How do you calculate the damping ratio of a mechanical system?

Damping ratio is a measure of how quickly a mechanical system returns to its equilibrium position after being disturbed. It is related to the amount of energy dissipated by the system due to friction, air resistance, or other damping forces. A high damping ratio means the system is over-damped and takes a long time to settle, while a low damping ratio means the system is under-damped and oscillates around the equilibrium point. A critical damping ratio of one means the system is critically damped and returns to the equilibrium position in the shortest possible time without overshooting.

Key takeaways from this article

Crunch the numbers:

To find the damping ratio, divide your system's damping coefficient by the critical damping coefficient. It's a bit like solving a puzzle where you have all the pieces; you just need to put them together correctly.
Mix it up:

Think of modeling damping forces as making a recipe. You choose your ingredients based on the 'flavor' of damping needed, then mix them with either math or software until they blend smoothly.

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1 What is a mechanical system?

A mechanical system is a collection of rigid or flexible bodies that are connected by joints, springs, dampers, or other elements that allow relative motion. Examples of mechanical systems include pendulums, mass-spring-damper systems, bridges, vehicles, robots, and machines. Mechanical systems can be modeled by differential equations that describe their dynamic behavior under external forces and torques.

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Mahesh Kumar Kandepi

Linkedin's Community Top Voice, Mechanical Engineering | Junior Associate at Quest Global | Facility Engineer | ExxonMobil | Oil & Gas | Piping
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The Damping ratio is a measure of how oscillations in a system decay after a disturbance. It is used to describe the behavior of a system in response to a sudden input. To calculate damping ratio : Damping Ratio= Damping Coefficient / Critical Damping Coefficient Critical damping coefficient = Two times the under root of m*k where m=mass of the system k= stiffness of the system

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Daniel Morais

Coordenador de Manutenção | Engenheiro de Confiabilidade | Especialista de Confiabilidade | Analista de PCM |
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Adicionalmente, um sistema mecânico é definido como um conjunto de corpos rígidos ou flexíveis interconectados por meio de juntas, molas, amortecedores ou outros elementos que possibilitam o movimento relativo entre eles. Exemplos comuns de sistemas mecânicos incluem pêndulos, sistemas massa-mola-amortecedor, pontes, veículos automotores, robôs industriais e diversas máquinas presentes na indústria e na vida cotidiana. É importante ressaltar que os sistemas mecânicos podem ser representados e analisados matematicamente por meio de equações diferenciais que descrevem seu comportamento dinâmico em resposta a forças e torques externos.

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2 How to model damping forces?

Damping forces are usually proportional to the velocity or displacement of the system, or a combination of both. The simplest way to model damping forces is to use a linear viscous damper, which exerts a force that is proportional to the relative velocity of the system. The damping coefficient is a constant that determines the strength of the damping force. For example, a mass-spring-damper system with a linear viscous damper can be modeled by the following equation: mx'' + cx' + kx = F(t) where x is the displacement of the mass, m is the mass, c is the damping coefficient, k is the spring constant, and F(t) is the external force.

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Bhavya Nandini Yarlagadda

Robotics Engineer | Associate Mechatronics Engineer at L&T | SUNY Buffalo Grad | GAT at Diageo
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Model damping like you're cooking: identify the flavor (viscous, solid, etc.), choose your technique (math, software, or real-world data), and stir it all together. Remember, the right ingredients for the right force make a smooth simulation! ️

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Daniel Morais

Coordenador de Manutenção | Engenheiro de Confiabilidade | Especialista de Confiabilidade | Analista de PCM |
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Adicionalmente, é importante destacar que as forças de amortecimento em um sistema mecânico são frequentemente proporcionais à velocidade ou ao deslocamento do sistema, ou uma combinação de ambos. Uma abordagem comum para modelar as forças de amortecimento é utilizar um amortecedor viscoso linear, que gera uma força proporcional à velocidade relativa do sistema. O coeficiente de amortecimento, denotado por "c", é uma constante que determina a magnitude da força de amortecimento.

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3 How to find the natural frequency?

The natural frequency of a mechanical system is the frequency at which it oscillates when it is free from external forces and damping. It depends on the mass and stiffness of the system, and it is also called the undamped natural frequency. The natural frequency can be found by solving the characteristic equation of the system, which is obtained by setting F(t) to zero and assuming a solution of the form x = Ae^(iwt), where A and w are constants. For example, the characteristic equation of a mass-spring-damper system with a linear viscous damper is: mw^2 + cw + k = 0 The natural frequency can be found by solving for w, which gives: w_n = sqrt(k/m) where w_n is the natural frequency.

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Daniel Morais

Coordenador de Manutenção | Engenheiro de Confiabilidade | Especialista de Confiabilidade | Analista de PCM |
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Adicionalmente, é relevante destacar que a frequência natural de um sistema mecânico representa a frequência na qual o sistema oscila quando está livre de forças externas e amortecimento. Essa frequência é determinada pela massa e rigidez do sistema e é também conhecida como frequência natural não amplificada. A frequência natural é fundamental para compreender o comportamento vibratório do sistema e pode ser encontrada resolvendo a equação característica associada ao sistema.

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4 How to find the damping ratio?

The damping ratio of a mechanical system is the ratio of the actual damping coefficient to the critical damping coefficient. The critical damping coefficient is the value of the damping coefficient that makes the system critically damped, which means it returns to the equilibrium position in the shortest possible time without overshooting. The critical damping coefficient can be found by setting the discriminant of the characteristic equation to zero, which gives: c_c = 2*sqrt(mk) where c_c is the critical damping coefficient. The damping ratio can then be found by dividing the actual damping coefficient by the critical damping coefficient, which gives: zeta = c/c_c where zeta is the damping ratio.

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5 How to use the damping ratio?

The damping ratio can be used to determine the type and behavior of the system response. Depending on the value of the damping ratio, the system can be classified as over-damped, critically damped, or under-damped. If zeta is greater than one, the system is over-damped and will return to the equilibrium position exponentially but slowly. If zeta is equal to one, the system is critically damped and will return to the equilibrium position exponentially but in the shortest possible time. On the other hand, if 0 is less than zeta which is less than one, then the system is under-damped and will have oscillations while returning to equilibrium position exponentially with a decaying amplitude that depends on the damping ratio. Moreover, we can use the damping ratio to calculate the damped natural frequency which is found by multiplying the natural frequency by the square root of one minus the square of the damping ratio.

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Bhavya Nandini Yarlagadda

Robotics Engineer | Associate Mechatronics Engineer at L&T | SUNY Buffalo Grad | GAT at Diageo
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Consider the damping ratio to be a system's wobbling volume knob: high = silent and steady (ideal for robots), low = joyful oscillations (like a swing), and just right (critical damping) = no irritating bounce-back.

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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How do you calculate the damping ratio of a mechanical system?

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3

4

5

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1 What is a mechanical system?

2 How to model damping forces?

3 How to find the natural frequency?

4 How to find the damping ratio?

5 How to use the damping ratio?

6 Here’s what else to consider

Mechanical Engineering

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How do you calculate the damping ratio of a mechanical system?

1

2

3

4

5

6

1 What is a mechanical system?

2 How to model damping forces?

3 How to find the natural frequency?

4 How to find the damping ratio?

5 How to use the damping ratio?

6 Here’s what else to consider

Mechanical Engineering

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