Dynamics and vibrations - Appunti

University of Padova

Dipartimento di Tecnica e Gestione dei Sistemi Industriali

Master’s Degree in Mechatronics Engineering

DYNAMICS AND VIBRATION

Contents

1 Introduction 3

1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Formulation of the Equation of Motion : Set of governing equations . . . . . . . . . . . . 4

1.3 Spring constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Definition of damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.1 Viscous Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4.2 Coulomb Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.3 Hysteretic Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.4 Viscous-elastic Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Free Vibrations of Single-Degree-of-Freedom Systems 11

2.1 Simple Harmonic Oscillator(SDOF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Homogeneous solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Damped Harmonic Oscillator (SDOF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 > 1 Overdamped system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

ξ

2.2.2 Critically damped system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

ξ=1

2.2.3 0<ξ<1 Under damped system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.4 Undamped system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

ξ=0

2.2.5 Under damped system response analyze . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.6 Estimation of value (can be an exercise in the quiz) . . . . . . . . . . . . . . . . 18

ξ

2.3 Proof of the no influence of g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Self excited vibration(stick-slip phenomenon) . . . . . . . . . . . . . . . . . . . . 20

3 Forced Vibrations of Single-Degree-of-Freedom Systems 23

3.1 Particular solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

x 2

3.1.1 Harmonic force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Structural damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Half power bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Response to a static unbalance system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.1 Transmissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5 Spring configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6.1 Sky hook damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.7 Accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.7.1 FRF of Accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.7.2 Vibrometer FRF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.8 Forced response to all the type of force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Equivalent System 54

4.1 Spring with non-negligible mass(distributed mass)(can be asked in the quiz)) . . . . . . . 56

4.1.1 Angular (rotational) vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 Exercise on Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.1 Vibration of a lifting system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.2 Vibration of a simple pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2.3 Vibration of a mechanism with gears . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2.4 Vibrations of Floating Cylinder (it can be found during the quiz) . . . . . . . . . . 64

5 Vibrations of Multi-Degree-Of-Freedom linear system 65

5.1 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.1.1 Decouple the equation of motion of an n-DOF linear system . . . . . . . . . . . . 69

5.1.2 2-DOF system exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.1.3 Half car like model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

1

5.1.4 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6 Beating Phenomenon 85

6.1 Two pendulum system (2-DOFs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2.1 Modal damping (viscous damping) . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2.2 Proportional damping-Rayleigh damping . . . . . . . . . . . . . . . . . . . . . . . 92

7 Forced Response of a multi degree of freedom system 94

7.0.1 Frequency response of a multi-degree-of-freedom (mdof) system by modal superpo-

sition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.1 Antiresonance Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.1.1 Dynamic vibration absorbers FRAHM . . . . . . . . . . . . . . . . . . . . . . . . 103

7.1.2 Centrifugal pendulum (tuned to excitation orders) . . . . . . . . . . . . . . . . . . 103

8 Torsional vibrations 105

8.0.1 Power transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

8.0.2 Geared torsional equivalent system . . . . . . . . . . . . . . . . . . . . . . . . . . 108

8.0.3 Inelastic impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

9 Vibrations of continuous system 113

9.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

9.1.1 Simply supported beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

9.1.2 Modal analysis beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

2

1 Introduction

All the topics inside this file can be found during the quiz and the open question during the exam. The

most common quiz exercises are highlighted with a phrase.

1.1 Definitions A system store energy in the form of Kinetic and potential energy (e.g. elastic or

Mechanical system:

potential energy) and exchanges energy with the environment through mechanical work and heat generated

by irreversible dissipation phenomena.

It’s state is described by mechanical variables (position velocity and acceleration and its evolution

x, ẋ, ẍ),

is governed by the principles of mechanics

1 kinetic energy

2

T mv

= 2

1 potential energy

2

−

U k(x x

= )

0

2

Typically in a mechanical system the only energy stored is the kinetic and elastic energy .

U

However, mechanical systems can not only store energy, but they can also exchange it with the environment.

Usually, the forms of energy exchanged are heat and work. Work can be both absorbed by the system

and released to the environment; on the contrary, heat always tends to decrease the internal energy of the

system. B

Z

Work F̄ ds̄

W

= = A

: The minimum of kinematically independents coordinates required to

Degrees of freedom (DOFs)

completely describe the configuration and motion of mechanical system.

Rigid body in a plane possible movement

Figure 1:

A set of n independent coordinates for a system with n DOFs is called a set of generalized coordinates (not

uniquely defined)

example: Rigid body in a plane: 3 DOFs (2 translations, 1 rotation Figure 1) Rigid body in a space: 6

DOFs(3 translations, 3 rotations) The translation is typically selected on orthogonal axis.

3

A mathematical formula, usually differential equation, that describes the relationship

Equation of Motion:

between the motion of a mechanical system and the force acting on it.

It establishes how position, velocity, and acceleration evolve over time as a consequence of applied forces

and initial conditions mẍ(t) kx(t) F

+ = (t)

The time evaluation of the mechanical variables of system as determined by its

Dynamic Behavior:

equations of motion.

In vibration analysis, understanding the dynamic behavior of the simple harmonic oscillator is fundamental

for interpreting the response of and continuous systems (e.g. rods, beams and strings).

SDOF, MDOF,

As long as the system is linear or can otherwise be linearized (i.e., small vibrations), the response of an

system can be obtained through the superposition of the effects of SDOF systems (an infinite

n-DOF n

number, instead, for continuous systems).

1.2 Formulation of the Equation of Motion : Set of governing equations

An equation of motion is a mathematical formula that describes how a physical system’s position, velocity,

or acceleration changes over time. In classical mechanics, the most fundamental root is Newton’s second

law, which connects force, mass, and acceleration.

• the balance of forces and moments acting on the system. typically

Equilibrium equations:Express

is referred to the static equilibrium which is when we have a specific position and 0 velocity . They

may be formulated:

In terms of force and moment balances trough Newton’s second law(linear balance, it governs

– the translation dynamics of rigid body) and Euler’s equation (angular momentum balance,

governs the rotational dynamics o fa rigid body)

In terms of energy balance trough an energy formulation (e.g. Lagrangian approach , Principle

– of Virtual work) In this course we will use more Newton’s second law:

m dQ̄ Z

Z

X F̄ F̄ dt dQ̄ impulse momentum theorem

F̄ dt dQ̄

= =

=⇒ = =⇒

i dt

i=1 Q̄ mv̄

=

m dv̄ cm Newton’s second law

X F̄ m mā

= =

i cm

dt

i=1

• Describe the relationship between forces and displacement (or velocities)

Constitutive equations:

as determined by the material properties and mechanical components of the system.

4

|F | |kx(t)| Hooke’s law

=

e |F | |C ẋ(t)|

=

v

Elastic force and viscous force is opposite respectively to the displacement and velocity(they

ALWAYS

try to stop the system).

• Relate displacements, velocities , and accelerations within the

Kinematic compatibility equations:

system, ensuring that the motion satisfies geometric constrains derived from Kinematic analysis of

the mechanism. Connecting rod and crank mechanism

Figure 2:

this type of equations are useful for system like the following one

System that exhibit angular vibration

Figure 3:

1.3 Spring constant

Typically indicate with the letter k, is the most important parameter that allows us to quantify the force

that a body oppose to an external force. 5

Simulation of a new piece

Figure 4:

We use some simulation, as the one in Figure 4 in order to find some parameter for example: Young’s

modulus, G shear modulus, I second moment of area, l length, d spring wire diameter, D coil diameter, n

number of active coils, that trough the following formulas give us the spring constant.:

4

• Coil Spring: Gd

k = 3

8nD

• Cantilever Beam (point load at end): 3EI

k = 3

l

• Simply Supported Beam (center load): 48EI

k = 3

l

• Simply Supported Beam (off-center load): 3EIl

k = 2 2

l l

1 2

• Clamped-Clamped Beam (center load): 192EI

k = 3

l

however in the course we will always know the k value.

1.4 Definition of damping

Before starting the analysis of the various cases of the damped system, it is important to know what is

meant by damping:

• Damping represents the mechanisms responsible for energy dissipation in vibrating systems (usually

conversion into heat typically generate from the movement of a fluid trough pins)

• Damping arises from a variety of physical phenomena:

Viscous friction (viscous effects in fluids)

– Coulomb (dry) friction

– Material damping (internal friction, mechanical hysteresis: relative motion between particles

– during deformation)

Impacts (conversion of energy into higher frequency vibration which is then dissipated)

– Acoustic radiation (vibration energy converted into sound)

– ...

–

• Different idealized damping models can be introduced

Viscous damping

– Coulomb damping

– Hysteretic (or structural) damping

– ...

– 6

1.4.1 Viscous Damping

The damping force is proportional to the relative velocity

Viscous damper

Figure 5:

The damping force is proportional to the relative velocity by the coefficient c:

− −

f c(

ẋ ẋ f c(

ẋ ẋ

= ), = )

1 1 2 2 2 1

typically one of the two is fixed to the ground make that velocity equal to zero.

Free vibration effect of viscous damping

Figure 6:

In Figure 6 it possible to see how a system respond to a step force with a damping factor lower than one.

We will discuss this case later but just for completeness the displacement varies as:

−ξω t

n

x Xe sin(ω t ϕ)

= +

d

| {z } | {z }

exponential decay term oscillatory term

Lets have a look to the operating principle of shock absorbers: Figure 7, presents two side-by-side cross-

sectional diagrams illustrating the internal fluid dynamics of a shock absorber during its two primary

operational phases. The left schematic demonstrates the deflective beam and washer method of orificing

during the compression stroke, where the downward movement of the piston rod forces fluid upward

through Valve 1 while simultaneously pushing the floating piston down into the variable-volume chamber.

Conversely, the right schematic details the rebound phase, showing the piston rod retracting upward and

drawing fluid back down through Valve 2. Both diagrams carefully label essential structural components,

such as the working cylinder, protective tube, piston valve, and mounting points, to clearly convey how the

internal mechanisms and fluid routing adapt to opposing directional forces. Keep in mind that the force is

always opposite to the velocity. 7

Internal components of a shock absorber

Figure 7:

(a) (b)

Example behavior of shock absorber

Figure 8:

Figure 8 displays three cross-sectional line drawings of shock absorbers paired with Force versus Velocity

graphs to illustrate distinct damping profiles. The leftmost schematic shows a fixed orifice mechanism

producing a progressive, exponential force curve, while the middle diagram features a spring-loaded deflec-

tive valve resulting in a digressive curve that flattens at higher velocities. The tilted diagram on the right

depicts a combined valving system, with its associated graph showing multiple curves to demonstrate how

blending these methods can yield tunable linear, progressive, or digressive suspension characteristics.

1.4.2 Coulomb Damping

The damping force has constant magnitude (independent of velocity once motion has started) and acts in

the direction opposite to the velocity. 8

Coulomb damping principle

Figure 9:

The damping force is proportional to the friction force which depend to the weight and the coefficient

F d

µ: 

−µF −µmg if ẋ >

= 0

n







F = if ẋ

0 = 0

d 



µF µmg if ẋ <

= 0

 n

Free vibration effect of Coulomb damping

Figure 10:

In Figure 10 it possible to see how in the free response, the amplitude decreases linearly with time. The

system may stop before reaching the equilibrium position and that is caused by the presence of the static

friction coefficient. The frequency is approximately the undamped natural frequency: Coulomb damping

does not significantly alter the frequency

1.4.3 Hysteretic Damping

In Hysteretic damping force is in-phase with displacement and proportional to velocity.

Difference between causal and acausal

Figure 11: 9

Defined in the complex domain via a loss factor (η), complex stiffness represents the energy dissipated per

cycle under harmonic excitation as the area enclosed by the load–displacement diagram. This model intro-

duces an equivalent viscous damping coefficient inversely proportional to the angular frequency; however,

because it is acausal, it is strictly unsuitable for describing transient responses. Due to its frequency-domain

definition, the hysteretic damping model is unsuitable for describing transient responses (cannot be view

on the free response of the system), as it introduces an acausal behavior where the system’s displacement

appears to anticipate the applied force. Hysteresis Loop

Figure 12:

This Figure 12 illustrates the a fundamental concept in structural dynamics and material

Hysteresis Loop,

science. It represents the relationship between and during a

Stress (Force) Strain (Displacement)

complete loading and unloading cycle.

1.4.4 Viscous-elastic Damping

(Materials plastics, polymers, rubbers, etc.) Viscous damping at the structural level can be interpreted as

the macroscopic effect of Kelvin-Voigt-type viscous-elastic behavior at the material level: σ Eϵ η ϵ̇.

= +

Basic viscous-elastic models and the Standard Linear Solid (Zener) model.

Figure 13:

Regarding the it effectively possesses an internal degree of freedom where the damping

Maxwell model,

coefficient becomes frequency-dependent: This model has limitations at low frequencies (ω →

η η(ω).

= 0)

because it behaves like a viscous fluid, meaning it cannot support a static load and fails to represent

static equilibrium correctly. By contrast, the model (or Zener model),

Standard Linear Solid (SLS)

which introduces an additional spring in parallel, remains physically consistent at low frequencies by

K

2

providing a proper static elastic response. Hysteretic damping models arise naturally when a material

exhibits frequency-dependent compl