2.1 Concept and representation of single degree-of-freedom systems
4 min readâĒjuly 30, 2024
systems are the building blocks of vibration analysis. They use one coordinate to describe motion, involving mass, spring, and damping elements. Understanding these systems is crucial for grasping more complex vibration problems.
The for SDOF systems relates displacement, velocity, and acceleration. Key concepts include and , which determine system behavior. These fundamentals apply to real-world applications like simple pendulums, car suspensions, and building seismic analysis.
Single Degree-of-Freedom Systems
Fundamentals of SDOF Systems
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- Single degree-of-freedom (SDOF) system describes motion using one coordinate or variable
- Key components include mass, spring element, and damping element
- Equation of motion relates displacement, velocity, and acceleration through second-order
- Natural frequency determined by system's mass and stiffness
- Higher mass decreases natural frequency
- Higher stiffness increases natural frequency
- Damping ratio influences system response to external forces and energy dissipation
- Low damping ratio results in prolonged oscillations
- High damping ratio leads to quick decay of motion
- Three types of motion based on damping ratio
- (oscillatory decay)
- (fastest return to equilibrium without oscillation)
- (slow return to equilibrium without oscillation)
- System response categorized as free or forced
- determined by initial conditions and system parameters
- depends on external excitations (periodic forces, impulses)
Mathematical Representation of SDOF Systems
- Equation of motion for SDOF system
- m: mass
- c: damping coefficient
- k: spring stiffness
- x: displacement
- F(t): external force
- Natural frequency calculation
- Damping ratio calculation
- General solution for free vibration
- A: amplitude
- :
- :
SDOF Systems in Applications
Real-World Examples of SDOF Systems
- Simple pendulum with angle of displacement as single coordinate
- Used in clocks, seismometers
- Mass suspended on vertical spring for
- Applied in vehicle seats, sensitive equipment mounts
- Car bouncing on suspension approximated as SDOF
- Helps in designing comfortable ride characteristics
- Single-story building under horizontal ground motion during earthquake
- Used for basic seismic analysis and design
- Torsional vibration of shaft with single disk
- Important in rotating machinery design (turbines, generators)
- in tall buildings to reduce wind-induced vibrations
- Examples include Taipei 101, John Hancock Tower
- Vertical motion of floating buoy in calm water
- Used in oceanographic studies, wave energy converters
SDOF Systems in Engineering Design
- Vibration isolators for sensitive equipment (microscopes, precision machinery)
- Reduce transmitted vibrations from environment
- in vehicles
- Improve ride quality and handling
- Seismic base isolation systems for buildings
- Protect structures from earthquake damage
- in power transmission systems
- Reduce harmful vibrations in rotating shafts
- in water towers
- Mitigate wind-induced oscillations
- Mass dampers in sports equipment (tennis rackets, golf clubs)
- Enhance performance by reducing vibrations
Spring-Mass-Damper Models for SDOF
Components of Spring-Mass-Damper Model
- Mass element represents system inertia
- Depicted as rigid block or point mass
- Determines of system
- Spring element represents system stiffness
- Usually shown as coil spring
- Stores
- Linear spring follows Hooke's Law: F = kx
- Damper element represents energy dissipation
- Illustrated as dashpot or viscous damper
- Dissipates energy through heat
- Linear damper force proportional to velocity: F = cv
- Free-body diagram includes
- Inertial force (ma)
- Spring force (kx)
- Damping force (cv)
- External forces (F(t))
Advanced Spring-Mass-Damper Models
- Nonlinear springs for large displacements
- Force-displacement relationship: F = kx + k2x^2 + k3x^3
- Multiple springs in series or parallel
- Series: 1/keq = 1/k1 + 1/k2
- Parallel: keq = k1 + k2
- Alternative damping mechanisms
- (dry friction): F = ΞN * sign(v)
- : F = jkx
- Rotational SDOF systems
- Torsional spring: T = kÎļ
- Rotational damper: T = cÏ
- Two-dimensional SDOF systems
- Planar motion with coupled x and y coordinates
Degrees of Freedom for Systems
Determining Degrees of Freedom
- Degrees of freedom equal minimum number of independent coordinates to define configuration
- Rigid body in 3D space has maximum six degrees of freedom
- Three translational (x, y, z)
- Three rotational (roll, pitch, yaw)
- Constraints reduce degrees of freedom
- Fixed support removes all degrees of freedom
- Pin joint allows rotation but restricts translation
- Calculate degrees of freedom
- Count possible independent motions
- Subtract number of constraints
- Planar motion of free rigid body has three degrees of freedom
- Two translational (x, y)
- One rotational (Îļ)
- Systems with multiple bodies
- Sum individual body degrees of freedom
- Subtract constraints between bodies
Examples of Degrees of Freedom Analysis
- Particle moving in straight line (1 DOF)
- Only x-coordinate needed to describe motion
- Simple pendulum (1 DOF)
- Angle Îļ fully defines position
- Mass sliding on inclined plane (1 DOF)
- Distance along plane describes motion
- Double pendulum (2 DOF)
- Two angles required to define configuration
- Planar four-bar linkage (1 DOF)
- One angle determines position of all links
- Spatial robot arm with 6 joints (6 DOF)
- Each joint angle contributes one degree of freedom
- Gyroscope (3 DOF)
- Three rotational degrees of freedom (precession, nutation, spin)
Key Terms to Review (27)
Amplitude Response: Amplitude response refers to how the amplitude of a system's output varies in relation to the amplitude of the input signal over a range of frequencies. It is crucial for understanding how mechanical systems behave under harmonic excitation, revealing how much a system will respond to different frequencies of input forces, particularly in single degree-of-freedom systems. The concept helps identify resonance and the frequency at which maximum response occurs, which is critical for designing stable and efficient systems.
Coulomb Damping: Coulomb damping refers to the type of damping that occurs due to the frictional forces between surfaces in contact. This form of damping is characterized by a constant resistive force that opposes the motion, regardless of the velocity of the system. It plays an essential role in understanding various mechanical systems, especially in relation to free vibrations, types of damping mechanisms, and the design of isolators.
Critically damped: Critically damped refers to a specific condition in a damping system where the damping is just enough to prevent oscillations while allowing the system to return to its equilibrium position in the shortest possible time. In this state, the system is on the verge of being overdamped and underdamped, leading to optimal performance in applications like suspension systems and control systems.
Damped Natural Frequency: Damped natural frequency refers to the frequency at which a damped system oscillates when disturbed from its equilibrium position, taking into account the effects of damping. It is an important parameter that reflects how quickly the oscillations of a system decay over time due to energy dissipation, and it is influenced by factors such as the mass, stiffness, and damping characteristics of the system. Understanding this frequency is crucial for analyzing the behavior of systems that experience damped vibrations, particularly in terms of how they respond to external forces.
Damping Ratio: The damping ratio is a dimensionless measure that describes how oscillations in a mechanical system decay after a disturbance. It indicates the level of damping present in the system and is crucial for understanding the system's response to vibrations and oscillatory motion.
Differential Equation: A differential equation is a mathematical equation that relates a function with its derivatives, expressing how a quantity changes in relation to another variable. In the study of mechanical systems, these equations help describe the dynamic behavior of systems under various conditions, providing insight into aspects such as motion, stability, and response characteristics.
Dynamic vibration absorbers: Dynamic vibration absorbers are devices used to reduce unwanted vibrations in mechanical systems by introducing a secondary mass-spring-damper system that absorbs the energy from the primary vibrating system. These absorbers work by tuning their natural frequency to match that of the disturbing vibrations, thus dissipating energy and minimizing the amplitude of oscillations in the main system. This principle is especially significant when discussing single degree-of-freedom systems, as it allows for a more efficient design and operation of mechanical structures.
Equation of Motion: An equation of motion describes the relationship between the forces acting on a system and its resultant motion, typically in the context of oscillatory systems. It provides a mathematical framework to analyze the dynamics of systems under various conditions, such as free and forced vibrations, damping, and external excitations. This concept is fundamental to understanding how mechanical systems respond to disturbances and is essential for designing effective vibration control strategies.
Forced response: Forced response refers to the steady-state behavior of a mechanical system when it is subjected to an external periodic excitation or input, which causes the system to vibrate. This concept highlights how a single degree-of-freedom system reacts to forces that are not natural frequencies of the system, emphasizing the relationship between external forces and the resulting motion. Understanding forced response is crucial in predicting how systems will behave under various operating conditions.
Free response: Free response refers to the motion of a system when it is allowed to oscillate without any external forces acting on it after initial conditions are set. This type of motion is important as it helps in understanding how mechanical systems behave under their own dynamics, particularly in single degree-of-freedom systems, which are simplified models that can be described by a single coordinate. This concept lays the groundwork for analyzing vibrations and response characteristics of various systems, giving insights into natural frequencies and mode shapes.
Frequency response analysis: Frequency response analysis is a method used to evaluate how a mechanical system responds to external forces across a range of frequencies. This technique helps in understanding the dynamic behavior of systems, particularly in terms of resonance, damping, and stability. It is essential for designing and optimizing systems to ensure they perform well under various operating conditions.
Kinetic Energy: Kinetic energy is the energy possessed by an object due to its motion, defined mathematically as $$KE = \frac{1}{2} mv^2$$, where 'm' is mass and 'v' is velocity. This energy plays a critical role in various mechanical systems, particularly during oscillations and vibrations, where it alternates with potential energy. Understanding how kinetic energy behaves in different contexts helps analyze the dynamics of vibrating systems and their responses to forces.
Modal analysis: Modal analysis is a technique used to determine the natural frequencies, mode shapes, and damping characteristics of a mechanical system. This method helps to understand how structures respond to dynamic loads and vibrations, providing insights that are crucial for design and performance optimization.
Natural Frequency: Natural frequency is the frequency at which a system tends to oscillate in the absence of any external forces. It is a fundamental characteristic of a mechanical system that describes how it responds to disturbances, and it plays a crucial role in the behavior of vibrating systems under various conditions.
Overdamped: Overdamped refers to a condition in a mechanical system where the damping force is so strong that the system returns to equilibrium without oscillating. This phenomenon occurs in systems with a damping ratio greater than one, leading to slower motion and longer settling times compared to critically damped or underdamped systems. Understanding overdamping is crucial as it relates to various damping mechanisms, how energy is dissipated in motion, and the behavior of single degree-of-freedom systems under external influences.
Phase Angle: Phase angle is a measure of the position of a point in time on a waveform cycle, typically expressed in degrees or radians. It is crucial in understanding the relationship between sinusoidal functions, especially when analyzing oscillatory motion and energy transfer in mechanical systems. The phase angle indicates how far along in its cycle an oscillating system is at a given moment, allowing for comparisons between multiple oscillations and insights into the system's dynamic behavior.
Potential Energy: Potential energy is the stored energy in a system that has the potential to do work due to its position or configuration. It is crucial for understanding how mechanical systems behave, especially when analyzing motion and vibrations. This concept is directly related to how energy is exchanged between kinetic and potential forms, and plays a key role in scenarios like undamped free vibrations, resonance, and energy methods for analyzing vibrations in single degree-of-freedom and coupled systems.
Resonant Frequency: Resonant frequency is the natural frequency at which a system tends to oscillate when not subjected to a continuous or repeated external force. At this frequency, even small periodic driving forces can cause the system to oscillate with increasing amplitude, leading to significant responses. This concept is vital in understanding how systems respond to different types of vibrations and is particularly important in mechanical systems, strings, cables, and single degree-of-freedom systems.
Shock Absorbers: Shock absorbers are mechanical devices designed to absorb and dissipate energy from vibrations and shocks, thereby improving comfort and stability in various systems, especially vehicles. They play a critical role in damped free vibrations by reducing oscillations, allowing for smoother operation. By utilizing different damping mechanisms, shock absorbers help control the motion of components in mechanical systems and contribute to vibration transmissibility, ensuring that disturbances are minimized across the system.
Single degree-of-freedom: A single degree-of-freedom (SDOF) system is a mechanical system that can move in only one independent direction or mode of motion. This concept simplifies the analysis of mechanical vibrations by reducing the complexity of multi-degree-of-freedom systems into manageable equations of motion. Understanding SDOF systems is fundamental as they serve as building blocks for more complex structures, allowing for straightforward calculations of natural frequencies, response to dynamic loads, and stability.
Spring-damper system: A spring-damper system is a mechanical model that combines a spring, which provides restorative force, and a damper, which dissipates energy to reduce oscillations. This system is commonly used to study the dynamic behavior of mechanical systems, particularly in single degree-of-freedom scenarios where motion is constrained to one direction. The interplay between the spring and damper elements allows for the analysis of vibration response and stability under various loading conditions.
Spring-mass system: A spring-mass system consists of a mass attached to a spring, which provides a restoring force when the mass is displaced from its equilibrium position. This setup is fundamental in understanding how mechanical vibrations and oscillatory motion work, where the mass and spring interact to produce periodic motion. The behavior of a spring-mass system is crucial in analyzing undamped free vibrations, where the system oscillates indefinitely without energy loss, and also serves as a classic example of a single degree-of-freedom system.
Structural Damping: Structural damping refers to the energy dissipation within a structure due to internal friction when subjected to vibrations. It plays a crucial role in the response of mechanical systems, particularly in reducing amplitude and enhancing stability by absorbing vibrational energy.
Tuned liquid dampers: Tuned liquid dampers are devices used to reduce vibrations in structures by utilizing a liquid mass that is tuned to resonate at the same frequency as the vibrating system. These dampers work by absorbing and dissipating energy from the vibrating structure, which helps to minimize the amplitude of oscillations. They are particularly effective in applications where traditional solid dampers may not be feasible or efficient, enhancing overall system stability and performance.
Tuned mass damper: A tuned mass damper is a device used to reduce vibrations in structures by utilizing a secondary mass that oscillates out of phase with the primary structure's motion. This system is particularly effective in controlling resonant vibrations, helping to stabilize buildings, bridges, and other structures subject to dynamic loads like wind or earthquakes. The design involves careful tuning of the mass and spring properties to ensure maximum effectiveness at specific frequencies.
Underdamped: Underdamped refers to a specific condition in a dynamic system where the system oscillates with decreasing amplitude over time, due to insufficient damping to prevent oscillation. This phenomenon is characterized by oscillations that occur before the system eventually comes to rest, typically resulting from a balance between inertia and restoring forces that isn't strong enough to eliminate motion quickly. Understanding underdamping is crucial for analyzing how systems respond to disturbances and can influence concepts like resonance, logarithmic decrement, and various types of damping mechanisms.
Vibration isolation: Vibration isolation is a technique used to reduce the transmission of vibrations from one object to another, thereby protecting sensitive equipment or structures from potentially damaging oscillations. This concept is important for minimizing the effects of vibrations generated by machinery, traffic, or environmental sources on adjacent structures and systems.