Structural Dynamics / Energy in vibrating structures
The energy balance
Multiply the equation of motion by velocity, integrate, and every term becomes an energy that has to be accounted for.
Lesson review: expert verification remains open
This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.
Checks needed for project use
- A dynamics reviewer must confirm excitation models, damping, modal assumptions and any comfort or serviceability interpretation. Small displacement alone is not an acceptance criterion.
- Independent validation of the associated interactive implementation, input domain and project applicability remains required; this review checks lesson context and selected calculations, not design certification.
- Verify energy definitions and dissipation models; energy closure alone does not prove response accuracy.
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesWhy this matters
The force view answers 'is it strong enough?'. The energy view answers questions the force view finds awkward.
Why does an undamped structure keep moving with nothing driving it? Why does a damper work, and why does it work better at high frequency? Why does a yielding structure survive forces several times its own strength? Why does base isolation help and a stiffer structure not?
All four have short answers in energy and long ones in force. And there is a practical payoff: the energy balance has to close, which makes it a useful check alongside equilibrium and response convergence.
By the end of this lesson you should be able to
- Derive the energy balance from the equation of motion
- Identify each term and say where the energy is at any instant
- Explain why damping energy only ever increases
- Use the residual to detect an inadequate time step
What you should already know
- The equation of motion (Module 3)
- Free vibration (Module 4)
- Work as force times distance
Four energies
Kinetic energy — energy the structure has because it is moving:
Ek = ½mẋ²
Strain energy — energy stored in the elastic deformation:
Es = ½kx²
Dissipated energy — energy the damper has converted into heat. It is an integral over time, because dissipation accumulates:
ED = ∫ cẋ² dt
Input energy — work done by the external force:
EI = ∫ Fẋ dt
Two of these are quantities the structure HAS at an instant; two are quantities that have accumulated since the start. That distinction matters. Kinetic and strain energy go up and down. Dissipated energy never goes down — a damper cannot give energy back, and if a computed ED ever decreases, the analysis is wrong.
Why the damping integral has ẋ squared
The damping force is cẋ and it acts through a distance ẋ dt in time dt, so the work it does is cẋ · ẋ dt = cẋ² dt.
The square is important for two reasons. It guarantees the integrand is never negative, which is why dissipation is monotonic. And it means dissipation is dominated by the fast parts of the motion — which is why a viscous damper is efficient at high frequency and poor at low frequency, and why a damper sized for a wind serviceability problem is not automatically the right damper for an earthquake.
From first principles
The energy balance
We want to show: Show that EI = Ek + Es + ED at every instant, by multiplying the equation of motion by velocity and integrating over time.
Work is force times distance. If we multiply every term in the equation of motion by the velocity, each term becomes a rate of doing work — power. Integrating over time turns each rate into an energy. Nothing new is being assumed; the equation of motion is simply being read in a different currency.
Try it
The energy account
Every joule the load puts in must be stored, carried or dissipated. Watch the account, and watch whether it balances.
Case
Coarsen it and watch the balance stop closing.
- Kinetic
- Strain
- Dissipated
- Total mechanical
Show the numbers behind this plot
Where the energy is, right now
Kinetic 0 J · Strain 592 J · Dissipated 0 J
- Mechanical energy now
- 592 J
- Cumulative dissipation
- 0 J
- Cumulative input work
- 0 J
- Balance residual
- -592 J
- Worst residual, as a fraction
- 100.0000%
With no damping the total mechanical energy is a constant: it merely moves between kinetic and strain twice per cycle. The residual is 100.0000% — which is measuring the integration error, not any physics.
What this shows: Energy balance is the strongest available check on a dynamic analysis. If input energy does not equal kinetic plus strain plus dissipated, the analysis is wrong — and the size of the gap tells you how wrong.
Energy and amplitude
At corresponding displacement extrema in a linear free vibration, mechanical energy is ½kx². Halving peak amplitude leaves one quarter of that peak energy; it does not establish whether the motion is harmless. Absolute energy, stress and acceleration still matter.
What an energy balance can establish
An energy account can reveal inconsistent signs, units or work calculations. Report the residual and how each term was integrated. A small residual is not proof of time-step accuracy: average-acceleration Newmark can conserve energy in undamped linear free vibration while retaining substantial phase error.
Refine the step and compare the displacement, drift, force or acceleration needed for the assessment. Check input assumptions separately. Neither a smooth plot nor a closing energy balance establishes that the physical model is correct.
Worked example
Where the energy is, at four instants of a free vibration
Given
- An undamped SDOF system: m = 8 000 kg, k = 2.0 MN/m
- Released from rest at x₀ = 50 mm
Find
The kinetic, strain and total energy at four points in the first cycle.
Assumptions
- Undamped, so total energy is constant and the arithmetic is checkable
Predict first
A decaying vibration has fallen from 20 mm to 10 mm amplitude. What fraction of its original energy remains?
Practice
A structure of stiffness 3.5 MN/m is displaced 40 mm and released from rest. What is its total mechanical energy, in J?
Practice
That structure has a mass of 14 000 kg. What is its maximum velocity, in m/s?
Practice
A vibration decays to 30% of its original amplitude. What percentage of the original energy remains?
Worked example
Where the energy goes in one cycle
Given
- A mass oscillating freely at 5 mm amplitude
- Stiffness 4.0 × 10⁶ N/m
- Damping 3 % of critical
Find
The stored energy and the fraction dissipated per cycle
Summary
- Multiply the equation of motion by velocity and integrate: that is the whole derivation
- EI = Ek + Es + ED, at every instant
- Kinetic and strain go up and down; dissipated energy only ever increases
- Undamped free vibration keeps a constant total, circulating twice per cycle
- Energy goes as amplitude squared — half the amplitude is a quarter of the energy
- The balance residual measures integration error that no displacement plot shows
- A closing balance is necessary for a correct answer and not sufficient