Behavior of Clamped-Edge Circular Membrane

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  • Thread starter Thread starter Chrono G. Xay
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SUMMARY

The discussion focuses on the behavior of a clamped-edge circular membrane, specifically in relation to drumhead tension and thickness selection for optimal tuning and feel. The tension is quantified using the equation $$T = σ \left( \frac {D f_1} {0.766} \right) ^2$$, where $$σ$$ represents the area mass unit and $$D$$ is the diameter. The constant 0.766 is derived from the first zero of the zero-order Bessel function, indicating its significance in the wave equation for circular membranes. The conversation also delves into the complexities of predicting diameter changes due to tension, utilizing equations such as $$ΔL = \frac{F L_0} {A E}$$ and exploring the implications of generalized Hooke's Law in cylindrical coordinates.

PREREQUISITES
  • Understanding of circular membrane dynamics and tension calculations
  • Familiarity with Bessel functions and their applications in wave equations
  • Knowledge of material properties, including modulus of elasticity and area mass unit
  • Basic calculus and mechanics of materials for stress and strain analysis
NEXT STEPS
  • Research the applications of Bessel functions in mechanical vibrations
  • Study the derivation and implications of the wave equation for circular membranes
  • Explore the relationship between tension, diameter, and frequency in drumheads
  • Investigate advanced topics in elasticity and stress analysis in cylindrical coordinates
USEFUL FOR

Musicians, particularly drummers and percussionists, mechanical engineers, and materials scientists interested in the physics of musical instruments and the mechanics of circular membranes.

  • #31
Chrono G. Xay said:
I’m interested in them because they’re a continuous portion of the membrane; as the rim pulls down on the hoop, which is clamped to the edge of the membrane’s collar, naturally the length of the collar affects the amount of elongation experienced until the desired musical pitch (and therefore tension) has been achieved.

(Up until now I thought there was no technical term for the portion of the membrane between the drumhead’s speaking diameter and the hoop, hence my use of “skirt”, but it’s actually called the ‘collar’.)

For this analysis I do not consider the added magnitude of elongation afforded by the collar to be insignificant.
It is not insignificant, but you also need to account for normal forces that result in frictional forces which affect the skin stresess and strains in membrane skin in this region. This is what allows the skin to be pulled down.
 
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  • #32
Chestermiller said:
It is not insignificant, but you also need to account for normal forces that result in frictional forces which affect the skin stresess and strains in membrane skin in this region. This is what allows the skin to be pulled down.
It’s for that very reason that when lowering the pitch of a drum you intentionally take the pitch slightly lower then what you want it, then tune it *up* to where you actually want it. 😊 The same is done for guitar strings.
The rest of it will basically work itself out on its own once someone starts striking it with a stick.

I also just realized that I had mistyped a few of the earlier functions. I’m sorry. $$ω \left( f_0, D_s \right) = \frac {πf_0 D_s} {J_0 \left( \frac 2 {D_s} {α_1} \right)}$$ $$ρ \left( ρ_M, h_m \right) = ρ_M h_m$$ $$σ \left( ω, ρ \right) = \frac {ω^2} ρ$$ $$D_0 \left( D_s, L_{c,0} \right) = {D_s} + 2L_{c,0}$$ $$D \left( D_0, ν, E \right) = D_0 \left[ 1 + \left( 1 + ν \right) \frac σ E \right]$$ $$L_c \left( D, D_s \right) = \frac {D - {D_s}} 2$$ $$δ \left( L_{c}, r_h, h_c, h_r \right) = \sqrt {L_{c}^2 - r_h^2} - \left( h_c + h_r \right)$$ $$n \left( δ, t \right) = \frac δ {t^{-1}}$$ $$Δδ = δ - δ_0$$ In order of appearance…

##f_0## = Desired fundamental frequency
##D_s## = Speaking diameter of membrane
##α_1## = 1st zero of First Order Bessel Function (2.4048)
##ρ## = Basis weight
##ρ_M## = 3D density of Mylar
##h_m## = Height (thickness) of membrane
##σ## = In-plane axisymmetric stress on membrane
##D_0## = Unstretched membrane overall diameter
##L_{c,0}## = Unstretched membrane collar length
##D## = Stretched membrane overall diameter
##ν## = Poisson’s ratio
##E## = Modulus of elasticity
##L_c## = Stretched membrane collar length
##δ## = Required displacement of membrane rim’s tension rods after tension rods have been taken as low as they will go without increasing the in-plane stress ##σ## above theoretical zero.
##r_h## = Extra radius of membrane hoop (from the center of the shell’s bearing edge to the inside diameter of the membrane’s hoop)
##h_c## = Height (depth) of membrane collar
##h_r## = Height (thickness) of membrane’s rim
##n## = Decimal number of revolutions tension rods must make to achieve needed in-plane stress
##t## = Number of threads per inch/mm on tension rods
##δ_0## = Initial displacement of tension rods (assuming the membrane is already tuned to some initial fundamental frequency ##f_{0,0}##; i.e. ##σ_0 > 0##)
 
Last edited:
  • #33
Chrono G. Xay said:
It’s for that very reason that when lowering the pitch of a drum you intentionally take the pitch slightly lower then what you want it, then tune it *up* to where you actually want it. 😊 The same is done for guitar strings.
The rest of it will basically work itself out on its own once someone starts striking it with a stick.

I also just realized that I had mistyped a few of the earlier functions. I’m sorry. $$ω \left( f_0, D_s \right) = \frac {πf_0 D_s} {J_0 \left( \frac 2 {D_s} {α_1} \right)}$$ $$ρ \left( ρ_M, h_m \right) = ρ_M h_m$$ $$σ \left( ω, ρ \right) = \frac {ω^2} ρ$$ $$D_0 \left( D_s, L_{c,0} \right) = {D_s} + 2L_{c,0}$$ $$D \left( D_0, ν, E \right) = D_0 \left[ 1 + \left( 1 + ν \right) \frac σ E \right]$$ $$L_c \left( D, D_s \right) = \frac {D - {D_s}} 2$$ $$δ \left( L_{c}, r_h, h_c, h_r \right) = \sqrt {L_{c}^2 - r_h^2} - \left( h_c + h_r \right)$$ $$n \left( δ, t \right) = \frac δ {t^{-1}}$$ $$Δδ = δ - δ_0$$ In order of appearance…

##f_0## = Desired fundamental frequency
##D_s## = Speaking diameter of membrane
##α_1## = 1st zero of First Order Bessel Function (2.4048)
##ρ## = Basis weight
##ρ_M## = 3D density of Mylar
##h_m## = Height (thickness) of membrane
##σ## = In-plane axisymmetric stress on membrane
##D_0## = Unstretched membrane overall diameter
##L_{c,0}## = Unstretched membrane collar length
##D## = Stretched membrane overall diameter
##ν## = Poisson’s ratio
##E## = Modulus of elasticity
##L_c## = Stretched membrane collar length
##δ## = Required displacement of membrane rim’s tension rods after tension rods have been taken as low as they will go without increasing the in-plane stress ##σ## above theoretical zero.
##r_h## = Extra radius of membrane hoop (from the center of the shell’s bearing edge to the inside diameter of the membrane’s hoop)
##h_c## = Height (depth) of membrane collar
##h_r## = Height (thickness) of membrane’s rim
##n## = Decimal number of revolutions tension rods must make to achieve needed in-plane stress
##t## = Number of threads per inch/mm on tension rods
##δ_0## = Initial displacement of tension rods (assuming the membrane is already tuned to some initial fundamental frequency ##f_{0,0}##; i.e. ##σ_0 > 0##)
I really don't have anything to add to what I said in my previous posts.
 

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