Shock Absorber Calculations – Impact Energy and Stopping Force

Shock absorbers reduce the forces produced when moving machinery or loads are brought to rest over a short distance. This page covers impact energy, stopping force, common energy-absorption methods and simplified calculations for selecting industrial shock absorbers.

The calculations include free-falling masses, horizontal motion, driven loads, inclined motion and swinging or rotating masses. They are intended as engineering reference methods for preliminary assessment and shock absorber selection.

• Shock absorber principles
• Types of shock absorber
• Impact-force comparison
• Shock absorber duty calculations

Shock Absorber Principles and Impact Loading

A moving object possesses kinetic energy determined by its mass and velocity. If its velocity changes significantly over a very short time or distance, high forces can result. Increasing the stopping distance and controlling the deceleration allows those forces to be reduced.

$$E=\frac{M{v}^2}{2}$$

Shock and impact behaviour can be considerably more complicated than a simple energy calculation. See the RoyMech Impact Loading reference for impact stresses and elastic response.

Energy can be absorbed or controlled using metal springs, elastomeric buffers, pneumatic springs, hydraulic dampers and purpose-designed industrial shock absorbers. The appropriate method depends on whether energy is to be stored temporarily and returned to the system or dissipated.

Types of Shock Absorber and Energy Absorber

Metal Springs

Metal springs provide a relatively simple and robust means of increasing the stopping distance and reducing impact loading. Helical springs, disc springs, leaf springs, ring springs and other arrangements can be used.

A spring primarily stores mechanical energy rather than dissipating it. Unless sufficient damping is present, the stored energy can cause rebound and oscillation. Spring systems are therefore frequently combined with damping.

Elastomeric Buffers

Elastomeric buffers provide compliance together with useful inherent damping. They can be moulded into compact shapes and are widely used for bump stops, buffers and vibration control.

Their performance depends strongly on material, temperature, loading rate, environment and chemical exposure, so allowable temperature and compatibility should be checked against the particular elastomer specification.

Hydraulic Dashpots

A hydraulic dashpot uses piston movement to force hydraulic fluid through a restriction. The resulting pressure provides resistance to motion and converts mechanical energy primarily into heat in the fluid.

A simple fixed-orifice arrangement does not necessarily produce constant deceleration. The resisting force varies with piston velocity and the hydraulic characteristics of the device.

Collapsing Safety Absorbers

Collapsing or sacrificial energy absorbers dissipate impact energy through permanent deformation. They can absorb substantial energy within a relatively compact stroke but are generally single-use devices and require replacement after operation.

Pneumatic and Air Springs

Air can provide substantial energy storage over a useful stroke. Because the gas is compressed as the device moves, the resisting force generally rises through the stroke. A simplified compression model may be represented by:

$$P{V}^n=\text{constant}$$

Actual behaviour depends on whether the compression is approximately isothermal, adiabatic or somewhere between the two.

Self-Compensating Hydraulic Shock Absorbers

Industrial hydraulic shock absorbers may use multiple metering orifices or other controlled-flow arrangements so that resistance changes as the piston moves through its stroke.

The objective is to use the available stroke efficiently and bring the moving load to rest with a relatively controlled deceleration, avoiding the large force peak associated with stopping the same load over a very short distance.

Impact Force Comparison

The following simplified example illustrates the importance of stopping distance. A 1 kg mass falls 0.5 m onto a short steel column having:

• Diameter = 0.1 m
• Length = 0.2 m
• Cross-sectional area = 0.00785 m²
• Elastic modulus = 210 GPa

Impact Without a Shock Absorber

The axial stiffness of the column is:

$$k=\frac{AE}{L}$$ $$k\approx 8.25\times {10}^9\text{N/m}$$

The gravitational potential energy immediately before impact is:

$$E_1=Mgh=4.905\text{J}$$

For the idealised elastic-column model, equating this energy to elastic strain energy gives:

$$E_1=\frac{{P_{\max }}^2}{2k}$$

Therefore:

$$P_{\max }=\sqrt{2E_{1}k}\approx 284\text{kN}$$

with an approximate maximum elastic deflection:

$${\delta }_{\max }=\frac{P_{\max }}{k}\approx 34\text{µm}$$

The very high calculated force results from stopping the moving mass over an extremely small elastic deflection.

The same idealised impact can be expressed in terms of static stress and static deflection:

$${\sigma }_{\max }={\sigma }_{\mathrm{st}}(1+\sqrt{1+\frac{2h}{{\delta }_{\mathrm{st}}}})$$

This is an idealised elastic impact calculation. Real systems lose energy through material damping, local deformation, sound, friction and other mechanisms.

Impact with a Spring

If a linear spring is introduced with a 0.1 m stroke, gravity continues to perform work while the mass moves through the spring stroke. The total energy that must be accommodated before the mass reaches maximum spring compression is therefore:

$$E_t=Mg(h+s)$$ $$E_t=1\times 9.81\times 0.6=5.886\text{J}$$

For a linear spring:

$$E_t=\frac{F_{\max }s}{2}$$ $$F_{\max }=\frac{2E_t}{s}=117.7\text{N}$$

This is dramatically below the idealised rigid-impact force. However, a spring stores the energy and can return it to the mass, producing rebound unless damping or another energy-dissipation mechanism is provided.

Impact with an Idealised Constant-Force Hydraulic Absorber

For an idealised absorber providing constant resisting force through the same 0.1 m stroke:

$$E_t=Fs$$ $$F=\frac{E_t}{s}=58.86\text{N}$$

This comparison is deliberately idealised. A real hydraulic absorber has a particular force-displacement and force-velocity characteristic, and the manufacturer's rated energy, impact velocity and effective-mass limits must be respected.

Shock Absorber Duty Calculations

Shock absorber selection is primarily an energy problem. For each operating cycle, determine the kinetic or potential energy present when the absorber is contacted and then add any work performed by gravity, a propelling force or a driving torque while the absorber moves through its stroke.

The following symbols are used:

For all cases:

$$E_t=E_1+E_2$$ $$E_{\mathrm{th}}=E_{t}n$$

For selection methods that use an equivalent mass at the absorber:

$$M_e=\frac{2E_t}{v_d^2}$$

1. Free-Falling Mass

For a mass falling vertically through height h, followed by absorber stroke s:

$$E_1=Mgh$$ $$E_2=Mgs$$ $$v_d=\sqrt{2gh}$$ $$M_e=\frac{2E_t}{v_d^2}$$

The last expression corrects a dimensional error in the original RoyMech transcription, which displayed 2M/v_d². Equivalent mass must be derived from energy and impact velocity.

2. Horizontal Moving Mass

Neglecting rolling or sliding resistance:

$$E_1=\frac{M{v}^2}{2}$$ $$E_2=0$$ $$v_d=v$$ $$M_e=M$$

3. Horizontal Moving Mass with Constant Propelling Force

For initial velocity v, constant propelling force F, and distance H over which that force acts before the absorber is contacted:

$$v_d=\sqrt{v^2+\frac{2FH}{M}}$$ $$E_1=\frac{M{v}_d^2}{2}$$

If the propelling force continues throughout the absorber stroke:

$$E_2=Fs$$ $$M_e=\frac{2E_t}{v_d^2}$$

4. Mass Moving Down an Incline

For a mass starting from rest and descending through vertical height h, with friction neglected:

$$E_1=Mgh=\frac{M{v}_d^2}{2}$$ $$v_d=\sqrt{2gh}$$

Gravity continues to add energy during absorber stroke:

$$E_2=Mgs\sin \theta$$ $$M_e=\frac{2E_t}{v_d^2}$$

5. Mass Moving Down an Incline with Propelling Force

For a mass starting from rest, with a constant force F acting down the incline over distance H before impact, and friction neglected:

$$E_1=(F+Mg\sin \theta )H$$ $$E_1=\frac{M{v}_d^2}{2}$$ $$v_d=\sqrt{\frac{2(F+Mg\sin \theta )H}{M}}$$

If the same force continues throughout absorber stroke:

$$E_2=(F+Mg\sin \theta )s$$ $$M_e=\frac{2E_t}{v_d^2}$$

6. Mass Moving Up an Incline with Propelling Force

For a mass starting from rest, with force F acting up the incline over distance H, friction neglected:

$$E_1=(F-Mg\sin \theta )H$$ $$E_1=\frac{M{v}_d^2}{2}$$ $$v_d=\sqrt{\frac{2(F-Mg\sin \theta )H}{M}}$$

If the driving force continues through the absorber stroke, gravity opposes the motion:

$$E_2=(F-Mg\sin \theta )s$$ $$M_e=\frac{2E_t}{v_d^2}$$

7. Swinging Mass Without Propelling Torque

For the simplified horizontal-plane arrangement shown, the kinetic energy at contact is:

$$E_1=\frac{M{v}^2}{2}=\frac{I{\omega }^2}{2}$$ $$E_2=0$$

If the concentrated mass travels at tangential velocity v at radius L, while the shock absorber acts at radius R:

$$v_d=\frac{vR}{L}=\omega R$$ $$M_e=\frac{2E_t}{v_d^2}$$

For a distributed rotating assembly, use the actual mass moment of inertia I rather than replacing the system by a concentrated mass unless that approximation is appropriate.

8. Swinging Mass with Constant Propelling Torque

The kinetic energy at first contact remains:

$$E_1=\frac{I{\omega }^2}{2}$$

For a small absorber stroke s acting at radius R, the approximate angular movement is s/R. If constant torque T continues to act:

$$E_2=\frac{Ts}{R}$$ $$v_d=\frac{vR}{L}=\omega R$$ $$M_e=\frac{2E_t}{v_d^2}$$

Practical Shock Absorber Selection

For a preliminary industrial shock absorber selection, establish the moving mass or rotational inertia, impact velocity, available stroke and all forces or torques that continue to add energy during the stopping stroke.

Calculate both the energy per operation and the total energy that must be dissipated over the operating period. The selected absorber should then be checked against the manufacturer's allowable energy per cycle, energy per hour, effective-mass range, impact-velocity range and installation requirements.

Related RoyMech References

• Impact Loading and Impact Stress
• Cams and Followers
• Flywheels
• Mechanisms
• Power Screws
• Springs

External Shock Absorber References

1. TPC Dayton – Anti-Vibration Mountings
2. Machine Design – Shock Absorbers
3. Monroe – Automotive Shock Absorbers
4. ACE Controls – Industrial Shock Absorbers
5. Enertrols – Shock Absorbers
6. Enidine – Industrial Shock Absorbers