Chapter 1 Waves and Vibration PDF

Chapter I Linear Systems with One Degree of Freedom I.1. Overview of Vibrations I.1.1. Definition of an Oscillation An oscillation refers to the movement of a body that alternates back and forth around an equilibrium position. Examples: the motion of a pe...

Chapter I Linear Systems with One Degree of Freedom I.1. Overview of Vibrations I.1.1. Definition of an Oscillation An oscillation refers to the movement of a body that alternates back and forth around an equilibrium position. Examples: the motion of a pendulum, a weight suspended from a spring, or a floating cylinder in a liquid, etc. Oscillations can be subdivided into:  Free oscillations: An oscillator is considered free if it oscillates without external intervention (i.e., without friction) as it returns to equilibrium.  Damped oscillations: The oscillator is subject to frictional forces that dissipate energy, causing the oscillations to gradually diminish and eventually stop.  Forced oscillations: An oscillator is forced when an external action supplies energy to it.  Damped forced oscillations: Here, a periodic external force (excitation) compensates for energy losses due to friction, allowing the oscillations to be sustained without damping. I.2. Vibration A motion that repeats itself over time is called vibratory or oscillatory motion. Any mechanical system with mass and a flexible element (such as a spring) or its equivalent in an electrical system (like inductance or capacitance) can undergo vibratory motion. In most cases, these systems include a damper in mechanical systems or a resistor in electrical systems. In general, a vibratory system can be represented by the following schematics:  Mechanical System: (M) Mass spring damper 1 Chapter I Linear Systems with One Degree of Freedom  Electrical System: Inductor Capacitor Resistor Figure I.1: Graphical Representation of the Vibratory System Elements In another way, it can be said that every vibratory system consists of three components:  Mechanical System  A means of storing kinetic energy, which is the mass.  A means of storing potential energy, which is the spring.  A means of dissipating energy, which is the damper.  Electrical System  A means of storing electrical energy, which is the capacitor.  A means of storing magnetic energy, which is the inductor.  A means of dissipating energy, which is the electrical resistor. During vibrations (oscillations), energy transforms from one form to another: from kinetic energy to potential energy and vice versa in a mechanical system, or from electrical energy to magnetic energy and vice versa in an electrical system. The simplest example of a mechanical vibratory system is the simple pendulum. In each oscillation, energy is converted from potential energy to kinetic energy and vice versa. Assuming that, at the initial moment, the mass is at position 1 (Fig. I.2), the total energy is in the form of potential energy. 2 Chapter I Linear Systems with One Degree of Freedom U  l1 cos   T 0 l U  l1 cos  T 0 (3) (2) (1) U 0 T=Tmax= Figure I.2: Transformation of potential energy to kinetic energy and vice versa in the vibrations of a simple pendulum. When the mass is released, the kinetic energy increases while the potential energy decreases until the complete transformation of potential energy into kinetic energy at position 2 (fig. I.2), where the mass has completed a quarter of the cycle. From position 2, potential energy increases and kinetic energy decreases until the complete transformation into potential energy at position 3 (the mass has completed half a cycle). The return of the mass to position 1 follows the same process, constituting a complete cycle. In a real scenario, during the movement of the mass, some energy is exchanged with the external environment. This portion is irrecoverable, leading to energy loss in the system with each cycle. I.3. Classification of Vibrations A vibration is a motion around the equilibrium position. It is characterized by a motion equation of the type of a second-order differential equation of the form: +2 + = (I.1) with: y: The displacement (m) : The velocity (m/s) : The acceleration (m/s²) δ: The damping coefficient : The natural frequency (rad/s) A(t): The forcing function. 3 Chapter I Linear Systems with One Degree of Freedom I.3.1 Free Vibrations and Forced Vibrations After an initial disturbance, a vibrating system is left without any action from an external force; in this case, the vibrations are known as free vibrations. The oscillation of a simple pendulum is an example of this type of vibration. If the system is subjected to an external force throughout these vibrations, the resulting vibrations are called forced vibrations. The simple pendulum can also serve as an example if the mass is subjected to an external periodic force. I.3.2 Damped Vibrations and Undamped Vibrations If the total energy of the system is conserved during the vibrations (meaning there is no energy dissipation), these vibrations are referred to as undamped vibrations. In contrast, if the system loses energy during these vibrations (considering the air resistance on the mass of the simple pendulum in the previous example), after a certain time, the mass stops due to energy dissipation. These vibrations are called damped vibrations. I.3.3 Regular Vibrations and Irregular Vibrations If the amplitude of the vibratory motion is known at all times, the resulting vibrations are called regular or deterministic. In other words, it is the vibratory motion for which the amplitude can be predicted at any moment. Conversely, if the amplitude cannot be predicted, the vibrations are referred to as irregular or non-deterministic. I.4 Definition of a Periodic Motion A periodic motion is one that repeats itself, with each cycle being identical. The duration of one cycle is called the period T, which is expressed in seconds (s).  The number of repetitions per second is called frequency (denoted as f, measured in Hertz (Hz) or (s-1) is related to the period by: (I.2) The number of rotations per second is called angular frequency (denoted as ω\omegaω, 4 Chapter I Linear Systems with One Degree of Freedom measured in radians per second (rad/s)). (I.3) Mathematically, the periodic motion with period T is defined by: x(t+T)=x(t). I.4.1 Vibratory Motion: A vibratory motion is a periodic sinusoidal motion, also referred to as harmonic motion, when the material body in motion reaches the same position and has the same velocity after regular time intervals of duration T. A vibratory motion can also be defined by its frequency f. The frequency indicates the number of complete oscillations (in the back-and-forth sense) occurring per second. I.4.2 Sinusoidal Vibratory Motion: If the displacement x (y or z) of a vibrating point is a simple sinusoidal function of time of the form: x(t) = A sin( ωt + φ) or x(t) = A cos( ωt + φ) x(t) is called the displacement (or position) at time t. A: the amplitude of a motion or the maximum displacement. ω: the angular frequency of the motion, expressed in (rad/s). φ: the initial phase, corresponding to the phase at time t=0s. Example: In Figure I.3, consider a sinusoidal vibratory motion of the form: g(t) A T(s) T Figure I.3 Example of a sinusoidal periodic motion. 5 Chapter I Linear Systems with One Degree of Freedom x(t)= Acos( ωt + φ), with a period T=1s and an amplitude of 1cm. A = 1 ⇒ x(t) = cos ( 2πt + φ). À t = 0 ⇒ x(0) = 0 ⇒ φ = ⇒X(t)=cos (2πt + ) (cm) I.4.3 Oscillatory Motion: An oscillating system is characterized by periodic movements around an equilibrium position due to an external disturbance. When the motion is sinusoidal, the oscillator is referred to as harmonic. Note: The oscillation is said to be anharmonic if the system evolves according to a periodic law of any non-sinusoidal form (Fig I.4). x(t) t T (b) Figure 1.4: Representation of an anharmonic oscillation. Two types of vibrations (oscillations) are distinguished:  Mechanical oscillations (simple pendulum, vibrating string, etc.)  Electromagnetic oscillations (light, radio waves, etc.) 6 Chapter I Linear Systems with One Degree of Freedom I.5 Generalized Coordinates of a Physical System I.5.1 Generalized Coordinates Generalized coordinates are any set of variables that can specify the state of a physical system. These coordinates are not always assumed to be independent, and their advantage over Cartesian coordinates is the ability to choose the most suitable coordinates to represent the system, taking its constraints into account. Example: In the case of a pendulum, it is advantageous to use the angle of the pendulum as one of the generalized coordinates. Generalized coordinates number n ≤3N, where N is the number of points needed to describe the system, and are often denoted as q 1(t), q2(t) … … … qN(t). I.5.2 Degrees of Freedom: The degree of freedom (DOF) of a system refers to the system's ability to perform translational and rotational motion relative to the axes. It is calculated as the number of related generalized coordinates needed to configure all elements of the system at any given moment minus the number of relationships that connect these coordinates: (I.4) d: Degree of freedom. N: Number of generalized coordinates. r: Number of relationships connecting these coordinates to each other.  Example: A homogeneous cylinder of mass M and radius R rolls without slipping on a horizontal platform. We have two generalized coordinates x and θ, so N=2. x and θ are related by the equation: x =rθ, therefore r=1. Thus, the number of degrees of freedom d=N−r=1. Figure I.5.a Cylinder 7 Chapter I Linear Systems with One Degree of Freedom It is also possible to determine the degree of freedom (DOF) using the following relation: DOF = [number of generalized coordinates] - [coordinates = 0 or constant]. We can list a few examples:  Example: (Particle in free fall) Out of the three coordinates x,y,z, the number of generalized coordinates is 3. We have two constant coordinates: y and z. DOF= 3-2 =1. Figure I.5.b Particle in Free Fall Example: Consider a mechanical system consisting of two points connected by a rod of length L. Find the number of degrees of freedom. M1(x1, y1, z1) so N=6 M1 l M2 M2(x2, y2, z2) Figure I.5.c Particle in Free Fall Constraint equation, ⇒ So d=5 (d=N-r=6-1=5) I.6 Choice of Method When choosing the calculation method to derive the equation of motion (EOM), we distinguish between:  Newton's equation.  Lagrange's equation. I.6.1 Newton's Equation This formalism is based on the fundamental principle of dynamics and is applied according to the type of motion, whether it be translation or rotation. 8 Chapter I Linear Systems with One Degree of Freedom I.6.1.1 Translational Motion: If a mass m system is subjected to external forces, the fundamental law of dynamics (F.L.D.) gives us: (I.5) I.6.1.2 Rotational Motion: The fundamental law of dynamics for rotational motion is expressed as: (1.6) The kinetic energy of rotation T of a body with moment of inertia J/Δ is: (I.7) (J is the moment of inertia with respect to the point on the axis of rotation) Table I.1 Moments of inertia of solids Shape Moment of inertia with respect to the center of gravity G (J/s) Rod (Length L, Mass M) Disk (Radius R, Mass M) Ring (Radius R, Mass M) Cylinder (Radius R, Mass M) Solid Sphere (Radius R, Mass M) Hollow Sphere (Radius R, Mass M) Point Mass m 0 Note: The moment of inertia of a mass M with any shape around a point A different from the center of gravity G is given by: J/A=J/G+M(AG)2 (Huygens-Steiner Theorem) I.6.2 Lagrange's Equation The Lagrangian function (Lagrangian of the system) is the difference between the kinetic energy and the potential energy of the system: 9 Chapter I Linear Systems with One Degree of Freedom (I.8) (I.9) : is the generalized coordinate that characterizes the vibratory motion. : The generalized external forces. D: is the dissipation function: , where is the coefficient of viscous friction. I.7 Free Undamped Systems (Free Oscillators) An oscillating system in the absence of any excitation force (external forces) is called a free (undamped free oscillator). The number of independent variables describing the system is called the degree of freedom (DOF). I.8 Harmonic Oscillator In mechanics, a harmonic oscillator is one that, when displaced from its equilibrium position by a distance (or angle ), is subjected to a restoring force that is opposite and proportional to the displacement (or θ). I.9 Equation of Motion The differential equation of an undamped free motion is in the form: The Lagrange equation for free oscillations of a conservative system is given by: (I.10)  Example 1: (Mass-Spring) Consider a mass attached to a spring with a spring constant K, applying a displacement to the mass in the x direction. F: Restoring force of the spring:  Newton's Dynamic Principle: Projection on the X-Axis 10 Chapter I Linear Systems with One Degree of Freedom ⇒ Figure I.6.a Mass-Spring At Equilibrium : ⇒ ⇒ ⇒ ⇒ is the Differential Equation of Motion ,  Calculation of the Lagrangian: = Lagrange's Equation: ⇒ ⇒ ⇒ The Ratio Being Positive and Setting: We obtain the differential equation of a harmonic vibration in the form: → x → Example 2: (Simple Pendulum) ⇒ ⇒ → y The kinetic energy: Figure I.6.b Simple Pendulum 11 Chapter I Linear Systems with One Degree of Freedom So ⇒ ⇒ Potential Energy of a System: ⇒ Calculation of the Lagrangian: L=T-U= Lagrange's Equation: ⇒ ⇒ ⇒ The Ratio Being Positive and Setting: We obtain the differential equation of a harmonic vibration in the form: This second-order, homogeneous differential equation, with being the natural frequency of the oscillator, must be positive for vibration to occur. I.10 Solution of the Differential Equation The solution of the equation is in the form: where r is a real number and A is a positive constant. We obtain two solutions: 12 Chapter I Linear Systems with One Degree of Freedom The general solution of the equation of motion According to Euler's formula: So ; Let's suppose: So: We find the general solution of equation , is and these are constants derived from the initial conditions. I.11 Electrical Harmonic Oscillator Let’s consider an LC electrical circuit consisting of a capacitor C and an inductor L. This circuit acts as an electrical harmonic oscillator, similar to a mechanical system like a spring with a mass.  Kirchhoff's Loop Law By applying Kirchhoff's loop law to this circuit, we get the following relation: Figure I.6.c LC Circuit 13 Chapter I Linear Systems with One Degree of Freedom ⇒ ⇒ =0 ⇒ q=0 with Calculation of the Lagrangian: = = U=Vc dq= L=T-U= Lagrange's Equation: ⇒ — ⇒ ⇒ ⇒ I.12 Mathematical System (Simple Model) The simplification of a complex vibratory system into a simple model representing the real case is done by determining the equivalent spring from all the existing springs and the equivalent mass from all the masses that make up the system. In what follows, we will provide simple examples to help understand the concept of the equivalent spring and the equivalent mass. I.12.1 Equivalent Spring In practice, springs are found arranged either in series or in parallel. I.12.1.a Springs in Parallel Would you like to continue from here or adjust the translation further? k1 k2 k eq x m x P  mg P  mg Physical System Mathematical Model Figure 1.8: Equivalent Spring for Two Springs in Series in a Mass-Spring System 14 Chapter I Linear Systems with One Degree of Freedom The equilibrium equations for the system are written as follows: For the real system: (I.11) (I.12) For the Mathematical Model: (I.13) From relations (I.12) and (I.13), the stiffness constant of the equivalent spring for two springs in series can be derived using the following relation: (I.14) If the system consists of multiple springs in series, the stiffness constant of the equivalent spring is given by the following relation: (I.15) I.12.1.b Springs in Series The suspension of mass m at the free end of two springs k1 and k2 causes elongations x1 and x2 in k1 and k2 , respectively. The total elongation is given by: (I.16) This relation reflects the fact that for springs in series, the total displacement is the sum of the individual displacements of each spring. k1 k1 keq x1 k2 m xt k2 P Physical system Mathematical Model (1) x2 m (3) P Physical system (2) Figure 1.9: Equivalent Spring of a Mass-Spring System with Two Springs in Series 15 Chapter I Linear Systems with One Degree of Freedom At mechanical equilibrium: If we consider the real system, the following can be written: (I.17) (I.18) For the mathematical Model: (I.19) From the previous relations, we can write: ⇒ (I.20) ⇒ (I.21) We sum equations (1.8) and (1.9), taking into account the following relation: (I.22) ⇒ In the end, we obtain the relation that gives the stiffness constant of the equivalent spring: (I.23) In the general case where the system consists of multiple springs in parallel, the stiffness constant of the equivalent spring can be expressed as follows: (I.24) 16