𝐿
𝑥
Vibrations 3.68 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s Equations
The governing equation of motion
𝑚 𝑥 + 𝑘2 +𝑇1
𝐿𝑥 = 0
The natural frequency
𝜔𝑛 = 𝑘𝑒/𝑚𝑒 = 𝑘2 + 𝑇1/𝐿 /𝑚
If the spring of constant 𝑘1 is compressed instead of being in
tension, then we can replace 𝑇1 by −𝑇1 , and the natural
frequency
𝜔𝑛 = 𝑘𝑒/𝑚𝑒 = 𝑘2 − 𝑇1/𝐿 /𝑚
The natural frequency 𝜔𝑛 can be made very low by adjusting
the compression of the spring with stiffness 𝑘1. At the same
time, the spring with stiffness 𝑘2 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
§6.Lagrange’s 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 𝑚
𝑣𝑚 =𝑑 𝑟𝑚𝑑𝑡
=𝑑
𝑑𝑡𝑥 + 𝐿𝑠𝑖𝑛𝜃 𝑖 + 𝐿 − 𝐿𝑐𝑜𝑠𝜃 𝑗
= −𝑅 𝜃 + 𝐿 𝜃𝑐𝑜𝑠𝜃 𝑖 + 𝐿 𝜃𝑠𝑖𝑛𝜃 𝑗
Vibrations 3.70 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s Equations
The kinetic energy of the system 𝑇 = 𝑇𝑑 + 𝑇𝑝
𝑇𝑑 =1
2𝑚𝑑 𝑥2 +
1
2𝐽𝐺 𝜃2
=1
2𝑚𝑑𝑅2 𝜃2 +
1
2𝐽𝐺 𝜃2
𝑇𝑝 =1
2𝑚𝑣𝑚
2
=1
2𝑚 −𝑅 𝜃 + 𝐿 𝜃𝑐𝑜𝑠𝜃 𝑖 + 𝐿 𝜃𝑠𝑖𝑛𝜃 𝑗
2
=1
2𝑚(𝑅2 + 𝐿2 − 2𝐿𝑅𝑐𝑜𝑠𝜃) 𝜃2
≈1
2𝑚 𝐿 − 𝑅 2 𝜃2
⟹ 𝑇 = 𝑇𝑑 + 𝑇𝑝 =1
2𝑚 𝐿 − 𝑅 2 + 𝑚𝑑𝑅2 + 𝐽𝐺 𝜃2 ≡
1
2𝑚𝑒
𝜃2
Vibrations 3.71 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s Equations
The potential energy of the system
𝑉 =1
2𝑘𝑥2 + 𝑚𝑔(𝐿 − 𝐿𝑐𝑜𝑠𝜃)
=1
2𝑘𝑅2𝜃2 + 𝑚𝑔𝐿 1 − 𝑐𝑜𝑠𝜃
⟹ 𝑉 =1
2𝑘𝑅2𝜃2 +
1
2𝑚𝑔𝐿𝜃2 𝑐𝑜𝑠𝜃 ≈ 1 −
𝜃2
2
=1
2𝑘𝑅2 + 𝑚𝑔𝐿 𝜃2 ≡
1
2𝑘𝑒𝜃
2
The dissipation function
𝐷 =1
2𝑐 𝑥2 =
1
2𝑐𝑅2 𝜃2 ≡
1
2𝑐𝑒
𝜃2
𝑚𝑒 𝜃 + 𝑐𝑒
𝜃 + 𝑘𝑒𝜃 = 0,𝜔𝑛 =𝑘𝑒
𝑚𝑒=
𝑘𝑅2 + 𝑚𝑔𝐿
𝑚(𝐿 − 𝑅)2+𝑚𝑑𝑅2 + 𝐽𝐺
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
§6.Lagrange’s 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
𝑉 =1
2𝑘𝑡𝜑
2 +1
2𝑘 𝑥0 𝑡 −𝑥1
2 +1
4𝑚2𝑔(𝐿2 −𝐿1)𝜑
2
=1
2𝑘𝑡𝜑
2 +1
2𝑘 𝑥0 𝑡 −𝐿2𝜑
2 +1
4𝑚2𝑔(𝐿2 −𝐿1)𝜑
2
The kinetic energy
𝑇 =1
2𝐽0 𝜑2 +
1
2𝑚1 𝑥1
2 =1
2𝐽0 +𝑚1𝐿2
2 𝜑2 ≡1
2𝑚𝑒 𝜑2
Dissipation function
Vibrations 3.73 Single DOF Systems: Governing Equations
𝐷 =1
2𝑐 𝑥2
2 =1
2𝑐𝐿1
2 𝜑2 ≡1
2𝑐𝑒 𝜑2
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s Equations
𝑉 =1
2𝑘𝑡𝜑
2 +1
2𝑘 𝑥0 𝑡 − 𝐿2𝜑
2 +1
4𝑚2𝑔(𝐿2 − 𝐿1)𝜑
2
The potential energy is not in the standard form ⟹ the
governing equation must be derived from Lagrange’s equation𝜕𝑉
𝜕𝜑= 𝑘𝑡 + 𝑘𝐿2
2 +1
2𝑚2𝑔(𝐿2 − 𝐿1) 𝜑 − 𝑘𝐿2𝑥0 𝑡
= 𝑘𝑒𝜑 − 𝑘𝐿2𝑥0 𝑡
The governing equation of motion
𝑚𝑒 𝜑 + 𝑐𝑒 𝜑 + 𝑘𝑒𝜑 = 𝑘𝐿2𝑥0(𝑡)
The natural frequency
𝜔𝑛 =𝑘𝑒
𝑚𝑒=
𝑘𝑡 + 𝑘𝐿22 + 𝑚2𝑔(𝐿2 − 𝐿1)/2
𝐽0 + 𝑚1𝐿22
Vibrations 3.74 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s Equations
Ex.3.16 Slider Mechanism
Obtain the equation of motion of the slider mechanism
Solution
The geometric constraints on the motion
𝑟2 𝜑 = 𝑎2 + 𝑏2 − 2𝑎𝑏𝑐𝑜𝑠𝜑 (a)
⟹ 𝑟 𝜑 =𝑎𝑏
𝑟(𝜑) 𝜑𝑠𝑖𝑛𝜑
𝑟 𝜑 𝑠𝑖𝑛𝛽 = 𝑏𝑠𝑖𝑛𝜑 (b)
𝑎 = 𝑟 𝜑 𝑐𝑜𝑠𝛽 + 𝑏𝑐𝑜𝑠𝜑 (c)
⟹ 𝑟 𝜑 𝑐𝑜𝑠𝛽 − 𝑟 𝜑 𝛽𝑠𝑖𝑛𝛽 − 𝑏 𝜑𝑠𝑖𝑛𝜑 = 0
⟹ 𝛽 = 𝑟 𝜑 𝑐𝑜𝑠𝛽 − 𝑏 𝜑𝑠𝑖𝑛𝜑
𝑟(𝜑)𝑠𝑖𝑛𝛽=
𝜑
𝑟2(𝜑)𝑎𝑏𝑐𝑜𝑠𝜑 − 𝑏2
Vibrations 3.75 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s Equations
System Kinetic Energy
𝑇 =1
2𝐽𝑚𝑏 +𝐽𝑚𝑒 𝜑2 +
1
2𝐽𝑚𝑙
𝛽2 +1
2𝑚𝑠 𝑟2 +
1
2𝑚𝑠𝑟
2 𝛽2
𝐽𝑚𝑏 =1
3𝑚𝑏𝑏
2
𝐽𝑚𝑒 =1
3𝑚𝑒𝑒
2
𝐽𝑚𝑙 =1
3𝑚𝑙𝑙
2
𝑚 𝜑 ≡ 𝐽𝑚𝑏 + 𝐽𝑚𝑒 + 𝐽𝑚𝑙 + 𝑚𝑠𝑟2
𝑎𝑏𝑐𝑜𝑠𝜑 − 𝑏2
𝑟2(𝜑)
2
+ 𝑚𝑠
𝑎𝑏𝑠𝑖𝑛𝜑
𝑟(𝜑)
2
⟹ 𝑇 =1
2𝑚(𝜑) 𝜑2
Vibrations 3.76 Single DOF Systems: Governing Equations
where
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
𝑑
𝑑𝑡
𝜕𝑇
𝜕 𝑞1−
𝜕𝑇
𝜕𝑞1+
𝜕𝐷
𝜕 𝑞1+
𝜕𝑉
𝜕𝑞1= 𝑄1 (3.44)
§6.Lagrange’s Equations
System Kinetic Energy
𝑇 =1
2𝑚(𝜑) 𝜑2
System Potential Energy
𝑉 =1
2𝑘𝑟2(𝜑) +
1
2𝑘𝑑 𝑑 𝑡 − 𝑒𝜑 2
Equation of motion
𝑑
𝑑𝑡
𝜕𝑇
𝜕 𝜑=
𝑑
𝑑𝑡𝑚(𝜑) 𝜑 = 𝑚(𝜑) 𝜑,
𝜕𝑇
𝜕𝜑= 𝑚′ 𝜑 𝜑2
𝜕𝑉
𝜕𝜑= 𝑘𝑟 𝜑 𝑟′ 𝜑 + 𝑘𝑑𝑒2𝜑 − 𝑘𝑑𝑒2𝑑(𝑡)
⟹ 𝑚 𝜑 𝜑 +1
2𝑚′ 𝜑 𝜑2 + 𝑘𝑟 𝜑 𝑟′ 𝜑 + 𝑘𝑑𝑒2𝜑 = 𝑘𝑑𝑒2𝑑(𝑡)
Vibrations 3.77 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s Equations
- Ex.3.17 Oscillations of A Crankshaft
Obtain the equation of motion of the crankshaft
Solution
• Kinematics
The position vector of the
slider mass 𝑚𝑝
𝑟𝑃 = 𝑟𝑐𝑜𝑠𝜃 + 𝑙𝑐𝑜𝑠𝛾 𝑖 + 𝑑 𝑗
The position vector of the center of mass 𝐺 of the crank
𝑟𝐺 = 𝑟𝑐𝑜𝑠𝜃 + 𝑎𝑐𝑜𝑠𝛾 𝑖 + 𝑟𝑠𝑖𝑛𝜃 + 𝑎𝑠𝑖𝑛𝛾 𝑗
From geometry
𝑟𝑠𝑖𝑛𝜃 = 𝑑 + 𝑙𝑠𝑖𝑛𝛾
The slider velocity
𝑣𝑃 = 𝑟𝑃 = −𝑟 𝜃𝑠𝑖𝑛𝜃 − 𝑙 𝛾𝑠𝑖𝑛𝛾 𝑖 = −𝑟 𝜃 𝑠𝑖𝑛𝜃 +𝑡𝑎𝑛𝛾𝑐𝑜𝑠𝜃 𝑖
Vibrations 3.78 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
⟹ 𝑟 𝜃𝑐𝑜𝑠𝜃 = 𝑙 𝛾𝑐𝑜𝑠𝛾 ⟹ 𝛾 =𝑟
𝑙
𝑐𝑜𝑠𝜃
𝑐𝑜𝑠𝛾 𝜃
2/7/2014
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§6.Lagrange’s Equations
The velocity of the center of mass 𝐺 of the crank
𝑣𝐺 = −𝑟 𝜃𝑠𝑖𝑛𝜃 − 𝑎 𝛾𝑠𝑖𝑛𝛾 𝑖 + 𝑟 𝜃𝑐𝑜𝑠𝜃 − 𝑎 𝛾𝑐𝑜𝑠𝛾 𝑗
⟹ 𝑣𝐺 = − 𝑠𝑖𝑛𝜃 +𝑎
𝑙𝑡𝑎𝑛𝛾𝑐𝑜𝑠𝜃 𝑟 𝜃 𝑖 +
𝑏
𝑙𝑐𝑜𝑠𝜃 𝑟 𝜃 𝑗
• System Kinetic Energy
The total kinetic energy of the system
𝑇 =1
2𝐽𝑑 𝜃2 +
1
2𝑚𝐺𝑣𝐺
2 +1
2𝐽𝐺 𝛾2 +
1
2𝑚𝑃𝑣𝑃
2 ≡1
2𝐽(𝜃) 𝜃2
𝐽 𝜃 = 𝐽𝑑 + 𝑟2𝑚𝐺 𝑠𝑖𝑛𝜃 +𝑎
𝑙𝑡𝑎𝑛𝛾𝑐𝑜𝑠𝜃
2
+𝑏
𝑙𝑐𝑜𝑠𝜃
2
+𝐽𝐺𝑟
𝑙
𝑐𝑜𝑠𝜃
𝑐𝑜𝑠𝛾
2
+ 𝑟2𝑚𝑃 𝑠𝑖𝑛𝜃 + 𝑡𝑎𝑛𝛾𝑐𝑜𝑠𝜃 2
Vibrations 3.79 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
where,
𝛾 = 𝑠𝑖𝑛−1𝑟
𝑙𝑠𝑖𝑛𝜃 −
𝑑
𝑙
§6.Lagrange’s Equations
• Equation of Motion
The governing equation of motion has the form
𝑑
𝑑𝑡
𝜕𝑇
𝜕 𝜃−
𝜕𝑇
𝜕𝜃= −𝑀(𝑡)
After performing the differentiation operations
𝐽 𝜃 𝜃 +1
2
𝜕𝐽(𝜃)
𝜕𝜃 𝜃2 = −𝑀(𝑡)
The angle 𝜃 can be expressed
𝜃 𝑡 = 𝜔𝑡 + 𝜙(𝑡)
Then
𝐽 𝜃 𝜙 +1
2
𝜕𝐽(𝜃)
𝜕𝜃𝜔 + 𝜙
2= −𝑀(𝑡)
Vibrations 3.80 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s Equations
- Ex.3.18 Vibration of A Centrifugal Governor
Derive the equation of motion of
governor by usingLagrange’sequation
Solution
The velocity vector relative to point
𝑂 of the left hand mass
𝑉𝑚 = −𝐿 𝜑𝑐𝑜𝑠𝜑 𝑖 + 𝐿 𝜑𝑠𝑖𝑛𝜑 𝑗
+(𝑟 + 𝐿𝑠𝑖𝑛𝜑)𝜔𝑘
The kinetic energy
𝑇 𝜑, 𝜑 = 21
2𝑚 𝑉𝑚𝑉𝑚
= 𝑚 −𝐿 𝜑𝑐𝑜𝑠𝜑 2 + 𝐿 𝜑𝑠𝑖𝑛𝜑 2 + 𝑟 + 𝐿𝑠𝑖𝑛𝜑 𝜔 2
= 𝑚𝜔2 𝑟 + 𝐿𝑠𝑖𝑛𝜑 2 + 𝑚 𝜑2𝐿2
Vibrations 3.81 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s Equations
The potential energy with respect to the static equilibrium position
𝑉 𝜑 =1
2𝑘 2𝐿 1−𝑐𝑜𝑠𝜑 2 −2𝑚𝑔𝐿𝑐𝑜𝑠𝜑
Using equation (3.44) with
𝑞1 = 𝜑
𝐷 = 0
𝑄1 = 0
and performing the required
operations, to obtain the following
governing equation
𝑚𝐿2 𝜑 − 𝑚𝑟𝐿𝜔2𝑐𝑜𝑠𝜑 − 𝑚𝜔2 + 2𝑘 𝐿2𝑠𝑖𝑛𝜑𝑐𝑜𝑠𝜑
+𝐿 𝑚𝑔 + 2𝑘𝐿 𝑠𝑖𝑛𝜑 = 0
Vibrations 3.82 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
𝑑
𝑑𝑡
𝜕𝑇
𝜕 𝑞1−
𝜕𝑇
𝜕𝑞1+
𝜕𝐷
𝜕 𝑞1+
𝜕𝑉
𝜕𝑞1= 𝑄1 (3.44)
§6.Lagrange’s Equations
𝑚𝐿2 𝜑 − 𝑚𝑟𝐿𝜔2𝑐𝑜𝑠𝜑 − 𝑚𝜔2 + 2𝑘 𝐿2𝑠𝑖𝑛𝜑𝑐𝑜𝑠𝜑
+𝐿 𝑚𝑔 + 2𝑘𝐿 𝑠𝑖𝑛𝜑 = 0
Introducing the quantities
𝛾 ≡𝑟
𝐿, 𝜔𝑝
2 ≡𝑔
𝐿, 𝜔𝑛
2 ≡2𝑘
𝑚Rewrite the equation
𝜑 − 𝛾𝜔2𝑐𝑜𝑠𝜑
− 𝜔2 + 𝜔𝑛2 𝑠𝑖𝑛𝜑𝑐𝑜𝑠𝜑
+ 𝜔𝑝2 + 𝜔𝑛
2 𝑠𝑖𝑛𝜑 = 0
Assume that the oscillation 𝜑 about 𝜑 = 0 are small (𝑐𝑜𝑠𝜑 ≈ 1,𝑠𝑖𝑛𝜑 ≈ 𝜑) to get the final equation
𝜑 + 𝜔𝑝2 − 𝜔2 𝜑 = 𝛾𝜔2
Vibrations 3.83 Single DOF Systems: Governing Equations
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
§6.Lagrange’s 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
𝑘𝛿 = 𝑚(𝑅 + 𝛿)Ω2 ⟹ 𝛿 =𝑅
𝜔1𝑛2
Ω2 − 1
, 𝜔1𝑛2 =
𝑘
𝑚
For small angles of rotation, the kinetic energy
𝑇 =1
2
1
2𝑚𝑟2
𝑥
𝑟
2
+1
2𝑚 𝑥2 =
1
2
3
2𝑚 𝑥2
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
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§6.Lagrange’s Equations
The potential energy for oscillations about the equilibrium position
𝑉 =1
2𝑘𝑥2
The Lagrange equation for this undamped system
𝑑
𝑑𝑡
𝜕𝑇
𝜕 𝑥−
𝜕𝑇
𝜕𝑥+
𝜕𝐷
𝜕 𝑥+
𝜕𝑉
𝜕𝑥= 𝑄𝑥 = 𝑚𝑥Ω2
where the centrifugal force 𝑚𝑥Ω2 is treated as an external force
The governing equation
3
2𝑚 𝑥 + 𝑘 − 𝑚Ω2 𝑥 = 0
The natural frequency
𝜔𝑛 =𝑘
𝑚=
2
3𝜔1𝑛
2 − Ω2
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
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