Energy Dissipated per Cycle
Calculate the energy dissipated per cycle in a viscous damper, ΔE = π·c·ω·X², from the viscous damping coefficient c, the excitation frequency ω (rad/s) and the vibration amplitude X. The result, in joules, is the mechanical energy converted to heat each oscillation cycle by the damper — proportional to frequency and to the square of amplitude. Quantifying it is essential to size dampers and dissipators (in suspensions, buildings under earthquakes, isolators) and to estimate the heat generated by vibration. The greater the dissipation, the faster free vibration decays. Enter the damping coefficient, the frequency and the amplitude.
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Energia dissipada por ciclo
Um amortecedor viscoso (como o de uma suspensão automotiva, cheio de óleo) converte energia mecânica de vibração em calor, e é justamente isso que faz a vibração livre decair e que limita a amplitude na ressonância. A energia dissipada a cada ciclo de oscilação harmônica é ΔE = π·c·ω·X², onde c é o coeficiente de amortecimento viscoso (a constante que relaciona força de amortecimento e velocidade, F = c·v), ω é a frequência da vibração (rad/s) e X é a amplitude. Essa fórmula vem da integral da força de amortecimento (proporcional à velocidade) ao longo de um ciclo completo, e representa a área do laço de histerese força-deslocamento — quanto maior o laço, mais energia perdida. Dois fatos saltam aos olhos. Primeiro, a energia dissipada cresce com o quadrado da amplitude: vibrações grandes dissipam muito mais (e por isso aquecem mais o amortecedor). Segundo, cresce linearmente com a frequência: amortecedores viscosos dissipam mais em vibrações rápidas. Quantificar ΔE é essencial em várias frentes: dimensionar amortecedores e dissipadores de energia (em suspensões, em edifícios e pontes projetados para dissipar energia sísmica ou de vento, em isoladores), estimar o aquecimento gerado (um amortecedor que dissipa muita potência precisa de área para trocar calor, senão o óleo superaquece e perde eficácia), e prever a taxa de decaimento da vibração livre. Em estruturas, a razão entre a energia dissipada por ciclo e a energia elástica armazenada define a capacidade de amortecimento do sistema. Informe o coeficiente de amortecimento, a frequência e a amplitude.
Related Tools
Damped Natural Frequency
Calculate the damped natural frequency of a vibrating system, ω_d = ω_n·√(1 − ζ²), from the undamped natural frequency ω_n and the damping ratio ζ. The result, in the unit of ω_n (rad/s or Hz), is the actual frequency at which an underdamped system oscillates freely after a disturbance — always lower than the undamped natural frequency, since damping slows the oscillation. For small ζ (lightly damped systems), ω_d ≈ ω_n; as ζ → 1 (critical damping), ω_d → 0 and the system stops oscillating. Enter the natural frequency and the damping ratio.
Amplification Factor (Q)
Calculate the resonance amplification factor (quality factor Q), Q = 1 ÷ (2·ζ), from the damping ratio ζ. The dimensionless result shows how many times the vibration amplitude at resonance exceeds the equivalent static deflection: lightly damped systems (small ζ) have high Q and sharp, dangerous resonance peaks; well-damped systems have low Q and smooth response. It is central to designing structures, machines and instruments to avoid destructive amplification and to characterizing the selectivity of filters and resonators. Enter the damping ratio.
Hydraulic Jump Energy Loss
Calculate the specific energy dissipated in a hydraulic jump, ΔE = (y₂ − y₁)³ ÷ (4·y₁·y₂), from the upstream y₁ (supercritical) and downstream y₂ (subcritical) sequent depths. The hydraulic jump is one of the most efficient energy dissipators in hydraulics: intense turbulence in the transition converts kinetic energy to heat and sound, removing excess flow energy. This head loss ΔE is exactly what is sought downstream of spillways, gates and bottom outlets — water arrives with very high energy (able to scour the riverbed and undermine the structure), and the stilling basin induces the jump to 'burn' that energy in a controlled way. The higher the incoming Froude number, the greater the dissipated fraction — jumps with Fr > 9 dissipate up to 85%. Enter the upstream and downstream sequent depths.
Arc Flash Incident Energy (Lee Method)
Computes arc flash incident energy by the Ralph Lee method, E = 5.12×10⁵ × V × I_bf × t ÷ d², with voltage in kV, bolted fault current in kA, fault clearing time in seconds and working distance in millimetres. Lee's method models an open-air arc as an ideal radiant heat source, ignoring the energy an enclosure reflects back; IEEE 1584 therefore keeps it only as the legacy model, recommended for open-air arcs and for voltages above 15 kV where the empirical equations do not apply. The result in cal/cm² sets the PPE category: 1.2 cal/cm² is the second-degree burn threshold and the value that bounds the arc flash boundary. The cal/cm² form is adopted (the J/cm² variant uses 2.142×10⁶ and is 4.184 times larger). Enter the voltage, the fault current, the clearing time and the working distance.
Modulus of Resilience
Calculate the modulus of resilience, U_r = σ_y² ÷ (2·E), from the yield strength σ_y (MPa) and the elastic modulus E (MPa). The result, in MJ/m³ (equivalent to MPa), is the strain energy the material absorbs per unit volume up to the elastic limit — the area under the linear part of the stress-strain curve. Materials with high yield strength and low modulus (like spring steels) have high resilience: they store much elastic energy and return it on unloading, with no permanent deformation. Enter the yield strength and the elastic modulus.
Critical Depth in Rectangular Channel
Calculate the critical depth of a rectangular channel, y_c = (q² ÷ g)^(1/3), from the unit discharge q (flow per unit width, m³/s/m) and gravity g. Critical depth is the depth at which specific energy is minimum, marking the boundary between the two open-flow regimes: above it the flow is subcritical (slow, deep, Fr < 1, downstream-controlled); below, supercritical (fast, shallow, Fr > 1, upstream-controlled); exactly at it, Fr = 1. Critical depth is central to channel and structure hydraulics: it defines the control section at spillways, weirs and flumes (Parshall), where flow passes through the critical regime stably and the stage-discharge relation is unique — allowing flow measurement from head. It also determines whether a hydraulic jump can form and guides water-surface profiles. Enter the channel's unit discharge.
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